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We consider the stochastic gradient descent (SGD) algorithm driven by a general stochastic sequence, including i.i.d noise and random walk on an arbitrary graph, among others; and analyze it in the asymptotic sense. Specifically, we employ…
Deep neural networks (DNN) are typically optimized using stochastic gradient descent (SGD). However, the estimation of the gradient using stochastic samples tends to be noisy and unreliable, resulting in large gradient variance and bad…
In this contribution, we present a full overview of the continuous stochastic gradient (CSG) method, including convergence results, step size rules and algorithmic insights. We consider optimization problems in which the objective function…
Stochastic gradient descent (SGD) is the main algorithm behind a large body of work in machine learning. In many cases, constraints are enforced via projections, leading to projected stochastic gradient algorithms. In recent years, a large…
(Mini-batch) Stochastic Gradient Descent is a popular optimization method which has been applied to many machine learning applications. But a rather high variance introduced by the stochastic gradient in each step may slow down the…
Classical stochastic gradient methods for optimization rely on noisy gradient approximations that become progressively less accurate as iterates approach a solution. The large noise and small signal in the resulting gradients makes it…
We present a multilevel stochastic gradient descent method for the optimal control of systems governed by partial differential equations under uncertain input data. The gradient descent method used to find the optimal control leverages a…
Stream stochastic gradient descent (SGD) is a simple and efficient method for solving online optimization problems in operations research (OR), where data is generated by parameter-dependent Markov chains. Unlike traditional approaches…
In this work, we propose an optimization algorithm which we call norm-adapted gradient descent. This algorithm is similar to other gradient-based optimization algorithms like Adam or Adagrad in that it adapts the learning rate of stochastic…
Stochastic Gradient Descent (SGD) is one of the simplest and most popular stochastic optimization methods. While it has already been theoretically studied for decades, the classical analysis usually required non-trivial smoothness…
We develop a new algorithm for non-convex stochastic optimization that finds an $\epsilon$-critical point in the optimal $O(\epsilon^{-3})$ stochastic gradient and Hessian-vector product computations. Our algorithm uses Hessian-vector…
In this dissertation we propose alternative analysis of distributed stochastic gradient descent (SGD) algorithms that rely on spectral properties of the data covariance. As a consequence we can relate questions pertaining to speedups and…
This paper introduces a novel approach to enhance the performance of the stochastic gradient descent (SGD) algorithm by incorporating a modified decay step size based on $\frac{1}{\sqrt{t}}$. The proposed step size integrates a logarithmic…
Many problems require to optimize empirical risk functions over large data sets. Gradient descent methods that calculate the full gradient in every descent step do not scale to such datasets. Various flavours of Stochastic Gradient Descent…
One of the major issues in stochastic gradient descent (SGD) methods is how to choose an appropriate step size while running the algorithm. Since the traditional line search technique does not apply for stochastic optimization algorithms,…
We study the generalization performance of gradient methods in the fundamental stochastic convex optimization setting, focusing on its dimension dependence. First, for full-batch gradient descent (GD) we give a construction of a learning…
The convergence of stochastic gradient descent is highly dependent on the step-size, especially on non-convex problems such as neural network training. Step decay step-size schedules (constant and then cut) are widely used in practice…
A variety of widely used optimization methods like SignSGD and Muon can be interpreted as instances of steepest descent under different norm-induced geometries. In this work, we study the implicit bias of mini-batch stochastic steepest…
Recent studies have shown that many nonconvex machine learning problems satisfy a generalized-smooth condition that extends beyond traditional smooth nonconvex optimization. However, the existing algorithms are not fully adapted to such…
Non-convex optimization problems are ubiquitous in machine learning, especially in Deep Learning. While such complex problems can often be successfully optimized in practice by using stochastic gradient descent (SGD), theoretical analysis…