Decentralized Stochastic Bilevel Optimization with Improved per-Iteration Complexity

TL;DR

This paper introduces a decentralized stochastic bilevel optimization algorithm that achieves optimal sample complexity without estimating full Hessian or Jacobian matrices, improving per-iteration efficiency in machine learning tasks.

Contribution

It presents a novel DSBO algorithm that matches best known sample complexity while reducing per-iteration computational costs by avoiding full Hessian and Jacobian matrix estimations.

Findings

Matches best known sample complexity for DSBO

Reduces per-iteration complexity by avoiding full Hessian/Jacobian estimation

Requires only first-order stochastic, Hessian-vector, and Jacobian-vector oracles

Abstract

Bilevel optimization recently has received tremendous attention due to its great success in solving important machine learning problems like meta learning, reinforcement learning, and hyperparameter optimization. Extending single-agent training on bilevel problems to the decentralized setting is a natural generalization, and there has been a flurry of work studying decentralized bilevel optimization algorithms. However, it remains unknown how to design the distributed algorithm with sample complexity and convergence rate comparable to SGD for stochastic optimization, and at the same time without directly computing the exact Hessian or Jacobian matrices. In this paper we propose such an algorithm. More specifically, we propose a novel decentralized stochastic bilevel optimization (DSBO) algorithm that only requires first order stochastic oracle, Hessian-vector product and Jacobian-vector product oracle. The sample complexity of our algorithm matches the currently best known results for DSBO, and the advantage of our algorithm is that it does not require estimating the full Hessian and Jacobian matrices, thereby having improved per-iteration complexity.