Nonlinear Gradient Mappings and Stochastic Optimization: A General Framework with Applications to Heavy-Tail Noise
Dušan Jakovetić, Dragana Bajović, Anit Kumar Sahu, Soummya Kar, Nemanja Milošević, Dušan Stamenković · SIAM Journal on Optimization · 2023
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 [Formula: see text], [Formula: see text]. In contrast, for the same noise setting, the linear SGD generates a sequence with unbounded variances. Furthermore, for general nonlinearities that can be decoupled componentwise and a class of joint nonlinearities, we show that the nonlinear SGD asymptotically (locally) achieves an [Formula: see text] 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 datasets with heavy-tail noises.