ICLR 2026 Papers

Conventional (contextual) MNL bandits model relevance-driven choice but ignore the potential benefit of within-assortment diversity, while submodular/combinatorial bandits encode diversity in rewards but lack structured choice probabilities. We bridge this gap with the *diversified multinomial logit* (DMNL) contextual bandit, which augments MNL choice probabilities with a generally submodular diversity function, formalizing the relevance--diversity relation in one model. Embedding diversity renders exact MNL assortment optimization intractable. We develop a *white-box* UCB-based algorithm, $\texttt{OFU-DMNL}$ that builds assortments item-wise by maximizing optimistic marginal gains, avoids black-box oracles, and provides end-to-end guarantees. We show that $\texttt{OFU-DMNL}$ achieves at least a $(1-\tfrac{1}{e+1})$-*approximate* regret bound $\tilde{O}\big(\tfrac{\sqrt{K}(d+1)}{K+1} \sqrt{T}\big)$, where $d$ is the context dimension, $K$ the maximum assortment size, and $T$ the horizon, and attains an improved approximation factor over standard submodular baselines. Experiments show consistent gains and, versus exhaustive enumeration, comparable regret with substantially lower runtime. DMNL bandits serves as a principled and practical basis for diversity-aware assortment optimization under uncertainty and our proposed algorithm offers a both statistically and computationally efficient solution.

Enforcing exact symmetry in machine learning models often yields significant gains in scientific applications, serving as a powerful inductive bias. However, recent work suggests that relying on approximate symmetry can offer greater flexibility and robustness. Despite promising empirical evidence, there has been little theoretical understanding, and in particular, a direct comparison between exact and approximate symmetry is missing from the literature. In this paper, we initiate this study by asking: What is the cost of enforcing exact versus approximate symmetry? To address this question, we introduce averaging complexity, a framework for quantifying the cost of enforcing symmetry via averaging. Our main result is an exponential separation: under standard conditions, achieving exact symmetry requires linear averaging complexity, whereas approximate symmetry can be attained with only logarithmic averaging complexity. To the best of our knowledge, this provides the first theoretical separation of these two cases, formally justifying why approximate symmetry may be preferable in practice. Beyond this, our tools and techniques may be of independent interest for the broader study of symmetries in machine learning.

Stochastic gradient descent (SGD) is a cornerstone algorithm for high-dimensional optimization, renowned for its empirical successes. Recent theoretical advances have provided a deep understanding of how SGD enables feature learning in high-dimensional nonlinear models, most notably the *single-index model* with i.i.d. data. In this work, we study the sequential learning problem for single-index models, also known as generalized linear bandits or ridge bandits, where SGD is a simple and natural solution, yet its learning dynamics remain largely unexplored. We show that, similar to the optimal interactive learner, SGD undergoes a distinct "burn-in" phase before entering the "learning" phase in this setting. Moreover, with an appropriately chosen learning rate schedule, a single SGD procedure simultaneously achieves near-optimal (or best-known) sample complexity and regret guarantees across both phases, for a broad class of link functions. Our results demonstrate that SGD remains highly competitive for learning single-index models under adaptive data.

We present GRAPE (Group RepresentAtional Position Encoding), a unified framework for positional encoding based on group actions. GRAPE brings together two families of mechanisms: (i) multiplicative rotations (Multiplicative GRAPE) in $\operatorname{SO}(d)$ and (ii) additive logit biases (Additive GRAPE) arising from unipotent actions in the general linear group $\mathrm{GL}$. In Multiplicative GRAPE, a position $n\in\mathbb{Z}$ (or $t\in\mathbb{R}$) acts as $\mathbf{G}(n)=\exp(n\,\omega\,\mathbf{L})$ with a rank‑2 skew generator $\mathbf{L} \in \mathbb{R}^{d \times d}$, yielding a relative, compositional, norm‑preserving map with a closed‑form matrix exponential. RoPE is recovered exactly when the $d/2$ planes are the canonical coordinate pairs with log‑uniform spectrum. Learned commuting subspaces and compact non‑commuting mixtures strictly extend this geometry to capture cross-subspace feature coupling at $O(d)$ and $O(rd)$ cost per head, respectively. In Additive GRAPE, additive logits arise as rank‑1 (or low‑rank) unipotent actions, recovering ALiBi and the Forgetting Transformer (FoX) as exact special cases while preserving an exact relative law and streaming cacheability. Altogether, GRAPE supplies a principled design space for positional geometry in long‑context models, subsuming RoPE and ALiBi as special cases.

Length generalization, the ability to solve problems of longer sequences than those observed during training, poses a core challenge of Transformer-based large language models (LLMs). Although existing studies have predominantly focused on data-driven approaches for particular arithmetic operations or symbolic manipulation tasks, these approaches tend to be task-specific with limited performance on individual tasks. To pursue a more general solution, this paper focuses on a broader case of reasoning problems that are *computable*, *i.e.*, problems that algorithms can solve, thus can be solved by the Turing machine, which operates over inputs of unbounded length. From this perspective, this paper proposes **T**uring m**A**chine **I**mitation **L**earning (**TAIL**) to improve the length generalization ability of LLMs. TAIL uses computer programs to directly synthesize chain-of-thought (CoT) data that imitate the execution process of a Turing machine, which *linearly* expands the reasoning steps into *atomic* states to alleviate shortcut pattern learning and explicit *memory* fetch mechanism to reduce the difficulties of dynamic and long-range data access. To validate the universality and reliability of TAIL, we construct a challenging synthetic dataset covering 8 classes of algorithms and 18 tasks. Without bells and whistles, TAIL significantly improves the length generalization ability as well as the performance of Qwen2.5-7B in individual tasks using only synthetic data, surpassing previous methods and DeepSeek-R1. The experimental results reveal that the key concepts in the Turing machine, instead of the human-like thinking styles, are indispensable for TAIL for length generalization, through which the model exhibits read-and-write behaviors consistent with the properties of the Turing machine in their attention layers. This work provides a promising direction for future research in the learning of LLM reasoning from synthetic data.

Metanetworks are neural architectures designed to operate directly on pretrained weights to perform downstream tasks. However, the parameter space serves only as a proxy for the underlying function class, and the parameter-function mapping is inherently non-injective: distinct parameter configurations may yield identical input-output behaviors. As a result, metanetworks that rely solely on raw parameters risk overlooking the intrinsic symmetries of the architecture. Reasoning about functional identity is therefore essential for effective metanetwork design, motivating the development of equivariant metanetworks, which incorporate equivariance principles to respect architectural symmetries. Existing approaches, however, typically enforce strict equivariance, which imposes rigid constraints and often leads to sparse and less expressive models. To address this limitation, we introduce the novel concept of quasi-equivariance, which allows metanetworks to move beyond the rigidity of strict equivariance while still preserving functional identity. We lay down a principled basis for this framework and demonstrate its broad applicability across diverse neural architectures, including feedforward, convolutional, and transformer networks. Through empirical evaluation, we show that quasi-equivariant metanetworks achieve good trade-offs between symmetry preservation and representational expressivity. These findings advance the theoretical understanding of weight-space learning and provide a principled foundation for the design of more expressive and functionally robust metanetworks.

It is well established that ReLU networks define continuous piecewise-linear functions, and that their linear regions are polyhedra in the input space. These regions form a complex that fully partitions the input space. The way these regions fit together is fundamental to the behavior of the network, as nonlinearities occur only at the boundaries where these regions connect. However, relatively little is known about the geometry of these complexes beyond bounds on the total number of regions, and calculating the complex exactly is intractable for most networks. In this work, we prove new theoretical results about these complexes that hold for all fully-connected ReLU networks, specifically about their connectivity graphs in which nodes correspond to regions and edges exist between each pair of regions connected by a face. We find that the average degree of this graph is upper bounded by twice the input dimension regardless of the width and depth of the network, and that the diameter of this graph has an upper bound that does not depend on input dimension, despite the number of regions increasing exponentially with input dimension. We corroborate our findings through experiments with networks trained on both synthetic and real-world data, which provide additional insight into the geometry of ReLU networks.

In this paper, we study a generalization of Markov games and pseudo-games that we call Markov pseudo-games, which like the former, captures time and uncertainty, and like the latter, allows for the players’ actions to determine the set of actions available to the other players. In the same vein as Arrow and Debreu, we intend for this model to be rich enough to encapsulate a broad mathematical framework for modeling economies. We then prove the existence of a game-theoretic equilibrium in our model, which in turn implies the existence of a general equilibrium in the corresponding economies. Finally, going beyond Arrow and Debreu, we introduce a solution method for Markov pseudo-games, and prove its polynomial-time convergence. We then provide an application of Markov pseudo-games to infinite-horizon Markov exchange economies, a stochastic economic model that extends Radner’s stochastic exchange economy and Magill and Quinzii’s infinite horizon incomplete markets model. We show that under suitable assumptions, the solutions of any infinite horizon Markov exchange economy (i.e., recursive Radner equilibria—RRE) can be formulated as the solution to a concave Markov pseudo-game, thus establishing the existence of RRE, and providing first-order methods for approximating RRE. Finally, we demonstrate the effectiveness of our approach in practice by building the corresponding generative adversarial policy neural network, and using it to compute RRE in a variety of infinite-horizon Markov exchange economies.

Universality results for equivariant neural networks remain rare. Those that do exist typically hold only in restrictive settings: either they rely on regular or higher-order tensor representations, leading to impractically high-dimensional hidden spaces, or they target specialized architectures, often confined to the invariant setting. This work develops a more general account. For invariant networks, we establish a universality theorem under separation constraints, showing that the addition of a fully connected readout layer secures approximation within the class of separation-constrained continuous functions. For equivariant networks, where results are even scarcer, we demonstrate that standard separability notions are inadequate and introduce the sharper criterion of *entry-wise separability*. We show that with sufficient depth or with the addition of appropriate readout layers, equivariant networks attain universality within the entry-wise separable regime. Together with prior results showing the failure of universality for shallow models, our findings identify depth and readout layers as a decisive mechanism for universality, additionally offering a unified perspective that subsumes and extends earlier specialized results.

We revisit FedExProx -- a recently proposed distributed optimization method designed to enhance convergence properties of parallel proximal algorithms via extrapolation. In the process, we uncover a surprising flaw: its known theoretical guarantees on quadratic optimization tasks are no better than those offered by the vanilla Gradient Descent (GD) method. Motivated by this observation, we develop a novel analysis framework, establishing a tighter linear convergence rate for non-strongly convex quadratic problems. By incorporating both computation and communication costs, we demonstrate that FedExProx can indeed provably outperform GD, in stark contrast to the original analysis. Furthermore, we consider partial participation scenarios and analyze two adaptive extrapolation strategies -- based on gradient diversity and Polyak stepsizes -- again significantly outperforming previous results. Moving beyond quadratics, we extend the applicability of our analysis to general functions satisfying the Polyak-Łojasiewicz condition, outperforming the previous strongly convex analysis while operating under weaker assumptions. Backed by empirical results, our findings point to a new and stronger potential of FedExProx, paving the way for further exploration of the benefits of extrapolation in federated learning.

Existing expert merging strategies for Sparse Mixture of Experts (SMoE) typically rely on input-dependent or input-independent averaging of expert parameters, but often lack a principled weighting mechanism. In this work, we reinterpret expert merging through the lens of game theory, revealing cooperative and competitive dynamics among experts. Based on this perspective, we introduce Nash Merging of Experts (NAMEx), a novel framework that incorporates Nash Bargaining into the merging process, enabling more balanced and efficient collaboration among experts. Additionally, we incorporate complex momentum into NAMEx to accelerate expert propagation with theoretical guarantees for convergence. Extensive experiments across language modeling, text classification, image classification, and zero-shot robustness under data corruption show that NAMEx consistently outperforms competing methods while integrating seamlessly with popular MoE architectures. Finally, we demonstrate NAMEx’s scalability by applying it to large-scale systems, including Qwen1.5-MoE (14B) and DeepSeek-MoE (16B), where it proves effective in both zero-shot and fine-tuning settings.

We propose a novel class of neural differential equation models called mean-field continuous sequence predictors (MFPs) for efficiently generating continuous sequences with potentially infinite-order complexity. To address complex inductive biases in time-series data, we employ mean-field dynamics structured through carefully designed graphons. By reframing continuous sequence prediction as mean-field games, we utilize a fictitious play strategy integrated with gradient-descent techniques. This approach exploits the stochastic maximum principle to determine the Nash equilibrium of the system. Both empirical evidence and theoretical analysis highlight the unique advantages of our framework, where a collective of continuous predictors achieves highly accurate predictions and consistently outperforms benchmark prior works.

Modern large language models (LLMs) excel at tasks that require storing and retrieving knowledge, such as factual recall and question answering. Transformers are central to this capability, thanks to their ability to encode information during training and retrieve it at inference. Existing theoretical analyses typically study transformers under idealized assumptions such as infinite data or orthogonal embeddings. In realistic settings, however, models are trained on finite datasets with non-orthogonal (random) embeddings. We address this gap by analyzing a single-layer transformer with random embeddings trained with (empirical) gradient descent on a simple token-retrieval task, where the model must identify an informative token within a length-$L$ sequence and learn a one-to-one mapping from tokens to labels. Our analysis tracks the "early phase" of gradient descent and yields explicit formulas for the model’s storage capacity—revealing a multiplicative dependence between sample size $N$, embedding dimension $d$, and sequence length $L$. We complement this with a lower bound for the statistical problem, showing that this multiplicative scaling is inherent under non-orthogonal embeddings.

While large language models (LLMs) demonstrate impressive performance across various tasks, their deployment in real-world scenarios is still constrained by high computational demands. Layer-wise pruning, a commonly employed strategy to mitigate inference costs, can partially address this challenge. However, existing approaches generally depend on static heuristic rules and fail to account for the interdependencies among layers, thereby limiting the effectiveness of the pruning process. To this end, this paper proposes a game-theoretic framework that formulates layer pruning as a cooperative game in which each layer acts as a player and model performance serves as the utility. As computing exact Shapley values is computationally infeasible for large language models (LLMs), we propose using a lightweight surrogate network to estimate layer-wise marginal contributions. This network can predict LLM performance for arbitrary layer combinations at a low computational cost. Additionally, we employ stratified Monte Carlo mask sampling to further reduce the cost of Sharpley value estimation. This approach captures inter-layer dependencies and dynamically identifies critical layers for pruning. Extensive experiments demonstrate the consistent superiority of our method in terms of perplexity and zero-shot accuracy, achieving more efficient and effective layer-wise pruning for large language models.

We study the problem of learning the utility functions of no-regret learning agents in a repeated normal-form game. Differing from most prior literature, we introduce a principal with the power to observe the agents playing the game, send agents signals, and give agents *payments* as a function of their actions. We show that the principal can, using a number of rounds polynomial in the size of the game, learn the utility functions of all agents to any desired precision $\varepsilon > 0$, for any no-regret learning algorithms of the agents. Our main technique is to formulate a zero-sum game between the principal and the agents, where the principal's strategy space is the set of all payment functions. Finally, we discuss implications for the problem of *steering* agents to a desired equilibrium: in particular, we introduce, using our utility-learning algorithm as a subroutine, the first algorithm for steering arbitrary no-regret learning agents without prior knowledge of their utilities.

In a multi-follower Bayesian Stackelberg game, a leader plays a mixed strategy over $L$ actions to which $n\ge 1$ followers, each having one of $K$ possible private types, best respond. The leader's optimal strategy depends on the distribution of the followers' private types. We study an online learning problem for Bayesian Stackelberg game, where a leader interacts for $T$ rounds with $n$ followers with types sampled from an unknown distribution every round. The leader's goal is to minimize regret, defined as the difference between the cumulative utility of the optimal strategy and that of the actually chosen strategies. We design learning algorithms for the leader under different settings. Under type feedback, where the leader observes the followers' types after each round, we design algorithms that achieve $\mathcal O\big(\sqrt{\min\{L\log(nKA T), ~ nK \} \cdot T} \big)$ regret for independent type distributions and $\mathcal O\big(\sqrt{\min\{L\log(nKA T), ~ K^n \} \cdot T} \big)$ regret for general type distributions. Interestingly, these bounds do not grow with $n$ at a polynomial rate. Under action feedback, where the leader only observes the followers' actions, we design algorithms with $\mathcal O( \min\{\sqrt{ n^L K^L A^{2L} L T \log T}, ~ K^n\sqrt{ T } \log T \} )$ regret. We also provide a lower bound of $\Omega(\sqrt{\min\{L, ~ nK\}T})$, almost matching the type-feedback upper bounds.

We study equilibrium finding in polymatrix games under differential privacy constraints. Prior work in this area fails to achieve both high-accuracy equilibria and a low privacy budget. To better understand the fundamental limitations of differential privacy in games, we show hardness results establishing that no algorithm can simultaneously obtain high accuracy and a vanishing privacy budget as the number of players tends to infinity. This impossibility holds in two regimes: (i) We seek to establish equilibrium approximation guarantees in terms of Euclidean \emph{distance} to the equilibrium set, and (ii) The adversary has access to all communication channels. We then consider the more realistic setting in which the adversary can access only a bounded number of channels and propose a new distributed algorithm that: recovers strategies with simultaneously vanishing \emph{Nash gap} (in expected utility, also referred to as \emph{exploitability}) and \emph{privacy budget} as the number of players increases. Our approach leverages structural properties of polymatrix games. To our knowledge, this is the first paper that can achieve this in equilibrium computation. Finally, we also provide numerical results to justify our algorithm.

We study the problem of learning to generate an answer (or completion) to a question (or prompt), where there could be multiple correct answers, any one of which is acceptable at test time. Learning is based on demonstrations of some correct answer to each training question, as in Supervised Fine Tuning (SFT). We formalize the problem as offline imitation learning in contextual bandits, with demonstrations from some optimal policy, without explicitly observed rewards. Prior work assumes that the demonstrator belongs to a low-complexity policy class, which motivates maximum likelihood estimation (i.e., log-loss minimization). In contrast, we propose relying only on the reward model (specifying which answers are correct) being in a low-cardinality class, which we argue is a weaker assumption. We show that likelihood maximization methods can fail in this case, and instead suggest an alternative novel approach that learns with sample complexity logarithmic in the cardinality of the reward class. Our work motivates looking beyond likelihood maximization when learning from demonstrations.

The nature of the relationship between Bayesian sampling and stochastic gradient descent in neural networks has been a long-standing open question in the theory of deep learning. We shed light on this question by modeling the long runtime behaviour of SGD as diffusion on porous media. Using singular learning theory, we show that the late stage dynamics are strongly impacted by the degeneracies of the loss surface. From this we are able to show that under reasonable choices of hyperparameters for vanilla SGD, the local steady state distribution of SGD (if it exists) is effectively a tempered version of the Bayesian posterior over the weights which accounts for local accessibility constraints.