Stochastic Multi-level Composition Optimization Algorithms with Level-Independent Convergence Rates

TL;DR

This paper introduces new stochastic algorithms for multi-level composition optimization problems, achieving level-independent convergence rates and improved sample complexity, with practical advantages like being parameter-free and easy to implement.

Contribution

The paper proposes two novel algorithms for stochastic multi-level optimization, with the second algorithm achieving improved sample complexity and practical benefits over previous methods.

Findings

First algorithm achieves $ ilde{O}(1/ ext{epsilon}^6)$ sample complexity.

Modified algorithm reduces complexity to $ ilde{O}(1/ ext{epsilon}^4)$ and is parameter-free.

First to match single-level convergence rates in multi-level stochastic optimization.

Abstract

In this paper, we study smooth stochastic multi-level composition optimization problems, where the objective function is a nested composition of $T$ functions. We assume access to noisy evaluations of the functions and their gradients, through a stochastic first-order oracle. For solving this class of problems, we propose two algorithms using moving-average stochastic estimates, and analyze their convergence to an $\epsilon$-stationary point of the problem. We show that the first algorithm, which is a generalization of \cite{GhaRuswan20} to the $T$ level case, can achieve a sample complexity of $\mathcal{O}(1/\epsilon^6)$ by using mini-batches of samples in each iteration. By modifying this algorithm using linearized stochastic estimates of the function values, we improve the sample complexity to $\mathcal{O}(1/\epsilon^4)$. {\color{black}This modification not only removes the requirement of having a mini-batch of samples in each iteration, but also makes the algorithm parameter-free and easy to implement}. To the best of our knowledge, this is the first time that such an online algorithm designed for the (un)constrained multi-level setting, obtains the same sample complexity of the smooth single-level setting, under standard assumptions (unbiasedness and boundedness of the second moments) on the stochastic first-order oracle.