Sign Operator for Coping with Heavy-Tailed Noise in Non-Convex Optimization: High Probability Bounds Under $(L_0,L_1)$-Smoothness

Abstract

In recent years, non-convex optimization problems are more often described by generalized $(L_0,L_1)$-smoothness assumption rather than standard one. Meanwhile, severely corrupted data used in these problems has increased the demand for methods capable of handling heavy-tailed noises, i.e., noises with bounded $\hat{I}º$-th moment. Motivated by these real-world trends and challenges, we explore sign-based methods in this setup and demonstrate their effectiveness in comparison with other popular solutions like clipping or normalization. In theory, we prove the first-known high probability convergence bounds under $(L_0,L_1)$-smoothness and heavy-tailed noises with mild parameter dependencies. In the case of standard smoothness, these bounds are novel for sign-based methods as well. In particular, SignSGD with batching achieves sample complexity $\tilde{O}\left(\left(\frac{\hat{I}”L_{0}d}{{\epsilon }^2}+\frac{\hat{I}”L_1{d}^{\frac{3}{2}}}{\epsilon }\right)\left[1+{\left(\frac{\ddot{I}}{ƒ}\epsilon \right)}^{\frac{\hat{I}}{º}}\hat{I}º-1\right]\right),\hat{I}º\in (1,2]$. Under the assumption of symmetric noises, SignSGD with Majority Voting can robustly work on the whole range of $\hat{I}º\in (0,2]$ with complexity $\tilde{O}\left(\left(\frac{\hat{I}”L_{0}d}{{\epsilon }^2}+\frac{\hat{I}”L_1{d}^{\frac{3}{2}}}{\epsilon }\right)\left[\frac{1}{\hat{I}{º}^2}+\frac{\ddot{I}{ƒ}^2}{{\epsilon }^2}\right]\right)$. We also obtain results for parameter-agnostic setups, Polyak-Lojasiewicz functions and momentum-based methods (in expectation). Our theoretical findings are supported by the superior performance of sign-based methods in training Large Language Models compared to clipping and normalization.