Nonlinear gradient mappings and stochastic optimization: A general framework with applications to heavytail noise
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 componentwise 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 heavytail noise, several easytoimplement nonlinearities from our framework are competitive with state of the art alternatives on real data sets with heavy tail noises.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.02593
 arXiv:
 arXiv:2204.02593
 Bibcode:
 2022arXiv220402593J
 Keywords:

 Mathematics  Optimization and Control;
 Computer Science  Information Theory;
 Computer Science  Machine Learning
 EPrint:
 Submitted for publication Nov 2021