Stochastic gradient descent (SGD) is a cornerstone algorithm for high-dimensional optimization, renowned for its empirical successes. Recent theoretical advances have provided a deep understanding of how SGD enables feature learning in high-dimensional nonlinear models, most notably the \textit{single-index model} with i.i.d. data. In this work, we study the sequential learning problem for single-index models, also known as generalized linear bandits or ridge bandits, where SGD is a simple and natural solution, yet its learning dynamics remain largely unexplored. We show that, similar to the optimal interactive learner, SGD undergoes a distinct ``burn-in'' phase before entering the ``learning'' phase in this setting. Moreover, with an appropriately chosen learning rate schedule, a single SGD procedure simultaneously achieves near-optimal (or best-known) sample complexity and regret guarantees across both phases, for a broad class of link functions. Our results demonstrate that SGD remains highly competitive for learning single-index models under adaptive data.
Gradient descent dynamics for deep equilibrium models
Carnegie Mellon University, Department of Statistics（卡内基梅隆大学统计系）；Carnegie Mellon University, Department of Statistics and Machine Learning（卡内基梅隆大学统计与机器学习系）
Emergence of heavy tails in homogenized stochastic gradient descent
School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）；TU Dresden（德累斯顿工业大学）；ScaDS.ai Center for scalable data analytics and artificial intelligence（ScaDS.ai可扩展数据分析与人工智能中心）
Institute of Science and Technology Austria（奥地利科学与技术研究所）；Sung Kyun Kwan University（顺天妇女大学）；New York University and Flatiron Institute（纽约大学和Flatiron研究所）；University of Southern California（南加州大学）
Precise gradient descent training dynamics for finite-width multi-layer neural networks
Department of Statistics, Rutgers University, Piscataway, NJ 08854, USA.（统计学系，罗格斯大学）；Graduate School of Arts and Sciences, The University of Tokyo, Meguro, Tokyo 153-8902, Japan. RIKEN Center for Advanced Intelligence Project, Chuo, Tokyo 103-0027, Japan.（艺术与科学研究生院，东京大学）
University of Oxford（牛津大学）；Instadeep Ltd（Instadeep有限公司）；University of Southern California（南加州大学）
We prove linear convergence of gradient descent to a global optimum for the training of deep residual networks with constant layer width and smooth activation function. We show that if the trained weights, as a function of the layer index, admit a scaling limit as the depth increases, then the limit has finite $p-$variation with $p=2$. Proofs are based on non-asymptotic estimates for the loss function and for norms of the network weights along the gradient descent path. We illustrate the relevance of our theoretical results to practical settings using detailed numerical experiments on supervis…
Despite their effective use in various fields, many aspects of neural networks are poorly understood. One important way to investigate the characteristics of neural networks is to explore the loss landscape. However, most models produce a high-dimensional non-convex landscape which is difficult to visualize. We discuss and extend existing visualization methods based on 1D- and 2D slicing with a novel method that approximates the actual loss landscape geometry by using charts with interpretable axes. Based on the assumption that observations on small neural networks can generalize to more compl…
Courant Institute of Mathematical Sciences（数学科学学院）；Center for Data Science, New York University（数据科学中心，纽约大学）；Department of Computer Science（计算机科学系）；Data Science Institute, Columbia University（数据科学学院，哥伦比亚大学）
University of Electronic Science and Technology of China（电子科技大学）
In this paper, we propose a physically inspired graph-theoretical clustering method, which first makes the data points organized into an attractive graph, called In-Tree, via a physically inspired rule, called Nearest Descent (ND). In particular, the rule of ND works to select the nearest node in the descending direction of potential as the parent node of each node, which is in essence different from the classical Gradient Descent or Steepest Descent. The constructed In-Tree proves a very good candidate for clustering due to its particular features and properties. In the In-Tree, the original …
Télécom Paris, Institut Polytechnique de Paris（巴黎理工学院巴黎电信学院）；University of Oxford（牛津大学）；Rutgers Business School（罗格斯商学院）；Facebook AI Research（脸书人工智能研究院）
The gradient noise (GN) in the stochastic gradient descent (SGD) algorithm is often considered to be Gaussian in the large data regime by assuming that the \emph{classical} central limit theorem (CLT) kicks in. This assumption is often made for mathematical convenience, since it enables SGD to be analyzed as a stochastic differential equation (SDE) driven by a Brownian motion. We argue that the Gaussianity assumption might fail to hold in deep learning settings and hence render the Brownian motion-based analyses inappropriate. Inspired by non-Gaussian natural phenomena, we consider the GN in a…
Korea University（高丽大学）；Ecole Normale Supérieure（巴黎高等师范学院）；Université PSL（巴黎文理研究大学）；CNRS（法国国家科学研究中心）；INRIA（法国国家信息与自动化研究所）；University of Toronto（多伦多大学）；Vector Institute（向量研究所）
In this paper, we propose a new covering technique localized for the trajectories of SGD. This localization provides an algorithm-specific complexity measured by the covering number, which can have dimension-independent cardinality in contrast to standard uniform covering arguments that result in exponential dimension dependency. Based on this localized construction, we show that if the objective function is a finite perturbation of a piecewise strongly convex and smooth function with $P$ pieces, i.e. non-convex and non-smooth in general, the generalization error can be upper bounded by $O(\sq…
Courant Institute of Mathematical Sciences, New York University（纽约大学库朗数学科学研究所）；Center for Data Science, New York University（纽约大学数据科学中心）；Center for Computational Mathematics, Flatiron Institute（熨斗研究所计算数学中心）
In shuffle privacy, each user sends a collection of randomized messages to a trusted shuffler, the shuffler randomly permutes these messages, and the resulting shuffled collection of messages must satisfy differential privacy. Prior work in this model has largely focused on protocols that use a single round of communication to compute algorithmic primitives like means, histograms, and counts. We present interactive shuffle protocols for stochastic convex optimization. Our protocols rely on a new noninteractive protocol for summing vectors of bounded $\ell_2$ norm. By combining this sum subrout…
Provable Benefits of Unsupervised Pre-training and Transfer Learning via Single-Index Models
Department of Statistics and Actuarial Science, University of Waterloo, Canada（滑铁卢大学统计与精算科学系）；Cheriton School of Computer Science, University of Waterloo, Canada（滑铁卢大学切尔顿计算机科学学院）；Department of Statistics, Harvard University, United States of America（哈佛大学统计系）
We consider expected risk minimization problems when the range of the estimator is required to be nonnegative, motivated by the settings of maximum likelihood estimation (MLE) and trajectory optimization. To facilitate nonlinear interpolation, we hypothesize that the search space is a Reproducing Kernel Hilbert Space (RKHS). We develop first and second-order variants of stochastic mirror descent employing (i) \emph{pseudo-gradients} and (ii) complexity-reducing projections. Compressive projection in the first-order scheme is executed via kernel orthogonal matching pursuit (KOMP), which overcom…
Inria（法国国家信息与自动化研究所）；PSL Research University（巴黎文理研究大学）；Ecole Polytechnique Fédérale de Lausanne（洛桑联邦理工学院）；Univ. Grenoble Alpes（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；MSR-Inria Joint Centre（微软研究院-法国国家信息与自动化研究所联合中心）
We introduce the continuized Nesterov acceleration, a close variant of Nesterov acceleration whose variables are indexed by a continuous time parameter. The two variables continuously mix following a linear ordinary differential equation and take gradient steps at random times. This continuized variant benefits from the best of the continuous and the discrete frameworks: as a continuous process, one can use differential calculus to analyze convergence and obtain analytical expressions for the parameters; and a discretization of the continuized process can be computed exactly with convergence r…
The proximal inertial gradient descent is efficient for the composite minimization and applicable for broad of machine learning problems. In this paper, we revisit the computational complexity of this algorithm and present other novel results, especially on the convergence rates of the objective function values. The non-ergodic O(1/k) rate is proved for proximal inertial gradient descent with constant stepzise when the objective function is coercive. When the objective function fails to promise coercivity, we prove the sublinear rate with diminishing inertial parameters. In the case that the o…
