ICLR 2025 Papers

Optimization methods are crucial to the success of machine learning, with Stochastic Gradient Descent (SGD) serving as a foundational algorithm for training models. However, SGD is often sensitive to the choice of the learning rate, which necessitates extensive hyperparameter tuning. In this work, we introduce a new variant of SGD that brings enhanced stability in two key aspects. First, our method allows the use of the same fixed learning rate to attain optimal convergence rates regardless of the noise magnitude, eliminating the need to adjust learning rates between noiseless and noisy settings. Second, our approach achieves these optimal rates over a wide range of learning rates, significantly reducing sensitivity compared to standard SGD, which requires precise learning rate selection. Our key innovation is a novel gradient estimator based on a double-momentum mechanism that combines two recent momentum-based techniques. Utilizing this estimator, we design both standard and accelerated algorithms that are robust to the choice of learning rate. Specifically, our methods attain optimal convergence rates in both noiseless and noisy stochastic convex optimization scenarios without the need for learning rate decay or fine-tuning. We also prove that our approach maintains optimal performance across a wide spectrum of learning rates, underscoring its stability and practicality. Empirical studies further validate the robustness and enhanced stability of our approach.

Machine Learning-based heuristics have recently shown impressive performance in solving a variety of hard combinatorial optimization problems (COPs). However they generally rely on a separate neural model, specialized and trained for each single problem. Any variation of a problem requires adjustment of its model and re-training from scratch. In this paper, we propose GOAL (for Generalist combinatorial Optimization Agent Learner), a generalist model capable of efficiently solving multiple COPs and which can be fine-tuned to solve new COPs. GOAL consists of a single backbone plus light-weight problem-specific adapters for input and output processing. The backbone is based on a new form of mixed-attention blocks which allows to handle problems defined on graphs with arbitrary combinations of node, edge and instance-level features. Additionally, problems which involve heterogeneous types of nodes or edges are handled through a novel multi-type transformer architecture, where the attention blocks are duplicated to attend the meaningful combinations of types while relying on the same shared parameters. We train GOAL on a set of routing, scheduling and classic graph problems and show that it is only slightly inferior to the specialized baselines while being the first multi-task model that solves a wide range of COPs. Finally we showcase the strong transfer learning capacity of GOAL by fine-tuning it on several new problems. Our code is available at https://github.com/naver/goal-co .

Given the limitations of backpropagation, perturbation-based gradient computation methods have recently gained focus for learning with only forward passes, also referred to as queries. Conventional forward learning consumes enormous queries on each data point for accurate gradient estimation through Monte Carlo sampling, which hinders the scalability of those algorithms. However, not all data points deserve equal queries for gradient estimation. In this paper, we study the problem of improving the forward learning efficiency from a novel perspective: how to reduce the gradient estimation variance with minimum cost? For this, we allocate the optimal number of queries within a set budget during training to balance estimation accuracy and computational efficiency. Specifically, with a simplified proxy objective and a reparameterization technique, we derive a novel plug-and-play query allocator with minimal parameters. Theoretical results are carried out to verify its optimality. We conduct extensive experiments for fine-tuning Vision Transformers on various datasets and further deploy the allocator to two black-box applications: prompt tuning and multimodal alignment for foundation models. All findings demonstrate that our proposed allocator significantly enhances the scalability of forward-learning algorithms, paving the way for real-world applications. The implementation is available at https://github.com/RTkenny/FLOPS-Forward-Learning-with-OPtimal-Sampling.

In this paper, we explore the two-point zeroth-order gradient estimator and identify the distribution of random perturbations that minimizes the estimator's asymptotic variance as the perturbation stepsize tends to zero. We formulate it as a constrained functional optimization problem over the space of perturbation distributions. Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length. While existing research has largely focused on fixed-length perturbations, the potential advantages of directional alignment have been overlooked. To address this gap, we delve into the theoretical and empirical properties of the directionally aligned perturbation (DAP) scheme, which adaptively offers higher accuracy along critical directions. Additionally, we provide a convergence analysis for stochastic gradient descent using $\delta$-unbiased random perturbations, extending existing complexity bounds to a wider range of perturbations. Through empirical evaluations on both synthetic problems and practical tasks, we demonstrate that DAPs outperform traditional methods under specific conditions.

Optimizing black-box functions is a fundamental problem in science and engineering. To solve this problem, many approaches learn a surrogate function that estimates the underlying objective from limited historical evaluations. Large Language Models (LLMs), with their strong pattern-matching capabilities via pretraining on vast amounts of data, stand out as a potential candidate for surrogate modeling. However, directly prompting a pretrained language model to produce predictions is not feasible in many scientific domains due to the scarcity of domain-specific data in the pretraining corpora and the challenges of articulating complex problems in natural language. In this work, we introduce LICO, a general-purpose model that extends arbitrary base LLMs for black-box optimization, with a particular application to the molecular domain. To achieve this, we equip the language model with a separate embedding layer and prediction layer, and train the model to perform in-context predictions on a diverse set of functions defined over the domain. Once trained, LICO can generalize to unseen molecule properties simply via in-context prompting. LICO performs competitively on PMO, a challenging molecular optimization benchmark comprising 23 objective functions, and achieves state-of-the-art performance on its low-budget version PMO-1K.

In a convergence of machine learning and biology, we reveal that diffusion models are evolutionary algorithms. By considering evolution as a denoising process and reversed evolution as diffusion, we mathematically demonstrate that diffusion models inherently perform evolutionary algorithms, naturally encompassing selection, mutation, and reproductive isolation. Building on this equivalence, we propose the Diffusion Evolution method: an evolutionary algorithm utilizing iterative denoising -- as originally introduced in the context of diffusion models -- to heuristically refine solutions in parameter spaces. Unlike traditional approaches, Diffusion Evolution efficiently identifies multiple optimal solutions and outperforms prominent mainstream evolutionary algorithms. Furthermore, leveraging advanced concepts from diffusion models, namely latent space diffusion and accelerated sampling, we introduce Latent Space Diffusion Evolution, which finds solutions for evolutionary tasks in high-dimensional complex parameter space while significantly reducing computational steps. This parallel between diffusion and evolution not only bridges two different fields but also opens new avenues for mutual enhancement, raising questions about open-ended evolution and potentially utilizing non-Gaussian or discrete diffusion models in the context of Diffusion Evolution.

Despite the vast empirical evidence supporting the efficacy of adaptive optimization methods in deep learning, their theoretical understanding is far from complete. This work introduces novel SDEs for commonly used adaptive optimizers: SignSGD, RMSprop(W), and Adam(W). These SDEs offer a quantitatively accurate description of these optimizers and help illuminate an intricate relationship between adaptivity, gradient noise, and curvature. Our novel analysis of SignSGD highlights a noteworthy and precise contrast to SGD in terms of convergence speed, stationary distribution, and robustness to heavy-tail noise. We extend this analysis to AdamW and RMSpropW, for which we observe that the role of noise is much more complex. Crucially, we support our theoretical analysis with experimental evidence by verifying our insights: this includes numerically integrating our SDEs using Euler-Maruyama discretization on various neural network architectures such as MLPs, CNNs, ResNets, and Transformers. Our SDEs accurately track the behavior of the respective optimizers, especially when compared to previous SDEs derived for Adam and RMSprop. We believe our approach can provide valuable insights into best training practices and novel scaling rules.

Multi-objective optimization (MOO) is receiving more attention in various fields such as multi-task learning. Recent works provide some effective algorithms with theoretical analysis but they are limited by the standard $L$-smooth or bounded-gradient assumptions, which typically do not hold for neural networks, such as Long short-term memory (LSTM) models and Transformers. In this paper, we study a more general and realistic class of generalized $\ell$-smooth loss functions, where $\ell$ is a general non-decreasing function of gradient norm. We revisit and analyze the fundamental multiple gradient descent algorithm (MGDA) and its stochastic version with double sampling for solving the generalized $\ell$-smooth MOO problems, which approximate the conflict-avoidant (CA) direction that maximizes the minimum improvement among objectives. We provide a comprehensive convergence analysis of these algorithms and show that they converge to an $\epsilon$-accurate Pareto stationary point with a guaranteed $\epsilon$-level average CA distance (i.e., the gap between the updating direction and the CA direction) over all iterations, where totally $\mathcal{O}(\epsilon^{-2})$ and $\mathcal{O}(\epsilon^{-4})$ samples are needed for deterministic and stochastic settings, respectively. We prove that they can also guarantee a tighter $\epsilon$-level CA distance in each iteration using more samples. Moreover, we analyze an efficient variant of MGDA named MGDA-FA using only $\mathcal{O}(1)$ time and space, while achieving the same performance guarantee as MGDA.

Bilevel optimization, with broad applications in machine learning, has an intricate hierarchical structure. Gradient-based methods have emerged as a common approach to large-scale bilevel problems. However, the computation of the hyper-gradient, which involves a Hessian inverse vector product, confines the efficiency and is regarded as a bottleneck. To circumvent the inverse, we construct a sequence of low-dimensional approximate Krylov subspaces with the aid of the Lanczos process. As a result, the constructed subspace is able to dynamically and incrementally approximate the Hessian inverse vector product with less effort and thus leads to a favorable estimate of the hyper-gradient. Moreover, we propose a provable subspace-based framework for bilevel problems where one central step is to solve a small-size tridiagonal linear system. To the best of our knowledge, this is the first time that subspace techniques are incorporated into bilevel optimization. This successful trial not only enjoys $\mathcal{O}(\epsilon^{-1})$ convergence rate but also demonstrates efficiency in a synthetic problem and two deep learning tasks.

Combinatorial optimization (CO) problems are pervasive across various domains, but their NP-hard nature often necessitates problem-specific heuristic algorithms. Recent advancements in deep learning have led to the development of learning-based heuristics, yet these approaches often struggle with limited search capabilities. We introduce Explore-and-Exploit GNN ($X^2$GNN, pronounced x-squared GNN), a novel unsupervised neural framework that combines exploration and exploitation for combinatorial search optimization: i) Exploration - $X^2$GNN generates multiple solutions simultaneously, promoting diversity in the search space; (ii) Exploitation - $X^2$GNN employs neural stochastic iterative refinement to exploit partial existing solutions, guiding the search toward promising regions and helping escape local optima. By balancing exploration and exploitation, $X^2$GNN achieves superior performance and generalization on several graph CO problems including Max Cut, Max Independent Set, and Max Clique. Notably, for large Max Clique problems, $X^2$GNN consistently generates solutions within 1.2\% of optimality, while other state-of-the-art learning-based approaches struggle to reach within 22\% of optimal. Moreover, $X^2$GNN consistently generates better solutions than Gurobi on large graphs for all three problems under reasonable time budgets. Furthermore, $X^2$GNN exhibits exceptional generalization capabilities. For the Maximum Independent Set problem, $X^2$GNN outperforms state-of-the-art methods even when trained on smaller or out-of-distribution graphs compared to the test set. Our framework offers a more effective and flexible approach to neural combinatorial optimization, addressing a key challenge in the field and providing a promising direction for future research in learning-based heuristics for combinatorial optimization.

Understanding the training dynamics of deep neural networks is challenging due to their high-dimensional nature and intricate loss landscapes. Recent studies have revealed that, along the training trajectory, the gradient approximately aligns with a low-rank top eigenspace of the training loss Hessian, referred to as the dominant subspace. Given this alignment, this paper explores whether neural networks can be trained within the dominant subspace, which, if feasible, could lead to more efficient training methods. Our primary observation is that when the SGD update is projected onto the dominant subspace, the training loss does not decrease further. This suggests that the observed alignment between the gradient and the dominant subspace is spurious. Surprisingly, projecting out the dominant subspace proves to be just as effective as the original update, despite removing the majority of the original update component. We observe similar behavior across practical setups, including the large learning rate regime (also known as Edge of Stability), Sharpness-Aware Minimization, momentum, and adaptive optimizers. We discuss the main causes and implications of this spurious alignment, shedding light on the dynamics of neural network training.

Optimization methods are widely employed in deep learning to address and mitigate undesired model responses. While gradient-based techniques have proven effective for image models, their application to language models is hindered by the discrete nature of the input space. This study introduces a novel optimization approach, termed the *functional homotopy* method, which leverages the functional duality between model training and input generation. By constructing a series of easy-to-hard optimization problems, we iteratively solve these using principles derived from established homotopy methods. We apply this approach to jailbreak attack synthesis for large language models (LLMs), achieving a 20%-30% improvement in success rate over existing methods in circumventing established safe open-source models such as Llama-2 and Llama-3.

Direct Preference Optimization (DPO) and its variants are increasingly used for aligning language models with human preferences. Although these methods are designed to teach a model to generate preferred responses more frequently relative to dispreferred responses, prior work has observed that the likelihood of preferred responses often decreases during training. The current work sheds light on the causes and implications of this counterintuitive phenomenon, which we term *likelihood displacement*. We demonstrate that likelihood displacement can be *catastrophic*, shifting probability mass from preferred responses to responses with an opposite meaning. As a simple example, training a model to prefer $\texttt{No}$ over $\texttt{Never}$ can sharply increase the probability of $\texttt{Yes}$. Moreover, when aligning the model to refuse unsafe prompts, we show that such displacement can *unintentionally lead to unalignment*, by shifting probability mass from preferred refusal responses to harmful responses (e.g., reducing the refusal rate of Llama-3-8B-Instruct from 74.4% to 33.4%). We theoretically characterize that likelihood displacement is driven by preferences that induce similar embeddings, as measured by a *centered hidden embedding similarity (CHES)* score. Empirically, the CHES score enables identifying which training samples contribute most to likelihood displacement in a given dataset. Filtering out these samples effectively mitigated unintentional unalignment in our experiments. More broadly, our results highlight the importance of curating data with sufficiently distinct preferences, for which we believe the CHES score may prove valuable.

Meta-learning aims to train models that can generalize to new tasks with limited labeled data by extracting shared features across diverse task datasets. Additionally, it accounts for prediction uncertainty during both training and evaluation, a concept known as uncertainty-aware meta-learning. Neural Process (NP) is a well-known uncertainty-aware meta-learning method that constructs implicit stochastic processes using parametric neural networks, enabling rapid adaptation to new tasks. However, existing NP methods face challenges in accommodating diverse input dimensions and learned features, limiting their broad applicability across regression tasks. To address these limitations and advance the utility of NP models as general regressors, we introduce Dimension Agnostic Neural Process (DANP). DANP incorporates Dimension Aggregator Block (DAB) to transform input features into a fixed-dimensional space, enhancing the model's ability to handle diverse datasets. Furthermore, leveraging the Transformer architecture and latent encoding layers, DANP learns a wider range of features that are generalizable across various tasks. Through comprehensive experimentation on various synthetic and practical regression tasks, we empirically show that DANP outperforms previous NP variations, showcasing its effectiveness in overcoming the limitations of traditional NP models and its potential for broader applicability in diverse regression scenarios.

Stochastic Variance Reduced Gradient (SVRG), introduced by Johnson & Zhang (2013), is a theoretically compelling optimization method. However, as Defazio & Bottou (2019) highlight, its effectiveness in deep learning is yet to be proven. In this work, we demonstrate the potential of SVRG in optimizing real-world neural networks. Our empirical analysis finds that, for deeper neural networks, the strength of the variance reduction term in SVRG should be smaller and decrease as training progresses. Inspired by this, we introduce a multiplicative coefficient $\alpha$ to control the strength and adjust it through a linear decay schedule. We name our method $\alpha$-SVRG. Our results show $\alpha$-SVRG better optimizes models, consistently reducing training loss compared to the baseline and standard SVRG across various model architectures and multiple image classification datasets. We hope our findings encourage further exploration into variance reduction techniques in deep learning. Code is available at github.com/davidyyd/alpha-SVRG.

In recent years, random subspace methods have been actively studied for large-dimensional nonconvex problems. Recent subspace methods have improved theoretical guarantees such as iteration complexity and local convergence rate while reducing computational costs by deriving descent directions in randomly selected low-dimensional subspaces. This paper proposes the Random Subspace Homogenized Trust Region (RSHTR) method with the best theoretical guarantees among random subspace algorithms for nonconvex optimization. RSHTR achieves an $\varepsilon$-approximate first-order stationary point in $O(\varepsilon^{-3/2})$ iterations, converging locally at a linear rate. Furthermore, under rank-deficient conditions, RSHTR satisfies $\varepsilon$-approximate second-order necessary conditions in $O(\varepsilon^{-3/2})$ iterations and exhibits a local quadratic convergence. Experiments on real-world datasets verify the benefits of RSHTR.

Density ratio estimation (DRE) is a core technique in machine learning used to capture relationships between two probability distributions. $f$-divergence loss functions, which are derived from variational representations of $f$-divergence, have become a standard choice in DRE for achieving cutting-edge performance. This study provides novel theoretical insights into DRE by deriving upper and lower bounds on the $L_p$ errors through $f$-divergence loss functions. These bounds apply to any estimator belonging to a class of Lipschitz continuous estimators, irrespective of the specific $f$-divergence loss function employed. The derived bounds are expressed as a product involving the data dimensionality and the expected value of the density ratio raised to the $p$-th power. Notably, the lower bound includes an exponential term that depends on the Kullback--Leibler (KL) divergence, revealing that the $L_p$ error increases significantly as the KL divergence grows when $p > 1$. This increase becomes even more pronounced as the value of $p$ grows. The theoretical insights are validated through numerical experiments.

Recently, numerous preference optimization algorithms have been introduced as extensions to the Direct Preference Optimization (DPO) family. While these methods have successfully aligned models with human preferences, there is a lack of understanding regarding the contributions of their additional components. Moreover, fair and consistent comparisons are scarce, making it difficult to discern which components genuinely enhance downstream performance. In this work, we propose RainbowPO, a unified framework that demystifies the effectiveness of existing DPO methods by categorizing their key components into seven broad directions. We integrate these components into a single cohesive objective, enhancing the performance of each individual element. Through extensive experiments, we demonstrate that RainbowPO outperforms existing DPO variants. Additionally, we provide insights to guide researchers in developing new DPO methods and assist practitioners in their implementations.

Wasserstein distributionally robust optimization (WDRO) optimizes against worst-case distributional shifts within a specified uncertainty set, leading to enhanced generalization on unseen adversarial examples, compared to standard adversarial training which focuses on pointwise adversarial perturbations. However, WDRO still suffers fundamentally from the robust overfitting problem, as it does not consider statistical error. We address this gap by proposing a novel robust optimization framework under a new uncertainty set for adversarial noise via Wasserstein distance and statistical error via Kullback-Leibler divergence, called the Statistically Robust WDRO. We establish a robust generalization bound for the new optimization framework, implying that out-of-distribution adversarial performance is at least as good as the statistically robust training loss with high probability. Furthermore, we derive conditions under which Stackelberg and Nash equilibria exist between the learner and the adversary, giving an optimal robust model in certain sense.Finally, through extensive experiments, we demonstrate that our method significantly mitigates robust overfitting and enhances robustness within the framework of WDRO.