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Asynchronous parallel implementations of stochastic gradient (SG) have been broadly used in solving deep neural network and received many successes in practice recently. However, existing theories cannot explain their convergence and…
We propose efficient methods for solving stochastic simple bilevel optimization problems with convex inner levels, where the goal is to minimize an outer stochastic objective function subject to the solution set of an inner stochastic…
In overparameterized logistic regression, gradient descent (GD) iterates diverge in norm while converging in direction to the maximum $\ell_2$-margin solution -- a phenomenon known as the implicit bias of GD. This work investigates…
We introduce \textit{basic inequalities} for first-order iterative optimization algorithms, forming a simple and versatile framework that connects implicit and explicit regularization. While related inequalities appear in the literature, we…
In this paper we consider the question of whether it is possible to apply a gradient averaging strategy to improve on the sublinear convergence rates without any increase in storage. Our analysis reveals that a positive answer requires an…
We introduce a proximal version of the stochastic dual coordinate ascent method and show how to accelerate the method using an inner-outer iteration procedure. We analyze the runtime of the framework and obtain rates that improve…
We study the implicit regularization of mini-batch stochastic gradient descent, when applied to the fundamental problem of least squares regression. We leverage a continuous-time stochastic differential equation having the same moments as…
In this paper, we utilize stochastic optimization to reduce the space complexity of convex composite optimization with a nuclear norm regularizer, where the variable is a matrix of size $m \times n$. By constructing a low-rank estimate of…
We investigate the Randomized Stochastic Accelerated Gradient (RSAG) method, utilizing either constant or adaptive step sizes, for stochastic optimization problems with generalized smooth objective functions. Under relaxed affine variance…
The simplicity of gradient descent (GD) made it the default method for training ever-deeper and complex neural networks. Both loss functions and architectures are often explicitly tuned to be amenable to this basic local optimization. In…
Mini-batch optimization has proven to be a powerful paradigm for large-scale learning. However, the state of the art parallel mini-batch algorithms assume synchronous operation or cyclic update orders. When worker nodes are heterogeneous…
Adversarial training is a common approach to improving the robustness of deep neural networks against adversarial examples. In this work, we propose a novel regularization approach as an alternative. To derive the regularizer, we formulate…
The goal of a learner in standard online learning is to maintain an average loss close to the loss of the best-performing single function in some class. In many real-world problems, such as rating or ranking items, there is no single best…
Stochastic variance reduced gradient (SVRG) is a popular variance reduction technique for accelerating stochastic gradient descent (SGD). We provide a first analysis of the method for solving a class of linear inverse problems in the lens…
Algorithmic stability is a classical approach to understanding and analysis of the generalization error of learning algorithms. A notable weakness of most stability-based generalization bounds is that they hold only in expectation.…
Stochastic gradient descent (SGD) for strongly convex functions converges at the rate $\bO(1/k)$. However, achieving good results in practice requires tuning the parameters (for example the learning rate) of the algorithm. In this paper we…
This paper generalizes the optimized gradient method (OGM) that achieves the optimal worst-case cost function bound of first-order methods for smooth convex minimization. Specifically, this paper studies a generalized formulation of OGM and…
The stochastic gradient descent (SGD) optimization algorithm plays a central role in a series of machine learning applications. The scientific literature provides a vast amount of upper error bounds for the SGD method. Much less attention…
Modern machine learning paradigms, such as deep learning, occur in or close to the interpolation regime, wherein the number of model parameters is much larger than the number of data samples. In this work, we propose a regularity condition…
Functional linear regression is one of the fundamental and well-studied methods in functional data analysis. In this work, we investigate the functional linear regression model within the context of reproducing kernel Hilbert space by…