ICLR 2025 Papers

In recent years, there has been a significant growth in research focusing on minimum $\ell_2$ norm (ridgeless) interpolation least squares estimators. However, the majority of these analyses have been limited to an unrealistic regression error structure, assuming independent and identically distributed errors with zero mean and common variance. In this paper, we explore prediction risk as well as estimation risk under more general regression error assumptions, highlighting the benefits of overparameterization in a more realistic setting that allows for clustered or serial dependence. Notably, we establish that the estimation difficulties associated with the variance components of both risks can be summarized through the trace of the variance-covariance matrix of the regression errors. Our findings suggest that the benefits of overparameterization can extend to time series, panel and grouped data.

Neural Ordinary Differential Equations (ODEs) have been successful in various applications due to their continuous nature and parameter-sharing efficiency. However, these unique characteristics also introduce challenges in training, particularly with respect to gradient computation accuracy and convergence analysis. In this paper, we address these challenges by investigating the impact of activation functions. We demonstrate that the properties of activation functions—specifically smoothness and nonlinearity—are critical to the training dynamics. Smooth activation functions guarantee globally unique solutions for both forward and backward ODEs, while sufficient nonlinearity is essential for maintaining the spectral properties of the Neural Tangent Kernel (NTK) during training. Together, these properties enable us to establish the global convergence of Neural ODEs under gradient descent in overparameterized regimes. Our theoretical findings are validated by numerical experiments, which not only support our analysis but also provide practical guidelines for scaling Neural ODEs, potentially leading to faster training and improved performance in real-world applications.

Self-supervised learning has become a cornerstone in computer vision, primarily divided into reconstruction-based methods like masked autoencoders (MAE) and discriminative methods such as contrastive learning (CL). Recent empirical observations reveal that MAE and CL capture different types of representations: CL tends to focus on global patterns, while MAE adeptly captures **both global and subtle local** information simultaneously. Despite a flurry of recent empirical investigations to shed light on this difference, theoretical understanding remains limited, especially on the dominant architecture **vision transformers** (ViTs). In this paper, to provide rigorous insights, we model the visual data distribution by considering two types of spatial features: dominant global features and comparatively minuscule local features, and study the impact of imbalance among these features. We analyze the training dynamics of one-layer softmax-based ViTs on both MAE and CL objectives using gradient descent. Our analysis shows that as the degree of feature imbalance varies, ViTs trained with the MAE objective effectively learn both global and local features to achieve near-optimal reconstruction, while the CL-trained ViTs favor predominantly global features, even under mild imbalance. These results provide a theoretical explanation for distinct behaviors of MAE and CL observed in empirical studies.

Chain-of-thought (CoT) significantly enhances the reasoning performance of large language models (LLM). While current theoretical studies often attribute this improvement to increased expressiveness and computational capacity, we argue that expressiveness is not the primary limitation in the LLM regime, as current large models will fail on simple tasks. Using a parity-learning setup, we demonstrate that CoT can substantially improve sample efficiency even when the representation power is sufficient. Specifically, with CoT, a transformer can learn the function within polynomial samples, whereas without CoT, the required sample size is exponential. Additionally, we show that CoT simplifies the learning process by introducing sparse sequential dependencies among input tokens, and leads to a sparse and interpretable attention. We validate our theoretical analysis with both synthetic and real-world experiments, confirming that sparsity in attention layers is a key factor of the improvement induced by CoT.

Since the success of GPT, large language models (LLMs) have revolutionized machine learning and have initiated the so-called *LLM prompting* paradigm. In the era of LLMs, people train a single general-purpose LLM and provide the LLM with different *prompts* to perform different tasks. However, such empirical success largely lacks theoretical understanding. Here, we present the first theoretical study on the LLM prompting paradigm to the best of our knowledge. In this work, we show that prompting is in fact Turing-complete: there exists a finite-size Transformer such that for any computable function, there exists a corresponding prompt following which the Transformer computes the function. Furthermore, we show that even though we use only a single finite-size Transformer, it can still achieve nearly the same complexity bounds as that of the class of all unbounded-size Transformers. Overall, our result reveals that prompting can enable a single finite-size Transformer to be efficiently universal, which establishes a theoretical underpinning for prompt engineering in practice.

The Adam optimizer is widely used for transformer optimization in practice, which makes understanding the underlying optimization mechanisms an important problem. However, due to the Adam's complexity, theoretical analysis of how it optimizes transformers remains a challenging task. Fortunately, Sign Gradient Descent (SignGD) serves as an effective surrogate for Adam. Despite its simplicity, theoretical understanding of how SignGD optimizes transformers still lags behind. In this work, we study how SignGD optimizes a two-layer transformer -- consisting of a softmax attention layer with trainable query-key parameterization followed by a linear layer -- on a linearly separable noisy dataset. We identify four stages in the training dynamics, each exhibiting intriguing behaviors. Based on the training dynamics, we prove the fast convergence but poor generalization of the learned transformer on the noisy dataset. We also show that Adam behaves similarly to SignGD in terms of both optimization and generalization in this setting. Additionally, we find that the poor generalization of SignGD is not solely due to data noise, suggesting that both SignGD and Adam requires high-quality data for real-world tasks. Finally, experiments on synthetic and real-world datasets empirically support our theoretical results.

The primary objective of learning methods is generalization. Classic generalization bounds, based on VC-dimension or Rademacher complexity, are uniformly applicable to all networks in the hypothesis space. On the other hand, algorithm-dependent generalization bounds, like stability bounds, address more practical scenarios and provide generalization conditions for neural networks trained using SGD. However, these bounds often rely on strict assumptions, such as the NTK hypothesis or convexity of the empirical loss, which are typically not met by neural networks. In order to establish generalizability under less stringent assumptions, this paper investigates generalizability of neural networks that minimize the empirical risk. A lower bound for population accuracy is established based on the expressiveness of these networks, which indicates that with adequately large training sample and network sizes, these networks can generalize effectively. Additionally, we provide a lower bound necessary for generalization, demonstrating that, for certain data distributions, the quantity of data required to ensure generalization exceeds the network size needed to represent that distribution. Finally, we provide theoretical insights into several phenomena in deep learning, including robust overfitting, importance of over-parameterization networks, and effects of loss functions.

Randomized algorithms play a crucial role in efficiently solving large-scale optimization problems. In this paper, we introduce Sketching for Regularized Optimization (SRO), a fast sketching algorithm designed for least squares problems with convex or nonconvex regularization. SRO operates by first creating a sketch of the original data matrix and then solving the sketched problem. We establish minimax optimal rates for sparse signal estimation by addressing the sketched sparse convex and nonconvex learning problems. Furthermore, we propose a novel Iterative SRO algorithm, which reduces the approximation error geometrically for sketched convex regularized problems. To the best of our knowledge, this work is among the first to provide a unified theoretical framework demonstrating minimax rates for convex and nonconvex sparse learning problems via sketching. Experimental results validate the efficiency and effectiveness of both the SRO and Iterative SRO algorithms.

Continual learning (CL) has garnered significant attention because of its ability to adapt to new tasks that arrive over time. Catastrophic forgetting (of old tasks) has been identified as a major issue in CL, as the model adapts to new tasks. The Mixture-of-Experts (MoE) model has recently been shown to effectively mitigate catastrophic forgetting in CL, by employing a gating network to sparsify and distribute diverse tasks among multiple experts. However, there is a lack of theoretical analysis of MoE and its impact on the learning performance in CL. This paper provides the first theoretical results to characterize the impact of MoE in CL via the lens of overparameterized linear regression tasks. We establish the benefit of MoE over a single expert by proving that the MoE model can diversify its experts to specialize in different tasks, while its router learns to select the right expert for each task and balance the loads across all experts. Our study further suggests an intriguing fact that the MoE in CL needs to terminate the update of the gating network after sufficient training rounds to attain system convergence, which is not needed in the existing MoE studies that do not consider the continual task arrival. Furthermore, we provide explicit expressions for the expected forgetting and overall generalization error to characterize the benefit of MoE in the learning performance in CL. Interestingly, adding more experts requires additional rounds before convergence, which may not enhance the learning performance. Finally, we conduct experiments on both synthetic and real datasets to extend these insights from linear models to deep neural networks (DNNs), which also shed light on the practical algorithm design for MoE in CL.

A rich line of recent works studies how participants in matching markets learn their unknown preferences through iterative interactions with each other. The two sides of participants in the market can be respectively formulated as players and arms in the bandit problem. To ensure market stability, the objective is to minimize the stable regret of each player. Though existing works provide significant theoretical upper bounds for players' stable regret, the results heavily rely on the assumption that each participant has a strict preference ranking. However, in real applications, multiple candidates (e.g., workers in the labor market and students in school admission) usually demonstrate comparable performance levels, making it challenging for participants (e.g., employers and schools) to differentiate and rank their preferences. To deal with the potential indifferent preferences, we propose an adaptive exploration algorithm based on arm-guided Gale-Shapley (AE-AGS). We show that its stable regret is of order $O(NK \log T / \Delta^2)$, where $N$ is the number of players, $K$ the number of arms, $T$ the total time horizon, and $\Delta$ the minimum non-zero preference gap. Extensive experiments demonstrate the algorithm's effectiveness in handling such complex situations and its consistent superiority over baselines.

Benign overfitting refers to the phenomenon where an over-parameterized model fits the training data perfectly, including noise in the data, but still generalizes well to the unseen test data. While prior work provides some theoretical understanding of this phenomenon under the in-distribution setup, modern machine learning often operates in a more challenging Out-of-Distribution (OOD) regime, where the target (test) distribution can be rather different from the source (training) distribution. In this work, we take an initial step towards understanding benign overfitting in the OOD regime by focusing on the basic setup of over-parameterized linear models under covariate shift. We provide non-asymptotic guarantees proving that benign overfitting occurs in standard ridge regression, even under the OOD regime when the target covariance satisfies certain structural conditions. We identify several vital quantities relating to source and target covariance, which govern the performance of OOD generalization. Our result is sharp, which provably recovers prior in-distribution benign overfitting guarantee (Tsigler & Bartlett, 2023), as well as under-parameterized OOD guarantee (Ge et al., 2024) when specializing to each setup. Moreover, we also present theoretical results for a more general family of target covariance matrix, where standard ridge regression only achieves a slow statistical rate of $\mathcal{O}(1/\sqrt{n})$ for the excess risk, while Principal Component Regression (PCR) is guaranteed to achieve the fast rate $\mathcal{O}(1/n)$, where $n$ is the number of samples.

Transformers are deep architectures that define ``in-context mappings'' which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for a vision transformer). In this work, we study in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically, uniformly address their expressivity, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens which becomes discrete for a finite number of these. The relevant notion of smoothness then corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. This result implies, as a special case, that transformers are universal approximators for continuous permutation-invariant mappings over a fixed number of tokens. It also establishes the universal approximation capability of transformers for certain in-context learning tasks, demonstrating in particular their ability to perform regression within context. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLPs between multi-head attention layers is also explicitly controlled. We consider both unmasked attentions (as used for the vision transformer) and masked causal attentions (as used for NLP and time series applications). We tackle the causal setting leveraging a space-time lifting to analyze causal attention as a mapping over probability distributions of tokens.

Motivated by the concept of satisficing in decision-making, we consider the problem of satisficing exploration in bandit optimization. In this setting, the learner aims at finding a satisficing arm whose mean reward exceeds a certain threshold. The performance is measured by satisficing regret, which is the cumulative deficit of the chosen arm's mean reward compared to the threshold. We propose $\texttt{SELECT}$, a general algorithmic template for Satisficing REgret Minimization via SampLing and LowEr Confidence bound Testing, that attains constant satisficing regret for a wide variety of bandit optimization problems in the realizable case (i.e., whenever a satisficing arm exists). Specifically, given a class of bandit optimization problems and a corresponding learning oracle with sub-linear (standard) regret upper bound, $\texttt{SELECT}$ iteratively makes use of the oracle to identify a potential satisficing arm. Then, it collects data samples from this arm, and continuously compares the lower confidence bound of the identified arm's mean reward against the threshold value to determine if it is a satisficing arm. As a complement, $\texttt{SELECT}$ also enjoys the same (standard) regret guarantee as the oracle in the non-realizable case. Finally, we conduct numerical experiments to validate the performance of $\texttt{SELECT}$ for several popular bandit optimization settings.

Learning with multiple modalities has recently demonstrated significant gains in many domains by maximizing the shared information across modalities. However, the current approaches strongly rely on high-quality paired datasets, which allow co-training from the paired labels from different modalities. In this context, we raise a pivotal question: Can a model with one modality synergize the training of other models with the different modalities, even without the paired multimodal labels? Our answer is 'Yes'. As a figurative description, we argue that a writer, i.e., a language model, can promote the training of a painter, i.e., a visual model, even without the paired ground truth of text and image. We theoretically argue that a superior representation can be achieved by the synergy between two different modalities without paired supervision. As proofs of concept, we broadly confirm the considerable performance gains from the synergy among visual, language, and audio models. From a theoretical viewpoint, we first establish a mathematical foundation of the synergy between two different modality models, where each one is trained with its own modality. From a practical viewpoint, our work aims to broaden the scope of multimodal learning to encompass the synergistic usage of single-modality models, relieving a strong limitation of paired supervision. The code is available at https://github.com/johnjaejunlee95/synergistic-multimodal.

Estimating the ratio of two probability densities from a finite number of observations is a central machine learning problem. A common approach is to construct estimators using binary classifiers that distinguish observations from the two densities. However, the accuracy of these estimators depends on the choice of the binary loss function, raising the question of which loss function to choose based on desired error properties. For example, traditional loss functions, such as logistic or boosting loss, prioritize accurate estimation of small density ratio values over large ones, even though the latter are more critical in many applications. In this work, we start with prescribed error measures in a class of Bregman divergences and characterize all loss functions that result in density ratio estimators with small error. Our characterization extends results on composite binary losses from Reid & Williamson (2010) and their connection to density ratio estimation as identified by Menon & Ong (2016). As a result, we obtain a simple recipe for constructing loss functions with certain properties, such as those that prioritize an accurate estimation of large density ratio values. Our novel loss functions outperform related approaches for resolving parameter choice issues of 11 deep domain adaptation algorithms in average performance across 484 real-world tasks including sensor signals, texts, and images.

We discuss several aspects of the loss landscape of regularized neural networks: the structure of stationary points, connectivity of optimal solutions, path with non-increasing loss to arbitrary global optimum, and the nonuniqueness of optimal solutions, by casting the problem into an equivalent convex problem and considering its dual. Starting from two-layer neural networks with scalar output, we first characterize the solution set of the convex problem using its dual and further characterize all stationary points. With the characterization, we show that the topology of the global optima goes through a phase transition as the width of the network changes, and construct counterexamples where the problem may have a continuum of optimal solutions. Finally, we show that the solution set characterization and connectivity results can be extended to different architectures, including two layer vector-valued neural networks and parallel three-layer neural networks.

The classic teacher-student model in machine learning posits that a strong teacher supervises a weak student to improve the student's capabilities. We instead consider the inverted situation, where a weak teacher supervises a strong student with imperfect pseudolabels. This paradigm was recently brought forth by \citet{burns2023weak} and termed \emph{weak-to-strong generalization}. We theoretically investigate weak-to-strong generalization for binary and multilabel classification in a stylized overparameterized spiked covariance model with Gaussian covariates where the weak teacher's pseudolabels are asymptotically like random guessing. Under these assumptions, we provably identify two asymptotic phases of the strong student's generalization after weak supervision: (1) successful generalization and (2) random guessing. Our techniques should eventually extend to weak-to-strong multiclass classification. Towards doing so, we prove a tight lower tail inequality for the maximum of correlated Gaussians, which may be of independent interest. Understanding the multilabel setting reinforces the value of using logits for weak supervision when they are available.

''Grokking'' is a phenomenon where a neural network first memorizes training data and generalizes poorly, but then suddenly transitions to near-perfect generalization after prolonged training. While intriguing, this delayed generalization phenomenon compromises predictability and efficiency. Ideally, models should generalize directly without delay. To this end, this paper proposes GrokTransfer, a simple and principled method for accelerating grokking in training neural networks, based on the key observation that data embedding plays a crucial role in determining whether generalization is delayed. GrokTransfer first trains a smaller, weaker model to reach a nontrivial (but far from optimal) test performance. Then, the learned input embedding from this weaker model is extracted and used to initialize the embedding in the target, stronger model. We rigorously prove that, on a synthetic XOR task where delayed generalization always occurs in normal training, GrokTransfer enables the target model to generalize directly without delay. Moreover, we demonstrate that, across empirical studies of different tasks, GrokTransfer effectively reshapes the training dynamics and eliminates delayed generalization, for both fully-connected neural networks and Transformers.

Score-based diffusion models, which generate new data by learning to reverse a diffusion process that perturbs data from the target distribution into noise, have achieved remarkable success across various generative tasks. Despite their superior empirical performance, existing theoretical guarantees are often constrained by stringent assumptions or suboptimal convergence rates. In this paper, we establish a fast convergence theory for the denoising diffusion probabilistic model (DDPM), a widely used SDE-based sampler, under minimal assumptions. Our analysis shows that, provided $\ell_{2}$-accurate estimates of the score functions, the total variation distance between the target and generated distributions is upper bounded by $O(d/T)$ (ignoring logarithmic factors), where $d$ is the data dimensionality and $T$ is the number of steps. This result holds for any target distribution with finite first-order moment. To our knowledge, this improves upon existing convergence theory for the DDPM sampler, while imposing minimal assumptions on the target data distribution and score estimates. This is achieved through a novel set of analytical tools that provides a fine-grained characterization of how the error propagates at each step of the reverse process.