Stochastic Gradient Descent (SGD) is one of the simplest and most popular algorithms in modern statistical and machine learning due to its computational and memory efficiency. Various averaging schemes have been proposed to accelerate the convergence of SGD in different settings. In this paper, we explore a general averaging scheme for SGD. Specifically, we establish the asymptotic normality of a broad range of weighted averaged SGD solutions and provide asymptotically valid online inference approaches. Furthermore, we propose an adaptive averaging scheme that exhibits both optimal statistical rate and favorable non-asymptotic convergence, drawing insights from the optimal weight for the linear model in terms of non-asymptotic mean squared error (MSE).

Feature learning, i.e. extracting meaningful representations of data, is quintessential to the practical success of neural networks trained with gradient descent, yet it is notoriously difficult to explain how and why it occurs. Recent theoretical studies have shown that shallow neural networks optimized on a single task with gradient-based methods can learn meaningful features, extending our understanding beyond the neural tangent kernel or random feature regime in which negligible feature learning occurs. But in practice, neural networks are increasingly often trained on {\em many} tasks simultaneously with differing loss functions, and these prior analyses do not generalize to such settings. In the multi-task learning setting, a variety of studies have shown effective feature learning by simple linear models. However, multi-task learning via {\em nonlinear} models, arguably the most common learning paradigm in practice, remains largely mysterious. In this work, we present the first results proving feature learning occurs in a multi-task setting with a nonlinear model. We show that when the tasks are binary classification problems with labels depending on only $r$ directions within the ambient $d\gg r$-dimensional input space, executing a simple gradient-based multitask learning algorithm on a two-layer ReLU neural network learns the ground-truth $r$ directions. In particular, any downstream task on the $r$ ground-truth coordinates can be solved by learning a linear classifier with sample and neuron complexity independent of the ambient dimension $d$, while a random feature model requires exponential complexity in $d$ for such a guarantee.

The nuclear fuel loading pattern optimization problem belongs to the class of large-scale combinatorial optimization. It is also characterized by multiple objectives and constraints, which makes it impossible to solve explicitly. Stochastic optimization methodologies including Genetic Algorithms and Simulated Annealing are used by different nuclear utilities and vendors, but hand-designed solutions continue to be the prevalent method in the industry. To improve the state-of-the-art, Deep Reinforcement Learning (RL), in particular, Proximal Policy Optimization is leveraged. This work presents a first-of-a-kind approach to utilize deep RL to solve the loading pattern problem and could be leveraged for any engineering design optimization. This paper is also to our knowledge the first to propose a study of the behavior of several hyper-parameters that influence the RL algorithm. The algorithm is highly dependent on multiple factors such as the shape of the objective function derived for the core design that behaves as a fudge factor that affects the stability of the learning. But also, an exploration/exploitation trade-off that manifests through different parameters such as the number of loading patterns seen by the agents per episode, the number of samples collected before a policy update nsteps, and an entropy factor ent_coef that increases the randomness of the policy during training. We found that RL must be applied similarly to a Gaussian Process in which the acquisition function is replaced by a parametrized policy. Then, once an initial set of hyper-parameters is found, reducing nsteps and ent_coef until no more learning is observed will result in the highest sample efficiency robustly and stably. This resulted in an economic benefit of 535,000- 642,000 $/year/plant.

A new variant of Newton's method for empirical risk minimization is studied, where at each iteration of the optimization algorithm, the gradient and Hessian of the objective function are replaced by robust estimators taken from existing literature on robust mean estimation for multivariate data. After proving a general theorem about the convergence of successive iterates to a small ball around the population-level minimizer, consequences of the theory in generalized linear models are studied when data are generated from Huber's epsilon-contamination model and/or heavytailed distributions. An algorithm for obtaining robust Newton directions based on the conjugate gradient method is also proposed, which may be more appropriate for high-dimensional settings, and conjectures about the convergence of the resulting algorithm are offered. Compared to robust gradient descent, the proposed algorithm enjoys the faster rates of convergence for successive iterates often achieved by second-order algorithms for convex problems, i.e., quadratic convergence in a neighborhood of the optimum, with a stepsize that may be chosen adaptively via backtracking linesearch.

Variational inference with Gaussian mixture models (GMMs) enables learning of highly tractable yet multi-modal approximations of intractable target distributions with up to a few hundred dimensions. The two currently most effective methods for GMM-based variational inference, VIPS and iBayes-GMM, both employ independent natural gradient updates for the individual components and their weights. We show for the first time, that their derived updates are equivalent, although their practical implementations and theoretical guarantees differ. We identify several design choices that distinguish both approaches, namely with respect to sample selection, natural gradient estimation, stepsize adaptation, and whether trust regions are enforced or the number of components adapted. We argue that for both approaches, the quality of the learned approximations can heavily suffer from the respective design choices: By updating the individual components using samples from the mixture model, iBayes-GMM often fails to produce meaningful updates to low-weight components, and by using a zero-order method for estimating the natural gradient, VIPS scales badly to higher-dimensional problems. Furthermore, we show that information-geometric trust-regions (used by VIPS) are effective even when using first-order natural gradient estimates, and often outperform the improved Bayesian learning rule (iBLR) update used by iBayes-GMM. We systematically evaluate the effects of design choices and show that a hybrid approach significantly outperforms both prior works. Along with this work, we publish our highly modular and efficient implementation for natural gradient variational inference with Gaussian mixture models, which supports 432 different combinations of design choices, facilitates the reproduction of all our experiments, and may prove valuable for the practitioner.

Stochastic optimization has found wide applications in minimizing objective functions in machine learning, which motivates a lot of theoretical studies to understand its practical success. Most of existing studies focus on the convergence of optimization errors, while the generalization analysis of stochastic optimization is much lagging behind. This is especially the case for nonconvex and nonsmooth problems often encountered in practice. In this paper, we initialize a systematic stability and generalization analysis of stochastic optimization on nonconvex and nonsmooth problems. We introduce novel algorithmic stability measures and establish their quantitative connection on the gap between population gradients and empirical gradients, which is then further extended to study the gap between the Moreau envelope of the empirical risk and that of the population risk. To our knowledge, these quantitative connection between stability and generalization in terms of either gradients or Moreau envelopes have not been studied in the literature. We introduce a class of sampling-determined algorithms, for which we develop bounds for three stability measures. Finally, we apply these discussions to derive error bounds for stochastic gradient descent and its adaptive variant, where we show how to achieve an implicit regularization by tuning the step sizes and the number of iterations.

Stochastic Approximation (SA) is a classical algorithm that has had since the early days a huge impact on signal processing, and nowadays on machine learning, due to the necessity to deal with a large amount of data observed with uncertainties. An exemplar special case of SA pertains to the popular stochastic (sub)gradient algorithm which is the working horse behind many important applications. A lesser-known fact is that the SA scheme also extends to non-stochastic-gradient algorithms such as compressed stochastic gradient, stochastic expectation-maximization, and a number of reinforcement learning algorithms. The aim of this article is to overview and introduce the non-stochastic-gradient perspectives of SA to the signal processing and machine learning audiences through presenting a design guideline of SA algorithms backed by theories. Our central theme is to propose a general framework that unifies existing theories of SA, including its non-asymptotic and asymptotic convergence results, and demonstrate their applications on popular non-stochastic-gradient algorithms. We build our analysis framework based on classes of Lyapunov functions that satisfy a variety of mild conditions. We draw connections between non-stochastic-gradient algorithms and scenarios when the Lyapunov function is smooth, convex, or strongly convex. Using the said framework, we illustrate the convergence properties of the non-stochastic-gradient algorithms using concrete examples. Extensions to the emerging variance reduction techniques for improved sample complexity will also be discussed.

Privacy leakage is a big problem in the big-data era. Solving a learning task based on big data intrinsically means that only through a collaborative effort sufficient data is available for training a global model with sufficient clean accuracy (utility). Federated learning is a framework where a learning task is solved by a loose federation of participating devices/clients which are coordinated by a central server [42, 8, 3, 33, 40, 9, 57, 29, 59, 10, 34, 36, 37, 39, 12, 30]. Clients, who use own local data to participate in a learning task by training a global model, want to have privacy guarantees for their local proprietary data. For this reason DP-SGD [1] was introduced as it adapts distributed Stochastic Gradient Descent (SGD)[55] with Differential Privacy (DP)[19, 15, 21, 18].

Ensemble Kalman inversion (EKI) is a sequential Monte Carlo method used to solve inverse problems within a Bayesian framework. Unlike backpropagation, EKI is a gradient-free optimization method that only necessitates the evaluation of artificial neural networks in forward passes. In this study, we examine the effectiveness of EKI in training neural ordinary differential equations (neural ODEs) for system identification and control tasks. To apply EKI to optimal control problems, we formulate inverse problems that incorporate a Tikhonov-type regularization term. Our numerical results demonstrate that EKI is an efficient method for training neural ODEs in system identification and optimal control problems, with runtime and quality of solutions that are competitive with commonly used gradient-based optimizers.

Phase association groups seismic wave arrivals according to their originating earthquakes. It is a fundamental task in a seismic data processing pipeline, but challenging to perform for smaller, high-rate seismic events which carry fundamental information about earthquake dynamics, especially with a commonly assumed inaccurate wave speed model. As a consequence, most association methods focus on larger events that occur at a lower rate and are thus easier to associate, even though microseismicity provides a valuable description of the elastic medium properties in the subsurface. In this paper, we show that association is possible at rates much higher than previously reported even when the wave speed is unknown. We propose Harpa, a high-rate seismic phase association method which leverages deep neural fields to build generative models of wave speeds and associated travel times, and first solves a joint spatio--temporal source localization and wave speed recovery problem, followed by association. We obviate the need for associated phases by interpreting arrival time data as probability measures and using an optimal transport loss to enforce data fidelity. The joint recovery problem is known to admit a unique solution under certain conditions but due to the non-convexity of the corresponding loss a simple gradient scheme converges to poor local minima. We show that this is effectively mitigated by stochastic gradient Langevin dynamics (SGLD). Numerical experiments show that \harpa~efficiently associates high-rate seismicity clouds over complex, unknown wave speeds and graciously handles noisy and missing picks.