Stochastic Composition Optimization of Functions Without Lipschitz Continuous Gradient

Abstract

In this paper, we study stochastic optimization of two-level composition of functions without Lipschitz continuous gradient. The smoothness property is generalized by the notion of relative smoothness which provokes the Bregman gradient method. We propose three stochastic composition Bregman gradient algorithms for the three possible relatively smooth compositional scenarios and provide their sample complexities to achieve an $$\epsilon $$ -approximate stationary point. For the smooth of relatively smooth composition, the first algorithm requires $$\mathcal {O}(\epsilon ^{-2})$$ calls to the stochastic oracles of the inner function value and gradient as well as the outer function gradient. When both functions are relatively smooth, the second algorithm requires $$\mathcal {O}(\epsilon ^{-3})$$ calls to the inner function value stochastic oracle and $$\mathcal {O}(\epsilon ^{-2})$$ calls to the inner and outer functions gradients stochastic oracles. We further improve the second algorithm by variance reduction for the setting where just the inner function is smooth. The resulting algorithm requires $$\mathcal {O}(\epsilon ^{-5/2})$$ calls to the inner function value stochastic oracle, $$\mathcal {O}(\epsilon ^{-3/2})$$ calls to the inner function gradient, and $$\mathcal {O}(\epsilon ^{-2})$$ calls to the outer function gradient stochastic oracles. Finally, we numerically evaluate the performance of these three algorithms over two different examples.