ICLR 2025 论文列表

Neural Combinatorial Optimization (NCO) methods have exhibited promising performance in solving Vehicle Routing Problems (VRPs). However, most NCO methods rely on the conventional self-attention mechanism that induces excessive computational complexity, thereby struggling to contend with large-scale VRPs and hindering their practical applicability. In this paper, we propose a lightweight cross-attention mechanism with linear complexity, by which a Transformer network is developed to learn efficient and favorable solutions for large-scale VRPs. We also propose a Self-Improved Training (SIT) algorithm that enables direct model training on large-scale VRP instances, bypassing extensive computational overhead for attaining labels. By iterating solution reconstruction, the Transformer network itself can generate improved partial solutions as pseudo-labels to guide the model training. Experimental results on the Travelling Salesman Problem (TSP) and the Capacitated Vehicle Routing Problem (CVRP) with up to 100K nodes indicate that our method consistently achieves superior performance for synthetic and real-world benchmarks, significantly boosting the scalability of NCO methods.

We address the problem of solving strongly convex and smooth minimization problems using stochastic gradient descent (SGD) algorithm with a constant step size. Previous works suggested to combine the Polyak-Ruppert averaging procedure with the Richardson-Romberg extrapolation to reduce the asymptotic bias of SGD at the expense of a mild increase of the variance. We significantly extend previous results by providing an expansion of the mean-squared error of the resulting estimator with respect to the number of iterations $n$. We show that the root mean-squared error can be decomposed into the sum of two terms: a leading one of order $\mathcal{O}(n^{-1/2})$ with explicit dependence on a minimax-optimal asymptotic covariance matrix, and a second-order term of order $\mathcal{O}(n^{-3/4})$, where the power $3/4$ is best known. We also extend this result to the higher-order moment bounds. Our analysis relies on the properties of the SGD iterates viewed as a time-homogeneous Markov chain. In particular, we establish that this chain is geometrically ergodic with respect to a suitably defined weighted Wasserstein semimetric.

Recently, the study of heavy-tailed noises in first-order nonconvex stochastic optimization has gotten a lot of attention since it was recognized as a more realistic condition as suggested by many empirical observations. Specifically, the stochastic noise (the difference between the stochastic and true gradient) is considered to have only a finite $\mathfrak{p}$-th moment where $\mathfrak{p}\in\left(1,2\right]$ instead of assuming it always satisfies the classical finite variance assumption. To deal with this more challenging setting, people have proposed different algorithms and proved them to converge at an optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate for smooth objectives after $T$ iterations. Notably, all these new-designed algorithms are based on the same technique – gradient clipping. Naturally, one may want to know whether the clipping method is a necessary ingredient and the only way to guarantee convergence under heavy-tailed noises. In this work, by revisiting the existing Batched Normalized Stochastic Gradient Descent with Momentum (Batched NSGDM) algorithm, we provide the first convergence result under heavy-tailed noises but without gradient clipping. Concretely, we prove that Batched NSGDM can achieve the optimal $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{3\mathfrak{p}-2}})$ rate even under the relaxed smooth condition. More interestingly, we also establish the first $\mathcal{O}(T^{\frac{1-\mathfrak{p}}{2\mathfrak{p}}})$ convergence rate in the case where the tail index $\mathfrak{p}$ is unknown in advance, which is arguably the common scenario in practice.

The stochastic three points (STP) algorithm is a derivative-free optimization technique designed for unconstrained optimization problems in $\mathbb{R}^d$. In this paper, we analyze this algorithm for three classes of functions: smooth functions that may lack convexity, smooth convex functions, and smooth functions that are strongly convex. Our work provides the first almost sure convergence results of the STP algorithm, alongside some convergence results in expectation. For the class of smooth functions, we establish that the best gradient iterate of the STP algorithm converges almost surely to zero at a rate of $o(1/{T^{\frac{1}{2}-\epsilon}})$ for any $\epsilon\in (0,\frac{1}{2})$, where $T$ is the number of iterations. Furthermore, within the same class of functions, we establish both almost sure convergence and convergence in expectation of the final gradient iterate towards zero. For the class of smooth convex functions, we establish that $f(\theta^T)$ converges to $\inf_{\theta \in \mathbb{R}^d} f(\theta)$ almost surely at a rate of $o(1/{T^{1-\epsilon}})$ for any $\epsilon\in (0,1)$, and in expectation at a rate of $O(\frac{d}{T})$ where $d$ is the dimension of the space. Finally, for the class of smooth functions that are strongly convex, we establish that when step sizes are obtained by approximating the directional derivatives of the function, $f(\theta^T)$ converges to $\inf_{\theta \in \mathbb{R}^d} f(\theta)$ in expectation at a rate of $O((1-\frac{\mu}{2\pi dL})^T)$, and almost surely at a rate of $o((1-s\frac{\mu}{2\pi dL})^T)$ for any $s\in (0,1)$, where $\mu$ and $L$ are the strong convexity and smoothness parameters of the function.

We introduce a novel class of regularization functions, called Cauchy–Schwarz (CS) regularizers, which can be designed to induce a wide range of properties in solution vectors of optimization problems. To demonstrate the versatility of CS regularizers, we derive regularization functions that promote discrete-valued vectors, eigenvectors of a given matrix, and orthogonal matrices. The resulting CS regularizers are simple, differentiable, and can be free of spurious stationary points, making them suitable for gradient-based solvers and large-scale optimization problems. In addition, CS regularizers automatically adapt to the appropriate scale, which is, for example, beneficial when discretizing the weights of neural networks. To demonstrate the efficacy of CS regularizers, we provide results for solving underdetermined systems of linear equations and weight quantization in neural networks. Furthermore, we discuss specializations, variations, and generalizations, which lead to an even broader class of new and possibly more powerful regularizers.

Distributionally robust optimization (DRO) is a powerful technique to train robust machine learning models that perform well under distribution shifts. Compared with empirical risk minimization (ERM), DRO optimizes the expected loss under the worst-case distribution in an uncertainty set of distributions. This paper revisits the important problem of DRO with non-convex smooth loss functions. For this problem, Jin et al. (2021) showed that its dual problem is generalized $(L_0, L_1)$-smooth condition and gradient noise satisfies the affine variance condition, designed an algorithm of mini-batch normalized gradient descent with momentum, and proved its convergence and complexity. In this paper, we show that the dual problem and the gradient noise satisfy simpler yet more precise partially generalized smoothness condition and partially affine variance condition by studying the optimization variable and dual variable separately, which further yields much simpler algorithm design and convergence analysis. We develop a double stochastic gradient descent with clipping (D-SGD-C) algorithm that converges to an $\epsilon$-stationary point with $\mathcal O(\epsilon^{-4})$ gradient complexity, which matches with results in Jin et al. (2021). Our algorithm does not need to use momentum, and the proof is much simpler, thanks to the more precise characterization of partially generalized smoothness and partially affine variance noise. We further design a variance-reduced method that achieves a lower gradient complexity of $\mathcal O(\epsilon^{-3})$. Our theoretical results and insights are further verified numerically on a number of tasks, and our algorithms outperform the existing DRO method (Jin et al., 2021).

The regularity or implicit bias in neural network optimization has been typically studied via the parameter norms or the landscape curvature, often overlooking the trajectory leading to these parameters. However, properties of the trajectory --- particularly its directionality --- capture critical aspects of how gradient descent navigates the landscape to converge to a solution. In this work, we introduce the notion of a Trajectory Map and derive natural complexity measures that highlight the directional characteristics of optimization trajectories. Our comprehensive analysis across vision and language modeling tasks reveals that (a) the trajectory's directionality at the macro-level saturates by the initial phase of training, wherein weight decay and momentum play a crucial but understated role; and (b) in subsequent training, trajectory directionality manifests in micro-level behaviors, such as oscillations, for which we also provide a theoretical analysis. This implies that neural optimization trajectories have, overall, a more linear form than zig-zaggy, as evident by high directional similarity, especially towards the end. To further hone this point, we show that when the trajectory direction gathers such an inertia, optimization proceeds largely unaltered even if the network is severely decapacitated (by freezing >99% of the parameters), --- thereby demonstrating the potential for significant computational and resource savings without compromising performance.

Training large-scale models under given resources requires careful design of parallelism strategies. In particular, the efficiency notion of critical batch size (CBS), concerning the compromise between time and compute, marks the threshold beyond which greater data parallelism leads to diminishing returns. To operationalize it, we propose a measure of CBS and pre-train a series of auto-regressive language models, ranging from 85 million to 1.2 billion parameters, on the C4 dataset. Through extensive hyper-parameter sweeps and careful control of factors such as batch size, momentum, and learning rate along with its scheduling, we systematically investigate the impact of scale on CBS. Then we fit scaling laws with respect to model and data sizes to decouple their effects. Overall, our results demonstrate that CBS scales primarily with data size rather than model size, a finding we justify theoretically through the analysis of infinite-width limits of neural networks and infinite-dimensional least squares regression. Of independent interest, we highlight the importance of common hyper-parameter choices and strategies for studying large-scale pre-training beyond fixed training durations.

Bayesian optimization devolves the global optimization of a costly objective function to the global optimization of a sequence of acquisition functions. This inner-loop optimization can be catastrophically difficult if it involves posterior sample paths, especially in higher dimensions. We introduce an efficient global optimization strategy for posterior samples based on global rootfinding. It provides gradient-based optimizers with two sets of judiciously selected starting points, designed to combine exploration and exploitation. The number of starting points can be kept small without sacrificing optimization quality. Remarkably, even with just one point from each set, the global optimum is discovered most of the time. The algorithm scales practically linearly to high dimensions, breaking the curse of dimensionality. For Gaussian process Thompson sampling (GP-TS), we demonstrate remarkable improvement in both inner- and outer-loop optimization, surprisingly outperforming alternatives like EI and GP-UCB in most cases. Our approach also improves the performance of other posterior sample-based acquisition functions, such as variants of entropy search. Furthermore, we propose a sample-average formulation of GP-TS, which has a parameter to explicitly control exploitation and can be computed at the cost of one posterior sample.

Natural language prompt optimization, or prompt engineering, has emerged as a powerful technique to unlock the potential of Large Language Models (LLMs) for various tasks. While existing methods primarily focus on maximizing a single task-specific performance metric for LLM outputs, real-world applications often require considering trade-offs between multiple objectives. In this work, we address this limitation by proposing an effective technique for multi-objective prompt optimization for LLMs. Specifically, we propose **ParetoPrompt**, a reinforcement learning~(RL) method that leverages dominance relationships between prompts to derive a policy model for prompts optimization using preference-based loss functions. By leveraging multi-objective dominance relationships, ParetoPrompt enables efficient exploration of the entire Pareto front without the need for a predefined scalarization of multiple objectives. Our experimental results show that ParetoPrompt consistently outperforms existing algorithms that use specific objective values. ParetoPrompt also yields robust performances when the objective metrics differ between training and testing.

This paper introduces a novel approach to solving multi-block nonconvex composite optimization problems through a proximal linearized Alternating Direction Method of Multipliers (ADMM). This method incorporates an Increasing Penalization and Decreasing Smoothing (IPDS) strategy. Distinguishing itself from existing ADMM-style algorithms, our approach (denoted IPDS-ADMM) imposes a less stringent condition, specifically requiring continuity in just one block of the objective function. IPDS-ADMM requires that the penalty increases and the smoothing parameter decreases, both at a controlled pace. When the associated linear operator is bijective, IPDS-ADMM uses an over-relaxation stepsize for faster convergence; however, when the linear operator is surjective, IPDS-ADMM uses an under-relaxation stepsize for global convergence. We devise a novel potential function to facilitate our convergence analysis and prove an oracle complexity $\mathcal{O}(\epsilon^{-3})$ to achieve an $\epsilon$-approximate critical point. To the best of our knowledge, this is the first complexity result for using ADMM to solve this class of nonsmooth nonconvex problems. Finally, some experiments on the sparse PCA problem are conducted to demonstrate the effectiveness of our approach.

Direct Preference Optimization (DPO) has emerged as a stable, scalable, and efficient solution for language model alignment. Despite its empirical success, the optimization properties, particularly the impact of samplers on its convergence rates, remain under-explored. In this paper, we provide a rigorous analysis of DPO's convergence rates with different sampling strategies under the exact gradient setting, revealing a surprising separation: uniform sampling achieves $\textbf{linear}$ convergence, while our proposed online sampler achieves $\textbf{quadratic}$ convergence. We further adapt the sampler to practical settings by incorporating posterior distributions and logit mixing, demonstrating improvements over previous methods. For example, it outperforms vanilla DPO by over $7.4$% on Safe-RLHF dataset. Our results not only offer insights into the theoretical understanding of DPO but also pave the way for further algorithm designs.

Recent research in Meta-Black-Box-Optimization (MetaBBO) have shown that meta-trained neural networks can effectively guide the design of black-box optimizers, significantly reducing the need for expert tuning and delivering robust performance across complex problem distributions. Despite their success, a paradox remains: MetaBBO still rely on human-crafted Exploratory Landscape Analysis features to inform the meta-level agent about the low-level optimization progress. To address the gap, this paper proposes Neural Exploratory Landscape Analysis (NeurELA), a novel framework that dynamically profiles landscape features through a two-stage, attention-based neural network, executed in an entirely end-to-end fashion. NeurELA is pre-trained over a variety of MetaBBO algorithms using a multi-task neuroevolution strategy. Extensive experiments show that NeurELA achieves consistently superior performance when integrated into different and even unseen MetaBBO tasks and can be efficiently fine-tuned for further performance boost. This advancement marks a pivotal step in making MetaBBO algorithms more autonomous and broadly applicable. The source code of NeurELA can be accessed at https://anonymous.4open.science/r/Neur-ELA-303C.

Second-order methods for convex optimization outperform first-order methods in terms of theoretical iteration convergence, achieving rates up to $O(k^{-5})$ for highly-smooth functions. However, their practical performance and applications are limited due to their multi-level structure and implementation complexity. In this paper, we present new results on high-order optimization methods, supported by their practical performance. First, we show that the basic high-order methods, such as the Cubic Regularized Newton Method, exhibit global superlinear convergence for $\mu$-strongly star-convex functions, a class that includes $\mu$-strongly convex functions and some non-convex functions. Theoretical convergence results are both inspired and supported by the practical performance of these methods. Secondly, we propose a practical version of the Nesterov Accelerated Tensor method, called NATA. It significantly outperforms the classical variant and other high-order acceleration techniques in practice. The convergence of NATA is also supported by theoretical results. Finally, we introduce an open-source computational library for high-order methods, called OPTAMI. This library includes various methods, acceleration techniques, and subproblem solvers, all implemented as PyTorch optimizers, thereby facilitating the practical application of high-order methods to a wide range of optimization problems. We hope this library will simplify research and practical comparison of methods beyond first-order.

This study addresses the challenge of noise in training datasets for Direct Preference Optimization (DPO), a method for aligning Large Language Models (LLMs) with human preferences. We categorize noise into pointwise noise, which includes low-quality data points, and pairwise noise, which encompasses erroneous data pair associations that affect preference rankings. Utilizing Distributionally Robust Optimization (DRO), we enhance DPO's resilience to these types of noise. Our theoretical insights reveal that DPO inherently embeds DRO principles, conferring robustness to pointwise noise, with the regularization coefficient $\beta$ playing a critical role in its noise resistance. Extending this framework, we introduce Distributionally Robustifying DPO (Dr. DPO), which integrates pairwise robustness by optimizing against worst-case pairwise scenarios. The novel hyperparameter $\beta'$ in Dr. DPO allows for fine-tuned control over data pair reliability, providing a strategic balance between exploration and exploitation in noisy training environments. Empirical evaluations demonstrate that Dr. DPO substantially improves the quality of generated text and response accuracy in preference datasets, showcasing enhanced performance in both noisy and noise-free settings.

Communication efficiency has garnered significant attention as it is considered the main bottleneck for large-scale decentralized Machine Learning applications in distributed and federated settings. In this regime, clients are restricted to transmitting small amounts of compressed information to their neighbors over a communication graph. Numerous endeavors have been made to address this challenging problem by developing algorithms with compressed communication for decentralized non-convex optimization problems. Despite considerable efforts, current theoretical understandings of the problem are still very limited, and existing algorithms all suffer from various limitations. In particular, these algorithms typically rely on strong, and often infeasible assumptions such as bounded data heterogeneity or require large batch access while failing to achieve linear speedup with the number of clients. In this paper, we introduce MoTEF, a novel approach that integrates communication compression with $\textbf{Mo}$mentum $\textbf{T}$racking and $\textbf{E}$rror $\textbf{F}$eedback. MoTEF is the first algorithm to achieve an asymptotic rate matching that of distributed SGD under arbitrary data heterogeneity, hence resolving a long-standing theoretical obstacle in decentralized optimization with compressed communication. We provide numerical experiments to validate our theoretical findings and confirm the practical superiority of MoTEF.

Bilevel optimization, which addresses challenges in hierarchical learning tasks, has gained significant interest in machine learning. Implementing gradient descent for bilevel optimization presents computational hurdles, notably the need to compute the exact lower-level solution and the inverse Hessian of the lower-level objective. While these two aspects are inherently connected, existing methods typically handle them separately by solving the lower-level problem and a linear system for the inverse Hessian-vector product. In this paper, we introduce a general framework to tackle these computational challenges in a coordinated manner. Specifically, we leverage quasi-Newton algorithms to accelerate the solution of the lower-level problem while efficiently approximating the inverse Hessian-vector product. Furthermore, by leveraging the superlinear convergence properties of BFGS, we establish a non-asymptotic convergence analysis for the BFGS adaptation within our framework. Numerical experiments demonstrate the comparable or superior performance of our proposed algorithms in real-world learning tasks, including hyperparameter optimization, data hyper-cleaning, and few-shot meta-learning.

Long-horizon combinatorial optimization problems (COPs), such as the Flexible Job-Shop Scheduling Problem (FJSP), often involve complex, interdependent decisions over extended time frames, posing significant challenges for existing solvers. While Rolling Horizon Optimization (RHO) addresses this by decomposing problems into overlapping shorter-horizon subproblems, such overlap often involves redundant computations. In this paper, we present L-RHO, the first learning-guided RHO framework for COPs. L-RHO employs a neural network to intelligently fix variables that in hindsight did not need to be re-optimized, resulting in smaller and thus easier-to-solve subproblems. For FJSP, this means identifying operations with unchanged machine assignments between consecutive subproblems. Applied to FJSP, L-RHO accelerates RHO by up to 54\% while significantly improving solution quality, outperforming other heuristic and learning-based baselines. We also provide in-depth discussions and verify the desirable adaptability and generalization of L-RHO across numerous FJSP variates, distributions, online scenarios and benchmark instances. Moreover, we provide a theoretical analysis to elucidate the conditions under which learning is beneficial.

We consider the decentralized minimization of a separable objective $\sum_{i=1}^{n} f_i(x_i)$, where the variables are coupled through an affine constraint $\sum_{i=1}^n\left(\mathbf{A}_i x_i - b_i\right) = 0$. We assume that the functions $f_i$, matrices $\mathbf{A}_i$, and vectors $b_i$ are stored locally by the nodes of a computational network, and that the functions $f_i$ are smooth and strongly convex. This problem has significant applications in resource allocation and systems control and can also arise in distributed machine learning. We propose lower complexity bounds for decentralized optimization problems with coupled constraints and a first-order algorithm achieving the lower bounds. To the best of our knowledge, our method is also the first linearly convergent first-order decentralized algorithm for problems with general affine coupled constraints.