Daily Papers

Reinforcement Learning (RL) has become a pivotal approach for enhancing the reasoning capabilities of Large Language Models (LLMs). However, a significant theoretical gap persists, as traditional token-level RL frameworks fail to align with the reasoning-level nature of complex, multi-step thought processes like Chain-of-Thought (CoT). To address this challenge, we introduce CoT-Space, a novel theoretical framework that recasts LLM reasoning from a discrete token-prediction task to an optimization process within a continuous, reasoning-level semantic space. By analyzing this process from both a noise perspective and a risk perspective, we demonstrate that the convergence to an optimal CoT length is a natural consequence of the fundamental trade-off between underfitting and overfitting. Furthermore, extensive experiments provide strong empirical validation for our theoretical findings. Our framework not only provides a coherent explanation for empirical phenomena such as overthinking but also offers a solid theoretical foundation to guide the future development of more effective and generalizable reasoning agents.

Zero-Input AI (ZIA) introduces a novel framework for human-computer interaction by enabling proactive intent prediction without explicit user commands. It integrates gaze tracking, bio-signals (EEG, heart rate), and contextual data (time, location, usage history) into a multi-modal model for real-time inference, targeting <100 ms latency. The proposed architecture employs a transformer-based model with cross-modal attention, variational Bayesian inference for uncertainty estimation, and reinforcement learning for adaptive optimization. To support deployment on edge devices (CPUs, TPUs, NPUs), ZIA utilizes quantization, weight pruning, and linear attention to reduce complexity from quadratic to linear with sequence length. Theoretical analysis establishes an information-theoretic bound on prediction error and demonstrates how multi-modal fusion improves accuracy over single-modal approaches. Expected performance suggests 85-90% accuracy with EEG integration and 60-100 ms inference latency. ZIA provides a scalable, privacy-preserving framework for accessibility, healthcare, and consumer applications, advancing AI toward anticipatory intelligence.

In large-scale AI training, Sparse Mixture-of-Experts (s-MoE) layers enable scaling by activating only a small subset of experts per token. An operational challenge in this design is load balancing: routing tokens to minimize the number of idle experts, which is important for the efficient utilization of (costly) GPUs. We provide a theoretical framework for analyzing the Auxiliary-Loss-Free Load Balancing (ALF-LB) procedure -- proposed by DeepSeek's Wang et al. (2024) -- by casting it as a one-step-per-iteration primal-dual method for an assignment problem. First, in a stylized deterministic setting, our framework yields several insightful structural properties: (i) a monotonic improvement of a Lagrangian objective, (ii) a preference rule that moves tokens from overloaded to underloaded experts, and (iii) an approximate-balancing guarantee. Then, we incorporate the stochastic and dynamic nature of AI training using a generalized online optimization formulation. In the online setting, we derive a strong convexity property of the objective that leads to a logarithmic expected regret bound under certain step-size choices. Additionally, we present real experiments on 1B-parameter DeepSeekMoE models to complement our theoretical findings. Together, these results build a principled framework for analyzing the Auxiliary-Loss-Free Load Balancing of s-MoE in AI models.

The reasoning abilities of large language models (LLMs) have improved with chain-of-thought (CoT) prompting, allowing models to solve complex tasks in a stepwise manner. However, training CoT capabilities requires detailed reasoning data, which is often scarce. The self-taught reasoner (STaR) framework addresses this by using reinforcement learning to automatically generate reasoning steps, reducing reliance on human-labeled data. Although STaR and its variants have demonstrated empirical success, a theoretical foundation explaining these improvements is lacking. This work provides a theoretical framework for understanding the effectiveness of reinforcement learning on CoT reasoning and STaR. Our contributions are: (1) an analysis of policy improvement, showing why LLM reasoning improves iteratively with STaR; (2) conditions for convergence to an optimal reasoning policy; (3) an examination of STaR's robustness, explaining how it can improve reasoning even when incorporating occasional incorrect steps; and (4) criteria for the quality of pre-trained models necessary to initiate effective reasoning improvement. This framework aims to bridge empirical findings with theoretical insights, advancing reinforcement learning approaches for reasoning in LLMs.

Backpropagation (BP) is the most successful and widely used algorithm in deep learning. However, the computations required by BP are challenging to reconcile with known neurobiology. This difficulty has stimulated interest in more biologically plausible alternatives to BP. One such algorithm is the inference learning algorithm (IL). IL has close connections to neurobiological models of cortical function and has achieved equal performance to BP on supervised learning and auto-associative tasks. In contrast to BP, however, the mathematical foundations of IL are not well-understood. Here, we develop a novel theoretical framework for IL. Our main result is that IL closely approximates an optimization method known as implicit stochastic gradient descent (implicit SGD), which is distinct from the explicit SGD implemented by BP. Our results further show how the standard implementation of IL can be altered to better approximate implicit SGD. Our novel implementation considerably improves the stability of IL across learning rates, which is consistent with our theory, as a key property of implicit SGD is its stability. We provide extensive simulation results that further support our theoretical interpretations and also demonstrate IL achieves quicker convergence when trained with small mini-batches while matching the performance of BP for large mini-batches.

Test-time scaling seeks to improve the reasoning performance of large language models (LLMs) by adding computational resources. A prevalent approach within the field is sampling-based test-time scaling methods, which enhance reasoning by generating multiple reasoning paths for a given input during inference. However, despite its practical success, the theoretical foundations remain underexplored. In this paper, we provide the first theoretical framework for analyzing sampling-based test-time scaling methods, grounded in the perspective of confidence estimation. Based on the framework, we analyze two dominant paradigms: self-consistency and perplexity, and reveal key limitations: self-consistency suffers from high estimation error while perplexity exhibits substantial modeling error and possible degradation of the estimation error convergence. To address these limitations, we introduce RPC, a hybrid method that leverages our theoretical insights through two key components: Perplexity Consistency and Reasoning Pruning. Perplexity Consistency combines the strengths of self-consistency and perplexity, boosting the convergence rate of estimation error from linear to exponential while preserving model error. Reasoning Pruning prevents degradation by eliminating low-probability reasoning paths. Both theoretical analysis and empirical results across seven benchmark datasets demonstrate that RPC has a strong potential for reducing reasoning error. Notably, RPC achieves reasoning performance comparable to self-consistency while not only enhancing confidence reliability but also reducing sampling costs by 50%. The code and resources are available at https://wnjxyk.github.io/RPC.

Reinforcement learning (RL) plays a crucial role in shaping the behavior of large language and reasoning models (LLMs/LRMs). However, it often produces brittle and unstable policies, leading to critical failures such as spurious reasoning, deceptive alignment, and instruction disobedience that undermine the trustworthiness and safety of LLMs/LRMs. Currently, these issues lack a unified theoretical explanation and are typically addressed using ad-hoc heuristics. This paper presents a rigorous mathematical framework for analyzing the stability of the mapping from a reward function to the optimal policy. We show that policy brittleness often stems from non-unique optimal actions, a common occurrence when multiple valid traces exist in a reasoning task. This theoretical lens provides a unified explanation for a range of seemingly disparate failures, reframing them as rational outcomes of optimizing rewards that may be incomplete or noisy, especially in the presence of action degeneracy. We extend this analysis from the fundamental single-reward setting to the more realistic multi-reward RL across diverse domains, showing how stability is governed by an "effective reward" aggregation mechanism. We also prove that entropy regularization restores policy stability at the cost of increased stochasticity. Our framework provides a unified explanation for recent empirical findings on deceptive reasoning, instruction-following trade-offs, and RLHF-induced sophistry, and is further validated through perturbation experiments in multi-reward RL. This work advances policy-stability analysis from empirical heuristics towards a principled theory, offering essential insights for designing safer and more trustworthy AI systems.

The field of explainable AI (XAI) has quickly become a thriving and prolific community. However, a silent, recurrent and acknowledged issue in this area is the lack of consensus regarding its terminology. In particular, each new contribution seems to rely on its own (and often intuitive) version of terms like "explanation" and "interpretation". Such disarray encumbers the consolidation of advances in the field towards the fulfillment of scientific and regulatory demands e.g., when comparing methods or establishing their compliance with respect to biases and fairness constraints. We propose a theoretical framework that not only provides concrete definitions for these terms, but it also outlines all steps necessary to produce explanations and interpretations. The framework also allows for existing contributions to be re-contextualized such that their scope can be measured, thus making them comparable to other methods. We show that this framework is compliant with desiderata on explanations, on interpretability and on evaluation metrics. We present a use-case showing how the framework can be used to compare LIME, SHAP and MDNet, establishing their advantages and shortcomings. Finally, we discuss relevant trends in XAI as well as recommendations for future work, all from the standpoint of our framework.

Algorithms for text-generation in dialogue can be misguided. For example, in task-oriented settings, reinforcement learning that optimizes only task-success can lead to abysmal lexical diversity. We hypothesize this is due to poor theoretical understanding of the objectives in text-generation and their relation to the learning process (i.e., model training). To this end, we propose a new theoretical framework for learning to generate text in dialogue. Compared to existing theories of learning, our framework allows for analysis of the multi-faceted goals inherent to text-generation. We use our framework to develop theoretical guarantees for learners that adapt to unseen data. As an example, we apply our theory to study data-shift within a cooperative learning algorithm proposed for the GuessWhat?! visual dialogue game. From this insight, we propose a new algorithm, and empirically, we demonstrate our proposal improves both task-success and human-likeness of the generated text. Finally, we show statistics from our theory are empirically predictive of multiple qualities of the generated dialogue, suggesting our theory is useful for model-selection when human evaluations are not available.

We develop a novel theoretical framework for understating OT schemes respecting a class structure. For this purpose, we propose a convex OT program with a sum-of-norms regularization term, which provably recovers the underlying class structure under geometric assumptions. Furthermore, we derive an accelerated proximal algorithm with a closed-form projection and proximal operator scheme, thereby affording a more scalable algorithm for computing optimal transport plans. We provide a novel argument for the uniqueness of the optimum even in the absence of strong convexity. Our experiments show that the new regularizer not only results in a better preservation of the class structure in the data but also yields additional robustness to the data geometry, compared to previous regularizers.

It is not fully understood why adversarial examples can deceive neural networks and transfer between different networks. To elucidate this, several studies have hypothesized that adversarial perturbations, while appearing as noises, contain class features. This is supported by empirical evidence showing that networks trained on mislabeled adversarial examples can still generalize well to correctly labeled test samples. However, a theoretical understanding of how perturbations include class features and contribute to generalization is limited. In this study, we provide a theoretical framework for understanding learning from perturbations using a one-hidden-layer network trained on mutually orthogonal samples. Our results highlight that various adversarial perturbations, even perturbations of a few pixels, contain sufficient class features for generalization. Moreover, we reveal that the decision boundary when learning from perturbations matches that from standard samples except for specific regions under mild conditions. The code is available at https://github.com/s-kumano/learning-from-adversarial-perturbations.

Recent empirical works have successfully used unlabeled data to learn feature representations that are broadly useful in downstream classification tasks. Several of these methods are reminiscent of the well-known word2vec embedding algorithm: leveraging availability of pairs of semantically "similar" data points and "negative samples," the learner forces the inner product of representations of similar pairs with each other to be higher on average than with negative samples. The current paper uses the term contrastive learning for such algorithms and presents a theoretical framework for analyzing them by introducing latent classes and hypothesizing that semantically similar points are sampled from the same latent class. This framework allows us to show provable guarantees on the performance of the learned representations on the average classification task that is comprised of a subset of the same set of latent classes. Our generalization bound also shows that learned representations can reduce (labeled) sample complexity on downstream tasks. We conduct controlled experiments in both the text and image domains to support the theory.

Transformer-based language models rely on positional encoding (PE) to handle token order and support context length extrapolation. However, existing PE methods lack theoretical clarity and rely on limited evaluation metrics to substantiate their extrapolation claims. We propose the Bayesian Attention Mechanism (BAM), a theoretical framework that formulates positional encoding as a prior within a probabilistic model. BAM unifies existing methods (e.g., NoPE and ALiBi) and motivates a new Generalized Gaussian positional prior that substantially improves long-context generalization. Empirically, BAM enables accurate information retrieval at 500times the training context length, outperforming previous state-of-the-art context length generalization in long context retrieval accuracy while maintaining comparable perplexity and introducing minimal additional parameters.

Automata learning is a popular technique used to automatically construct an automaton model from queries. Much research went into devising ad hoc adaptations of algorithms for different types of automata. The CALF project seeks to unify these using category theory in order to ease correctness proofs and guide the design of new algorithms. In this paper, we extend CALF to cover learning of algebraic structures that may not have a coalgebraic presentation. Furthermore, we provide a detailed algorithmic account of an abstract version of the popular L* algorithm, which was missing from CALF. We instantiate the abstract theory to a large class of Set functors, by which we recover for the first time practical tree automata learning algorithms from an abstract framework and at the same time obtain new algorithms to learn algebras of quotiented polynomial functors.

Large language models (LLMs) are expected to respond accurately but often exhibit deficient reasoning or generate hallucinatory content. To address these, studies prefixed with ``Self-'' such as Self-Consistency, Self-Improve, and Self-Refine have been initiated. They share a commonality: involving LLMs evaluating and updating itself to mitigate the issues. Nonetheless, these efforts lack a unified perspective on summarization, as existing surveys predominantly focus on categorization without examining the motivations behind these works. In this paper, we summarize a theoretical framework, termed Internal Consistency, which offers unified explanations for phenomena such as the lack of reasoning and the presence of hallucinations. Internal Consistency assesses the coherence among LLMs' latent layer, decoding layer, and response layer based on sampling methodologies. Expanding upon the Internal Consistency framework, we introduce a streamlined yet effective theoretical framework capable of mining Internal Consistency, named Self-Feedback. The Self-Feedback framework consists of two modules: Self-Evaluation and Self-Update. This framework has been employed in numerous studies. We systematically classify these studies by tasks and lines of work; summarize relevant evaluation methods and benchmarks; and delve into the concern, ``Does Self-Feedback Really Work?'' We propose several critical viewpoints, including the ``Hourglass Evolution of Internal Consistency'', ``Consistency Is (Almost) Correctness'' hypothesis, and ``The Paradox of Latent and Explicit Reasoning''. Furthermore, we outline promising directions for future research. We have open-sourced the experimental code, reference list, and statistical data, available at https://github.com/IAAR-Shanghai/ICSFSurvey.

Increasing the size of a Transformer model does not always lead to enhanced performance. This phenomenon cannot be explained by the empirical scaling laws. Furthermore, improved generalization ability occurs as the model memorizes the training samples. We present a theoretical framework that sheds light on the memorization process and performance dynamics of transformer-based language models. We model the behavior of Transformers with associative memories using Hopfield networks, such that each transformer block effectively conducts an approximate nearest-neighbor search. Based on this, we design an energy function analogous to that in the modern continuous Hopfield network which provides an insightful explanation for the attention mechanism. Using the majorization-minimization technique, we construct a global energy function that captures the layered architecture of the Transformer. Under specific conditions, we show that the minimum achievable cross-entropy loss is bounded from below by a constant approximately equal to 1. We substantiate our theoretical results by conducting experiments with GPT-2 on various data sizes, as well as training vanilla Transformers on a dataset of 2M tokens.

Text-to-image (T2I) models excel on single-entity prompts but struggle with multi-subject descriptions, often showing attribute leakage, identity entanglement, and subject omissions. We introduce the first theoretical framework with a principled, optimizable objective for steering sampling dynamics toward multi-subject fidelity. Viewing flow matching (FM) through stochastic optimal control (SOC), we formulate subject disentanglement as control over a trained FM sampler. This yields two architecture-agnostic algorithms: (i) a training-free test-time controller that perturbs the base velocity with a single-pass update, and (ii) Adjoint Matching, a lightweight fine-tuning rule that regresses a control network to a backward adjoint signal while preserving base-model capabilities. The same formulation unifies prior attention heuristics, extends to diffusion models via a flow-diffusion correspondence, and provides the first fine-tuning route explicitly designed for multi-subject fidelity. Empirically, on Stable Diffusion 3.5, FLUX, and Stable Diffusion XL, both algorithms consistently improve multi-subject alignment while maintaining base-model style. Test-time control runs efficiently on commodity GPUs, and fine-tuned controllers trained on limited prompts generalize to unseen ones. We further highlight FOCUS (Flow Optimal Control for Unentangled Subjects), which achieves state-of-the-art multi-subject fidelity across models.

3D Gaussian Splatting (3DGS) has emerged as a powerful technique for real-time, high-resolution novel view synthesis. By representing scenes as a mixture of Gaussian primitives, 3DGS leverages GPU rasterization pipelines for efficient rendering and reconstruction. To optimize scene coverage and capture fine details, 3DGS employs a densification algorithm to generate additional points. However, this process often leads to redundant point clouds, resulting in excessive memory usage, slower performance, and substantial storage demands - posing significant challenges for deployment on resource-constrained devices. To address this limitation, we propose a theoretical framework that demystifies and improves density control in 3DGS. Our analysis reveals that splitting is crucial for escaping saddle points. Through an optimization-theoretic approach, we establish the necessary conditions for densification, determine the minimal number of offspring Gaussians, identify the optimal parameter update direction, and provide an analytical solution for normalizing off-spring opacity. Building on these insights, we introduce SteepGS, incorporating steepest density control, a principled strategy that minimizes loss while maintaining a compact point cloud. SteepGS achieves a ~50% reduction in Gaussian points without compromising rendering quality, significantly enhancing both efficiency and scalability.

Post-hoc out-of-distribution (OOD) detection has garnered intensive attention in reliable machine learning. Many efforts have been dedicated to deriving score functions based on logits, distances, or rigorous data distribution assumptions to identify low-scoring OOD samples. Nevertheless, these estimate scores may fail to accurately reflect the true data density or impose impractical constraints. To provide a unified perspective on density-based score design, we propose a novel theoretical framework grounded in Bregman divergence, which extends distribution considerations to encompass an exponential family of distributions. Leveraging the conjugation constraint revealed in our theorem, we introduce a ConjNorm method, reframing density function design as a search for the optimal norm coefficient p against the given dataset. In light of the computational challenges of normalization, we devise an unbiased and analytically tractable estimator of the partition function using the Monte Carlo-based importance sampling technique. Extensive experiments across OOD detection benchmarks empirically demonstrate that our proposed ConjNorm has established a new state-of-the-art in a variety of OOD detection setups, outperforming the current best method by up to 13.25% and 28.19% (FPR95) on CIFAR-100 and ImageNet-1K, respectively.

New light particles have received considerable attention in recent years. Baryon-number-violating (BNV) nucleon decays involving such light particles are able to provide stringent constraints. They exhibit distinctive experimental signatures that merit thorough investigation. We systematically investigate BNV nucleon decay with a light scalar in an effective field theory framework. Within this framework, we set stringent bounds on BNV operators using available experimental data and predict the occurrence of several BNV three-body nucleon decays. We further study contributions to dinucleon to dilepton transitions in a nucleus mediated by the scalar, which complements single nucleon decay. Finally, we provide three ultraviolet-complete models that can generate different subsets of BNV operators in leading order. Our theoretical framework will facilitate experimental searches for those exotic nucleon decays.

We establish a fundamental connection between score-based diffusion models and non-equilibrium thermodynamics by deriving performance limits based on entropy rates. Our main theoretical contribution is a lower bound on the negative log-likelihood of the data that relates model performance to entropy rates of diffusion processes. We numerically validate this bound on a synthetic dataset and investigate its tightness. By building a bridge to entropy rates - system, intrinsic, and exchange entropy - we provide new insights into the thermodynamic operation of these models, drawing parallels to Maxwell's demon and implications for thermodynamic computing hardware. Our framework connects generative modeling performance to fundamental physical principles through stochastic thermodynamics.

We are born with the ability to learn concepts by comparing diverse observations. This helps us to understand the new world in a compositional manner and facilitates extrapolation, as objects naturally consist of multiple concepts. In this work, we argue that the cognitive mechanism of comparison, fundamental to human learning, is also vital for machines to recover true concepts underlying the data. This offers correctness guarantees for the field of concept learning, which, despite its impressive empirical successes, still lacks general theoretical support. Specifically, we aim to develop a theoretical framework for the identifiability of concepts with multiple classes of observations. We show that with sufficient diversity across classes, hidden concepts can be identified without assuming specific concept types, functional relations, or parametric generative models. Interestingly, even when conditions are not globally satisfied, we can still provide alternative guarantees for as many concepts as possible based on local comparisons, thereby extending the applicability of our theory to more flexible scenarios. Moreover, the hidden structure between classes and concepts can also be identified nonparametrically. We validate our theoretical results in both synthetic and real-world settings.

Transformer architectures have revolutionized artificial intelligence (AI) through their attention mechanisms, yet the computational principles underlying their success remain opaque. We present a novel theoretical framework that reinterprets the core computation of attention as Pavlovian conditioning. Our model finds a direct mathematical analogue in linear attention, which simplifies the analysis of the underlying associative process. We demonstrate that attention's queries, keys, and values can be mapped to the three elements of classical conditioning: test stimuli that probe associations, conditional stimuli (CS) that serve as retrieval cues, and unconditional stimuli (US) that contain response information. Through this lens, we suggest that each attention operation constructs a transient associative memory via a Hebbian rule, where CS-US pairs form dynamic associations that test stimuli can later retrieve. Our framework yields several theoretical insights grounded in this linearized model: (1) a capacity theorem showing that attention heads can store O(d_k) associations before interference degrades retrieval; (2) an error propagation analysis revealing fundamental architectural trade-offs of balancing model depth, width, and head redundancy to maintain reliability; and (3) an understanding of how biologically plausible learning rules could enhance transformer architectures. By establishing this deep connection, we suggest that the success of modern AI may stem not from architectural novelty alone, but from implementing computational principles that biology optimized over millions of years of evolution.

A modern paradigm for generalization in machine learning and AI consists of pre-training a task-agnostic foundation model, generally obtained using self-supervised and multimodal contrastive learning. The resulting representations can be used for prediction on a downstream task for which no labeled data is available. We present a theoretical framework to better understand this approach, called zero-shot prediction. We identify the target quantities that zero-shot prediction aims to learn, or learns in passing, and the key conditional independence relationships that enable its generalization ability.

Top-k decoding is a widely used method for sampling from LLMs: at each token, only the largest k next-token-probabilities are kept, and the next token is sampled after re-normalizing them to sum to unity. Top-k and other sampling methods are motivated by the intuition that true next-token distributions are sparse, and the noisy LLM probabilities need to be truncated. However, to our knowledge, a precise theoretical motivation for the use of top-k decoding is missing. In this work, we develop a theoretical framework that both explains and generalizes top-k decoding. We view decoding at a fixed token as the recovery of a sparse probability distribution. We consider Bregman decoders obtained by minimizing a separable Bregman divergence (for both the primal and dual cases) with a sparsity-inducing ell_0 regularization. Despite the combinatorial nature of the objective, we show how to optimize it efficiently for a large class of divergences. We show that the optimal decoding strategies are greedy, and further that the loss function is discretely convex in k, so that binary search provably and efficiently finds the optimal k. We show that top-k decoding arises as a special case for the KL divergence, and identify new decoding strategies that have distinct behaviors (e.g., non-linearly up-weighting larger probabilities after re-normalization).

This paper formalizes an emerging learning paradigm that uses a trained model as a reference to guide and enhance the training of a target model through strategic data selection or weighting, named model steering. While ad-hoc methods have been used in various contexts, including the training of large foundation models, its underlying principles remain insufficiently understood, leading to sub-optimal performance. In this work, we propose a theory-driven framework for model steering called DRRho risk minimization, which is rooted in Distributionally Robust Optimization (DRO). Through a generalization analysis, we provide theoretical insights into why this approach improves generalization and data efficiency compared to training without a reference model. To the best of our knowledge, this is the first time such theoretical insights are provided for the new learning paradigm, which significantly enhance our understanding and practice of model steering. Building on these insights and the connection between contrastive learning and DRO, we introduce a novel method for Contrastive Language-Image Pretraining (CLIP) with a reference model, termed DRRho-CLIP. Extensive experiments validate the theoretical insights, reveal a superior scaling law compared to CLIP without a reference model, and demonstrate its strength over existing heuristic approaches.

Contrastive learning -- a modern approach to extract useful representations from unlabeled data by training models to distinguish similar samples from dissimilar ones -- has driven significant progress in foundation models. In this work, we develop a new theoretical framework for analyzing data augmentation-based contrastive learning, with a focus on SimCLR as a representative example. Our approach is based on the concept of approximate sufficient statistics, which we extend beyond its original definition in oko2025statistical for contrastive language-image pretraining (CLIP) using KL-divergence. We generalize it to equivalent forms and general f-divergences, and show that minimizing SimCLR and other contrastive losses yields encoders that are approximately sufficient. Furthermore, we demonstrate that these near-sufficient encoders can be effectively adapted to downstream regression and classification tasks, with performance depending on their sufficiency and the error induced by data augmentation in contrastive learning. Concrete examples in linear regression and topic classification are provided to illustrate the broad applicability of our results.

We propose a theoretical framework demonstrating that popular Large Language Model (LLM) alignment methods, including Reinforcement Learning from Human Feedback (RLHF) and alternatives, fundamentally function as divergence estimators between aligned (preferred or safe) and unaligned (less-preferred or harmful) distributions. This explains the separation phenomenon between safe and harmful prompts in the model hidden representation after alignment. Inspired by the theoretical results, we identify that some alignment methods are better than others in terms of separation and, introduce a new method, KLDO, and further demonstrate the implication of our theories. We advocate for compliance-refusal datasets over preference datasets to enhance safety alignment, supported by both theoretical reasoning and empirical evidence. Additionally, to quantify safety separation, we leverage a distance metric in the representation space and statistically validate its efficacy as a statistical significant indicator of LLM resilience against jailbreak attacks.

Class imbalance (CI) is a longstanding problem in machine learning, slowing down training and reducing performances. Although empirical remedies exist, it is often unclear which ones work best and when, due to the lack of an overarching theory. We address a common case of imbalance, that of anomaly (or outlier) detection. We provide a theoretical framework to analyze, interpret and address CI. It is based on an exact solution of the teacher-student perceptron model, through replica theory. Within this framework, one can distinguish several sources of CI: either intrinsic, train or test imbalance. Our analysis reveals that the optimal train imbalance is generally different from 50%, with a non trivial dependence on the intrinsic imbalance, the abundance of data and on the noise in the learning. Moreover, there is a crossover between a small noise training regime where results are independent of the noise level to a high noise regime where performances quickly degrade with noise. Our results challenge some of the conventional wisdom on CI and offer practical guidelines to address it.

Transformers have achieved remarkable success across diverse domains, but their monolithic architecture presents challenges in interpretability, adaptability, and scalability. This paper introduces a novel modular Transformer architecture that explicitly decouples knowledge and reasoning through a generalized cross-attention mechanism to a globally shared knowledge base with layer-specific transformations, specifically designed for effective knowledge retrieval. Critically, we provide a rigorous mathematical derivation demonstrating that the Feed-Forward Network (FFN) in a standard Transformer is a specialized case (a closure) of this generalized cross-attention, revealing its role in implicit knowledge retrieval and validating our design. This theoretical framework provides a new lens for understanding FFNs and lays the foundation for future research exploring enhanced interpretability, adaptability, and scalability, enabling richer interplay with external knowledge bases and other systems.

We introduce a new analytical framework to quantify the changes in a machine learning algorithm's output distribution following the inclusion of a few data points in its training set, a notion we define as leave-one-out distinguishability (LOOD). This problem is key to measuring data **memorization** and **information leakage** in machine learning, and the **influence** of training data points on model predictions. We illustrate how our method broadens and refines existing empirical measures of memorization and privacy risks associated with training data. We use Gaussian processes to model the randomness of machine learning algorithms, and validate LOOD with extensive empirical analysis of information leakage using membership inference attacks. Our theoretical framework enables us to investigate the causes of information leakage and where the leakage is high. For example, we analyze the influence of activation functions, on data memorization. Additionally, our method allows us to optimize queries that disclose the most significant information about the training data in the leave-one-out setting. We illustrate how optimal queries can be used for accurate **reconstruction** of training data.

We introduce a theoretical framework for differentiable surface evolution that allows discrete topology changes through the use of topological derivatives for variational optimization of image functionals. While prior methods for inverse rendering of geometry rely on silhouette gradients for topology changes, such signals are sparse. In contrast, our theory derives topological derivatives that relate the introduction of vanishing holes and phases to changes in image intensity. As a result, we enable differentiable shape perturbations in the form of hole or phase nucleation. We validate the proposed theory with optimization of closed curves in 2D and surfaces in 3D to lend insights into limitations of current methods and enable improved applications such as image vectorization, vector-graphics generation from text prompts, single-image reconstruction of shape ambigrams and multi-view 3D reconstruction.

Although diffusion models (DMs) have shown promising performances in a number of tasks (e.g., speech synthesis and image generation), they might suffer from error propagation because of their sequential structure. However, this is not certain because some sequential models, such as Conditional Random Field (CRF), are free from this problem. To address this issue, we develop a theoretical framework to mathematically formulate error propagation in the architecture of DMs, The framework contains three elements, including modular error, cumulative error, and propagation equation. The modular and cumulative errors are related by the equation, which interprets that DMs are indeed affected by error propagation. Our theoretical study also suggests that the cumulative error is closely related to the generation quality of DMs. Based on this finding, we apply the cumulative error as a regularization term to reduce error propagation. Because the term is computationally intractable, we derive its upper bound and design a bootstrap algorithm to efficiently estimate the bound for optimization. We have conducted extensive experiments on multiple image datasets, showing that our proposed regularization reduces error propagation, significantly improves vanilla DMs, and outperforms previous baselines.

In recent years, concept-based approaches have emerged as some of the most promising explainability methods to help us interpret the decisions of Artificial Neural Networks (ANNs). These methods seek to discover intelligible visual 'concepts' buried within the complex patterns of ANN activations in two key steps: (1) concept extraction followed by (2) importance estimation. While these two steps are shared across methods, they all differ in their specific implementations. Here, we introduce a unifying theoretical framework that comprehensively defines and clarifies these two steps. This framework offers several advantages as it allows us: (i) to propose new evaluation metrics for comparing different concept extraction approaches; (ii) to leverage modern attribution methods and evaluation metrics to extend and systematically evaluate state-of-the-art concept-based approaches and importance estimation techniques; (iii) to derive theoretical guarantees regarding the optimality of such methods. We further leverage our framework to try to tackle a crucial question in explainability: how to efficiently identify clusters of data points that are classified based on a similar shared strategy. To illustrate these findings and to highlight the main strategies of a model, we introduce a visual representation called the strategic cluster graph. Finally, we present https://serre-lab.github.io/Lens, a dedicated website that offers a complete compilation of these visualizations for all classes of the ImageNet dataset.

We introduce a theoretical framework for sampling from unnormalized densities based on a smoothing scheme that uses an isotropic Gaussian kernel with a single fixed noise scale. We prove one can decompose sampling from a density (minimal assumptions made on the density) into a sequence of sampling from log-concave conditional densities via accumulation of noisy measurements with equal noise levels. Our construction is unique in that it keeps track of a history of samples, making it non-Markovian as a whole, but it is lightweight algorithmically as the history only shows up in the form of a running empirical mean of samples. Our sampling algorithm generalizes walk-jump sampling (Saremi & Hyv\"arinen, 2019). The "walk" phase becomes a (non-Markovian) chain of (log-concave) Markov chains. The "jump" from the accumulated measurements is obtained by empirical Bayes. We study our sampling algorithm quantitatively using the 2-Wasserstein metric and compare it with various Langevin MCMC algorithms. We also report a remarkable capacity of our algorithm to "tunnel" between modes of a distribution.

Contrastive learning (CL) has emerged as a powerful technique for representation learning, with or without label supervision. However, supervised CL is prone to collapsing representations of subclasses within a class by not capturing all their features, and unsupervised CL may suppress harder class-relevant features by focusing on learning easy class-irrelevant features; both significantly compromise representation quality. Yet, there is no theoretical understanding of class collapse or feature suppression at test time. We provide the first unified theoretically rigorous framework to determine which features are learnt by CL. Our analysis indicate that, perhaps surprisingly, bias of (stochastic) gradient descent towards finding simpler solutions is a key factor in collapsing subclass representations and suppressing harder class-relevant features. Moreover, we present increasing embedding dimensionality and improving the quality of data augmentations as two theoretically motivated solutions to {feature suppression}. We also provide the first theoretical explanation for why employing supervised and unsupervised CL together yields higher-quality representations, even when using commonly-used stochastic gradient methods.

Reinforcement learning (RL) problems can be challenging without well-shaped rewards. Prior work on provably efficient RL methods generally proposes to address this issue with dedicated exploration strategies. However, another way to tackle this challenge is to reformulate it as a multi-task RL problem, where the task space contains not only the challenging task of interest but also easier tasks that implicitly function as a curriculum. Such a reformulation opens up the possibility of running existing multi-task RL methods as a more efficient alternative to solving a single challenging task from scratch. In this work, we provide a theoretical framework that reformulates a single-task RL problem as a multi-task RL problem defined by a curriculum. Under mild regularity conditions on the curriculum, we show that sequentially solving each task in the multi-task RL problem is more computationally efficient than solving the original single-task problem, without any explicit exploration bonuses or other exploration strategies. We also show that our theoretical insights can be translated into an effective practical learning algorithm that can accelerate curriculum learning on simulated robotic tasks.

Neural networks are capable of translating between languages -- in some cases even between two languages where there is little or no access to parallel translations, in what is known as Unsupervised Machine Translation (UMT). Given this progress, it is intriguing to ask whether machine learning tools can ultimately enable understanding animal communication, particularly that of highly intelligent animals. We propose a theoretical framework for analyzing UMT when no parallel translations are available and when it cannot be assumed that the source and target corpora address related subject domains or posses similar linguistic structure. We exemplify this theory with two stylized models of language, for which our framework provides bounds on necessary sample complexity; the bounds are formally proven and experimentally verified on synthetic data. These bounds show that the error rates are inversely related to the language complexity and amount of common ground. This suggests that unsupervised translation of animal communication may be feasible if the communication system is sufficiently complex.

Linear time-invariant state space models (SSM) are a classical model from engineering and statistics, that have recently been shown to be very promising in machine learning through the Structured State Space sequence model (S4). A core component of S4 involves initializing the SSM state matrix to a particular matrix called a HiPPO matrix, which was empirically important for S4's ability to handle long sequences. However, the specific matrix that S4 uses was actually derived in previous work for a particular time-varying dynamical system, and the use of this matrix as a time-invariant SSM had no known mathematical interpretation. Consequently, the theoretical mechanism by which S4 models long-range dependencies actually remains unexplained. We derive a more general and intuitive formulation of the HiPPO framework, which provides a simple mathematical interpretation of S4 as a decomposition onto exponentially-warped Legendre polynomials, explaining its ability to capture long dependencies. Our generalization introduces a theoretically rich class of SSMs that also lets us derive more intuitive S4 variants for other bases such as the Fourier basis, and explains other aspects of training S4, such as how to initialize the important timescale parameter. These insights improve S4's performance to 86% on the Long Range Arena benchmark, with 96% on the most difficult Path-X task.

We propose a novel framework to study asynchronous federated learning optimization with delays in gradient updates. Our theoretical framework extends the standard FedAvg aggregation scheme by introducing stochastic aggregation weights to represent the variability of the clients update time, due for example to heterogeneous hardware capabilities. Our formalism applies to the general federated setting where clients have heterogeneous datasets and perform at least one step of stochastic gradient descent (SGD). We demonstrate convergence for such a scheme and provide sufficient conditions for the related minimum to be the optimum of the federated problem. We show that our general framework applies to existing optimization schemes including centralized learning, FedAvg, asynchronous FedAvg, and FedBuff. The theory here provided allows drawing meaningful guidelines for designing a federated learning experiment in heterogeneous conditions. In particular, we develop in this work FedFix, a novel extension of FedAvg enabling efficient asynchronous federated training while preserving the convergence stability of synchronous aggregation. We empirically demonstrate our theory on a series of experiments showing that asynchronous FedAvg leads to fast convergence at the expense of stability, and we finally demonstrate the improvements of FedFix over synchronous and asynchronous FedAvg.

The reverse derivative is a fundamental operation in machine learning and automatic differentiation. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by Cartesian differential categories for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

Deep convolutional neural networks have led to breakthrough results in numerous practical machine learning tasks such as classification of images in the ImageNet data set, control-policy-learning to play Atari games or the board game Go, and image captioning. Many of these applications first perform feature extraction and then feed the results thereof into a trainable classifier. The mathematical analysis of deep convolutional neural networks for feature extraction was initiated by Mallat, 2012. Specifically, Mallat considered so-called scattering networks based on a wavelet transform followed by the modulus non-linearity in each network layer, and proved translation invariance (asymptotically in the wavelet scale parameter) and deformation stability of the corresponding feature extractor. This paper complements Mallat's results by developing a theory that encompasses general convolutional transforms, or in more technical parlance, general semi-discrete frames (including Weyl-Heisenberg filters, curvelets, shearlets, ridgelets, wavelets, and learned filters), general Lipschitz-continuous non-linearities (e.g., rectified linear units, shifted logistic sigmoids, hyperbolic tangents, and modulus functions), and general Lipschitz-continuous pooling operators emulating, e.g., sub-sampling and averaging. In addition, all of these elements can be different in different network layers. For the resulting feature extractor we prove a translation invariance result of vertical nature in the sense of the features becoming progressively more translation-invariant with increasing network depth, and we establish deformation sensitivity bounds that apply to signal classes such as, e.g., band-limited functions, cartoon functions, and Lipschitz functions.

Consistency models (CMs) are a powerful class of diffusion-based generative models optimized for fast sampling. Most existing CMs are trained using discretized timesteps, which introduce additional hyperparameters and are prone to discretization errors. While continuous-time formulations can mitigate these issues, their success has been limited by training instability. To address this, we propose a simplified theoretical framework that unifies previous parameterizations of diffusion models and CMs, identifying the root causes of instability. Based on this analysis, we introduce key improvements in diffusion process parameterization, network architecture, and training objectives. These changes enable us to train continuous-time CMs at an unprecedented scale, reaching 1.5B parameters on ImageNet 512x512. Our proposed training algorithm, using only two sampling steps, achieves FID scores of 2.06 on CIFAR-10, 1.48 on ImageNet 64x64, and 1.88 on ImageNet 512x512, narrowing the gap in FID scores with the best existing diffusion models to within 10%.

As AI model size grows, neural scaling laws have become a crucial tool to predict the improvements of large models when increasing capacity and the size of original (human or natural) training data. Yet, the widespread use of popular models means that the ecosystem of online data and text will co-evolve to progressively contain increased amounts of synthesized data. In this paper we ask: How will the scaling laws change in the inevitable regime where synthetic data makes its way into the training corpus? Will future models, still improve, or be doomed to degenerate up to total (model) collapse? We develop a theoretical framework of model collapse through the lens of scaling laws. We discover a wide range of decay phenomena, analyzing loss of scaling, shifted scaling with number of generations, the ''un-learning" of skills, and grokking when mixing human and synthesized data. Our theory is validated by large-scale experiments with a transformer on an arithmetic task and text generation using the large language model Llama2.

This paper investigates the limitations of the normalization in attention mechanisms. We begin with a theoretical framework that enables the identification of the model's selective ability and the geometric separation involved in token selection. Our analysis includes explicit bounds on distances and separation criteria for token vectors under softmax scaling. Through experiments with pre-trained GPT-2 model, we empirically validate our theoretical results and analyze key behaviors of the attention mechanism. Notably, we demonstrate that as the number of selected tokens increases, the model's ability to distinguish informative tokens declines, often converging toward a uniform selection pattern. We also show that gradient sensitivity under softmax normalization presents challenges during training, especially at low temperature settings. These findings advance current understanding of softmax-based attention mechanism and motivate the need for more robust normalization and selection strategies in future attention architectures.

We introduce a theoretical framework for diffusion-based image editing by formulating it as a reverse-time bridge modeling problem. This approach modifies the backward process of a pretrained diffusion model to construct a bridge that converges to an implicit distribution associated with the editing target at time 0. Building on this framework, we propose h-Edit, a novel editing method that utilizes Doob's h-transform and Langevin Monte Carlo to decompose the update of an intermediate edited sample into two components: a "reconstruction" term and an "editing" term. This decomposition provides flexibility, allowing the reconstruction term to be computed via existing inversion techniques and enabling the combination of multiple editing terms to handle complex editing tasks. To our knowledge, h-Edit is the first training-free method capable of performing simultaneous text-guided and reward-model-based editing. Extensive experiments, both quantitative and qualitative, show that h-Edit outperforms state-of-the-art baselines in terms of editing effectiveness and faithfulness. Our source code is available at https://github.com/nktoan/h-edit.

As the range of applications for Large Language Models (LLMs) continues to grow, the demand for effective serving solutions becomes increasingly critical. Despite the versatility of LLMs, no single model can optimally address all tasks and applications, particularly when balancing performance with cost. This limitation has led to the development of LLM routing systems, which combine the strengths of various models to overcome the constraints of individual LLMs. Yet, the absence of a standardized benchmark for evaluating the performance of LLM routers hinders progress in this area. To bridge this gap, we present RouterBench, a novel evaluation framework designed to systematically assess the efficacy of LLM routing systems, along with a comprehensive dataset comprising over 405k inference outcomes from representative LLMs to support the development of routing strategies. We further propose a theoretical framework for LLM routing, and deliver a comparative analysis of various routing approaches through RouterBench, highlighting their potentials and limitations within our evaluation framework. This work not only formalizes and advances the development of LLM routing systems but also sets a standard for their assessment, paving the way for more accessible and economically viable LLM deployments. The code and data are available at https://github.com/withmartian/routerbench.

Understanding Transformer-based models has attracted significant attention, as they lie at the heart of recent technological advances across machine learning. While most interpretability methods rely on running models over inputs, recent work has shown that a zero-pass approach, where parameters are interpreted directly without a forward/backward pass is feasible for some Transformer parameters, and for two-layer attention networks. In this work, we present a theoretical analysis where all parameters of a trained Transformer are interpreted by projecting them into the embedding space, that is, the space of vocabulary items they operate on. We derive a simple theoretical framework to support our arguments and provide ample evidence for its validity. First, an empirical analysis showing that parameters of both pretrained and fine-tuned models can be interpreted in embedding space. Second, we present two applications of our framework: (a) aligning the parameters of different models that share a vocabulary, and (b) constructing a classifier without training by ``translating'' the parameters of a fine-tuned classifier to parameters of a different model that was only pretrained. Overall, our findings open the door to interpretation methods that, at least in part, abstract away from model specifics and operate in the embedding space only.

We provide a theoretical framework for Reinforcement Learning with Human Feedback (RLHF). Our analysis shows that when the true reward function is linear, the widely used maximum likelihood estimator (MLE) converges under both the Bradley-Terry-Luce (BTL) model and the Plackett-Luce (PL) model. However, we show that when training a policy based on the learned reward model, MLE fails while a pessimistic MLE provides policies with improved performance under certain coverage assumptions. Additionally, we demonstrate that under the PL model, the true MLE and an alternative MLE that splits the K-wise comparison into pairwise comparisons both converge. Moreover, the true MLE is asymptotically more efficient. Our results validate the empirical success of existing RLHF algorithms in InstructGPT and provide new insights for algorithm design. Furthermore, our results unify the problem of RLHF and max-entropy Inverse Reinforcement Learning (IRL), and provide the first sample complexity bound for max-entropy IRL.

In recent years, diffusion models have emerged as powerful tools for generating ensemble members in meteorology. In this work, we demonstrate that a Denoising Diffusion Implicit Model (DDIM) can effectively control ensemble variance by varying the number of diffusion steps. Introducing a theoretical framework, we relate diffusion steps to the variance expressed by the reverse diffusion process. Focusing on reanalysis downscaling, we propose an ensemble diffusion model for the full ERA5-to-CERRA domain, generating variance-calibrated ensemble members for wind speed at full spatial and temporal resolution. Our method aligns global mean variance with a reference ensemble dataset and ensures spatial variance is distributed in accordance with observed meteorological variability. Additionally, we address the lack of ensemble information in the CARRA dataset, showcasing the utility of our approach for efficient, high-resolution ensemble generation.

With the rapid advancement of AI systems, their abilities to store, retrieve, and utilize information over the long term - referred to as long-term memory - have become increasingly significant. These capabilities are crucial for enhancing the performance of AI systems across a wide range of tasks. However, there is currently no comprehensive survey that systematically investigates AI's long-term memory capabilities, formulates a theoretical framework, and inspires the development of next-generation AI long-term memory systems. This paper begins by systematically introducing the mechanisms of human long-term memory, then explores AI long-term memory mechanisms, establishing a mapping between the two. Based on the mapping relationships identified, we extend the current cognitive architectures and propose the Cognitive Architecture of Self-Adaptive Long-term Memory (SALM). SALM provides a theoretical framework for the practice of AI long-term memory and holds potential for guiding the creation of next-generation long-term memory driven AI systems. Finally, we delve into the future directions and application prospects of AI long-term memory.

We establish a new theoretical framework for learning under multi-class, instance-dependent label noise. This framework casts learning with label noise as a form of domain adaptation, in particular, domain adaptation under posterior drift. We introduce the concept of relative signal strength (RSS), a pointwise measure that quantifies the transferability from noisy to clean posterior. Using RSS, we establish nearly matching upper and lower bounds on the excess risk. Our theoretical findings support the simple Noise Ignorant Empirical Risk Minimization (NI-ERM) principle, which minimizes empirical risk while ignoring label noise. Finally, we translate this theoretical insight into practice: by using NI-ERM to fit a linear classifier on top of a self-supervised feature extractor, we achieve state-of-the-art performance on the CIFAR-N data challenge.

A primary challenge in large language model (LLM) development is their onerous pre-training cost. Typically, such pre-training involves optimizing a self-supervised objective (such as next-token prediction) over a large corpus. This paper explores a promising paradigm to improve LLM pre-training efficiency and quality by suitably leveraging a small language model (SLM). In particular, this paradigm relies on an SLM to both (1) provide soft labels as additional training supervision, and (2) select a small subset of valuable ("informative" and "hard") training examples. Put together, this enables an effective transfer of the SLM's predictive distribution to the LLM, while prioritizing specific regions of the training data distribution. Empirically, this leads to reduced LLM training time compared to standard training, while improving the overall quality. Theoretically, we develop a statistical framework to systematically study the utility of SLMs in enabling efficient training of high-quality LLMs. In particular, our framework characterizes how the SLM's seemingly low-quality supervision can enhance the training of a much more capable LLM. Furthermore, it also highlights the need for an adaptive utilization of such supervision, by striking a balance between the bias and variance introduced by the SLM-provided soft labels. We corroborate our theoretical framework by improving the pre-training of an LLM with 2.8B parameters by utilizing a smaller LM with 1.5B parameters on the Pile dataset.

Facing the current debate on whether Large Language Models (LLMs) attain near-human intelligence levels (Mitchell & Krakauer, 2023; Bubeck et al., 2023; Kosinski, 2023; Shiffrin & Mitchell, 2023; Ullman, 2023), the current study introduces a benchmark for evaluating social intelligence, one of the most distinctive aspects of human cognition. We developed a comprehensive theoretical framework for social dynamics and introduced two evaluation tasks: Inverse Reasoning (IR) and Inverse Inverse Planning (IIP). Our approach also encompassed a computational model based on recursive Bayesian inference, adept at elucidating diverse human behavioral patterns. Extensive experiments and detailed analyses revealed that humans surpassed the latest GPT models in overall performance, zero-shot learning, one-shot generalization, and adaptability to multi-modalities. Notably, GPT models demonstrated social intelligence only at the most basic order (order = 0), in stark contrast to human social intelligence (order >= 2). Further examination indicated a propensity of LLMs to rely on pattern recognition for shortcuts, casting doubt on their possession of authentic human-level social intelligence. Our codes, dataset, appendix and human data are released at https://github.com/bigai-ai/Evaluate-n-Model-Social-Intelligence.

This paper examines the challenges associated with achieving life-long superalignment in AI systems, particularly large language models (LLMs). Superalignment is a theoretical framework that aspires to ensure that superintelligent AI systems act in accordance with human values and goals. Despite its promising vision, we argue that achieving superalignment requires substantial changes in the current LLM architectures due to their inherent limitations in comprehending and adapting to the dynamic nature of these human ethics and evolving global scenarios. We dissect the challenges of encoding an ever-changing spectrum of human values into LLMs, highlighting the discrepancies between static AI models and the dynamic nature of human societies. To illustrate these challenges, we analyze two distinct examples: one demonstrates a qualitative shift in human values, while the other presents a quantifiable change. Through these examples, we illustrate how LLMs, constrained by their training data, fail to align with contemporary human values and scenarios. The paper concludes by exploring potential strategies to address and possibly mitigate these alignment discrepancies, suggesting a path forward in the pursuit of more adaptable and responsive AI systems.

Representation learning is all about discovering the hidden modular attributes that generate the data faithfully. We explore the potential of Denoising Diffusion Probabilistic Model (DM) in unsupervised learning of the modular attributes. We build a theoretical framework that connects the diffusion time-steps and the hidden attributes, which serves as an effective inductive bias for unsupervised learning. Specifically, the forward diffusion process incrementally adds Gaussian noise to samples at each time-step, which essentially collapses different samples into similar ones by losing attributes, e.g., fine-grained attributes such as texture are lost with less noise added (i.e., early time-steps), while coarse-grained ones such as shape are lost by adding more noise (i.e., late time-steps). To disentangle the modular attributes, at each time-step t, we learn a t-specific feature to compensate for the newly lost attribute, and the set of all 1,...,t-specific features, corresponding to the cumulative set of lost attributes, are trained to make up for the reconstruction error of a pre-trained DM at time-step t. On CelebA, FFHQ, and Bedroom datasets, the learned feature significantly improves attribute classification and enables faithful counterfactual generation, e.g., interpolating only one specified attribute between two images, validating the disentanglement quality. Codes are in https://github.com/yue-zhongqi/diti.

This paper studies the theoretical framework of the alignment process of generative models with Reinforcement Learning from Human Feedback (RLHF). We consider a standard mathematical formulation, the reverse-KL regularized contextual bandit for RLHF. Despite its widespread practical application, a rigorous theoretical analysis of this formulation remains open. We investigate its behavior in three distinct settings -- offline, online, and hybrid -- and propose efficient algorithms with finite-sample theoretical guarantees. Moving towards practical applications, our framework, with a robust approximation of the information-theoretical policy improvement oracle, naturally gives rise to several novel RLHF algorithms. This includes an iterative version of the Direct Preference Optimization (DPO) algorithm for online settings, and a multi-step rejection sampling strategy for offline scenarios. Our empirical evaluations on real-world alignment experiment of large language model demonstrate that these proposed methods significantly surpass existing strong baselines, such as DPO and Rejection Sampling Optimization (RSO), showcasing the connections between solid theoretical foundations and their powerful practical implementations.

Certain statistical models are capable of interpreting input strings as instructions, or prompts, and carry out tasks based on them. Many approaches to prompting and pre-training these models involve the automated generation of these prompts. We call these approaches meta-prompting, or prompting to obtain prompts. We propose a theoretical framework based on category theory to generalize and describe them. This framework is flexible enough to account for LLM stochasticity; and allows us to obtain formal results around task agnosticity and equivalence of various meta-prompting approaches. We experiment with meta-prompting in two active areas of model research: creativity and ideation. We find that user preference favors (p < 0.01) the prompts generated under meta-prompting, as well as their corresponding outputs, over a series of hardcoded baseline prompts that include the original task prompt. Using our framework, we argue that meta-prompting is more effective than basic prompting at generating desirable outputs.

We propose a conceptually simple and lightweight framework for improving the robustness of vision models through the combination of knowledge distillation and data augmentation. We address the conjecture that larger models do not make for better teachers by showing strong gains in out-of-distribution robustness when distilling from pretrained foundation models. Following this finding, we propose Discrete Adversarial Distillation (DAD), which leverages a robust teacher to generate adversarial examples and a VQGAN to discretize them, creating more informative samples than standard data augmentation techniques. We provide a theoretical framework for the use of a robust teacher in the knowledge distillation with data augmentation setting and demonstrate strong gains in out-of-distribution robustness and clean accuracy across different student architectures. Notably, our method adds minor computational overhead compared to similar techniques and can be easily combined with other data augmentations for further improvements.

Emotion regulation is a crucial element in dealing with emotional events and has positive effects on mental health. This paper aims to provide a more comprehensive understanding of emotional events by introducing a new French corpus of emotional narratives collected using a questionnaire for emotion regulation. We follow the theoretical framework of the Component Process Model which considers emotions as dynamic processes composed of four interrelated components (behavior, feeling, thinking and territory). Each narrative is related to a discrete emotion and is structured based on all emotion components by the writers. We study the interaction of components and their impact on emotion classification with machine learning methods and pre-trained language models. Our results show that each component improves prediction performance, and that the best results are achieved by jointly considering all components. Our results also show the effectiveness of pre-trained language models in predicting discrete emotion from certain components, which reveal differences in how emotion components are expressed.

Machine learning interatomic potentials (MLIPs) balance high accuracy and lower costs compared to density functional theory calculations, but their performance often depends on the size and diversity of training datasets. Large datasets improve model accuracy and generalization but are computationally expensive to produce and train on, while smaller datasets risk discarding rare but important atomic environments and compromising MLIP accuracy/reliability. Here, we develop an information-theoretical framework to quantify the efficiency of dataset compression methods and propose an algorithm that maximizes this efficiency. By framing atomistic dataset compression as an instance of the minimum set cover (MSC) problem over atom-centered environments, our method identifies the smallest subset of structures that contains as much information as possible from the original dataset while pruning redundant information. The approach is extensively demonstrated on the GAP-20 and TM23 datasets, and validated on 64 varied datasets from the ColabFit repository. Across all cases, MSC consistently retains outliers, preserves dataset diversity, and reproduces the long-tail distributions of forces even at high compression rates, outperforming other subsampling methods. Furthermore, MLIPs trained on MSC-compressed datasets exhibit reduced error for out-of-distribution data even in low-data regimes. We explain these results using an outlier analysis and show that such quantitative conclusions could not be achieved with conventional dimensionality reduction methods. The algorithm is implemented in the open-source QUESTS package and can be used for several tasks in atomistic modeling, from data subsampling, outlier detection, and training improved MLIPs at a lower cost.

In this work, we prove existence and uniqueness of a bounded viscosity solution for the Cauchy problem of degenerate parabolic equations with variable exponent coefficients. We construct the solution directly using the stochastic representation, then verify it satisfies the Cauchy problem. The corresponding SDE, on the other hand, allows the drift and diffusion coefficients to respond nonlinearly to the current state through the state-dependent variable exponents, and thus, extends the expressive power of classical SDEs to better capture complex dynamics. To validate our theoretical framework, we conduct comprehensive numerical experiments comparing finite difference solutions (Crank-Nicolson on logarithmic grids) with Monte Carlo simulations of the SDE.

Aligning large language models with human preferences is critical for creating reliable and controllable AI systems. A human preference can be visualized as a high-dimensional vector where different directions represent trade-offs between desired attributes (e.g., helpfulness vs. verbosity). Yet, because the training data often reflects dominant, average preferences, LLMs tend to perform well on common requests but fall short in specific, individual needs. This mismatch creates a preference coverage gap. Existing methods often address this through costly retraining, which may not be generalized to the full spectrum of diverse preferences. This brittleness means that when a user's request reflects a nuanced preference deviating from the training data's central tendency, model performance can degrade unpredictably. To address this challenge, we introduce Robust Preference Selection (RPS), a post-hoc, training-free method by leveraging directional neighborhood consensus. Instead of forcing a model to generate a response from a single, highly specific preference, RPS samples multiple responses from a local neighborhood of related preferences to create a superior candidate pool. It then selects the response that best aligns with the user's original intent. We provide a theoretical framework showing our neighborhood generation strategy is provably superior to a strong baseline that also samples multiple candidates. Comprehensive experiments across three distinct alignment paradigms (DPA, DPO, and SFT) demonstrate that RPS consistently improves robustness against this baseline, achieving win rates of up to 69% on challenging preferences from under-represented regions of the space without any model retraining. Our work presents a practical, theoretically-grounded solution for enhancing the reliability of preference-aligned models.

Post-training processes are essential phases in grounding pre-trained language models to real-world tasks, with learning from demonstrations or preference signals playing a crucial role in this adaptation. We present a unified theoretical framework bridging Supervised Fine-Tuning (SFT) and preference learning in Large Language Model (LLM) post-training. Through rigorous mathematical derivation, we demonstrate that both SFT and preference learning methods like Direct Preference Optimization (DPO) operate within the same optimal policy-reward subspace, with SFT representing a special case of implicit reward learning. Our analysis reveals a critical limitation in conventional SFT: the KL divergence term in distribution matching becomes constant with respect to the policy during optimization, failing to constrain model updates. To address this, we propose a simple yet effective learning rate reduction approach that yields significant performance improvements (up to 25\% relative gain and 6\% absolute win rate increase in instruction following tasks. Additionally, we derive alternative SFT objectives from various f-divergence functions that preserve the KL term during optimization, further enhancing post-DPO model performance. Finally, we extend the theoretical relationship between LLM logits and Q-functions from preference learning to the SFT context, providing mathematical derivations and experimental validation.

Reliable machine learning and statistical analysis rely on diverse, well-distributed training data. However, real-world datasets are often limited in size and exhibit underrepresentation across key subpopulations, leading to biased predictions and reduced performance, particularly in supervised tasks such as classification. To address these challenges, we propose Conditional Data Synthesis Augmentation (CoDSA), a novel framework that leverages generative models, such as diffusion models, to synthesize high-fidelity data for improving model performance across multimodal domains including tabular, textual, and image data. CoDSA generates synthetic samples that faithfully capture the conditional distributions of the original data, with a focus on under-sampled or high-interest regions. Through transfer learning, CoDSA fine-tunes pre-trained generative models to enhance the realism of synthetic data and increase sample density in sparse areas. This process preserves inter-modal relationships, mitigates data imbalance, improves domain adaptation, and boosts generalization. We also introduce a theoretical framework that quantifies the statistical accuracy improvements enabled by CoDSA as a function of synthetic sample volume and targeted region allocation, providing formal guarantees of its effectiveness. Extensive experiments demonstrate that CoDSA consistently outperforms non-adaptive augmentation strategies and state-of-the-art baselines in both supervised and unsupervised settings.

ExpertRAG is a novel theoretical framework that integrates Mixture-of-Experts (MoE) architectures with Retrieval Augmented Generation (RAG) to advance the efficiency and accuracy of knowledge-intensive language modeling. We propose a dynamic retrieval gating mechanism coupled with expert routing, enabling the model to selectively consult an external knowledge store or rely on specialized internal experts based on the query's needs. The paper lays out the theoretical foundations of ExpertRAG, including a probabilistic formulation that treats retrieval and expert selection as latent decisions, and mathematical justifications for its efficiency in both computation and knowledge utilization. We derive formulae to quantify the expected computational cost savings from selective retrieval and the capacity gains from sparse expert utilization. A comparative analysis positions ExpertRAG against standard RAG (with always-on retrieval) and pure MoE models (e.g., Switch Transformer, Mixtral) to highlight its unique balance between parametric knowledge and non-parametric retrieval. We also outline an experimental validation strategy, proposing benchmarks and evaluation protocols to test ExpertRAG's performance on factual recall, generalization, and inference efficiency. The proposed framework, although presented theoretically, is supported by insights from prior work in RAG and MoE, and is poised to provide more factual, efficient, and adaptive generation by leveraging the best of both paradigms. In summary, ExpertRAG contributes a new perspective on scaling and augmenting language models, backed by a thorough analysis and a roadmap for empirical validation.

The correctness of computations remains a significant challenge in computer science, with traditional approaches relying on automated testing or formal verification. Self-testing/correcting programs introduce an alternative paradigm, allowing a program to verify and correct its own outputs via randomized reductions, a concept that previously required manual derivation. In this paper, we present Bitween, a method and tool for automated learning of randomized (self)-reductions and program properties in numerical programs. Bitween combines symbolic analysis and machine learning, with a surprising finding: polynomial-time linear regression, a basic optimization method, is not only sufficient but also highly effective for deriving complex randomized self-reductions and program invariants, often outperforming sophisticated mixed-integer linear programming solvers. We establish a theoretical framework for learning these reductions and introduce RSR-Bench, a benchmark suite for evaluating Bitween's capabilities on scientific and machine learning functions. Our empirical results show that Bitween surpasses state-of-the-art tools in scalability, stability, and sample efficiency when evaluated on nonlinear invariant benchmarks like NLA-DigBench. Bitween is open-source as a Python package and accessible via a web interface that supports C language programs.

This paper presents a comprehensive analysis of quantization techniques for optimizing Large Language Models (LLMs), specifically focusing on Post-Training Quantization (PTQ) and Quantization-Aware Training (QAT). Through empirical evaluation across models ranging from 10M to 1B parameters, we demonstrate that quantization can achieve up to 68% reduction in model size while maintaining performance within 6% of full-precision baselines when utilizing our proposed scaling factor {\gamma}. Our experiments show that INT8 quantization delivers a 40% reduction in computational cost and power consumption, while INT4 quantization further improves these metrics by 60%. We introduce a novel theoretical framework for mixed-precision quantization, deriving optimal bit allocation strategies based on layer sensitivity and weight variance. Hardware efficiency evaluations on edge devices reveal that our quantization approach enables up to 2.4x throughput improvement for INT8 and 3x for INT4, with 60% power reduction compared to full-precision models.

This study examines the performance of finite spin quantum batteries (QBs) using Heisenberg spin models with Dzyaloshinsky-Moriya (DM) and Kaplan--Shekhtman--Entin-Wohlman--Aharony (KSEA) interactions. The QBs are modeled as interacting quantum spins in local inhomogeneous magnetic fields, inducing variable Zeeman splitting. We derive analytical expressions for the maximal extractable work, ergotropy and the capacity of QBs, as recently examined by Yang et al. [Phys. Rev. Lett. 131, 030402 (2023)]. These quantities are analytically linked through certain quantum correlations, as posited in the aforementioned study. Different Heisenberg spin chain models exhibit distinct behaviors under varying conditions, emphasizing the importance of model selection for optimizing QB performance. In antiferromagnetic (AFM) systems, maximum ergotropy occurs with a Zeeman splitting field applied to either spin, while ferromagnetic (FM) systems benefit from a uniform Zeeman field. Temperature significantly impacts QB performance, with ergotropy in the AFM case being generally more robust against temperature increases compared to the FM case. Incorporating DM and KSEA couplings can significantly enhance the capacity and ergotropy extraction of QBs. However, there exists a threshold beyond which additional increases in these interactions cause a sharp decline in capacity and ergotropy. This behavior is influenced by temperature and quantum coherence, which signal the occurrence of a sudden phase transition. The resource theory of quantum coherence proposed by Baumgratz et al. [Phys. Rev. Lett. 113, 140401 (2014)] plays a crucial role in enhancing ergotropy and capacity. However, ergotropy is limited by both the system's capacity and the amount of coherence. These findings support the theoretical framework of spin-based QBs and may benefit future research on quantum energy storage devices.

We introduce a formal information-theoretic framework for image captioning by regarding it as a representation learning task. Our framework defines three key objectives: task sufficiency, minimal redundancy, and human interpretability. Building upon this foundation, we propose a novel Pyramid of Captions (PoCa) method, which constructs caption pyramids by generating localized captions for zoomed-in image patches and integrating them with global caption information using large language models. This approach leverages intuition that the detailed examination of local patches can reduce error risks and address inaccuracies in global captions, either by correcting the hallucination or adding missing details. Based on our theoretical framework, we formalize this intuition and provide formal proof demonstrating the effectiveness of PoCa under certain assumptions. Empirical tests with various image captioning models and large language models show that PoCa consistently yields more informative and semantically aligned captions, maintaining brevity and interpretability.

RNA design aims to find a sequence that folds with highest probability into a designated target structure. However, certain structures are undesignable, meaning no sequence can fold into the target structure under the default (Turner) RNA folding model. Understanding the specific local structures (i.e., "motifs") that contribute to undesignability is crucial for refining RNA folding models and determining the limits of RNA designability. Despite its importance, this problem has received very little attention, and previous efforts are neither scalable nor interpretable. We develop a new theoretical framework for motif (un-)designability, and design scalable and interpretable algorithms to identify minimal undesignable motifs within a given RNA secondary structure. Our approach establishes motif undesignability by searching for rival motifs, rather than exhaustively enumerating all (partial) sequences that could potentially fold into the motif. Furthermore, we exploit rotational invariance in RNA structures to detect, group, and reuse equivalent motifs and to construct a database of unique minimal undesignable motifs. To achieve that, we propose a loop-pair graph representation for motifs and a recursive graph isomorphism algorithm for motif equivalence. Our algorithms successfully identify 24 unique minimal undesignable motifs among 18 undesignable puzzles from the Eterna100 benchmark. Surprisingly, we also find over 350 unique minimal undesignable motifs and 663 undesignable native structures in the ArchiveII dataset, drawn from a diverse set of RNA families. Our source code is available at https://github.com/shanry/RNA-Undesign and our web server is available at http://linearfold.org/motifs.

Existing alignment methods share a common topology of information flow, where reward information is collected from humans, modeled with preference learning, and used to tune language models. However, this shared topology has not been systematically characterized, nor have its alternatives been thoroughly explored, leaving the problems of low data efficiency and unreliable generalization unaddressed. As a solution, we introduce a theoretical framework for investigating reward generalization in reinforcement learning from human feedback (RLHF), focusing on the topology of information flow at both macro and micro levels. At the macro level, we portray the RLHF information flow as an autoencoding process over behavior distributions, formalizing the RLHF objective of distributional consistency between human preference and model behavior. At the micro level, we present induced Bayesian networks as a theory of reward generalization in RLHF, introducing fine-grained dataset topologies into generalization bounds. Combining analysis on both levels, we propose reward modeling from tree-structured preference information. It is shown to reduce reward uncertainty by up to Theta(log n/loglog n) times compared to baselines, where n is the dataset size. Validation on three NLP tasks shows that our tree-based reward model achieves an average win rate of 65% against baseline methods, thus improving reward generalization for free via topology design.

We provide a new theoretical framework for the variable-step deferred correction (DC) methods based on the well-known BDF2 formula. By using the discrete orthogonal convolution kernels, some high-order BDF2-DC methods are proven to be stable on arbitrary time grids according to the recent definition of stability (SINUM, 60: 2253-2272). It significantly relaxes the existing step-ratio restrictions for the BDF2-DC methods (BIT, 62: 1789-1822). The associated sharp error estimates are established by taking the numerical effects of the starting approximations into account, and they suggest that the BDF2-DC methods have no aftereffect, that is, the lower-order starting scheme for the BDF2 scheme will not cause a loss in the accuracy of the high-order BDF2-DC methods. Extensive tests on the graded and random time meshes are presented to support the new theory.

We study the probabilistic modeling performed by Autoregressive Large Language Models (LLMs) through the angle of time directionality, addressing a question first raised in (Shannon, 1951). For large enough models, we empirically find a time asymmetry in their ability to learn natural language: a difference in the average log-perplexity when trying to predict the next token versus when trying to predict the previous one. This difference is at the same time subtle and very consistent across various modalities (language, model size, training time, ...). Theoretically, this is surprising: from an information-theoretic point of view, there should be no such difference. We provide a theoretical framework to explain how such an asymmetry can appear from sparsity and computational complexity considerations, and outline a number of perspectives opened by our results.

Language model alignment has become an important component of AI safety, allowing safe interactions between humans and language models, by enhancing desired behaviors and inhibiting undesired ones. It is often done by tuning the model or inserting preset aligning prompts. Recently, representation engineering, a method which alters the model's behavior via changing its representations post-training, was shown to be effective in aligning LLMs (Zou et al., 2023a). Representation engineering yields gains in alignment oriented tasks such as resistance to adversarial attacks and reduction of social biases, but was also shown to cause a decrease in the ability of the model to perform basic tasks. In this paper we study the tradeoff between the increase in alignment and decrease in helpfulness of the model. We propose a theoretical framework which provides bounds for these two quantities, and demonstrate their relevance empirically. First, we find that under the conditions of our framework, alignment can be guaranteed with representation engineering, and at the same time that helpfulness is harmed in the process. Second, we show that helpfulness is harmed quadratically with the norm of the representation engineering vector, while the alignment increases linearly with it, indicating a regime in which it is efficient to use representation engineering. We validate our findings empirically, and chart the boundaries to the usefulness of representation engineering for alignment.

This article presents a theoretical framework for adapting the Common Model of Cognition to large generative network models within the field of artificial intelligence. This can be accomplished by restructuring modules within the Common Model into shadow production systems that are peripheral to a central production system, which handles higher-level reasoning based on the shadow productions' output. Implementing this novel structure within the Common Model allows for a seamless connection between cognitive architectures and generative neural networks.

Large transformer models pretrained on offline reinforcement learning datasets have demonstrated remarkable in-context reinforcement learning (ICRL) capabilities, where they can make good decisions when prompted with interaction trajectories from unseen environments. However, when and how transformers can be trained to perform ICRL have not been theoretically well-understood. In particular, it is unclear which reinforcement-learning algorithms transformers can perform in context, and how distribution mismatch in offline training data affects the learned algorithms. This paper provides a theoretical framework that analyzes supervised pretraining for ICRL. This includes two recently proposed training methods -- algorithm distillation and decision-pretrained transformers. First, assuming model realizability, we prove the supervised-pretrained transformer will imitate the conditional expectation of the expert algorithm given the observed trajectory. The generalization error will scale with model capacity and a distribution divergence factor between the expert and offline algorithms. Second, we show transformers with ReLU attention can efficiently approximate near-optimal online reinforcement learning algorithms like LinUCB and Thompson sampling for stochastic linear bandits, and UCB-VI for tabular Markov decision processes. This provides the first quantitative analysis of the ICRL capabilities of transformers pretrained from offline trajectories.

Preference-based Reinforcement Learning (PbRL) is a paradigm in which an RL agent learns to optimize a task using pair-wise preference-based feedback over trajectories, rather than explicit reward signals. While PbRL has demonstrated practical success in fine-tuning language models, existing theoretical work focuses on regret minimization and fails to capture most of the practical frameworks. In this study, we fill in such a gap between theoretical PbRL and practical algorithms by proposing a theoretical reward-agnostic PbRL framework where exploratory trajectories that enable accurate learning of hidden reward functions are acquired before collecting any human feedback. Theoretical analysis demonstrates that our algorithm requires less human feedback for learning the optimal policy under preference-based models with linear parameterization and unknown transitions, compared to the existing theoretical literature. Specifically, our framework can incorporate linear and low-rank MDPs with efficient sample complexity. Additionally, we investigate reward-agnostic RL with action-based comparison feedback and introduce an efficient querying algorithm tailored to this scenario.

The goal of continual learning is to find a model that solves multiple learning tasks which are presented sequentially to the learner. A key challenge in this setting is that the learner may forget how to solve a previous task when learning a new task, a phenomenon known as catastrophic forgetting. To address this challenge, many practical methods have been proposed, including memory-based, regularization-based, and expansion-based methods. However, a rigorous theoretical understanding of these methods remains elusive. This paper aims to bridge this gap between theory and practice by proposing a new continual learning framework called Ideal Continual Learner (ICL), which is guaranteed to avoid catastrophic forgetting by construction. We show that ICL unifies multiple well-established continual learning methods and gives new theoretical insights into the strengths and weaknesses of these methods. We also derive generalization bounds for ICL which allow us to theoretically quantify how rehearsal affects generalization. Finally, we connect ICL to several classic subjects and research topics of modern interest, which allows us to make historical remarks and inspire future directions.

Contemporary predictive models are hard to interpret as their deep nets exploit numerous complex relations between input elements. This work suggests a theoretical framework for model interpretability by measuring the contribution of relevant features to the functional entropy of the network with respect to the input. We rely on the log-Sobolev inequality that bounds the functional entropy by the functional Fisher information with respect to the covariance of the data. This provides a principled way to measure the amount of information contribution of a subset of features to the decision function. Through extensive experiments, we show that our method surpasses existing interpretability sampling-based methods on various data signals such as image, text, and audio.

This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.

We propose a novel theoretical framework of analysis for Generative Adversarial Networks (GANs). We reveal a fundamental flaw of previous analyses which, by incorrectly modeling GANs' training scheme, are subject to ill-defined discriminator gradients. We overcome this issue which impedes a principled study of GAN training, solving it within our framework by taking into account the discriminator's architecture. To this end, we leverage the theory of infinite-width neural networks for the discriminator via its Neural Tangent Kernel. We characterize the trained discriminator for a wide range of losses and establish general differentiability properties of the network. From this, we derive new insights about the convergence of the generated distribution, advancing our understanding of GANs' training dynamics. We empirically corroborate these results via an analysis toolkit based on our framework, unveiling intuitions that are consistent with GAN practice.

We propose a novel theoretical framework to understand contrastive self-supervised learning (SSL) methods that employ dual pairs of deep ReLU networks (e.g., SimCLR). First, we prove that in each SGD update of SimCLR with various loss functions, including simple contrastive loss, soft Triplet loss and InfoNCE loss, the weights at each layer are updated by a covariance operator that specifically amplifies initial random selectivities that vary across data samples but survive averages over data augmentations. To further study what role the covariance operator plays and which features are learned in such a process, we model data generation and augmentation processes through a hierarchical latent tree model (HLTM) and prove that the hidden neurons of deep ReLU networks can learn the latent variables in HLTM, despite the fact that the network receives no direct supervision from these unobserved latent variables. This leads to a provable emergence of hierarchical features through the amplification of initially random selectivities through contrastive SSL. Extensive numerical studies justify our theoretical findings. Code is released in https://github.com/facebookresearch/luckmatters/tree/master/ssl.

Graph Neural Networks (GNNs) have been shown to be effective models for different predictive tasks on graph-structured data. Recent work on their expressive power has focused on isomorphism tasks and countable feature spaces. We extend this theoretical framework to include continuous features - which occur regularly in real-world input domains and within the hidden layers of GNNs - and we demonstrate the requirement for multiple aggregation functions in this context. Accordingly, we propose Principal Neighbourhood Aggregation (PNA), a novel architecture combining multiple aggregators with degree-scalers (which generalize the sum aggregator). Finally, we compare the capacity of different models to capture and exploit the graph structure via a novel benchmark containing multiple tasks taken from classical graph theory, alongside existing benchmarks from real-world domains, all of which demonstrate the strength of our model. With this work, we hope to steer some of the GNN research towards new aggregation methods which we believe are essential in the search for powerful and robust models.

We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.

Users like sharing personal photos with others through social media. At the same time, they might want to make automatic identification in such photos difficult or even impossible. Classic obfuscation methods such as blurring are not only unpleasant but also not as effective as one would expect. Recent studies on adversarial image perturbations (AIP) suggest that it is possible to confuse recognition systems effectively without unpleasant artifacts. However, in the presence of counter measures against AIPs, it is unclear how effective AIP would be in particular when the choice of counter measure is unknown. Game theory provides tools for studying the interaction between agents with uncertainties in the strategies. We introduce a general game theoretical framework for the user-recogniser dynamics, and present a case study that involves current state of the art AIP and person recognition techniques. We derive the optimal strategy for the user that assures an upper bound on the recognition rate independent of the recogniser's counter measure. Code is available at https://goo.gl/hgvbNK.

This paper develops a comprehensive theoretical framework that imports concepts from stochastic thermodynamics to model price impact and characterize the feasibility of round-trip arbitrage in financial markets. A trading cycle is treated as a non-equilibrium thermodynamic process, where price impact represents dissipative work and market noise plays the role of thermal fluctuations. The paper proves a Financial Second Law: under general convex impact functionals, any round-trip trading strategy yields non-positive expected profit. This structural constraint is complemented by a fluctuation theorem that bounds the probability of profitable cycles in terms of dissipated work and market volatility. The framework introduces a statistical ensemble of trading strategies governed by a Gibbs measure, leading to a free energy decomposition that connects expected cost, strategy entropy, and a market temperature parameter. The framework provides rigorous, testable inequalities linking microstructural impact to macroscopic no-arbitrage conditions, offering a novel physics-inspired perspective on market efficiency. The paper derives explicit analytical results for prototypical trading strategies and discusses empirical validation protocols.

Causal abstraction is a promising theoretical framework for explainable artificial intelligence that defines when an interpretable high-level causal model is a faithful simplification of a low-level deep learning system. However, existing causal abstraction methods have two major limitations: they require a brute-force search over alignments between the high-level model and the low-level one, and they presuppose that variables in the high-level model will align with disjoint sets of neurons in the low-level one. In this paper, we present distributed alignment search (DAS), which overcomes these limitations. In DAS, we find the alignment between high-level and low-level models using gradient descent rather than conducting a brute-force search, and we allow individual neurons to play multiple distinct roles by analyzing representations in non-standard bases-distributed representations. Our experiments show that DAS can discover internal structure that prior approaches miss. Overall, DAS removes previous obstacles to conducting causal abstraction analyses and allows us to find conceptual structure in trained neural nets.

Federated Averaging (FedAvg) and its variants are the most popular optimization algorithms in federated learning (FL). Previous convergence analyses of FedAvg either assume full client participation or partial client participation where the clients can be uniformly sampled. However, in practical cross-device FL systems, only a subset of clients that satisfy local criteria such as battery status, network connectivity, and maximum participation frequency requirements (to ensure privacy) are available for training at a given time. As a result, client availability follows a natural cyclic pattern. We provide (to our knowledge) the first theoretical framework to analyze the convergence of FedAvg with cyclic client participation with several different client optimizers such as GD, SGD, and shuffled SGD. Our analysis discovers that cyclic client participation can achieve a faster asymptotic convergence rate than vanilla FedAvg with uniform client participation under suitable conditions, providing valuable insights into the design of client sampling protocols.

Reinforcement learning (RL) provides a theoretical framework for continuously improving an agent's behavior via trial and error. However, efficiently learning policies from scratch can be very difficult, particularly for tasks with exploration challenges. In such settings, it might be desirable to initialize RL with an existing policy, offline data, or demonstrations. However, naively performing such initialization in RL often works poorly, especially for value-based methods. In this paper, we present a meta algorithm that can use offline data, demonstrations, or a pre-existing policy to initialize an RL policy, and is compatible with any RL approach. In particular, we propose Jump-Start Reinforcement Learning (JSRL), an algorithm that employs two policies to solve tasks: a guide-policy, and an exploration-policy. By using the guide-policy to form a curriculum of starting states for the exploration-policy, we are able to efficiently improve performance on a set of simulated robotic tasks. We show via experiments that JSRL is able to significantly outperform existing imitation and reinforcement learning algorithms, particularly in the small-data regime. In addition, we provide an upper bound on the sample complexity of JSRL and show that with the help of a guide-policy, one can improve the sample complexity for non-optimism exploration methods from exponential in horizon to polynomial.

UMAP (Uniform Manifold Approximation and Projection) is a novel manifold learning technique for dimension reduction. UMAP is constructed from a theoretical framework based in Riemannian geometry and algebraic topology. The result is a practical scalable algorithm that applies to real world data. The UMAP algorithm is competitive with t-SNE for visualization quality, and arguably preserves more of the global structure with superior run time performance. Furthermore, UMAP has no computational restrictions on embedding dimension, making it viable as a general purpose dimension reduction technique for machine learning.

In Reinforcement Learning (RL), regularization has emerged as a popular tool both in theory and practice, typically based either on an entropy bonus or a Kullback-Leibler divergence that constrains successive policies. In practice, these approaches have been shown to improve exploration, robustness and stability, giving rise to popular Deep RL algorithms such as SAC and TRPO. Policy Mirror Descent (PMD) is a theoretical framework that solves this general regularized policy optimization problem, however the closed-form solution involves the sum of all past Q-functions, which is intractable in practice. We propose and analyze PMD-like algorithms that only keep the last M Q-functions in memory, and show that for finite and large enough M, a convergent algorithm can be derived, introducing no error in the policy update, unlike prior deep RL PMD implementations. StaQ, the resulting algorithm, enjoys strong theoretical guarantees and is competitive with deep RL baselines, while exhibiting less performance oscillation, paving the way for fully stable deep RL algorithms and providing a testbed for experimentation with Policy Mirror Descent.

The transfer of a star's angular momentum to its atmosphere is a topic of considerable and wide-ranging interest in astrophysics. This letter considers the effect of kinetic and magnetic turbulence on the solar wind's angular momentum. The effects are quantified in a theoretical framework that employs Reynolds-averaged mean field magnetohydrodynamics, allowing for fluctuations of arbitrary amplitude. The model is restricted to the solar equatorial (\(r-\phi\)) plane with axial symmetry, which permits the effect of turbulence to be expressed in analytical form as a modification to the classic Weber & Davis (1967) theory, dependent on the \(r,\phi\) shear component of the Reynolds stress tensor. A solar wind simulation with turbulence transport modeling and Parker Solar Probe observations at the Alfv\'en surface are employed to quantify this turbulent modification to the solar wind's angular momentum, which is found to be ~ 3% - 10% and tends to be negative. Implications for solar and stellar rotational evolution are discussed.

Large Language Models (LLMs) have reshaped the landscape of artificial intelligence by demonstrating exceptional performance across various tasks. However, substantial computational requirements make their deployment challenging on devices with limited resources. Recently, compression methods using low-rank matrix techniques have shown promise, yet these often lead to degraded accuracy or introduce significant overhead in parameters and inference latency. This paper introduces Modular Decomposition (MoDeGPT), a novel structured compression framework that does not need recovery fine-tuning while resolving the above drawbacks. MoDeGPT partitions the Transformer block into modules comprised of matrix pairs and reduces the hidden dimensions via reconstructing the module-level outputs. MoDeGPT is developed based on a theoretical framework that utilizes three well-established matrix decomposition algorithms -- Nystr\"om approximation, CR decomposition, and SVD -- and applies them to our redefined transformer modules. Our comprehensive experiments show MoDeGPT, without backward propagation, matches or surpasses previous structured compression methods that rely on gradient information, and saves 98% of compute costs on compressing a 13B model. On Llama-2/3 and OPT models, MoDeGPT maintains 90-95% zero-shot performance with 25-30% compression rates. Moreover, the compression can be done on a single GPU within a few hours and increases the inference throughput by up to 46%.

Predictive Coding (PC) is a theoretical framework in cognitive science suggesting that the human brain processes cognition through spatiotemporal prediction of the visual world. Existing studies have developed spatiotemporal prediction neural networks based on the PC theory, emulating its two core mechanisms: Correcting predictions from residuals and hierarchical learning. However, these models do not show the enhancement of prediction skills on real-world forecasting tasks and ignore the Precision Weighting mechanism of PC theory. The precision weighting mechanism posits that the brain allocates more attention to signals with lower precision, contributing to the cognitive ability of human brains. This work introduces the Cognitive Diffusion Probabilistic Models (CogDPM), which demonstrate the connection between diffusion probabilistic models and PC theory. CogDPM features a precision estimation method based on the hierarchical sampling capabilities of diffusion models and weight the guidance with precision weights estimated by the inherent property of diffusion models. We experimentally show that the precision weights effectively estimate the data predictability. We apply CogDPM to real-world prediction tasks using the United Kindom precipitation and ERA surface wind datasets. Our results demonstrate that CogDPM outperforms both existing domain-specific operational models and general deep prediction models by providing more proficient forecasting.

Distributed representations of words encode lexical semantic information, but what type of information is encoded and how? Focusing on the skip-gram with negative-sampling method, we found that the squared norm of static word embedding encodes the information gain conveyed by the word; the information gain is defined by the Kullback-Leibler divergence of the co-occurrence distribution of the word to the unigram distribution. Our findings are explained by the theoretical framework of the exponential family of probability distributions and confirmed through precise experiments that remove spurious correlations arising from word frequency. This theory also extends to contextualized word embeddings in language models or any neural networks with the softmax output layer. We also demonstrate that both the KL divergence and the squared norm of embedding provide a useful metric of the informativeness of a word in tasks such as keyword extraction, proper-noun discrimination, and hypernym discrimination.

Models that can predict the occurrence of events ahead of time with low false-alarm rates are critical to the acceptance of decision support systems in the medical community. This challenging task is typically treated as a simple binary classification, ignoring temporal dependencies between samples, whereas we propose to exploit this structure. We first introduce a common theoretical framework unifying dynamic survival analysis and early event prediction. Following an analysis of objectives from both fields, we propose Temporal Label Smoothing (TLS), a simpler, yet best-performing method that preserves prediction monotonicity over time. By focusing the objective on areas with a stronger predictive signal, TLS improves performance over all baselines on two large-scale benchmark tasks. Gains are particularly notable along clinically relevant measures, such as event recall at low false-alarm rates. TLS reduces the number of missed events by up to a factor of two over previously used approaches in early event prediction.

This research introduces a novel evaluation framework designed to assess large language models' (LLMs) ability to acknowledge uncertainty on 675 fundamentally unsolvable problems. Using a curated dataset of graduate-level grand challenge questions with intentionally unknowable answers, we evaluated twelve state-of-the-art LLMs, including both open and closed-source models, on their propensity to admit ignorance rather than generate plausible but incorrect responses. The best models scored in 62-68% accuracy ranges for admitting the problem solution was unknown in fields ranging from biology to philosophy and mathematics. We observed an inverse relationship between problem difficulty and model accuracy, with GPT-4 demonstrating higher rates of uncertainty acknowledgment on more challenging problems (35.8%) compared to simpler ones (20.0%). This pattern indicates that models may be more prone to generate speculative answers when problems appear more tractable. The study also revealed significant variations across problem categories, with models showing difficulty in acknowledging uncertainty in invention and NP-hard problems while performing relatively better on philosophical and psychological challenges. These results contribute to the growing body of research on artificial general intelligence (AGI) assessment by highlighting the importance of uncertainty recognition as a critical component of future machine intelligence evaluation. This impossibility test thus extends previous theoretical frameworks for universal intelligence testing by providing empirical evidence of current limitations in LLMs' ability to recognize their own knowledge boundaries, suggesting new directions for improving model training architectures and evaluation approaches.

Sequential dependencies present a fundamental bottleneck in deploying large-scale autoregressive models, particularly for real-time applications. While traditional optimization approaches like pruning and quantization often compromise model quality, recent advances in generation-refinement frameworks demonstrate that this trade-off can be significantly mitigated. This survey presents a comprehensive taxonomy of generation-refinement frameworks, analyzing methods across autoregressive sequence tasks. We categorize methods based on their generation strategies (from simple n-gram prediction to sophisticated draft models) and refinement mechanisms (including single-pass verification and iterative approaches). Through systematic analysis of both algorithmic innovations and system-level implementations, we examine deployment strategies across computing environments and explore applications spanning text, images, and speech generation. This systematic examination of both theoretical frameworks and practical implementations provides a foundation for future research in efficient autoregressive decoding.

Representations are at the core of all deep reinforcement learning (RL) methods for both Markov decision processes (MDPs) and partially observable Markov decision processes (POMDPs). Many representation learning methods and theoretical frameworks have been developed to understand what constitutes an effective representation. However, the relationships between these methods and the shared properties among them remain unclear. In this paper, we show that many of these seemingly distinct methods and frameworks for state and history abstractions are, in fact, based on a common idea of self-predictive abstraction. Furthermore, we provide theoretical insights into the widely adopted objectives and optimization, such as the stop-gradient technique, in learning self-predictive representations. These findings together yield a minimalist algorithm to learn self-predictive representations for states and histories. We validate our theories by applying our algorithm to standard MDPs, MDPs with distractors, and POMDPs with sparse rewards. These findings culminate in a set of preliminary guidelines for RL practitioners.

Generating functions, which are widely used in combinatorics and probability theory, encode function values into the coefficients of a polynomial. In this paper, we explore their use as a tractable probabilistic model, and propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are strictly more expressive efficient than many existing tractable probabilistic models, including determinantal point processes (DPPs), probabilistic circuits (PCs) such as sum-product networks, and tractable graphical models. We contend that PGCs are not just a theoretical framework that unifies vastly different existing models, but also show great potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks. We also highlight PGCs' connection to the theory of strongly Rayleigh distributions.

We present Free energy Estimators with Adaptive Transport (FEAT), a novel framework for free energy estimation -- a critical challenge across scientific domains. FEAT leverages learned transports implemented via stochastic interpolants and provides consistent, minimum-variance estimators based on escorted Jarzynski equality and controlled Crooks theorem, alongside variational upper and lower bounds on free energy differences. Unifying equilibrium and non-equilibrium methods under a single theoretical framework, FEAT establishes a principled foundation for neural free energy calculations. Experimental validation on toy examples, molecular simulations, and quantum field theory demonstrates improvements over existing learning-based methods.

Instruction tuning -- tuning large language models on instruction-output pairs -- is a promising technique for making models better adapted to the real world. Yet, the key factors driving the model's capability to understand and follow instructions not seen during training remain under-explored. Our investigation begins with a series of synthetic experiments within the theoretical framework of a Turing-complete algorithm called Markov algorithm, which allows fine-grained control over the instruction-tuning data. Generalization and robustness with respect to the training distribution emerge once a diverse enough set of tasks is provided, even though very few examples are provided for each task. We extend these initial results to a real-world application scenario of code generation and find that a more diverse instruction set, extending beyond code-related tasks, improves the performance of code generation. Our observations suggest that a more diverse semantic space for instruction-tuning sets greatly improves the model's ability to follow instructions and perform tasks.

Dynamic and static components in scenes often exhibit distinct properties, yet most 4D reconstruction methods treat them indiscriminately, leading to suboptimal performance in both cases. This work introduces SDD-4DGS, the first framework for static-dynamic decoupled 4D scene reconstruction based on Gaussian Splatting. Our approach is built upon a novel probabilistic dynamic perception coefficient that is naturally integrated into the Gaussian reconstruction pipeline, enabling adaptive separation of static and dynamic components. With carefully designed implementation strategies to realize this theoretical framework, our method effectively facilitates explicit learning of motion patterns for dynamic elements while maintaining geometric stability for static structures. Extensive experiments on five benchmark datasets demonstrate that SDD-4DGS consistently outperforms state-of-the-art methods in reconstruction fidelity, with enhanced detail restoration for static structures and precise modeling of dynamic motions. The code will be released.

We propose Embodied AI as the next fundamental step in the pursuit of Artificial General Intelligence, juxtaposing it against current AI advancements, particularly Large Language Models. We traverse the evolution of the embodiment concept across diverse fields - philosophy, psychology, neuroscience, and robotics - to highlight how EAI distinguishes itself from the classical paradigm of static learning. By broadening the scope of Embodied AI, we introduce a theoretical framework based on cognitive architectures, emphasizing perception, action, memory, and learning as essential components of an embodied agent. This framework is aligned with Friston's active inference principle, offering a comprehensive approach to EAI development. Despite the progress made in the field of AI, substantial challenges, such as the formulation of a novel AI learning theory and the innovation of advanced hardware, persist. Our discussion lays down a foundational guideline for future Embodied AI research. Highlighting the importance of creating Embodied AI agents capable of seamless communication, collaboration, and coexistence with humans and other intelligent entities within real-world environments, we aim to steer the AI community towards addressing the multifaceted challenges and seizing the opportunities that lie ahead in the quest for AGI.

Generating entanglement between distant quantum systems is at the core of quantum networking. In recent years, numerous theoretical protocols for remote entanglement generation have been proposed, of which many have been experimentally realized. Here, we provide a modular theoretical framework to elucidate the general mechanisms of photon-mediated entanglement generation between single spins in atomic or solid-state systems. Our framework categorizes existing protocols at various levels of abstraction and allows for combining the elements of different schemes in new ways. These abstraction layers make it possible to readily compare protocols for different quantum hardware. To enable the practical evaluation of protocols tailored to specific experimental parameters, we have devised numerical simulations based on the framework with our codes available online.

Reinforcement learning (RL) allows an agent interacting sequentially with an environment to maximize its long-term expected return. In the distributional RL (DistrRL) paradigm, the agent goes beyond the limit of the expected value, to capture the underlying probability distribution of the return across all time steps. The set of DistrRL algorithms has led to improved empirical performance. Nevertheless, the theory of DistrRL is still not fully understood, especially in the control case. In this paper, we present the simpler one-step distributional reinforcement learning (OS-DistrRL) framework encompassing only the randomness induced by the one-step dynamics of the environment. Contrary to DistrRL, we show that our approach comes with a unified theory for both policy evaluation and control. Indeed, we propose two OS-DistrRL algorithms for which we provide an almost sure convergence analysis. The proposed approach compares favorably with categorical DistrRL on various environments.

Question answering (QA) is a fundamental means to facilitate assessment and training of narrative comprehension skills for both machines and young children, yet there is scarcity of high-quality QA datasets carefully designed to serve this purpose. In particular, existing datasets rarely distinguish fine-grained reading skills, such as the understanding of varying narrative elements. Drawing on the reading education research, we introduce FairytaleQA, a dataset focusing on narrative comprehension of kindergarten to eighth-grade students. Generated by educational experts based on an evidence-based theoretical framework, FairytaleQA consists of 10,580 explicit and implicit questions derived from 278 children-friendly stories, covering seven types of narrative elements or relations. Our dataset is valuable in two folds: First, we ran existing QA models on our dataset and confirmed that this annotation helps assess models' fine-grained learning skills. Second, the dataset supports question generation (QG) task in the education domain. Through benchmarking with QG models, we show that the QG model trained on FairytaleQA is capable of asking high-quality and more diverse questions.

Digital Twin was introduced over a decade ago, as an innovative all-encompassing tool, with perceived benefits including real-time monitoring, simulation and forecasting. However, the theoretical framework and practical implementations of digital twins (DT) are still far from this vision. Although successful implementations exist, sufficient implementation details are not publicly available, therefore it is difficult to assess their effectiveness, draw comparisons and jointly advance the DT methodology. This work explores the various DT features and current approaches, the shortcomings and reasons behind the delay in the implementation and adoption of digital twin. Advancements in machine learning, internet of things and big data have contributed hugely to the improvements in DT with regards to its real-time monitoring and forecasting properties. Despite this progress and individual company-based efforts, certain research gaps exist in the field, which have caused delay in the widespread adoption of this concept. We reviewed relevant works and identified that the major reasons for this delay are the lack of a universal reference framework, domain dependence, security concerns of shared data, reliance of digital twin on other technologies, and lack of quantitative metrics. We define the necessary components of a digital twin required for a universal reference framework, which also validate its uniqueness as a concept compared to similar concepts like simulation, autonomous systems, etc. This work further assesses the digital twin applications in different domains and the current state of machine learning and big data in it. It thus answers and identifies novel research questions, both of which will help to better understand and advance the theory and practice of digital twins.

Graph Neural Networks (GNNs) are an effective framework for representation learning of graphs. GNNs follow a neighborhood aggregation scheme, where the representation vector of a node is computed by recursively aggregating and transforming representation vectors of its neighboring nodes. Many GNN variants have been proposed and have achieved state-of-the-art results on both node and graph classification tasks. However, despite GNNs revolutionizing graph representation learning, there is limited understanding of their representational properties and limitations. Here, we present a theoretical framework for analyzing the expressive power of GNNs to capture different graph structures. Our results characterize the discriminative power of popular GNN variants, such as Graph Convolutional Networks and GraphSAGE, and show that they cannot learn to distinguish certain simple graph structures. We then develop a simple architecture that is provably the most expressive among the class of GNNs and is as powerful as the Weisfeiler-Lehman graph isomorphism test. We empirically validate our theoretical findings on a number of graph classification benchmarks, and demonstrate that our model achieves state-of-the-art performance.

Reasoning is central to human intelligence, enabling structured problem-solving across diverse tasks. Recent advances in large language models (LLMs) have greatly enhanced their reasoning abilities in arithmetic, commonsense, and symbolic domains. However, effectively extending these capabilities into multimodal contexts-where models must integrate both visual and textual inputs-continues to be a significant challenge. Multimodal reasoning introduces complexities, such as handling conflicting information across modalities, which require models to adopt advanced interpretative strategies. Addressing these challenges involves not only sophisticated algorithms but also robust methodologies for evaluating reasoning accuracy and coherence. This paper offers a concise yet insightful overview of reasoning techniques in both textual and multimodal LLMs. Through a thorough and up-to-date comparison, we clearly formulate core reasoning challenges and opportunities, highlighting practical methods for post-training optimization and test-time inference. Our work provides valuable insights and guidance, bridging theoretical frameworks and practical implementations, and sets clear directions for future research.

This paper establishes a mathematical foundation for the Adam optimizer, elucidating its connection to natural gradient descent through Riemannian and information geometry. We rigorously analyze the diagonal empirical Fisher information matrix (FIM) in Adam, clarifying all detailed approximations and advocating for the use of log probability functions as loss, which should be based on discrete distributions, due to the limitations of empirical FIM. Our analysis uncovers flaws in the original Adam algorithm, leading to proposed corrections such as enhanced momentum calculations, adjusted bias corrections, and gradient clipping. We refine the weight decay term based on our theoretical framework. Our modified algorithm, Fisher Adam (FAdam), demonstrates superior performance across diverse domains including LLM, ASR, and VQ-VAE, achieving state-of-the-art results in ASR.

Large Language Models (LLMs) are revolutionizing several areas of Artificial Intelligence. One of the most remarkable applications is creative writing, e.g., poetry or storytelling: the generated outputs are often of astonishing quality. However, a natural question arises: can LLMs be really considered creative? In this article, we first analyze the development of LLMs under the lens of creativity theories, investigating the key open questions and challenges. In particular, we focus our discussion on the dimensions of value, novelty, and surprise as proposed by Margaret Boden in her work. Then, we consider different classic perspectives, namely product, process, press, and person. We discuss a set of ``easy'' and ``hard'' problems in machine creativity, presenting them in relation to LLMs. Finally, we examine the societal impact of these technologies with a particular focus on the creative industries, analyzing the opportunities offered, the challenges arising from them, and the potential associated risks, from both legal and ethical points of view.

Can classical game-theoretic frameworks be extended to capture the bounded rationality and causal reasoning of AI agents? We investigate this question by extending Causal Normal Form Games (CNFGs) to sequential settings, introducing Sequential Causal Multi-Agent Systems (S-CMAS) that incorporate Pearl's Causal Hierarchy across leader-follower interactions. While theoretically elegant -- we prove PSPACE-completeness, develop equilibrium refinements, and establish connections to signaling theory -- our comprehensive empirical investigation reveals a critical limitation: S-CNE provides zero welfare improvement over classical Stackelberg equilibrium across all tested scenarios. Through 50+ Monte Carlo simulations and hand-crafted synthetic examples, we demonstrate that backward induction with rational best-response eliminates any strategic advantage from causal layer distinctions. We construct a theoretical example illustrating conditions where benefits could emerge (ε-rational satisficing followers), though implementation confirms that even relaxed rationality assumptions prove insufficient when good instincts align with optimal play. This negative result provides valuable insight: classical game-theoretic extensions grounded in rational choice are fundamentally incompatible with causal reasoning advantages, motivating new theoretical frameworks beyond standard Nash equilibrium for agentic AI.

Large language models display remarkable capabilities in logical and mathematical reasoning, allowing them to solve complex tasks. Interestingly, these abilities emerge in networks trained on the simple task of next-token prediction. In this work, we present a theoretical framework for studying auto-regressive next-token predictors. We demonstrate that even simple models such as linear next-token predictors, trained on Chain-of-Thought (CoT) data, can approximate any function efficiently computed by a Turing machine. We introduce a new complexity measure -- length complexity -- which measures the number of intermediate tokens in a CoT sequence required to approximate some target function, and analyze the interplay between length complexity and other notions of complexity. Finally, we show experimentally that simple next-token predictors, such as linear networks and shallow Multi-Layer Perceptrons (MLPs), display non-trivial performance on text generation and arithmetic tasks. Our results demonstrate that the power of language models can be attributed, to a great extent, to the auto-regressive next-token training scheme, and not necessarily to a particular choice of architecture.

In this paper, we introduce a novel theoretical framework for multi-task regression, applying random matrix theory to provide precise performance estimations, under high-dimensional, non-Gaussian data distributions. We formulate a multi-task optimization problem as a regularization technique to enable single-task models to leverage multi-task learning information. We derive a closed-form solution for multi-task optimization in the context of linear models. Our analysis provides valuable insights by linking the multi-task learning performance to various model statistics such as raw data covariances, signal-generating hyperplanes, noise levels, as well as the size and number of datasets. We finally propose a consistent estimation of training and testing errors, thereby offering a robust foundation for hyperparameter optimization in multi-task regression scenarios. Experimental validations on both synthetic and real-world datasets in regression and multivariate time series forecasting demonstrate improvements on univariate models, incorporating our method into the training loss and thus leveraging multivariate information.

We study the problem of watermarking large language models (LLMs) generated text -- one of the most promising approaches for addressing the safety challenges of LLM usage. In this paper, we propose a rigorous theoretical framework to quantify the effectiveness and robustness of LLM watermarks. We propose a robust and high-quality watermark method, Unigram-Watermark, by extending an existing approach with a simplified fixed grouping strategy. We prove that our watermark method enjoys guaranteed generation quality, correctness in watermark detection, and is robust against text editing and paraphrasing. Experiments on three varying LLMs and two datasets verify that our Unigram-Watermark achieves superior detection accuracy and comparable generation quality in perplexity, thus promoting the responsible use of LLMs. Code is available at https://github.com/XuandongZhao/Unigram-Watermark.

Two main challenges in Reinforcement Learning (RL) are designing appropriate reward functions and ensuring the safety of the learned policy. To address these challenges, we present a theoretical framework for Inverse Reinforcement Learning (IRL) in constrained Markov decision processes. From a convex-analytic perspective, we extend prior results on reward identifiability and generalizability to both the constrained setting and a more general class of regularizations. In particular, we show that identifiability up to potential shaping (Cao et al., 2021) is a consequence of entropy regularization and may generally no longer hold for other regularizations or in the presence of safety constraints. We also show that to ensure generalizability to new transition laws and constraints, the true reward must be identified up to a constant. Additionally, we derive a finite sample guarantee for the suboptimality of the learned rewards, and validate our results in a gridworld environment.

Recently, Diffusion Models (DMs) boost a wave in AI for Art yet raise new copyright concerns, where infringers benefit from using unauthorized paintings to train DMs to generate novel paintings in a similar style. To address these emerging copyright violations, in this paper, we are the first to explore and propose to utilize adversarial examples for DMs to protect human-created artworks. Specifically, we first build a theoretical framework to define and evaluate the adversarial examples for DMs. Then, based on this framework, we design a novel algorithm, named AdvDM, which exploits a Monte-Carlo estimation of adversarial examples for DMs by optimizing upon different latent variables sampled from the reverse process of DMs. Extensive experiments show that the generated adversarial examples can effectively hinder DMs from extracting their features. Therefore, our method can be a powerful tool for human artists to protect their copyright against infringers equipped with DM-based AI-for-Art applications. The code of our method is available on GitHub: https://github.com/mist-project/mist.git.

The last decade has witnessed a boom of deep learning research and applications achieving state-of-the-art results in various domains. However, most advances have been established empirically, and their theoretical analysis remains lacking. One major issue is that our current interpretation of neural networks (NNs) as function approximators is too generic to support in-depth analysis. In this paper, we remedy this by proposing an alternative interpretation that identifies NNs as chain graphs (CGs) and feed-forward as an approximate inference procedure. The CG interpretation specifies the nature of each NN component within the rich theoretical framework of probabilistic graphical models, while at the same time remains general enough to cover real-world NNs with arbitrary depth, multi-branching and varied activations, as well as common structures including convolution / recurrent layers, residual block and dropout. We demonstrate with concrete examples that the CG interpretation can provide novel theoretical support and insights for various NN techniques, as well as derive new deep learning approaches such as the concept of partially collapsed feed-forward inference. It is thus a promising framework that deepens our understanding of neural networks and provides a coherent theoretical formulation for future deep learning research.