NeurIPS 2024 论文列表

Uniformity testing is arguably one of the most fundamental distribution testing problems. Given sample access to an unknown distribution $\mathbf{p}$ on $[n]$, one must decide if $\mathbf{p}$ is uniform or $\varepsilon$-far from uniform (in total variation distance). A long line of work established that uniformity testing has sample complexity $\Theta(\sqrt{n}\varepsilon^{-2})$. However, when the input distribution is neither uniform nor far from uniform, known algorithms may have highly non-replicable behavior. Consequently, if these algorithms are applied in scientific studies, they may lead to contradictory results that erode public trust in science. In this work, we revisit uniformity testing under the framework of algorithmic replicability [STOC '22], requiring the algorithm to be replicable under arbitrary distributions. While replicability typically incurs a $\rho^{-2}$ factor overhead in sample complexity, we obtain a replicable uniformity tester using only $\tilde{O}(\sqrt{n} \varepsilon^{-2} \rho^{-1})$ samples. To our knowledge, this is the first replicable learning algorithm with (nearly) linear dependence on $\rho$. Lastly, we consider a class of ``symmetric" algorithms [FOCS '00] whose outputs are invariant under relabeling of the domain $[n]$, which includes all existing uniformity testers (including ours). For this natural class of algorithms, we prove a nearly matching sample complexity lower bound for replicable uniformity testing.

We aim to understand the optimal PAC sample complexity in multiclass learning. While finiteness of the Daniely-Shalev-Shwartz (DS) dimension has been shown to characterize the PAC learnability of a concept class [Brukhim, Carmon, Dinur, Moran, and Yehudayoff, 2022], there exist polylog factor gaps in the leading term of the sample complexity. In this paper, we reduce the gap in terms of the dependence on the error parameter to a single log factor and also propose two possible routes towards completely resolving the optimal sample complexity, each based on a key open question we formulate: one concerning list learning with bounded list size, the other concerning a new type of shifting for multiclass concept classes. We prove that a positive answer to either of the two questions would completely resolve the optimal sample complexity up to log factors of the DS dimension.

We study the problem of online multi-group learning, a learning model in which an online learner must simultaneously achieve small prediction regret on a large collection of (possibly overlapping) subsequences corresponding to a family of groups. Groups are subsets of the context space, and in fairness applications, they may correspond to subpopulations defined by expressive functions of demographic attributes. In this paper, we design such oracle-efficient algorithms with sublinear regret under a variety of settings, including: (i) the i.i.d. setting, (ii) the adversarial setting with smoothed context distributions, and (iii) the adversarial transductive setting.

Multi-Output Regression (MOR) has been widely used in scientific data analysis for decision-making. Unlike traditional regression models, MOR aims to simultaneously predict multiple real-valued outputs given an input. However, the increasing dimensionality of the outputs poses significant challenges regarding interpretability and computational scalability for modern MOR applications. As a first step to address these challenges, this paper proposes a Sparse \& High-dimensional-Output REgression (SHORE) model by incorporating additional sparsity requirements to resolve the output interpretability, and then designs a computationally efficient two-stage optimization framework capable of solving SHORE with provable accuracy via compression on outputs. Theoretically, we show that the proposed framework is computationally scalable while maintaining the same order of training loss and prediction loss before-and-after compression under arbitrary or relatively weak sample set conditions. Empirically, numerical results further validate the theoretical findings, showcasing the efficiency and accuracy of the proposed framework.

Convolution is a fundamental operation in the 3D backbone. However, under certain conditions, the feature extraction ability of traditional convolution methods may be weakened. In this paper, we introduce a new convolution method based on $\ell_p$-norm. For theoretical support, we prove the universal approximation theorem for $\ell_p$-norm based convolution, and analyze the robustness and feasibility of $\ell_p$-norms in 3D point cloud tasks. Concretely, $\ell_{\infty}$-norm based convolution is prone to feature loss. $\ell_2$-norm based convolution is essentially a linear transformation of the traditional convolution. $\ell_1$-norm based convolution is an economical and effective feature extractor. We propose customized optimization strategies to accelerate the training process of $\ell_1$-norm based Nets and enhance the performance. Besides, a theoretical guarantee is given for the convergence by \textit{regret} argument. We apply our methods to classic networks and conduct related experiments. Experimental results indicate that our approach exhibits competitive performance with traditional CNNs, with lower energy consumption and instruction latency.

In a graph bisection problem, we are given a graph $G$ with two equally-sized unlabeled communities, and the goal is to recover the vertices in these communities. A popular heuristic, known as spectral clustering, is to output an estimated community assignment based on the eigenvector corresponding to the second-smallest eigenvalue of the Laplacian of $G$. Spectral algorithms can be shown to provably recover the cluster structure for graphs generated from probabilistic models, such as the Stochastic Block Model (SBM). However, spectral clustering is known to be non-robust to model mis-specification. Techniques based on semidefinite programming have been shown to be more robust, but they incur significant computational overheads. In this work, we study the robustness of spectral algorithms against semirandom adversaries. Informally, a semirandom adversary is allowed to ``helpfully'' change the specification of the model in a way that is consistent with the ground-truth solution. Our semirandom adversaries in particular are allowed to add edges inside clusters or increase the probability that an edge appears inside a cluster. Semirandom adversaries are a useful tool to determine the extent to which an algorithm has overfit to statistical assumptions on the input. On the positive side, we identify a wide range of semirandom adversaries under which spectral bisection using the _unnormalized_ Laplacian is strongly consistent, i.e., it exactly recovers the planted partitioning. On the negative side, we show that in many of these settings, _normalized_ spectral bisection outputs a partitioning that makes a classification mistake on a constant fraction of the vertices. Finally, we demonstrate numerical experiments that complement our theoretical findings.

The saturation effects, which originally refer to the fact that kernel ridge regression (KRR) fails to achieve the information-theoretical lower bound when the regression function is over-smooth, have been observed for almost 20 years and were rigorously proved recently for kernel ridge regression and some other spectral algorithms over a fixed dimensional domain. The main focus of this paper is to explore the saturation effects for a large class of spectral algorithms (including the KRR, gradient descent, etc.) in large dimensional settings where $n \asymp d^{\gamma}$. More precisely, we first propose an improved minimax lower bound for the kernel regression problem in large dimensional settings and show that the gradient flow with early stopping strategy will result in an estimator achieving this lower bound (up to a logarithmic factor). Similar to the results in KRR, we can further determine the exact convergence rates (both upper and lower bounds) of a large class of (optimal tuned) spectral algorithms with different qualification $\tau$'s. In particular, we find that these exact rate curves (varying along $\gamma$) exhibit the periodic plateau behavior and the polynomial approximation barrier. Consequently, we can fully depict the saturation effects of the spectral algorithms and reveal a new phenomenon in large dimensional settings (i.e., the saturation effect occurs in large dimensional setting as long as the source condition $s>\tau$ while it occurs in fixed dimensional setting as long as $s>2\tau$).

We introduce a new linear-algebraic tool based on group representation theory, and use it to address three key problems in machine learning. 1. Past researchers have proposed fast attention algorithms for LLMs by approximating or replace softmax attention with other functions, such as low-degree polynomials. The key property of these functions is that, when applied entry-wise to the matrix $QK^{\top}$, the result is a low rank matrix when $Q$ and $K$ are $n \times d$ matrices and $n \gg d$. This suggests a natural question: what are all functions $f$ with this property? If other $f$ exist and are quickly computable, they can be used in place of softmax for fast subquadratic attention algorithms. It was previously known that low-degree polynomials have this property. We prove that low-degree polynomials are the only piecewise continuous functions with this property. This suggests that the low-rank fast attention only works for functions approximable by polynomials. Our work gives a converse to the polynomial method in algorithm design. 2. We prove the first full classification of all positive definite kernels that are functions of Manhattan or $\ell_1$ distance. Our work generalizes an existing theorem at the heart of all kernel methods in machine learning: the classification of all positive definite kernels that are functions of Euclidean distance. 3. The key problem in metric transforms, a mathematical theory used in geometry and machine learning, asks what functions transform pairwise distances in semi-metric space $M$ to semi-metric space $N$ for specified $M$ and $N$. We provide the first full classification of functions that transform Manhattan distances to Manhattan distances. Our work generalizes the foundational work of Schoenberg, which fully classifies functions that transform Euclidean to Euclidean distances. We additionally prove results about stable-rank preserving functions that are potentially useful in algorithmic design, and more. Our core new tool is called the representation theory of the hyperrectangle.

Given the vast number of classifiers that have been (and continue to be) proposed, reliable methods for comparing them are becoming increasingly important. The desire for reliability is broken down into three main aspects: (1) Comparisons should allow for different quality metrics simultaneously. (2) Comparisons should take into account the statistical uncertainty induced by the choice of benchmark suite. (3) The robustness of the comparisons under small deviations in the underlying assumptions should be verifiable. To address (1), we propose to compare classifiers using a generalized stochastic dominance ordering (GSD) and present the GSD-front as an information-efficient alternative to the classical Pareto-front. For (2), we propose a consistent statistical estimator for the GSD-front and construct a statistical test for whether a (potentially new) classifier lies in the GSD-front of a set of state-of-the-art classifiers. For (3), we relax our proposed test using techniques from robust statistics and imprecise probabilities. We illustrate our concepts on the benchmark suite PMLB and on the platform OpenML.

The implicit bias of gradient descent has long been considered the primary mechanism explaining the superior generalization of over-parameterized neural networks without overfitting, even when the training error is zero. However, the implicit bias toward adversarial robustness has rarely been considered in the research community, although it is crucial for the trustworthiness of machine learning models. To fill this gap, in this paper, we explore whether consecutive layers collaborate to strengthen adversarial robustness during gradient descent. By quantifying this collaboration between layers using our proposed concept, co-correlation, we demonstrate a monotonically increasing trend in co-correlation, which implies a decreasing trend in adversarial robustness during gradient descent. Additionally, we observe different behaviours between narrow and wide neural networks during gradient descent. We conducted extensive experiments that verified our proposed theorems.

How to solve high-dimensional linear programs (LPs) efficiently is a fundamental question. Recently, there has been a surge of interest in reducing LP sizes using *random projections*, which can accelerate solving LPs independently of improving LP solvers. This paper explores a new direction of *data-driven projections*, which use projection matrices learned from data instead of random projection matrices. Given training data of $n$-dimensional LPs, we learn an $n\times k$ projection matrix with $n > k$. When addressing a future LP instance, we reduce its dimensionality from $n$ to $k$ via the learned projection matrix, solve the resulting LP to obtain a $k$-dimensional solution, and apply the learned matrix to it to recover an $n$-dimensional solution. On the theoretical side, a natural question is: how much data is sufficient to ensure the quality of recovered solutions? We address this question based on the framework of *data-driven algorithm design*, which connects the amount of data sufficient for establishing generalization bounds to the *pseudo-dimension* of performance metrics. We obtain an $\tilde{\mathrm{O}}(nk^2)$ upper bound on the pseudo-dimension, where $\tilde{\mathrm{O}}$ compresses logarithmic factors. We also provide an $\Omega(nk)$ lower bound, implying our result is tight up to an $\tilde{\mathrm{O}}(k)$ factor. On the practical side, we explore two simple methods for learning projection matrices: PCA- and gradient-based methods. While the former is relatively efficient, the latter can sometimes achieve better solution quality. Experiments demonstrate that learning projection matrices from data is indeed beneficial: it leads to significantly higher solution quality than the existing random projection while greatly reducing the time for solving LPs.

Distributed learning is essential to train machine learning algorithms across *heterogeneous* agents while maintaining data privacy. We conduct an asymptotic analysis of Unified Distributed SGD (UD-SGD), exploring a variety of communication patterns, including decentralized SGD and local SGD within Federated Learning (FL), as well as the increasing communication interval in the FL setting. In this study, we assess how different sampling strategies, such as *i.i.d.* sampling, shuffling, and Markovian sampling, affect the convergence speed of UD-SGD by considering the impact of agent dynamics on the limiting covariance matrix as described in the Central Limit Theorem (CLT). Our findings not only support existing theories on linear speedup and asymptotic network independence, but also theoretically and empirically show how efficient sampling strategies employed by individual agents contribute to overall convergence in UD-SGD. Simulations reveal that a few agents using highly efficient sampling can achieve or surpass the performance of the majority employing moderately improved strategies, providing new insights beyond traditional analyses focusing on the worst-performing agent.

This paper presents a comprehensive analysis of the growth rate of $H$-consistency bounds (and excess error bounds) for various surrogate losses used in classification. We prove a square-root growth rate near zero for smooth margin-based surrogate losses in binary classification, providing both upper and lower bounds under mild assumptions. This result also translates to excess error bounds. Our lower bound requires weaker conditions than those in previous work for excess error bounds, and our upper bound is entirely novel. Moreover, we extend this analysis to multi-class classification with a series of novel results, demonstrating a universal square-root growth rate for smooth *comp-sum* and *constrained losses*, covering common choices for training neural networks in multi-class classification. Given this universal rate, we turn to the question of choosing among different surrogate losses. We first examine how $H$-consistency bounds vary across surrogates based on the number of classes. Next, ignoring constants and focusing on behavior near zero, we identify *minimizability gaps* as the key differentiating factor in these bounds. Thus, we thoroughly analyze these gaps, to guide surrogate loss selection, covering: comparisons across different comp-sum losses, conditions where gaps become zero, and general conditions leading to small gaps. Additionally, we demonstrate the key role of minimizability gaps in comparing excess error bounds and $H$-consistency bounds.

Kernel methods are widely utilized in machine learning field to learn, from training data, a latent function in a reproducing kernel Hilbert space. It is well known that the approximator thus obtained usually achieves a linear representation, which brings various computational benefits, while maintaining great representation power (i.e., universal approximation). However, when non-negativity constraints are imposed on the function's outputs, the literature usually takes the kernel method-based approximators as offering linear representations at the expense of limited model flexibility or good representation power by allowing for their nonlinear forms. The main contribution of this paper is to derive a sufficient condition for a positive definite kernel so that it may construct flexible and linear approximators of non-negative functions. We call a kernel function that offers these attributes an *inverse M-kernel*; it is reminiscent of the inverse M-matrix. Furthermore, we show that for a one-dimensional input space, universal exponential/Abel kernels are inverse M-kernels and construct linear universal approximators of non-negative functions. To the best of our knowledge, it is the first time that the existence of linear universal approximators of non-negative functions has been elucidated. We confirm the effectiveness of our results by experiments on the problems of non-negativity-constrained regression, density estimation, and intensity estimation. Finally, we discuss issues and perspectives on multi-dimensional input settings.

Interactive decision making, encompassing bandits, contextual bandits, and reinforcement learning, has recently been of interest to theoretical studies of experimentation design and recommender system algorithm research. One recent finding in this area is that the well-known Graves-Lai constant being zero is a necessary and sufficient condition for achieving bounded (or constant) regret in interactive decision-making. As this condition may be a strong requirement for many applications, the practical usefulness of pursuing bounded regret has been questioned. In this paper, we show that the condition of the Graves-Lai constant being zero is also necessary for a consistent algorithm to achieve delay model robustness when reward delays are unknown (i.e., when feedback is anonymous). Here, model robustness is measured in terms of $\epsilon$-robustness, one of the most widely used and one of the least adversarial robustness concepts in the robust statistics literature. In particular, we show that $\epsilon$-robustness cannot be achieved for a consistent (i.e., uniformly sub-polynomial regret) algorithm, however small the nonzero $\epsilon$ value is, when the Grave-Lai constant is not zero. While this is a strongly negative result, we also provide a positive result for linear rewards models (contextual linear bandits, reinforcement learning with linear MDP) that the Grave-Lai constant being zero is also sufficient for achieving bounded regret without any knowledge of delay models, i.e., the best of both the efficiency world and the delay robustness world.

We develop a unifying framework for information-theoretic lower bound in statistical estimation and interactive decision making. Classical lower bound techniques---such as Fano's method, Le Cam's method, and Assouad's lemma---are central to the study of minimax risk in statistical estimation, yet are insufficient to provide tight lower bounds for \emph{interactive decision making} algorithms that collect data interactively (e.g., algorithms for bandits and reinforcement learning). Recent work of Foster et al. provides minimax lower bounds for interactive decision making using seemingly different analysis techniques from the classical methods. These results---which are proven using a complexity measure known as the \emph{Decision-Estimation Coefficient} (DEC)---capture difficulties unique to interactive learning, yet do not recover the tightest known lower bounds for passive estimation. We propose a unified view of these distinct methodologies through a new lower bound approach called \emph{interactive Fano method}. As an application, we introduce a novel complexity measure, the \emph{Fractional Covering Number}, which facilitates the new lower bounds for interactive decision making that extend the DEC methodology by incorporating the complexity of estimation. Using the fractional covering number, we (i) provide a unified characterization of learnability for \emph{any} stochastic bandit problem, (ii) close the remaining gap between the upper and lower bounds in Foster et al. (up to polynomial factors) for any interactive decision making problem in which the underlying model class is convex.

This paper investigates the impact of alignment between the target function of interest and the kernel matrix on a variety of kernel-based methods based on a general loss belonging to a rich loss function family, which covers many commonly used methods in regression and classification problems. We consider the truncated kernel-based method (TKM) which is estimated within a reduced function space constructed by using the spectral truncation of the kernel matrix and compare its theoretical behavior to that of the standard kernel-based method (KM) under various settings. By using the kernel complexity function that quantifies the complexity of the induced function space, we derive the upper bounds for both TKM and KM, and further reveal their dependencies on the degree of target-kernel alignment. Specifically, for the alignment with polynomial decay, the established results indicate that under the just-aligned and weakly-aligned regimes, TKM and KM share the same learning rate. Yet, under the strongly-aligned regime, KM suffers the saturation effect, while TKM can be continuously improved as the alignment becomes stronger. This further implies that TKM has a strong ability to capture the strong alignment and provide a theoretically guaranteed solution to eliminate the phenomena of saturation effect. The minimax lower bound is also established for the squared loss to confirm the optimality of TKM. Extensive numerical experiments further support our theoretical findings. The Python code for reproducing the numerical experiments is available at https://github.com/wywangen.

In-context learning (ICL) of large language models has proven to be a surprisingly effective method of learning a new task from only a few demonstrative examples. In this paper, we shed light on the efficacy of ICL from the viewpoint of statistical learning theory. We develop approximation and generalization error analyses for a transformer model composed of a deep neural network and one linear attention layer, pretrained on nonparametric regression tasks sampled from general function spaces including the Besov space and piecewise $\gamma$-smooth class. In particular, we show that sufficiently trained transformers can achieve -- and even improve upon -- the minimax optimal estimation risk in context by encoding the most relevant basis representations during pretraining. Our analysis extends to high-dimensional or sequential data and distinguishes the \emph{pretraining} and \emph{in-context} generalization gaps, establishing upper and lower bounds w.r.t. both the number of tasks and in-context examples. These findings shed light on the effectiveness of few-shot prompting and the roles of task diversity and representation learning for ICL.

In current deep learning tasks, Adam-style optimizers—such as Adam, Adagrad, RMSprop, Adafactor, and Lion—have been widely used as alternatives to SGD-style optimizers. These optimizers typically update model parameters using the sign of gradients, resulting in more stable convergence curves. The learning rate and the batch size are the most critical hyperparameters for optimizers, which require careful tuning to enable effective convergence. Previous research has shown that the optimal learning rate increases linearly (or follows similar rules) with batch size for SGD-style optimizers. However, this conclusion is not applicable to Adam-style optimizers. In this paper, we elucidate the connection between optimal learning rates and batch sizes for Adam-style optimizers through both theoretical analysis and extensive experiments. First, we raise the scaling law between batch sizes and optimal learning rates in the “sign of gradient” case, in which we prove that the optimal learning rate first rises and then falls as the batch size increases. Moreover, the peak value of the surge will gradually move toward the larger batch size as training progresses. Second, we conduct experiments on various CV and NLP tasks and verify the correctness of the scaling law.