ICLR 2026 论文列表

Communication efficiency is a central challenge in distributed machine learning training, and message compression is a widely used solution. However, standard Error Feedback (EF) methods (Seide et al., 2014), though effective for smooth unconstrained optimization with compression (Karimireddy et al., 2019), fail in the broader and practically important setting of composite optimization, which captures, e.g., objectives consisting of a smooth loss combined with a non-smooth regularizer or constraints. The theoretical foundation and behavior of EF in the context of the general composite setting remain largely unexplored. In this work, we consider composite optimization with EF. We point out that the basic EF mechanism and its analysis no longer stand when a composite part is involved. We argue that this is because of a fundamental limitation in the method and its analysis technique. We propose a novel method that combines _Dual Averaging_ with EControl (Gao et al., 2024), a state-of-the-art variant of the EF mechanism, and achieves for the first time a strong convergence analysis for composite optimization with error feedback. Along with our new algorithm, we also provide a new and novel analysis template for inexact dual averaging method, which might be of independent interest. We also provide experimental results to complement our theoretical findings.

Physics-Informed Neural Networks (PINNs) have emerged as a powerful tool for solving PDEs, yet training is difficult due to a multi-objective loss that couples PDE residuals, initial/boundary conditions, and auxiliary physics terms. Existing remedies often yield infeasible scaling factors or conflicting update directions, resulting in degraded performance. In this paper, we show that training PINNs requires jointly considering feasible scaling factors and a non-conflict direction. Through a geometric analysis of per-loss gradients, we define the $\textit{harmonized cone}$ as the intersection of their primal and dual cones, which characterizes directions that are simultaneously feasible and non-conflicting. Building on this, we propose $HARMONIC$ (HARMONIzed Cone gradient descent), a training procedure that computes updates within the harmonized cone by leveraging the Double Description method to aggregate extreme rays. Theoretically, we establish convergence guarantees in nonconvex settings and prove the existence of a nontrivial harmonized cone. Across standard PDE benchmarks, $HARMONIC$ generally outperforms state-of-the-art methods while ensuring feasible and non-conflict updates.

Diffusion models have recently emerged as a powerful class of generative models and have achieved state-of-the-art performance in various image synthesis tasks. However, training diffusion models generally requires large amounts of data and suffer from overfitting when the dataset size is limited. To address these limitations, we propose a novel method called WILD-Diffusion, which is inspired by Wasserstein Distributionally Robust Optimization (WDRO), an important and elegant mathematical formulation from robust optimization area. Specifically, WILD-Diffusion utilizes WDRO to iteratively generate new training samples within a Wasserstein distance based uncertainty set centered at the limited data data distribution. This carefully designed method can progressively augment the training set throughout the training process and effectively overcome the obstacles caused by the limited data issue. Moreover, we establish the convergence guarantee for our algorithm even though the mixture of diffusion process and WDRO brings significant challenges to our analysis in theory. Finally, we conduct a set of experiments to verify the effectiveness of our proposed method. With WILD-Diffusion, we can achieve more than a $10$% reduction in FID using only $20$% of the training data across different datasets. Moreover, our method can attain state-of-the-art FID with as few as $100$ images, both in pretrained and non-pretrained settings.

Considerable efforts have been devoted to machine learning (ML) for combinatorial optimization (CO) problems, especially on graphs. Compared to the active and well-established research for representation learning of text and vision, etc., it remains under-studied for the representation learning of CO problems, especially across different types. In this paper, we try to fill this gap (especially for NP-complete (NPC) problems, as they, in fact, can be reduced to one another). Our so-called ConRep4CO framework, performs contrastive learning by first transforming CO instances in various original forms into the form of Boolean satisfiability (SAT). This scheme is readily doable, especially for NPC problems, including those practical graph decision problems (GDPs) which are inherently related to their NP-hard optimization versions. Specifically, each positive pair of instances for contrasting consists of an instance in its original form and its corresponding transformed SAT form, while the negative samples are other instances not in correspondence. Extensive experiments on seven GDPs (most of which are NPC) show that ConRep4CO significantly improves the representation quality and generalizability to problem scale. Furthermore, we conduct extensive experiments on NP-hard optimization versions of the GDPs, including MVC, MIS, MC and MDS. The results show that introducing ConRep4CO can yield performance improvements of 61.27%, 32.20%, 36.46%, and 45.29% in objective value gaps compared to problem-specific baselines, highlighting the potential of ConRep4CO as a unified pre-training paradigm for CO problems.

As Large Language Models (LLMs) are increasingly used, their security risks have drawn increasing attention. Existing research reveals that LLMs are highly susceptible to jailbreak attacks, with effectiveness varying across language contexts. This paper investigates the role of classical Chinese in jailbreak attacks. Owing to its conciseness and obscurity, classical Chinese can partially bypass existing safety constraints, exposing notable vulnerabilities in LLMs. Based on this observation, this paper proposes a framework, CC-BOS, for the automatic generation of classical Chinese adversarial prompts based on multi-dimensional fruit fly optimization, facilitating efficient and automated jailbreak attacks in black-box settings. Prompts are encoded into eight policy dimensions—covering role, behavior, mechanism, metaphor, expression, knowledge, trigger pattern and context; and iteratively refined via smell search, visual search, and cauchy mutation. This design enables efficient exploration of the search space, thereby enhancing the effectiveness of black-box jailbreak attacks. To enhance readability and evaluation accuracy, we further design a classical Chinese to English translation module. Extensive experiments demonstrate that effectiveness of the proposed CC-BOS, consistently outperforming state-of-the-art jailbreak attack methods.

Large language models (LLMs) are increasingly applied to software engineering (SWE), but they struggle on real-world tasks that are long-horizon and often out of distribution. Current systems typically adopt monolithic designs where a single model attempts to interpret ambiguous issues, navigate large codebases, and implement fixes in one extended reasoning chain. This design makes it difficult to generalize beyond training data. Inspired by how human engineers decompose problems into sub-tasks, we argue that SWE agents should be structured as orchestrators coordinating specialized sub-agents, each responsible for a specific sub-task such as bug reproduction, fault localization, code modification, or validation. The central challenge is how to design these hierarchies effectively. Manual decompositions follow human workflows but often mismatch LLM capabilities, while automated search methods such as evolutionary strategies require evaluating a very large number of candidates, making them prohibitively expensive for SWE. We show that formulating hierarchy discovery as a multi-armed bandit problem enables efficient exploration of sub-agent designs under limited budgets. On SWE-bench-Verified, this approach outperforms single-agent systems and manually designed multi-agent systems. On SWE-bench-Live, which features recent and out-of-distribution issues, our system ranks 2nd on the leaderboard with a 36B model, surpassing larger systems such as GPT-4 and Claude. This provides the first evidence that hierarchical multi-agent systems improves generalization on challenging long-horizon SWE tasks.

Designing robust reinforcement learning (RL) agents in the presence of imperfect reward signals remains a core challenge. In practice, agents are often trained with proxy rewards that only approximate the true objective, leaving them vulnerable to reward hacking, where high proxy returns arise from unintended or exploitative behaviors. Recent work formalizes this issue using r-correlation between proxy and true rewards, but existing methods like occupancy-regularized policy optimization (ORPO) optimize against a fixed proxy and do not provide strong guarantees against broader classes of correlated proxies. In this work, we formulate reward hacking as a robust policy optimization problem over the space of all r-correlated proxy rewards. We derive a tractable max-min formulation, where the agent maximizes performance under the worst-case proxy consistent with the correlation constraint. We further show that when the reward is a linear function of known features, our approach can be adapted to incorporate this prior knowledge, yielding both improved policies and interpretable worst-case rewards. Experiments across several environments show that our algorithms consistently outperform ORPO in worst-case returns, and offer improved robustness and stability across different levels of proxy–true reward correlation. These results show that our approach provides both robustness and transparency in settings where reward design is inherently uncertain.

Increasing the batch size during training --- a “batch ramp'' --- is a promising strategy to accelerate large language model pretraining. While for SGD, doubling the batch size can be equivalent to halving the learning rate, the optimal strategy for adaptive optimizers like Adam is less clear. As a result, any batch-ramp scheduling, if used at all, is typically tuned heuristically. This work develops a principled framework for batch-size scheduling and introduces Seesaw: whenever a standard scheduler would halve the learning rate, Seesaw instead multiplies it by $1/\sqrt{2}$ and doubles the batch size, preserving loss dynamics while reducing serial steps. Theoretically, we provide, to our knowledge, the first finite-sample proof of equivalence between learning-rate decay and batch-size ramp-up for SGD on noisy linear regression, and we extend this equivalence to normalized SGD, a tractable proxy for Adam, under a variance-dominated regime observed in practice. Empirically, on 150M/300M/600M-parameter models trained at Chinchilla scale using a constant (critical) batch size, Seesaw matches cosine decay at equal FLOPs while reducing wall-clock time by $\approx 36\%$, approaching the theoretical limit implied by our analysis.

We study matrix completion via deep matrix factorization (a.k.a. deep linear neural networks) as a simplified testbed to examine how network depth influences training dynamics. Despite the simplicity and importance of the problem, prior theory largely focuses on shallow (depth-2) models and does not fully explain the implicit low-rank bias observed in deeper networks. We identify coupled dynamics as a key mechanism behind this bias and show that it intensifies with increasing depth. Focusing on gradient flow under diagonal observations, we prove: (a) networks of depth $\geq 3$ exhibit coupling unless initialized diagonally, and (b) convergence to rank-1 occurs if and only if the dynamics is coupled—resolving an open question by Menon (2024) for a family of initializations. We also revisit the loss of plasticity phenomenon in matrix completion (Kleinman et al., 2024), where pre-training on few observations and resuming with more degrades performance. We show that deep models avoid plasticity loss due to their low-rank bias, whereas depth-2 networks pre-trained under decoupled dynamics fail to converge to low-rank, even when resumed training (with additional data) satisfies the coupling condition—shedding light on the mechanism behind this phenomenon.

We study the Nearest Neighbor Search (NNS) problem in a high-dimensional setting where data originates from a low-dimensional subspace and is corrupted by Gaussian noise. Specifically, we consider a semi-random model where $n$ points from an unknown $k$-dimensional subspace of $\mathbb{R}^d$ ($k \ll d$) are perturbed by zero-mean $d$-dimensional Gaussian noise with variance $\sigma^2$ on each coordinate. Without loss of generality, we may assume the nearest neighbor is at distance $1$ from the query, and that all other points are at distance at least $1+\varepsilon$. We assume we are given only the noisy data and are required to find NN of the uncorrupted data. We prove the following results: 1. For $\sigma \in O(1/k^{1/4})$, we show that simply performing SVD denoises the data; namely, we provably recover accurate NN of uncorrupted data (Theorem 1.1). 2. For $\sigma \gg 1/k^{1/4}$, NN in uncorrupted data is not even {\bf identifiable} from the noisy data in general. This is a matching lower bound on $\sigma$ with the above result, demonstrating the necessity of this threshold for NNS (Lemma 3.1). 3. For $\sigma \gg 1/\sqrt k$, the noise magnitude ($\sigma \sqrt{d}$) is significantly exceeds the inter-point distances in the unperturbed data. Moreover, NN in noisy data is different from NN in the uncorrupted data in general. \end{enumerate} Note that (1) and (3) together imply SVD identifies correct NN in uncorrupted data even in a regime where it is different from NN in noisy data. This was not the case in existing literature (see e.g. (Abdullah et al., 2014)). Another comparison with (Abdullah et al., 2014) is that it requires $\sigma$ to be at least an inverse polynomial in the ambient dimension $d$. The proof of (1) above uses upper bounds on perturbations of singular spaces of matrices as well as concentration and spherical symmetry of Gaussians. We thus give theoretical justification for the performance of spectral methods in practice. We also provide empirical results on real datasets to corroborate our findings.

Foundation models update slowly due to resource-intensive training, whereas domain-specific models evolve rapidly between releases. Model merging seeks to combine multiple expert models into a single, more capable model, reducing storage and serving costs while supporting decentralized development. Despite its potential, previous studies have primarily focused on merging visual classification models or Large Language Models (LLMs) for code and math tasks. Recently, Multimodal LLMs (MLLMs) that extend LLMs through large-scale multimodal training have gained traction. However, no benchmark exists for model merging research that clearly divides the tasks of MLLM training and evaluation. In this paper, $(i)$ we introduce a model merging benchmark for MLLMs, which includes multiple tasks such as VQA, Geometry, Chart, OCR, and Grounding, studying both LoRA and full fine-tuning models. Moreover, we explore how model merging can combine different modalities (e.g., vision-language, audio-language, and video-language models), moving toward the Omni-language model. $(ii)$ We implement 10 model merging algorithms on the benchmark. Furthermore, we propose a novel method that removes noise from task vectors and robustly optimizes the merged vector based on a loss defined over task vector interactions, achieving an average performance gain of 2.48\%. $(iii)$ We find that model merging offers a promising way for building improved MLLMs without requiring training data. Our results also demonstrate that the complementarity among multiple modalities outperforms individual modalities.

Prompt engineering has become central to eliciting the capabilities of large language models (LLMs). At its core lies prompt selection - efficiently identifying the most effective prompts. However, most prior investigations overlook a key challenge: the inherently multi-faceted nature of prompt performance, which cannot be captured by a single metric. To fill this gap, we study the multi-objective prompt selection problem under two practical settings: Pareto prompt set recovery and best feasible prompt identification. Casting the problem into the pure-exploration bandits framework, we adapt provably efficient algorithms from multi-objective bandits and further introduce a novel design for best feasible arm identification in structured bandits, with theoretical guarantees on the identification error in the linear case. Extensive experiments across multiple LLMs show that the bandit-based approaches yield significant improvements over baselines, establishing a principled and efficient framework for multi-objective prompt optimization.

Direct Preference Optimization (DPO) has emerged as a powerful framework for aligning large language models (LLMs) with human preferences via pairwise comparisons. However, its performance is highly sensitive to the quality of training samples: when the reference policy is poorly aligned with human preferences, ambiguous pairs can dominate the gradient signal and degrade generalization. To address this, we propose RAPPO($\textbf{R}$eliable $\textbf{A}$lignment for $\textbf{P}$reference $\textbf{P}$olicy $\textbf{O}$ptimization), a simple sample-aware modification of the DPO loss that mitigates reference-policy misalignment by filtering out the hardest, most ambiguous samples. We theoretically show that RAPPO yields improved generalization guarantees. RAPPO is lightweight and requires only a few lines of code to be integrated into any existing DPO-type algorithm. Surprisingly, With this simple modification, our simulations across a broad suite of alignment tasks and benchmarks show consistent gains over DPO and recent state-of-the-art baselines. On the PKU-SafeRLHF benchmark, RAPPO attains helpfulness $0.693$ ($+34.8\%$ over DPO) and harmlessness $0.357$ ($-21.0\%$ vs DPO).

Quality-Diversity (QD) algorithms constitute a branch of optimization that is concerned with discovering a diverse and high-quality set of solutions to an optimization problem. Current QD methods commonly maintain diversity by dividing the behavior space into discrete regions, ensuring that solutions are distributed across different parts of the space. The QD problem is then solved by searching for the best solution in each region. This approach to QD optimization poses challenges in large solution spaces, where storing many solutions is impractical, and in high-dimensional behavior spaces, where discretization becomes ineffective due to the curse of dimensionality. We present an alternative framing of the QD problem, called \emph{Soft QD}, that sidesteps the need for discretizations. We validate this formulation by demonstrating its desirable properties, such as monotonicity, and by relating its limiting behavior to the widely used QD Score metric. Furthermore, we leverage it to derive a novel differentiable QD algorithm, \emph{Soft QD Using Approximated Diversity (SQUAD)}, and demonstrate empirically that it is competitive with current state of the art methods on standard benchmarks while offering better scalability to higher dimensional problems.

Nonsmooth composite optimization with orthogonality constraints has a wide range of applications in statistical learning and data science. However, this problem is challenging due to its nonsmooth objective and computationally expensive, non-convex constraints. In this paper, we propose a new approach called \textbf{OBCD}, which leverages Block Coordinate Descent to address these challenges. \textbf{OBCD} is a feasible method with a small computational footprint. In each iteration, it updates $k$ rows of the solution matrix, where $k \geq 2$, by globally solving a small nonsmooth optimization problem under orthogonality constraints. We prove that the limiting points of \textbf{OBCD}, referred to as (global) block-$k$ stationary points, offer stronger optimality than standard critical points. Furthermore, we show that \textbf{OBCD} converges to $\epsilon$-block-$k$ stationary points with an iteration complexity of $\mathcal{O}(1/\epsilon)$. Additionally, under the Kurdyka-Lojasiewicz (KL) inequality, we establish the non-ergodic convergence rate of \textbf{OBCD}. We also demonstrate how novel breakpoint search methods can be used to solve the subproblem in \textbf{OBCD}. Empirical results show that our approach consistently outperforms existing methods.

Reinforcement learning with verifiable reward has recently emerged as a central paradigm for post-training large language models (LLMs); however, prevailing mean-based methods, such as Group Relative Policy Optimization (GRPO), suffer from entropy collapse and limited reasoning gains. We argue that these issues stem from overemphasizing high-probability output sequences while neglecting rare but informative reasoning paths. To address these challenges, we propose Risk-based Policy Optimization (RiskPO), which substitutes classical mean-based objectives with principled risk measures. Specifically, we introduce a Mixed Value-at-Risk objective that integrates weighted attention over multiple regions of the reward distribution, thereby amplifying gradient signals on challenging instances and preventing overconfident convergence. We further design a bundling scheme that aggregates multiple questions into bundles, thus enriching the feedback signal and yielding more stable and informative training dynamics. Theoretically, we prove that the risk-averse update alleviates entropy collapse and promotes exploration. Numerically, RiskPO achieves consistent and significant improvements in mathematical reasoning, multi-modal reasoning, and code generation benchmarks, surpassing GRPO and its variants on both Pass@1 and Pass@k metrics. Our results demonstrate that risk-based optimization provides a rigorous and effective paradigm for enhancing LLM reasoning capabilities.

Stochastic gradient descent with momentum (SGDM) is widely used in machine learning, yet high-probability learning bounds for SGDM in non-convex settings remain scarce. In this paper, we provide high-probability convergence bounds and generalization bounds for SGDM. First, we establish such bounds for the gradient norm in the general non-convex case. The resulting convergence bounds are tighter than existing theoretical results, and to the best of our knowledge, the obtained generalization bounds are the first ones for SGDM. Next, under the Polyak-{\L}ojasiewicz condition, we derive bounds for the function-value error instead of the gradient norm, and the corresponding learning rates are faster than in the general non-convex case. Finally, by additionally assuming a mild Bernstein condition on the gradient, we obtain even sharper generalization bounds whose learning rates can reach $\widetilde{\mathcal{O}}(1/n^2)$ in the low-noise regime, where $n$ is the sample size. Overall, we provide a systematic study of high-probability learning bounds for non-convex SGDM.

Autoformalization, which translates natural language mathematics into machine-verifiable formal statements, is critical for using formal mathematical reasoning to solve math problems stated in natural language. While Large Language Models can generate syntactically correct formal statements, they often fail to preserve the original problem's semantic intent. This limitation arises from the LLM approaches' treating autoformalization as a simplistic translation task which lacks mechanisms for self-reflection and iterative refinement that human experts naturally employ. To address these issues, we propose ReForm, a Reflective Autoformalization method that tightly integrates semantic consistency evaluation into the autoformalization process. This enables the model to iteratively generate formal statements, assess its semantic fidelity, and self-correct identified errors through progressive refinement. To effectively train this reflective model, we introduce Prospective Bounded Sequence Optimization (PBSO), which employs different rewards at different sequence positions to ensure that the model develops both accurate autoformalization and correct semantic validations, preventing superficial critiques that would undermine the purpose of reflection. Extensive experiments across four autoformalization benchmarks demonstrate that ReForm achieves an average improvement of 22.6 percentage points over the strongest baselines. To further ensure evaluation reliability, we introduce ConsistencyCheck, a benchmark of 859 expert-annotated items that not only validates LLMs as judges but also reveals that autoformalization is inherently difficult: even human experts produce semantic errors in up to 38.5\% of cases.

We prove the stability and global convergence of a coupled actor-critic gradient flow for infinite-horizon and entropy-regularised Markov decision processes (MDPs) in continuous state and action space with linear function approximation under Q-function realisability. We consider a version of the actor critic gradient flow where the critic is updated using temporal difference (TD) learning while the policy is updated using a policy mirror descent method on a separate timescale. For general action spaces, the relative entropy regularizer is unbounded and thus it is not clear a priori that the actor-critc flow does not suffer from finite-time blow-up. Therefore we first demonstrate stability which in turn enables us obtain a convergence rate of the actor critic flow to the optimal regularised value function. The arguments presented show that timescale separation is crucial for stability and convergence in this setting.