Nonasymptotic Analysis of Accelerated Methods With Inexact Oracle Under Absolute Error Bound

Abstract

Performance analysis of first-order algorithms with inexact oracles has gained recent attention due to various emerging applications in which obtaining exact gradients is impossible or computationally expensive. Previous research has demonstrated that the performance of accelerated first-order methods is more sensitive to gradient errors compared with non-accelerated ones. This paper investigates the nonasymptotic convergence bound of two accelerated methods with inexact gradients to solve deterministic smooth convex problems. Performance Estimation Problem (PEP) is used as the primary tool to analyze the convergence bounds of the underlying algorithms. By finding an analytical solution to PEP, we derive novel convergence bounds of the Generalized Optimized Gradient Method (GOGM) and Generalized Fast Gradient Method (GFGM) with inexact gradient oracles following the absolute error bound. Next, we analyze the tradeoff between the vanishing term and the accumulated error in the convergence bound that guides finding the optimal stepsize. Furthermore, we determine the optimal strategy to set the gradient inexactness along iterations, ensuring that the accumulated error remains subordinate to the vanishing term. Finally, we establish a lower bound for accumulated error for a subclass of first-order methods satisfying two properties, which then motivated us to develop PEP solutions satisfying this lower bound for GOGM and GFGM methods.