ICLR 2025 Papers

The success of modern machine learning is due in part to the adaptive optimization methods that have been developed to deal with the difficulties of training large models over complex datasets. One such method is gradient clipping: a practical procedure with limited theoretical underpinnings. In this work, we study clipping in a least squares problem under streaming SGD. We develop a theoretical analysis of the learning dynamics in the limit of large intrinsic dimension—a model and dataset dependent notion of dimensionality. In this limit we find a deterministic equation that describes the evolution of the loss and demonstrate that this equation predicts the path of clipped SGD on synthetic, CIFAR10, and Wikitext2 data. We show that with Gaussian noise clipping cannot improve SGD performance. Yet, in other noisy settings, clipping can provide benefits with tuning of the clipping threshold. We propose a simple heuristic for near optimal scheduling of the clipping threshold which requires the tuning of only one hyperparameter. We conclude with a discussion about the links between high-dimensional clipping and neural network training.

Momentum based optimizers are central to a wide range of machine learning applications. These typically rely on an Exponential Moving Average (EMA) of gradients, which decays exponentially the present contribution of older gradients. This accounts for gradients being local linear approximations which lose their relevance as the iterate moves along the loss landscape. This work questions the use of a single EMA to accumulate past gradients and empirically demonstrates how this choice can be sub-optimal: a single EMA cannot simultaneously give a high weight to the immediate past, and a non-negligible weight to older gradients. Building on this observation, we propose AdEMAMix, a simple modification of the Adam optimizer with a mixture of two EMAs to better take advantage of past gradients. Our experiments on language modeling and image classification show---quite surprisingly---that gradients can stay relevant for tens of thousands of steps. They help to converge faster, and often to lower minima: e.g., a $1.3$B parameter AdEMAMix LLM trained on $101$B tokens performs comparably to an AdamW model trained on $197$B tokens ($+95\%$). Moreover, our method significantly slows-down model forgetting during training. Our work motivates further exploration of different types of functions to leverage past gradients, beyond EMAs.

A distribution shift between the training and test data can severely harm performance of machine learning models. Importance weighting addresses this issue by assigning different weights to data points during training. We argue that existing heuristics for determining the weights are suboptimal, as they neglect the increase of the variance of the estimated model due to the limited sample size of the training data. We interpret the optimal weights in terms of a bias-variance trade-off, and propose a bi-level optimization procedure in which the weights and model parameters are optimized simultaneously. We apply this framework to existing importance weighting techniques for last-layer retraining of deep neural networks in the presence of sub-population shifts and show empirically that optimizing weights significantly improves generalization performance.

Temporal difference (TD) learning represents a fascinating paradox: It is the prime example of a divergent algorithm that has not vanished after its instability was proven. On the contrary, TD continues to thrive in reinforcement learning (RL), suggesting that it provides significant compensatory benefits. Empirical evidence supports this, as many RL tasks require substantial computational resources, and TD delivers a crucial speed advantage that makes these tasks solvable. However, it is limited to cases where the divergence issues are absent or negligible for unknown reasons. So far, the theoretical foundations behind the speed-up are also unclear. In our work, we address these shortcomings of TD by employing techniques for analyzing iterative schemes developed over the past century. Our analysis reveals that TD possesses a mechanism that enables efficient mapping into the smallest eigenspace—an operation previously thought to necessitate costly matrix inversion. Notably, this effect is independent of the conditioning of the problem, making it particularly well-suited for RL tasks characterized by rapidly increasing condition numbers, e.g. through delayed rewards. Our novel theoretical understanding allows us to develop a scalable algorithm that integrates TD’s speed with the reliable convergence of gradient descent (GD). We additionally validate these improvements through a rigorous mathematical proof in two dimensions, as well as experiments on problems where TD and GD falter, providing valuable insights into the future of optimization techniques in artificial intelligence

This paper studies second-order methods for convex-concave minimax optimization. Monteiro & Svaiter (2012) proposed a method to solve the problem with an optimal iteration complexity of $\mathcal{O}(\epsilon^{-3/2})$ to find an $\epsilon$-saddle point. However, it is unclear whether the computational complexity, $\mathcal{O}((N+ d^2) d \epsilon^{-2/3})$, can be improved. In the above, we follow Doikov et al. (2023) and assume the complexity of obtaining a first-order oracle as $N$ and the complexity of obtaining a second-order oracle as $dN$. In this paper, we show that the computation cost can be reduced by reusing Hessian across iterations. Our methods take the overall computational complexity of $\tilde{\mathcal{O}}( (N+d^2)(d+ d^{2/3}\epsilon^{-2/3}))$, which improves those of previous methods by a factor of $d^{1/3}$. Furthermore, we generalize our method to strongly-convex-strongly-concave minimax problems and establish the complexity of $\tilde{\mathcal{O}}((N+d^2) (d + d^{2/3} \kappa^{2/3}) )$ when the condition number of the problem is $\kappa$, enjoying a similar speedup upon the state-of-the-art method. Numerical experiments on both real and synthetic datasets also verify the efficiency of our method.

We consider neural networks (NNs) where the final layer is down-scaled by a fixed hyperparameter $\gamma$. Recent work has identified $\gamma$ as controlling the strength of feature learning. As $\gamma$ increases, network evolution changes from "lazy" kernel dynamics to "rich" feature-learning dynamics, with a host of associated benefits including improved performance on common tasks. In this work, we conduct a thorough empirical investigation of the effect of scaling $\gamma$ across a variety of models and datasets in the online training setting. We first examine the interaction of $\gamma$ with the learning rate $\eta$, identifying several scaling regimes in the $\gamma$-$\eta$ plane which we explain theoretically using a simple model. We find that the optimal learning rate $\eta^*$ scales non-trivially with $\gamma$. In particular, $\eta^* \propto \gamma^2$ when $\gamma \ll 1$ and $\eta^* \propto \gamma^{2/L}$ when $\gamma \gg 1$ for a feed-forward network of depth $L$. Using this optimal learning rate scaling, we proceed with an empirical study of the under-explored ``ultra-rich'' $\gamma \gg 1$ regime. We find that networks in this regime display characteristic loss curves, starting with a long plateau followed by a drop-off, sometimes followed by one or more additional staircase steps. We find networks of different large $\gamma$ values optimize along similar trajectories up to a reparameterization of time. We further find that optimal online performance is often found at large $\gamma$ and could be missed if this hyperparameter is not tuned. Our findings indicate that analytical study of the large-$\gamma$ limit may yield useful insights into the dynamics of representation learning in performant models.

Pretraining large language models often requires significant computational resources and memory due to their vast parameter amount. An effective approach to enhance parameter efficiency in both training and inference is to parameterize each full-size weight as the product of two trainable low-rank factors. While low-rank fine-tuning has achieved great success, low-rank pretraining remains challenging as it requires learning extensive knowledge from scratch under the restrictive low-rank parameterization. During standard low-rank pretraining, separately optimizing the low-rank factors introduces redundant information from the full gradient, which hinders the learning process. To achieve efficient yet effective low-rank pretraining, we propose a **Lo**w-rank **R**iemannian **O**ptimizer (**LORO**). At each LORO update step, the low-rank factor pairs are jointly updated to ensure their full-size product moves along the steepest descent direction on the low-rank manifold, without the need to compute any memory-intensive full-size matrices or gradients. Hence, our LORO finds low-rank models that achieve high performance comparable to full-size pretrained models, while significantly reducing memory usage and accelerating both training and inference. A LLaMA 1B model pretrained with LORO achieves a perplexity score of 2\% better than the full-size baseline, with a 54\% reduction in model memory, a $\times1.8$ speedup in training, and a $\times2.2$ speedup in inference. The code is available on https://github.com/mzf666/LORO-main.

Quadratic programming (QP) forms a crucial foundation in optimization, appearing in a broad spectrum of domains and serving as the basis for more advanced algorithms. Consequently, as the scale and complexity of modern applications continue to grow, the development of efficient and reliable QP algorithms becomes increasingly vital. In this context, this paper introduces a novel deep learning-aided distributed optimization architecture designed for tackling large-scale QP problems. First, we combine the state-of-the-art Operator Splitting QP (OSQP) method with a consensus approach to derive **DistributedQP**, a new method tailored for network-structured problems, with convergence guarantees to optimality. Subsequently, we unfold this optimizer into a deep learning framework, leading to **DeepDistributedQP**, which leverages learned policies to accelerate reaching to desired accuracy within a restricted amount of iterations. Our approach is also theoretically grounded through Probably Approximately Correct (PAC)-Bayes theory, providing generalization bounds on the expected optimality gap for unseen problems. The proposed framework, as well as its centralized version **DeepQP**, significantly outperform their standard optimization counterparts on a variety of tasks such as randomly generated problems, optimal control, linear regression, transportation networks and others. Notably, DeepDistributedQP demonstrates strong generalization by training on small problems and scaling to solve much larger ones (up to 50K variables and 150K constraints) using the same policy. Moreover, it achieves orders-of-magnitude improvements in wall-clock time compared to OSQP. The certifiable performance guarantees of our approach are also demonstrated, ensuring higher-quality solutions over traditional optimizers.

Large Language Models (LLMs) are increasingly used in applications where the model selects from competing third-party content, such as in LLM-powered search engines or chatbot plugins. In this paper, we introduce _Preference Manipulation Attacks_, a new class of attacks that manipulate an LLM's selections to favor the attacker. We demonstrate that carefully crafted website content or plugin documentations can trick an LLM to promote the attacker products and discredit competitors, thereby increasing user traffic and monetization (a form of adversarial Search Engine Optimization). We show this can lead to a _prisoner's dilemma_, where all parties are incentivized to launch attacks, but this collectively degrades the LLM's outputs for everyone. We demonstrate our attacks on production LLM search engines (Bing and Perplexity) and plugin APIs (for GPT-4 and Claude). As LLMs are increasingly used to rank third-party content, we expect Preference Manipulation Attacks to emerge as a significant threat.

In this paper, we study the data-dependent convergence and generalization behavior of gradient methods for neural networks with smooth activation. Our first result is a novel bound on the excess risk of deep networks trained by the logistic loss via an alogirthmic stability analysis. Compared to previous works, our results improve upon the shortcomings of the well-established Rademacher complexity-based bounds. Importantly, the bounds we derive in this paper are tighter, hold even for neural networks of small width, do not scale unfavorably with width, are algorithm-dependent, and consequently capture the role of initialization on the sample complexity of gradient descent for deep nets. Specialized to noiseless data separable with margin $\gamma$ by neural tangent kernel (NTK) features of a network of width $\Omega(poly(\log(n)))$, we show the test-error rate $e^{O(L)}/{\gamma^2 n}$, where $n$ is the training set size and $L$ denotes the number of hidden layers. This results in an improvement in the test loss bound compared to previous works while maintaining the poly-logarithmic width conditions. We further investigate excess risk bounds for deep nets trained with noisy data, establishing that under a polynomial condition on the network width, gradient descent can achieve the optimal excess risk. Finally, we show that a large step-size significantly improves upon the NTK regime's results in classifying the XOR distribution. In particular, we show for a one-hidden layer neural network of constant width $m$ with quadratic activation and standard Gaussian initialization that SGD with linear sample complexity and with a large step-size $\eta=m$ reaches the perfect test accuracy after only $\lceil\log(d)\rceil$ iterations, where $d$ is the data dimension.

Black-box optimization algorithms have been widely used in various machine learning problems, including reinforcement learning and prompt fine-tuning. However, directly optimizing the training loss value, as commonly done in existing black-box optimization methods, could lead to suboptimal model quality and generalization performance. To address those problems in black-box optimization, we propose a novel Sharpness-Aware Black-box Optimization (SABO) algorithm, which applies a sharpness-aware minimization strategy to improve the model generalization. Specifically, the proposed SABO method first reparameterizes the objective function by its expectation over a Gaussian distribution. Then it iteratively updates the parameterized distribution by approximated stochastic gradients of the maximum objective value within a small neighborhood around the current solution in the Gaussian distribution space. Theoretically, we prove the convergence rate and generalization bound of the proposed SABO algorithm. Empirically, extensive experiments on the black-box prompt fine-tuning tasks demonstrate the effectiveness of the proposed SABO method in improving model generalization performance.

Developing a contemporary optimal transport (OT) solver requires navigating trade-offs among several critical requirements: GPU parallelization, scalability to high-dimensional problems, theoretical convergence guarantees, empirical performance in terms of precision versus runtime, and numerical stability in practice. With these challenges in mind, we introduce a specialized truncated Newton algorithm for entropic-regularized OT. In addition to proving that locally quadratic convergence is possible without assuming a Lipschitz Hessian, we provide strategies to maximally exploit the high rate of local convergence in practice. Our GPU-parallel algorithm exhibits exceptionally favorable runtime performance, achieving high precision orders of magnitude faster than many existing alternatives. This is evidenced by wall-clock time experiments on 24 problem sets (12 datasets $\times$ 2 cost functions). The scalability of the algorithm is showcased on an extremely large OT problem with $n \approx 10^6$, solved approximately under weak entropic regularization.

Local learning, which trains a network through layer-wise local targets and losses, has been studied as an alternative to backpropagation (BP) in neural computation. However, its algorithms often become more complex or require additional hyperparameters due to the locality, making it challenging to identify desirable settings where the algorithm progresses in a stable manner. To provide theoretical and quantitative insights, we introduce maximal update parameterization ($\mu$P) in the infinite-width limit for two representative designs of local targets: predictive coding (PC) and target propagation (TP). We verify that $\mu$P enables hyperparameter transfer across models of different widths. Furthermore, our analysis reveals unique and intriguing properties of $\mu$P that are not present in conventional BP. By analyzing deep linear networks, we find that PC's gradients interpolate between first-order and Gauss-Newton-like gradients, depending on the parameterization. We demonstrate that, in specific standard settings, PC in the infinite-width limit behaves more similarly to the first-order gradient. For TP, even with the standard scaling of the last layer differing from classical $\mu$P, its local loss optimization favors the feature learning regime over the kernel regime.

Classical neural networks with random initialization famously behave as Gaussian processes in the limit of many neurons, which allows one to completely characterize their training and generalization behavior. No such general understanding exists for quantum neural networks (QNNs), which—outside of certain special cases—are known to not behave as Gaussian processes when randomly initialized. We here prove that QNNs and their first two derivatives instead generally form what we call "Wishart processes," where certain algebraic properties of the network determine the hyperparameters of the process. This Wishart process description allows us to, for the first time: give necessary and sufficient conditions for a QNN architecture to have a Gaussian process limit; calculate the full gradient distribution, generalizing previously known barren plateau results; and calculate the local minima distribution of algebraically constrained QNNs. Our unified framework suggests a certain simple operational definition for the "trainability" of a given QNN model using a newly introduced, experimentally accessible quantity we call the "degrees of freedom" of the network architecture.

While fusing heterogeneous open-source LLMs with varying architectures and sizes can potentially integrate the strengths of different models, existing fusion methods face significant challenges, such as vocabulary alignment and merging distribution matrices. These procedures are not only complex but also prone to introducing noise and errors. In this paper, we propose an implicit fusion method, Weighted-Reward Preference Optimization (WRPO), which leverages preference optimization between the source LLMs and the target LLM to transfer their capabilities effectively. WRPO eliminates the need for vocabulary alignment and matrix fusion and can be efficiently scaled to accommodate various LLMs. To address distributional deviations between the source and target LLMs, WRPO introduces a progressive adaptation strategy that gradually shifts reliance on preferred examples from the target LLM to the source LLMs. Extensive experiments on the MT-Bench, AlpacaEval-2, and Arena-Hard benchmarks demonstrate that WRPO consistently outperforms existing knowledge fusion methods and various fine-tuning baselines. When applied to LLaMA3-8B-Instruct as the target model, WRPO achieves a length-controlled win rate of 55.9\% against GPT-4-Preview-1106 on AlpacaEval-2 and a win rate of 46.2\% against GPT-4-0314 on Arena-Hard. Our code is available at https://github.com/SLIT-AI/WRPO.

Inspired by the similar update pattern of softmax natural policy gradient and Hadamard policy gradient, we propose to study a general policy update rule called $\phi$-update, where $\phi$ refers to a scaling function on advantage functions. Under very mild conditions on $\phi$, the global asymptotic state value convergence of $\phi$-update is firstly established. Then we show that the policy produced by $\phi$-update indeed converges, even when there are multiple optimal policies. This is in stark contrast to existing results where explicit regularizations are required to guarantee the convergence of the policy. Since softmax natural policy gradient is an instance of $\phi$-update, it provides an affirmative answer to the question whether the policy produced by softmax natural policy gradient converges. The exact asymptotic convergence rate of state values is further established based on the policy convergence. Lastly, we establish the global linear convergence of $\phi$-update.

Recent advancements in timestep-distilled diffusion models have enabled high-quality image generation that rivals non-distilled multi-step models, but with significantly fewer inference steps. While such models are attractive for applications due to the low inference cost and latency, fine-tuning them with a naive diffusion objective would result in degraded and blurry outputs. An intuitive alternative is to repeat the diffusion distillation process with a fine-tuned teacher model, which produces good results but is cumbersome and computationally intensive: the distillation training usually requires magnitude higher of training compute compared to fine-tuning for specific image styles. In this paper, we present an algorithm named pairwise sample optimization (PSO), which enables the direct fine-tuning of an arbitrary timestep-distilled diffusion model. PSO introduces additional reference images sampled from the current time-step distilled model, and increases the relative likelihood margin between the training images and reference images. This enables the model to retain its few-step generation ability, while allowing for fine-tuning of its output distribution. We also demonstrate that PSO is a generalized formulation which be flexible extended to both offline-sampled and online-sampled pairwise data, covering various popular objectives for diffusion model preference optimization. We evaluate PSO in both preference optimization and other fine-tuning tasks, including style transfer and concept customization. We show that PSO can directly adapt distilled models to human-preferred generation with both offline and online-generated pairwise preference image data. PSO also demonstrates effectiveness in style transfer and concept customization by directly tuning timestep-distilled diffusion models.

In this paper, we provide tight lower bounds for the oracle complexity of minimizing high-order Hölder smooth and uniformly convex functions. Specifically, for a function whose $p^{th}$-order derivatives are Hölder continuous with degree $\nu$ and parameter $H$, and that is uniformly convex with degree $q$ and parameter $\sigma$, we focus on two asymmetric cases: (1) $q > p + \nu$, and (2) $q < p+\nu$. Given up to $p^{th}$-order oracle access, we establish worst-case oracle complexities of $\Omega\left( \left( \frac{H}{\sigma}\right)^\frac{2}{3(p+\nu)-2}\left( \frac{\sigma}{\epsilon}\right)^\frac{2(q-p-\nu)}{q(3(p+\nu)-2)}\right)$ in the first case with an $\ell_\infty$-ball-truncated-Gaussian smoothed hard function and $\Omega\left(\left(\frac{H}{\sigma}\right)^\frac{2}{3(p+\nu)-2}+ \log\log\left(\left(\frac{\sigma^{p+\nu}}{H^q}\right)^\frac{1}{p+\nu-q}\frac{1}{\epsilon}\right)\right)$ in the second case, for reaching an $\epsilon$-approximate solution in terms of the optimality gap. Our analysis generalizes previous lower bounds for functions under first- and second-order smoothness as well as those for uniformly convex functions, and furthermore our results match the corresponding upper bounds in this general setting.

Aligning language models (LMs) with human preferences has become a key area of research, enabling these models to meet diverse user needs better. Inspired by weak-to-strong generalization, where a strong LM fine-tuned on labels generated by a weaker model can consistently outperform its weak supervisor, we extend this idea to model alignment. In this work, we observe that the alignment behavior in weaker models can be effectively transferred to stronger models and even exhibit an amplification effect. Based on this insight, we propose a method called Weak-to-Strong Preference Optimization (WSPO), which achieves strong model alignment by learning the distribution differences before and after the alignment of the weak model. Experiments demonstrate that WSPO delivers outstanding performance, improving the win rate of Qwen2-7B-Instruct on Arena-Hard from 39.70 to 49.60, achieving a remarkable 47.04 length-controlled win rate on AlpacaEval 2, and scoring 7.33 on MT-bench. Our results suggest that using the weak model to elicit a strong model with a high alignment ability is feasible. The code is available at https://github.com/zwhong714/weak-to-strong-preference-optimization.