Solving Stochastic Compositional Optimization is Nearly as Easy as Solving Stochastic Optimization

TL;DR

This paper introduces a new stochastic compositional gradient method (SCSC) that converges at the same rate as standard SGD, simplifying and accelerating compositional optimization in applications like reinforcement learning.

Contribution

The paper proposes SCSC, a single-loop, fixed batch size method that achieves SGD-like convergence rates for stochastic compositional optimization, with easy integration of acceleration techniques.

Findings

SCSC converges at the same rate as SGD for non-compositional optimization.

Applying Adam to SCSC yields state-of-the-art performance.

SCSC performs well on portfolio management and meta-learning tasks.

Abstract

Stochastic compositional optimization generalizes classic (non-compositional) stochastic optimization to the minimization of compositions of functions. Each composition may introduce an additional expectation. The series of expectations may be nested. Stochastic compositional optimization is gaining popularity in applications such as reinforcement learning and meta learning. This paper presents a new Stochastically Corrected Stochastic Compositional gradient method (SCSC). SCSC runs in a single-time scale with a single loop, uses a fixed batch size, and guarantees to converge at the same rate as the stochastic gradient descent (SGD) method for non-compositional stochastic optimization. This is achieved by making a careful improvement to a popular stochastic compositional gradient method. It is easy to apply SGD-improvement techniques to accelerate SCSC. This helps SCSC achieve state-of-the-art performance for stochastic compositional optimization. In particular, we apply Adam to SCSC, and the exhibited rate of convergence matches that of the original Adam on non-compositional stochastic optimization. We test SCSC using the portfolio management and model-agnostic meta-learning tasks.