ICLR 2024 Papers

In our era of enormous neural networks, empirical progress has been driven by the philosophy that *more is better.* Recent deep learning practice has found repeatedly that larger model size, more data, and more computation (resulting in lower training loss) optimizing to near-interpolation improves performance. In this paper, we give theoretical backing to these empirical observations by showing that these three properties hold in random feature (RF) regression, a class of models equivalent to shallow networks with only the last layer trained. Concretely, we first show that the test risk of RF regression decreases monotonically with both the number of features and samples, provided the ridge penalty is tuned optimally. In particular, this implies that infinite width RF architectures are preferable to those of any finite width. We then proceed to demonstrate that, for a large class of tasks characterized by powerlaw eigenstructure, training to near-zero training loss is *obligatory:* near-optimal performance can *only* be achieved when the training error is much smaller than the test error. Grounding our theory in real-world data, we find empirically that standard computer vision tasks with convolutional neural kernels clearly fall into this class. Taken together, our results tell a simple, testable story of the benefits of overparameterization and overfitting in random feature models.

We study the notion of a generalization bound being _uniformly tight_, meaning that the difference between the bound and the population loss is small for all learning algorithms and all population distributions. Numerous generalization bounds have been proposed in the literature as potential explanations for the ability of neural networks to generalize in the overparameterized setting. However, in their paper "Fantastic Generalization Measures and Where to Find Them," Jiang et al. (2020) examine more than a dozen generalization bounds, and show empirically that none of them are uniformly tight. This raises the question of whether uniformly-tight generalization bounds are at all possible in the overparameterized setting. We consider two types of generalization bounds: (1) bounds that may depend on the training set and the learned hypothesis (e.g., margin bounds). We prove mathematically that no such bound can be uniformly tight in the overparameterized setting; (2) bounds that may in addition also depend on the learning algorithm (e.g., stability bounds). For these bounds, we show a trade-off between the algorithm's performance and the bound's tightness. Namely, if the algorithm achieves good accuracy on certain distributions, then no generalization bound can be uniformly tight for it in the overparameterized setting. We explain how these formal results can, in our view, inform research on generalization bounds for neural networks, while stressing that other interpretations of these results are also possible.

Contrastive learning is a highly successful technique for learning representations of data from labeled tuples, specifying the distance relations within the tuple. We study the sample complexity of contrastive learning, i.e. the minimum number of labeled tuples sufficient for getting high generalization accuracy. We give tight bounds on the sample complexity in a variety of settings, focusing on arbitrary distance functions, $\ell_p$-distances, and tree metrics. Our main result is an (almost) optimal bound on the sample complexity of learning $\ell_p$-distances for integer $p$. For any $p \ge 1$, we show that $\tilde \Theta(nd)$ labeled tuples are necessary and sufficient for learning $d$-dimensional representations of $n$-point datasets. Our results hold for an arbitrary distribution of the input samples and are based on giving the corresponding bounds on the Vapnik-Chervonenkis/Natarajan dimension of the associated problems. We further show that the theoretical bounds on sample complexity obtained via VC/Natarajan dimension can have strong predictive power for experimental results, in contrast with the folklore belief about a substantial gap between the statistical learning theory and the practice of deep learning.

Adversarial training is a widely used strategy for making neural networks resistant to adversarial perturbations. For a neural network of width $m$, $n$ input training data in $d$ dimension, it takes $\Omega(mnd)$ time cost per training iteration for the forward and backward computation. In this paper we analyze the convergence guarantee of adversarial training procedure on a two-layer neural network with shifted ReLU activation, and shows that only $o(m)$ neurons will be activated for each input data per iteration. Furthermore, we develop an algorithm for adversarial training with time cost $o(m n d)$ per iteration by applying half-space reporting data structure.

Transformers have become the go-to architecture for language and vision tasks, yet their theoretical properties, especially memorization capacity, remain elusive. This paper investigates the memorization abilities of multi-head attention mechanisms, examining how many example sequences they can memorize, as a function of the number of heads and sequence length. Motivated by experimental findings on vision transformers, we introduce novel assumptions about the linear independence of input data, distinct from the commonly used general-position assumption. Under these assumptions, we demonstrate that an attention layer with $H$ heads, dimension $d$, and context size $n < d,$ featuring $\Theta(Hd^2)$ parameters, can memorize $\Omega(Hn)$ examples. Our analysis sheds light on how different attention heads handle various example sequences, aided by the softmax operator’s saturation property. We validate our findings through experiments on synthetic data.

We rigorously study the joint evolution of training dynamics via stochastic gradient descent (SGD) and the spectra of empirical Hessian and gradient matrices. We prove that in two canonical classification tasks for multi-class high-dimensional mixtures and either 1 or 2-layer neural networks, the SGD trajectory rapidly aligns with emerging low-rank outlier eigenspaces of the Hessian and gradient matrices. Moreover, in multi-layer settings this alignment occurs per layer, with the final layer's outlier eigenspace evolving over the course of training, and exhibiting rank deficiency when the SGD converges to sub-optimal classifiers. This establishes some of the rich predictions that have arisen from extensive numerical studies in the last decade about the spectra of Hessian and information matrices over the course of training in overparametrized networks.

We investigate learning the equilibria in non-stationary multi-agent systems and address the challenges that differentiate multi-agent learning from single-agent learning. Specifically, we focus on games with bandit feedback, where testing an equilibrium can result in substantial regret even when the gap to be tested is small, and the existence of multiple optimal solutions (equilibria) in stationary games poses extra challenges. To overcome these obstacles, we propose a versatile black-box approach applicable to a broad spectrum of problems, such as general-sum games, potential games, and Markov games, when equipped with appropriate learning and testing oracles for stationary environments. Our algorithms can achieve $\widetilde{O}\left(\Delta^{1/4}T^{3/4}\right)$ regret when the degree of nonstationarity, as measured by total variation $\Delta$, is known, and $\widetilde{O}\left(\Delta^{1/5}T^{4/5}\right)$ regret when $\Delta$ is unknown, where $T$ is the number of rounds. Meanwhile, our algorithm inherits the favorable dependence on number of agents from the oracles. As a side contribution that may be independent of interest, we show how to test for various types of equilibria by a black-box reduction to single-agent learning, which includes Nash equilibria, correlated equilibria, and coarse correlated equilibria.

Kernel methods are widely used in machine learning, especially for classification problems. However, the theoretical analysis of kernel classification is still limited. This paper investigates the statistical performances of kernel classifiers. With some mild assumptions on the conditional probability $\eta(x)=\mathbb{P}(Y=1\mid X=x)$, we derive an upper bound on the classification excess risk of a kernel classifier using recent advances in the theory of kernel regression. We also obtain a minimax lower bound for Sobolev spaces, which shows the optimality of the proposed classifier. Our theoretical results can be extended to the generalization error of overparameterized neural network classifiers. To make our theoretical results more applicable in realistic settings, we also propose a simple method to estimate the interpolation smoothness of $2\eta(x)-1$ and apply the method to real datasets.

Deep neural networks with feature learning have shown surprising generalization performance in high dimensional settings, but it has not been fully understood how and when they enjoy the benefit of feature learning. In this paper, we theoretically analyze the statistical properties of the benefits from feature learning in a two-layer linear neural network with multiple outputs in a high-dimensional setting. For that purpose, we propose a new criterion that allows feature learning of a two-layer linear neural network in a high-dimensional setting. Interestingly, we can show that models with smaller values of the criterion generalize even in situations where normal ridge regression fails to generalize. This is because the proposed criterion contains a proper regularization for the feature mapping and acts as an upper bound on the predictive risk. As an important characterization of the criterion, the two-layer linear neural network that minimizes this criterion can achieve the optimal Bayes risk that is determined by the distribution of the true signals across the multiple outputs. To the best of our knowledge, this is the first study to specifically identify the conditions under which a model obtained by proper feature learning can outperform normal ridge regression in a high-dimensional multiple-output linear regression problem.

We propose a new bound for generalization of neural networks using Koopman operators. Whereas most of existing works focus on low-rank weight matrices, we focus on full-rank weight matrices. Our bound is tighter than existing norm-based bounds when the condition numbers of weight matrices are small. Especially, it is completely independent of the width of the network if the weight matrices are orthogonal. Our bound does not contradict to the existing bounds but is a complement to the existing bounds. As supported by several existing empirical results, low-rankness is not the only reason for generalization. Furthermore, our bound can be combined with the existing bounds to obtain a tighter bound. Our result sheds new light on understanding generalization of neural networks with full-rank weight matrices, and it provides a connection between operator-theoretic analysis and generalization of neural networks.

We study the problem of online prediction, in which at each time step $t \in \{1,2, \cdots T\}$, an individual $x_t$ arrives, whose label we must predict. Each individual is associated with various groups, defined based on their features such as age, sex, race etc., which may intersect. Our goal is to make predictions that have regret guarantees not just overall but also simultaneously on each sub-sequence comprised of the members of any single group. Previous work such as [Blum & Lykouris][1] and [Lee et al][2] provide attractive regret guarantees for these problems; however, these are computationally intractable on large model classes (e.g., the set of all linear models, as used in linear regression). We show that a simple modification of the sleeping experts technique of [Blum & Lykouris][1] yields an efficient *reduction* to the well-understood problem of obtaining diminishing external regret *absent group considerations*. Our approach gives similar regret guarantees compared to [Blum & Lykouris][1]; however, we run in time linear in the number of groups, and are oracle-efficient in the hypothesis class. This in particular implies that our algorithm is efficient whenever the number of groups is polynomially bounded and the external-regret problem can be solved efficiently, an improvement on [Blum & Lykouris][1]'s stronger condition that the model class must be small. Our approach can handle online linear regression and online combinatorial optimization problems like online shortest paths. Beyond providing theoretical regret bounds, we evaluate this algorithm with an extensive set of experiments on synthetic data and on two real data sets --- Medical costs and the Adult income dataset, both instantiated with intersecting groups defined in terms of race, sex, and other demographic characteristics. We find that uniformly across groups, our algorithm gives substantial error improvements compared to running a standard online linear regression algorithm with no groupwise regret guarantees.

In this work, we consider the optimization process of minibatch stochastic gradient descent (SGD) on a 2-layer neural network with data separated by a quadratic ground truth function. We prove that with data drawn from the Boolean hypercube labeled by the quadratic ``XOR'' function $y = -x_ix_j$ , it is possible to train to a population error $o(1)$ with $\Theta(d\text{polylog}(d))$ samples. Our result considers simultaneously training both layers of the two-layer-neural network with ReLU activations via standard minibatch SGD on the logistic loss. To our knowledge, this work is the first to give a sample complexity of for efficiently learning the XOR function on isotropic data on a standard neural network with standard training. Our main technique is showing that the network evolves in two phases: a \em signal-finding \em phase where the network is small and many of the neurons evolve independently to find features, and a \em signal-heavy \em phase, where SGD maintains and balances the features. We leverage the simultaneous training of the layers to show that it is sufficient for only a small fraction of the neurons to learn features, since those neurons will be amplified by the simultaneous growth of their second layer weights.

Information-theoretic generalization analysis has achieved astonishing success in characterizing the generalization capabilities of noisy and iterative learning algorithms. However, current advancements are mostly restricted to average-case scenarios and necessitate the stringent bounded loss assumption, leaving a gap with regard to computationally tractable PAC generalization analysis, especially for long-tailed loss distributions. In this paper, we bridge this gap by introducing a novel class of PAC bounds through leveraging loss entropies. These bounds simplify the computation of key information metrics in previous PAC information-theoretic bounds to one-dimensional variables, thereby enhancing computational tractability. Moreover, our data-independent bounds provide novel insights into the generalization behavior of the minimum error entropy criterion, while our data-dependent bounds improve over previous results by alleviating the bounded loss assumption under both leave-one-out and supersample settings. Extensive numerical studies indicate strong correlations between the generalization error and the induced loss entropy, showing that the presented bounds adeptly capture the patterns of the true generalization gap under various learning scenarios.

Understanding how overparameterized neural networks generalize despite perfect interpolation of noisy training data is a fundamental question. Mallinar et. al. (2022) noted that neural networks seem to often exhibit ``tempered overfitting'', wherein the population risk does not converge to the Bayes optimal error, but neither does it approach infinity, yielding non-trivial generalization. However, this has not been studied rigorously. We provide the first rigorous analysis of the overfiting behaviour of regression with minimum norm ($\ell_2$ of weights), focusing on univariate two-layer ReLU networks. We show overfitting is tempered (with high probability) when measured with respect to the $L_1$ loss, but also show that the situation is more complex than suggested by Mallinar et. al., and overfitting is catastrophic with respect to the $L_2$ loss, or when taking an expectation over the training set.

Grokking is the intriguing phenomenon where a model learns to generalize long after it has fit the training data. We show both analytically and numerically that grokking can surprisingly occur in linear networks performing linear tasks in a simple teacher-student setup. In this setting, the full training dynamics is derived in terms of the expected training and generalization data covariance matrix. We present exact predictions on how the grokking time depends on input and output dimensionality, train sample size, regularization, and network parameters initialization. The key findings are that late generalization increase may not imply a transition from "memorization" to "understanding", but can simply be an artifact of the accuracy measure. We provide empirical verification for these propositions, along with preliminary results indicating that some predictions also hold for deeper networks, with non-linear activations.

Recent deep learning applications, exemplified by text-to-image tasks, often involve high-dimensional inputs and outputs. While several studies have investigated the function estimation capabilities of deep learning, research on dilated convolutional neural networks (CNNs) has mainly focused on cases where input dimensions are infinite but output dimensions are one-dimensional, similar to many other studies. However, many practical deep learning tasks involve high-dimensional (or even infinite dimensional) inputs and outputs. In this paper, we investigate the optimality of dilated CNNs for estimating a map between infinite-dimensional input and output spaces by analyzing their approximation and estimation abilities. For that purpose, we first show that approximation and estimation errors depend only on the smoothness and decay rate with respect to the infinity norm of the output, and their estimation accuracy actually achieve the {\it minimax optimal} rate of convergence. Second, we demonstrate that the dilated CNNs outperform {\it any} linear estimators including kernel ridge regression and $k$-NN estimators in a minimax error sense, highlighting the usefulness of feature learning realized by deep neural networks. Our theoretical analysis particularly explains the success of deep learning in recent high-dimensional input-output tasks.

Due to the rise of privacy concerns, in many practical applications, the training data is aggregated before being shared with the learner to protect the privacy of users' sensitive responses. In an aggregate learning framework, the dataset is grouped into bags of samples, where each bag is available only with an aggregate response, providing a summary of individuals' responses in that bag. In this paper, we study two natural loss functions for learning from aggregate responses: the bag-level loss and the instance-level loss. In the former, the model is learned by minimizing a loss between the aggregate responses and aggregate model predictions, while in the latter, the model aims to fit individual predictions to the aggregate responses. In this work, we show that the instance-level loss can be perceived as a regularized form of the bag-level loss. This observation allows us to compare the two approaches with respect to the bias and variance of the resulting estimators and to introduce a novel interpolating estimator that combines the two approaches. For linear regression tasks, we provide a precise characterization of the risk of the interpolating estimator in an asymptotic regime where the size of the training set grows in proportion to the feature dimension. Our analysis enables us to theoretically understand the effect of different factors, such as bag size, on the model's prediction risk. Additionally, we propose a mechanism for differentially private learning from aggregate responses and derive the optimal bag size in terms of the prediction risk-privacy trade-off. We also carry out thorough experiments to corroborate our theory and show the efficacy of the interpolating estimator.

As a novel and effective fine-tuning paradigm based on large-scale pre-trained language models (PLMs), prompt-tuning aims to reduce the gap between downstream tasks and pre-training objectives. While prompt-tuning has yielded continuous advancements in various tasks, such an approach still remains a persistent defect: prompt-tuning methods fail to generalize to specific few-shot patterns. From the perspective of distribution analyses, we disclose that the intrinsic issues behind the phenomenon are the over-multitudinous conceptual knowledge contained in PLMs and the abridged knowledge for target downstream domains, which jointly result in that PLMs mis-locate the knowledge distributions corresponding to the target domains in the universal knowledge embedding space. To this end, we intuitively explore to approximate the unabridged target domains of downstream tasks in a debiased manner, and then abstract such domains to generate discriminative prompts, thereby providing the de-ambiguous guidance for PLMs. Guided by such an intuition, we propose a simple yet effective approach, namely BayesPrompt, to learn prompts that contain the domain discriminative information against the interference from domain-irrelevant knowledge. BayesPrompt primitively leverages known distributions to approximate the debiased factual distributions of target domains and further uniformly samples certain representative features from the approximated distributions to generate the ultimate prompts for PLMs. We provide theoretical insights with the connection to domain adaptation. Empirically, our method achieves state-of-the-art performance on benchmarks.

Real-world classifiers can benefit from the option of abstaining from predicting on samples where they have low confidence. Such abstention is particularly useful on samples which are close to the learned decision boundary, or which are outliers with respect to the training sample. These settings have been the subject of extensive but disjoint study in the selective classification (SC) and out-of-distribution (OOD) detection literature. Recent work on selective classification with OOD detection (SCOD) has argued for the unified study of these problems; however, the formal underpinnings of this problem are still nascent, and existing techniques are heuristic in nature. In this paper, we propose new plugin estimators for SCOD that are theoretically grounded, effective, and generalise existing approaches from the SC and OOD detection literature. In the course of our analysis, we formally explicate how naïve use of existing SC and OOD detection baselines may be inadequate for SCOD. We empirically demonstrate that our approaches yields competitive SC and OOD detection trade-offs compared to common baselines.