High-Probability Bounds for Stochastic Optimization and Variational Inequalities: the Case of Unbounded Variance
This work addresses a gap in stochastic optimization theory by providing more general convergence guarantees, which is incremental but important for researchers and practitioners dealing with problems outside standard assumptions.
The paper tackles the problem of deriving high-probability convergence bounds for stochastic optimization and variational inequalities under less restrictive assumptions, such as unbounded variance, and achieves results for various problem classes including non-convex and strongly convex setups.
During recent years the interest of optimization and machine learning communities in high-probability convergence of stochastic optimization methods has been growing. One of the main reasons for this is that high-probability complexity bounds are more accurate and less studied than in-expectation ones. However, SOTA high-probability non-asymptotic convergence results are derived under strong assumptions such as the boundedness of the gradient noise variance or of the objective's gradient itself. In this paper, we propose several algorithms with high-probability convergence results under less restrictive assumptions. In particular, we derive new high-probability convergence results under the assumption that the gradient/operator noise has bounded central $α$-th moment for $α\in (1,2]$ in the following setups: (i) smooth non-convex / Polyak-Lojasiewicz / convex / strongly convex / quasi-strongly convex minimization problems, (ii) Lipschitz / star-cocoercive and monotone / quasi-strongly monotone variational inequalities. These results justify the usage of the considered methods for solving problems that do not fit standard functional classes studied in stochastic optimization.