Nonlinear gradient mappings and stochastic optimization: A general framework with applications to heavy-tail noise

TL;DR

This paper presents a comprehensive framework for nonlinear stochastic gradient descent tailored for heavy-tail noise scenarios, providing convergence guarantees and demonstrating practical effectiveness of various nonlinearities.

Contribution

It introduces a unified nonlinear SGD framework with strong convergence guarantees under heavy-tail noise, including novel nonlinearity options and explicit convergence rates.

Findings

Nonlinear SGD converges to zero MSE at rate O(1/t^ζ) under heavy-tail noise.

Linear SGD can have unbounded variance in the same noise setting.

Experimental results show nonlinearities are competitive with state-of-the-art methods.

Abstract

We introduce a general framework for nonlinear stochastic gradient descent (SGD) for the scenarios when gradient noise exhibits heavy tails. The proposed framework subsumes several popular nonlinearity choices, like clipped, normalized, signed or quantized gradient, but we also consider novel nonlinearity choices. We establish for the considered class of methods strong convergence guarantees assuming a strongly convex cost function with Lipschitz continuous gradients under very general assumptions on the gradient noise. Most notably, we show that, for a nonlinearity with bounded outputs and for the gradient noise that may not have finite moments of order greater than one, the nonlinear SGD's mean squared error (MSE), or equivalently, the expected cost function's optimality gap, converges to zero at rate~$O(1/t^\zeta)$, $\zeta \in (0,1)$. In contrast, for the same noise setting, the linear SGD generates a sequence with unbounded variances. Furthermore, for the nonlinearities that can be decoupled component wise, like, e.g., sign gradient or component-wise clipping, we show that the nonlinear SGD asymptotically (locally) achieves a $O(1/t)$ rate in the weak convergence sense and explicitly quantify the corresponding asymptotic variance. Experiments show that, while our framework is more general than existing studies of SGD under heavy-tail noise, several easy-to-implement nonlinearities from our framework are competitive with state of the art alternatives on real data sets with heavy tail noises.