ICLR 2026 论文列表

This paper studies online bidding subject to simultaneous budget and return-on-investment (ROI) constraints, which encodes the goal of balancing high volume and profitability. We formulate the problem as a general constrained online learning problem that can be applied to diverse bidding settings (e.g., first-price or second-price auctions) and feedback regimes (e.g., full or partial information), among others. We introduce L2FOB, a Look-ahead Lyapunov Framework for Online Bidding with strong empirical and theoretical performance. By combining optimistic reward and pessimistic cost estimation with the look-ahead virtual queue mechanism, L2FOB delivers safe and optimal bidding decisions. We provide adaptive guarantees: L2FOB achieves $O (\mathcal{E}\_r(T,p)+(\nu^* / \rho) \mathcal{E}\_c(T,p))$ regret and $O (\mathcal{E}\_r(T,p)+\mathcal{E}\_c(T,p))$ anytime ROI constraint violation, where $\mathcal{E}_r(T,p)$ and $\mathcal{E}_c(T,p)$ are cumulative estimation errors over $T$ rounds, $\rho$ is the average per-round budget, and $\nu^*$ is the offline optimal average reward. We instantiate L2FOB in several online bidding settings, demonstrating guarantees that match or improve upon the best-known results. These results are derived from the novel look-ahead design and Lyapunov stability analysis. Numerical experiments further validate our theoretical guarantees.

Multi-task Vehicle Routing Problems (VRPs) aim to minimize routing costs while satisfying diverse constraints. Existing solvers typically adopt a unified reinforcement learning (RL) framework to learn generalizable patterns across tasks. However, they often overlook the constraint and node dynamics during the decision process, making the model fail to accurately react to the current context. To address this limitation, we propose Chain-of-Context Learning (CCL), a novel framework that progressively captures the evolving context to guide fine-grained node adaptation. Specifically, CCL constructs step-wise contextual information via a Relevance-Guided Context Reformulation (RGCR) module, which adaptively prioritizes salient constraints. This context then guides node updates through a Trajectory-Shared Node Re-embedding (TSNR) module, which aggregates shared node features from all trajectories' contexts and uses them to update inputs for the next step. By modeling evolving preferences of the RL agent, CCL captures step-by-step dependencies in sequential decision-making. We evaluate CCL on 48 diverse VRP variants, including 16 in-distribution and 32 out-of-distribution (with unseen constraints) tasks. Experimental results show that CCL performs favorably against the state-of-the-art baselines, achieving the best performance on all in-distribution tasks and the majority of out-of-distribution tasks.

The probabilistic diffusion model (DM), generating content by inferencing through a recursive chain structure, has emerged as a powerful framework for visual generation. After pre-training on enormous data, the model needs to be properly aligned to meet requirements for downstream applications. How to efficiently align the foundation DM is a crucial task. Contemporary methods are either based on Reinforcement Learning (RL) or truncated Backpropagation (BP). However, RL and truncated BP suffer from low sample efficiency and biased gradient estimation, respectively, resulting in limited improvement or, even worse, complete training failure. To overcome the challenges, we propose the Recursive Likelihood Ratio (RLR) optimizer, a Half-Order (HO) fine-tuning paradigm for DM. The HO gradient estimator enables the computation graph rearrangement within the recursive diffusive chain, making the RLR's gradient estimator **an unbiased one with lower variance** than other methods. We theoretically investigate the bias, variance, and convergence of our method. Extensive experiments are conducted on image and video generation to validate the superiority of the RLR. Furthermore, we propose a novel prompt technique that is natural for the RLR to achieve a synergistic effect. The implementation is available at https://github.com/RTkenny/RLR-Optimizer.

Optimizers are critical for training deep neural networks. Existing training processes rely on a single static optimizer (e.g., SGD) or a simple hybrid of two optimizers, which miss the opportunity to exploit evolving dynamics in different training states, degrading model quality and convergence. In this paper, we propose a novel dynamic optimizer switching method called **D**ynamic **O**ptimizer **I**nterleaving **T**raining (DOIT) method, which builds surrogate models to predict different optimizers' performance from current parameter states. DOIT uses an acquisition function that combines the results from surrogate models with transferability assessments and process information to select a suitable optimizer for the subsequent training. Experiments on various models and tasks (e.g., image and text classification, machine translation, and object detection) show that DOIT effectively enhances the training, achieving faster convergence (i.e., 2\% to 10\% faster) and higher accuracy (i.e., 1\% to 3\% improvement). Additional independent experiments and case studies further validate DOIT's effectiveness.

Recently, a new optimization method based on the linear minimization oracle (LMO), called Muon, has been attracting increasing attention since it can train neural networks faster than the existing adaptive optimization methods, such as Adam. In this paper, we study how Muon can be utilized in federated learning. We first show that straightforwardly using Muon as the local optimizer of FedAvg does not work since the LMO is a biased operator. We then propose FedMuon, which can mitigate this issue and can converge to the stationary point. We also analyze how solving the LMO approximately affects the convergence rate and find that, surprisingly, FedMuon can converge for any number of Newton-Schulz iterations, while it can converge faster as we solve the LMO more accurately. Through experiments, we demonstrated that FedMuon can outperform the state-of-the-art federated learning methods.

We propose Frank–Wolfe (FW) algorithms with an adaptive Bregman step-size strategy for smooth adaptable (also called: relatively smooth) (weakly-) convex functions. This means that the gradient of the objective function is not necessarily Lipschitz continuous, and we only require the smooth adaptable property. Compared with existing FW algorithms, our assumptions are less restrictive. We establish convergence guarantees in various settings, including convergence rates ranging from sublinear to linear, depending on the assumptions for convex and nonconvex objective functions. Assuming that the objective function is weakly convex and satisfies the local quadratic growth condition, we provide both local sublinear and local linear convergence with respect to the primal gap. We also propose a variant of the away-step FW algorithm using Bregman distances over polytopes. We establish faster global convergence (up to a linear rate) for convex optimization under the Hölder error bound condition and local linear convergence for nonconvex optimization under the local quadratic growth condition. Numerical experiments demonstrate that our proposed FW algorithms outperform existing methods.

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.

Fast convergence and high-quality image recovery are two essential features of algorithms for solving ill-posed imaging inverse problems. Existing methods, such as regularization by denoising (RED), often focus on designing sophisticated image priors to improve reconstruction quality, while leaving convergence acceleration to heuristics. To bridge the gap, we propose Restarted Inertia with Score-based Priors (RISP) as a principled extension of RED. RISP incorporates a restarting inertia for fast convergence, while still allowing score-based image priors for high-quality reconstruction. We prove that RISP attains a faster stationary-point convergence rate than RED, without requiring the convexity of the image prior. We further derive and analyze the associated continuous-time dynamical system, offering insight into the connection between RISP and the heavy-ball ordinary differential equation (ODE). Experiments across a range of imaging inverse problems demonstrate that RISP enables fast convergence while achieving high-quality reconstructions.

This paper introduces a theoretical framework to resolve a central paradox in modern machine learning: When is it better to use less data? This question has become critical as classical scaling laws suggesting ``more is more'' (Sun et al., 2025) are challenged by methods like LIMO (``less is more'') and s1 (Ye et al., 2025; Muenighoff et al., 2025), which achieve superior performance with small, aggressively curated datasets. Here, we study data curation strategies where an imperfect oracle selects the training examples according to their difficulty and correctness. Our results provide exact scaling law curves for test error under both label-agnostic and label-aware curation rules, revealing when and why keeping only a subset of data can improve generalization. In contrast to classical scaling laws, we show that under certain conditions, small curated datasets can outperform full datasets, and we provide analytical conditions for this by deriving precise phase transition curves tied to data size and quality. We validate these theoretical claims with empirical results on ImageNet, confirming our predictions about when curation improves accuracy and can even mitigate model collapse. Furthermore, our framework provides a principled explanation for the contradictory curation strategies recently observed in LLM mathematical reasoning.

Multimodal Dataset Distillation (MDD) has emerged as a vital paradigm for enabling efficient training of vision-language models (VLMs) in the era of multimodal data proliferation. Unlike traditional dataset distillation methods that focus on single-modal tasks, MDD presents distinct challenges: (i) the effective distillation of heterogeneous multimodal knowledge, complicated by feature space misalignment and asynchronous optimization dynamics; and (ii) the lack of discrete class guidance, which hinders the distribution coverage and representativeness of synthetic data due to the vastness and continuity of the semantic space. To address these challenges, this paper proposes an Asynchronous Matching with Dynamic sampling (AMD) framework. AMD enables asynchronous trajectory matching by decoupling the selection of starting points for image and text trajectories. Additionally, a Semantics-Aware Prototype Mining module is introduced, which replaces random initialization by leveraging feature-space clustering to identify representative prototypes, enhancing the coverage and representativeness of the distilled samples. Extensive experiments demonstrate that AMD achieves superior distillation performance on Flickr30k and COCO (e.g., IR@1, IR@5, and IR@10 \textbf{gains of 4.5\%, 9.6\%, and 10.9\%}, respectively, on Flickr30k 200 pairs.) with negligible computational overhead.

We study the problem of recovering a low-tubal-rank tensor $\mathcal{X}\_\star\in \mathbb{R}^{n \times n \times k}$ from noisy linear measurements under the t-product framework. A widely adopted strategy involves factorizing the optimization variable as $\mathcal{U} * \mathcal{U}^\top$, where $\mathcal{U} \in \mathbb{R}^{n \times R \times k}$, followed by applying factorized gradient descent (FGD) to solve the resulting optimization problem. Since the tubal-rank $r$ of the underlying tensor $\mathcal{X}_\star$ is typically unknown, this method often assumes $r < R \le n$, a regime known as over-parameterization. However, when the measurements are corrupted by some dense noise (e.g., sub-Gaussian noise), FGD with the commonly used spectral initialization yields a recovery error that grows linearly with the over-estimated tubal-rank $R$. To address this issue, we show that using a small initialization enables FGD to achieve a nearly minimax optimal recovery error, even when the tubal-rank $R$ is significantly overestimated. Using a four-stage analytic framework, we analyze this phenomenon and establish the sharpest known error bound to date, which is independent of the overestimated tubal-rank $R$. Furthermore, we provide a theoretical guarantee showing that an easy-to-use early stopping strategy can achieve the best known result in practice. All these theoretical findings are validated through a series of simulations and real-data experiments.

We prove the first guarantees of sparse recovery for ReLU neural networks, where the sparse network weights constitute the signal to be recovered. Specifically, we study structural properties of the sparse network weights for two-layer, scalar-output networks under which a simple iterative hard thresholding algorithm recovers these weights exactly, using memory that grows linearly in the number of nonzero weights. We validate this theoretical result with simple experiments on recovery of sparse planted MLPs, MNIST classification, and implicit neural representations. Experimentally, we find performance that is competitive with, and often exceeds, a high-performing but memory-inefficient baseline based on iterative magnitude pruning.

Scaling foundation model training with Distributed Data Parallel~(DDP) methods is bandwidth-limited. Existing infrequent communication methods like Local SGD were designed to synchronize model parameters only and cannot be trivially applied to adaptive optimizers due to additional optimizer states. Heuristic approaches that keep states local or reset them lack guarantees and can be unstable in compute‑efficient batch regimes; conversely, Local Adam synchronizes all states uniformly and is provably convergent but triples communication costs. We propose Desynced Low Communication Adaptive Optimizers (DES-LOC), a family of optimizers assigning independent synchronization periods to parameters and momenta, enabling lower communication costs while preserving convergence. Our theoretical analysis shows that while parameter synchronization dominates the asymptotic rate in-expectation, high-probability convergence guarantees require at least infrequent synchronization of the second momentum. Furthermore, we prove that more frequent momentum sync permits larger stable step sizes. Experiments on language models of up to 1.7B show that DES-LOC can communicate 170x less than DDP and 2x less than the previous state-of-the-art Local Adam, enabling 1.3x–2.1x wall‑clock speedups over DDP for 1-13B models on 100Gb/s links. Furthermore, unlike previous heuristic methods, DES-LOC is robust to worker failures offering a scalable, efficient, and fault-tolerant solution for foundation model training.

We introduce Velocity-Regularized Adam (VRAdam), a physics-inspired optimizer for training deep neural networks that draws on ideas from quartic terms for kinetic energy with its stabilizing effects on various system dynamics. Previous algorithms, including the ubiquitous Adam, operate at the so-called adaptive edge of stability regime during training, leading to rapid oscillations and slowed convergence of loss. However, VRAdam adds a higher order penalty on the learning rate based on the velocity such that the algorithm automatically slows down whenever weight updates become large. In practice, we observe that the effective dynamic learning rate shrinks in high-velocity regimes, and damping oscillations. By combining this velocity‑based regularizer for global damping with Adam’s per‑parameter scaling, we create a powerful hybrid optimizer. For this optimizer, we provide rigorous theoretical analysis of operation at the edge of stability from a physical and control perspective for the momentum. Furthermore, we derive convergence bounds with the rate $\mathcal{O}(\ln(N)/\sqrt{N})$ for a stochastic non‑convex objective under mild assumptions. We demonstrate that VRAdam exceeds the performance against standard optimizers including AdamW. We benchmark various tasks such as image classification, language modeling, and generative modeling using diverse architectures and training methodologies including Convolutional Neural Networks (CNNs), Transformers, and GFlowNets.

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.

Reflecting the greater significance of recent history over the distant past in non-stationary environments, $\lambda$-discounted regret has been introduced in online convex optimization (OCO) to gracefully forget past data as new information arrives. When the discount factor $\lambda$ is given, online gradient descent with an appropriate step size achieves an $O(1/\sqrt{1-\lambda})$ discounted regret. However, the value of $\lambda$ is often not predetermined in real-world scenarios. This gives rise to a significant \emph{open question}: is it possible to develop a discounted algorithm that adapts to an unknown discount factor. In this paper, we affirmatively answer this question by providing a novel analysis to demonstrate that smoothed OGD (SOGD) achieves a uniform $O(\sqrt{\log T/1-\lambda})$ discounted regret, holding for all values of $\lambda$ across a continuous interval simultaneously. The basic idea is to maintain multiple OGD instances to handle different discount factors, and aggregate their outputs sequentially by an online prediction algorithm named as Discounted-Normal-Predictor (DNP). Our analysis reveals that DNP can combine the decisions of two experts, even when they operate on discounted regret with different discount factors.

Transformers have become the de facto backbone of modern deep learning, yet their training typically demands an advanced optimizer with adaptive learning rate like AdamW, rather than a momentum SGDW (mSGDW). Previous works show that it is mainly due to a heavy-tailed distribution of the gradients. In this paper, we introduce a Deeply Normalized Transformer (DNT), which is meticulously engineered to overcome this limitation enabling seamless training with vanilla mSGDW while yielding comparable performance to the Transformers trained via AdamW. To be specific, in DNT, we strategically integrate normalization techniques at proper positions in the Transformers to effectively modulate the Jacobian matrices of each layer, balance the influence of weights, activations, and their interactions, and thus enable the distributions of gradients concentrated. We provide both theoretical justifications of the normalization technique used in our DNT and extensive empirical evaluation on two popular Transformer architectures to validate that: a) DNT outperforms its counterparts (i.e., ViT and GPT), and b) DNT can be effectively trained with vanilla mSGDW.

Training large models with distributed data parallelism (DDP) requires frequent communication of gradients across workers, which can saturate bandwidth. Infrequent communication strategies (e.g., Local SGD) reduce this overhead but, when applied to adaptive optimizers, often suffer a performance gap relative to fully synchronous DDP. We trace this gap to a time-scale mismatch: the optimizer's fast-moving momentum, tuned for frequent updates, decays too quickly to smooth gradients over long intervals, leading to noise-dominated optimization. To address this, we propose MT-DAO, a family of optimizers that employs multiple slow- and fast-moving first momenta or the gradient to track update dynamics across different time scales, for which we provide the first convergence guarantees. Empirically, for language-model pre-training, this eliminates the performance gap with DDP, outperforming infrequent-communication baselines in perplexity and reducing iso-token wall-clock time by 6-27% on Ethernet interconnects. At the 720M scale, MT-DAO reaches a target perplexity in 24% fewer steps and 35% less time than the single-momentum DDP baseline. MT-DAO enables effective cross-datacenter training and training over wide geographic areas.

Asynchronous stochastic gradient methods are central to scalable distributed optimization, particularly when devices differ in computational capabilities. Such settings arise naturally in federated learning, where training takes place on smartphones and other heterogeneous edge devices. In addition to varying computation speeds, these devices often hold data from different distributions. However, existing asynchronous SGD methods struggle in such heterogeneous settings and face two key limitations. First, many rely on unrealistic assumptions of similarity across workers' data distributions. Second, methods that relax this assumption still fail to achieve theoretically optimal performance under heterogeneous computation times. We introduce Ringleader ASGD, the first asynchronous SGD algorithm that attains the theoretical lower bounds for parallel first-order stochastic methods in the smooth nonconvex regime, thereby achieving optimal time complexity under data heterogeneity and without restrictive similarity assumptions. Our analysis further establishes that Ringleader ASGD remains optimal under arbitrary and even time-varying worker computation speeds, closing a fundamental gap in the theory of asynchronous optimization.