相关论文: Implicit Regularization of Accelerated Methods in …

In this work, we analyze the regularizing property of the stochastic gradient descent for the efficient numerical solution of a class of nonlinear ill-posed inverse problems in Hilbert spaces. At each step of the iteration, the method…

We examine the behavior of accelerated gradient methods in smooth nonconvex unconstrained optimization, focusing in particular on their behavior near strict saddle points. Accelerated methods are iterative methods that typically step along…

While momentum-based optimization algorithms are commonly used in the notoriously non-convex optimization problems of deep learning, their analysis has historically been restricted to the convex and strongly convex setting. In this article,…

Current state-of-the-art analyses on the convergence of gradient descent for training neural networks focus on characterizing properties of the loss landscape, such as the Polyak-Lojaciewicz (PL) condition and the restricted strong…

In this paper we propose a new class of iterative regularization methods for solving ill-posed linear operator equations. The prototype of these iterative regularization methods is in the form of second order evolution equation with a…

Many problems in science and engineering involve, as part of their solution process, the consideration of a separable function which is the sum of two convex functions, one of them possibly non-smooth. Recently a few works have discussed…

Randomized-subspace methods reduce the cost of first-order optimization by using only low-dimensional projected-gradient information, a feature that is attractive in forward-mode automatic differentiation and communication-limited settings.…

We consider stochastic gradient methods under the interpolation regime where a perfect fit can be obtained (minimum loss at each observation). While previous work highlighted the implicit regularization of such algorithms, we consider an…

For infinitesimal learning rates, stochastic gradient descent (SGD) follows the path of gradient flow on the full batch loss function. However moderately large learning rates can achieve higher test accuracies, and this generalization…

We consider the problem of supervised learning with convex loss functions and propose a new form of iterative regularization based on the subgradient method. Unlike other regularization approaches, in iterative regularization no constraint…

How to find flat minima? We propose running normalized gradient descent, usually reserved for nonsmooth optimization, with sufficiently slowly diminishing step sizes. This induces implicit regularization towards flat minima if an…

Recent years have seen a flurry of activities in designing provably efficient nonconvex procedures for solving statistical estimation problems. Due to the highly nonconvex nature of the empirical loss, state-of-the-art procedures often…

Large learning rates, when applied to gradient descent for nonconvex optimization, yield various implicit biases including the edge of stability (Cohen et al., 2021), balancing (Wang et al., 2022), and catapult (Lewkowycz et al., 2020).…

Many important machine learning applications involve regularized nonconvex bi-level optimization. However, the existing gradient-based bi-level optimization algorithms cannot handle nonconvex or nonsmooth regularizers, and they suffer from…

Momentum based stochastic gradient methods such as heavy ball (HB) and Nesterov's accelerated gradient descent (NAG) method are widely used in practice for training deep networks and other supervised learning models, as they often provide…

Optimization algorithms for solving nonconvex inverse problem have attracted significant interests recently. However, existing methods require the nonconvex regularization to be smooth or simple to ensure convergence. In this paper, we…

Stochastic Gradient Descent (SGD) has become the method of choice for solving a broad range of machine learning problems. However, some of its learning properties are still not fully understood. We consider least squares learning in…

Gradient restarting has been shown to improve the numerical performance of accelerated gradient methods. This paper provides a mathematical analysis to understand these advantages. First, we establish global linear convergence guarantees…

Arguably, the two most popular accelerated or momentum-based optimization methods in machine learning are Nesterov's accelerated gradient and Polyaks's heavy ball, both corresponding to different discretizations of a particular second order…

It is well known that the finite step-size ($h$) in Gradient Descent (GD) implicitly regularizes solutions to flatter minima. A natural question to ask is "Does the momentum parameter $\beta$ play a role in implicit regularization in…