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A Parameter-Free Zeroth-Order Algorithm for Decentralized Stochastic Convex Optimization

Jiawei Chen, Alexander Rogozin

Abstract

We consider decentralized stochastic convex optimization on connected network, in which gradients of agents are unavailable and each agent can query only noisy function values of its own local objective. The goal is to minimize the average objective over a compact convex domain using only local two point zeroth-order oracles and peer-to-peer communication. We propose a decentralized POEM method (D-POEM) that combines symmetric two point smoothing with adaptive radius and stepsize rules, thereby avoiding prior knowledge of the Lipschitz constant and diameter. For convex Lipschitz continuous objectives, we prove an convergence rate that separates a centralized optimization term from a network disagreement term. We further conduct the numerical experiments to demonstrate POEM outperforms existing distributed zeroth-order method.

A Parameter-Free Zeroth-Order Algorithm for Decentralized Stochastic Convex Optimization

Abstract

We consider decentralized stochastic convex optimization on connected network, in which gradients of agents are unavailable and each agent can query only noisy function values of its own local objective. The goal is to minimize the average objective over a compact convex domain using only local two point zeroth-order oracles and peer-to-peer communication. We propose a decentralized POEM method (D-POEM) that combines symmetric two point smoothing with adaptive radius and stepsize rules, thereby avoiding prior knowledge of the Lipschitz constant and diameter. For convex Lipschitz continuous objectives, we prove an convergence rate that separates a centralized optimization term from a network disagreement term. We further conduct the numerical experiments to demonstrate POEM outperforms existing distributed zeroth-order method.
Paper Structure (14 sections, 11 theorems, 63 equations, 1 figure, 1 algorithm)

This paper contains 14 sections, 11 theorems, 63 equations, 1 figure, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Method
  4. Main Result
  5. Numerical Experiments
  6. Conclusion and Discussion
  7. Discussion.
  9. Proofs
  10. Auxilary Lemma
  11. Decomposition and primary bounds
  12. Consensus error
  13. Proof of the main theorems
  14. Expected-rate proof

Key Result

lemma 1

For every $t\ge 1$,

Figures (1)

  • Figure 1: Twenty-agent benchmark on mushrooms, a9a, and w8a. The top row plots objective at the network average versus total network-wide zeroth-order oracle calls; the bottom row plots the same objective versus communication rounds.

Theorems & Definitions (21)

  • lemma 1: Weighted regret bound
  • lemma 2: Estimator noise
  • lemma 3: Smoothing bias
  • lemma 4: Time-averaged consensus error
  • lemma 5: Consensus error
  • theorem 1: Function gap
  • theorem 2: Conditional convergence rate
  • remark 1
  • corollary 1: Stochastic zeroth-order and communication complexity
  • lemma 6: shamir2017optimal, Lemma 10
  • ...and 11 more