Linear Convergence of the Proximal Incremental Aggregated Gradient Method under Quadratic Growth Condition
It relaxes the strong convexity assumption for linear convergence of PIAG, benefiting optimization for large-scale sums with non-smooth terms.
The paper proves that the proximal incremental aggregated gradient method achieves global linear convergence under the quadratic growth condition, which is weaker than strong convexity, with linear rates in both function value and iterate errors.
Under the strongly convex assumption, several recent works studied the global linear convergence rate of the proximal incremental aggregated gradient (PIAG) method for minimizing the sum of a large number of smooth component functions and a non-smooth convex function. In this paper, under \textsl{the quadratic growth condition}--a strictly weaker condition than the strongly convex assumption, we derive a new global linear convergence rate result, which implies that the PIAG method attains global linear convergence rates in both the function value and iterate point errors. The main idea behind is to construct a certain Lyapunov function.