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Distributed Gradient Tracking Methods with Guarantees for Computing a Solution to Stochastic MPECs

Mohammadjavad Ebrahimi, Uday V. Shanbhag, Farzad Yousefian

TL;DR

The goal in this work is to provably resolve stochastic MPECs in distributed regimes where the agents only have access to their local objectives and an inexact best-response to the lower-level equilibrium problem by devise a new method called randomized smoothed distributed zeroth-order gradient tracking (rs-DZGT).

Abstract

We consider a class of hierarchical multi-agent optimization problems over networks where agents seek to compute an approximate solution to a single-stage stochastic mathematical program with equilibrium constraints (MPEC). MPECs subsume several important problem classes including Stackelberg games, bilevel programs, and traffic equilibrium problems, to name a few. Our goal in this work is to provably resolve stochastic MPECs in distributed regimes where the agents only have access to their local objectives and an inexact best-response to the lower-level equilibrium problem. To this end, we devise a new method called randomized smoothed distributed zeroth-order gradient tracking (rs-DZGT). This is a novel gradient tracking scheme where agents employ a zeroth-order implicit scheme to approximate their (unavailable) local gradients. Leveraging the properties of a randomized smoothing technique, we establish the convergence of the method and derive complexity guarantees for computing a stationary point of an optimization problem with a smoothed implicit global objective. We also provide preliminary numerical experiments where we compare the performance of rs-DZGT on networks under different settings with that of its centralized counterpart.

Distributed Gradient Tracking Methods with Guarantees for Computing a Solution to Stochastic MPECs

TL;DR

The goal in this work is to provably resolve stochastic MPECs in distributed regimes where the agents only have access to their local objectives and an inexact best-response to the lower-level equilibrium problem by devise a new method called randomized smoothed distributed zeroth-order gradient tracking (rs-DZGT).

Abstract

We consider a class of hierarchical multi-agent optimization problems over networks where agents seek to compute an approximate solution to a single-stage stochastic mathematical program with equilibrium constraints (MPEC). MPECs subsume several important problem classes including Stackelberg games, bilevel programs, and traffic equilibrium problems, to name a few. Our goal in this work is to provably resolve stochastic MPECs in distributed regimes where the agents only have access to their local objectives and an inexact best-response to the lower-level equilibrium problem. To this end, we devise a new method called randomized smoothed distributed zeroth-order gradient tracking (rs-DZGT). This is a novel gradient tracking scheme where agents employ a zeroth-order implicit scheme to approximate their (unavailable) local gradients. Leveraging the properties of a randomized smoothing technique, we establish the convergence of the method and derive complexity guarantees for computing a stationary point of an optimization problem with a smoothed implicit global objective. We also provide preliminary numerical experiments where we compare the performance of rs-DZGT on networks under different settings with that of its centralized counterpart.
Paper Structure (6 sections, 9 theorems, 57 equations, 1 figure, 2 tables, 2 algorithms)

This paper contains 6 sections, 9 theorems, 57 equations, 1 figure, 2 tables, 2 algorithms.

Table of Contents

  1. INTRODUCTION
  2. Preliminaries
  3. Algorithm outline
  4. Convergence theory
  5. Numerical Results
  6. CONCLUSIONS

Key Result

Lemma 1

Consider $h^{\eta}$ as defined above. Then, the following results hold. (i) The smoothed function $h^{\eta}$ is continuously differentiable and $\nabla_x h^{\eta}(x)=\left(\tfrac{n}{\eta}\right)\mathbb{E}_{v\in \eta \mathbb{S}}\left[\left(h(x+v)-h(x)\right)\tfrac{v}{\|v\|}\right].$ (ii) Suppose $h$

Figures (1)

  • Figure 1: Sample average implicit objective function computed by Algorithm \ref{['alg:DZGT']} for various network sizes compared to sample average implicit objective function computed by ZSOL-ncvx. The stepsize in the two rows are different.

Theorems & Definitions (22)

  • Lemma 1: cui2104complexity
  • Remark 1
  • Remark 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 3
  • ...and 12 more