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Hessian-Free Distributed Bilevel Optimization via Penalization with Time-Scale Separation

Youcheng Niu, Jinming Xu, Ying Sun, Li Chai, Jiming Chen

TL;DR

A loopless distributed algorithm is proposed, AHEAD, that employs multiple-timescale updates to solve the DBO problem asymptotically without requiring Hessian computation and reveals a clear dependence of convergence performance on node heterogeneity, penalty parameters, and network connectivity.

Abstract

This paper considers a class of distributed bilevel optimization (DBO) problems with a coupled inner-level subproblem. Existing approaches typically rely on hypergradient estimations involving computationally expensive Hessian evaluation. To address this, we approximate the DBO problem as a minimax problem by properly designing a penalty term that enforces both the constraint imposed by the inner-level subproblem and the consensus among the decision variables of agents. Moreover, we propose a loopless distributed algorithm, AHEAD, that employs multiple-timescale updates to solve the approximate problem asymptotically without requiring Hessian computation. Theoretically, we establish sharp convergence rates for nonconvex-strongly-convex settings and for distributed minimax problems as special cases. Our analysis reveals a clear dependence of convergence performance on node heterogeneity, penalty parameters, and network connectivity, with a weaker assumption on heterogeneity that only requires bounded gradients at the optimum. Numerical experiments corroborate our theoretical results.

Hessian-Free Distributed Bilevel Optimization via Penalization with Time-Scale Separation

TL;DR

A loopless distributed algorithm is proposed, AHEAD, that employs multiple-timescale updates to solve the DBO problem asymptotically without requiring Hessian computation and reveals a clear dependence of convergence performance on node heterogeneity, penalty parameters, and network connectivity.

Abstract

This paper considers a class of distributed bilevel optimization (DBO) problems with a coupled inner-level subproblem. Existing approaches typically rely on hypergradient estimations involving computationally expensive Hessian evaluation. To address this, we approximate the DBO problem as a minimax problem by properly designing a penalty term that enforces both the constraint imposed by the inner-level subproblem and the consensus among the decision variables of agents. Moreover, we propose a loopless distributed algorithm, AHEAD, that employs multiple-timescale updates to solve the approximate problem asymptotically without requiring Hessian computation. Theoretically, we establish sharp convergence rates for nonconvex-strongly-convex settings and for distributed minimax problems as special cases. Our analysis reveals a clear dependence of convergence performance on node heterogeneity, penalty parameters, and network connectivity, with a weaker assumption on heterogeneity that only requires bounded gradients at the optimum. Numerical experiments corroborate our theoretical results.

Paper Structure

This paper contains 21 sections, 11 theorems, 57 equations, 4 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Penalty-based Single-Level Approximation
  4. Algorithm and Main Results
  5. AHEAD Algorithm
  6. Convergence Results
  7. Applications to Distributed Minimax Problems
  8. Convergence Analysis
  9. Technical Preliminaries
  10. Supporting Lemmas
  11. Proofs of The Main Results
  12. Numerical Experiments
  13. Synthetic Example
  14. Hyperparameter Optimization
  15. Conclusions
  16. ...and 6 more sections

Key Result

Theorem 1

Consider the sequence $\{x_i^k, y_i^k, z_i^k\}$ generated by Algorithm alg:1. Suppose Assumptions ASS-outer-level-ASS-heterogeneity hold. If the penalty parameter $\lambda$ satisfies $\lambda>\frac{2L_{f,1}}{\mu_g}$ and the step sizes $\alpha$, $\beta$, $\gamma$ respectively satisfy then, for any total number of iterations $K$, we have where $\mathcal{C}(\cdot)$ and $\mathcal{B}(\cdot)$ represen

Figures (4)

  • Figure 1: The performance of the proposed AHEAD algorithm: (a) The loss $f,g$ and constraint tolerance with respect to iterations; (b) The optimal gap and consensus errors with respect to the number of iterations; (c) and (d): The trajectories of the individual state $(x_i^k, y_i^k)$ and the average state $(\bar{x}^k, \bar{y}^k)$ within the contours of the outer- and inner-level objectives, respectively.
  • Figure 2: The performance of the proposed AHEAD algorithm under networks with different connectivity levels, where high, medium, and low connectivity correspond to spectral gaps of $\rho = 0.274$, $\rho = 0.644$, and $\rho = 0.923$, respectively.
  • Figure 3: The performance of the proposed AHEAD algorithm under different node heterogeneity: (a) Data distribution; (b) Testing accuracy.
  • Figure 4: Performance comparison of MA-DSBO, SLDBO, and the proposed AHEAD algorithm: (a) Testing accuracy; (b) Training loss.

Theorems & Definitions (30)

  • Remark 1: Hessian-free property
  • Definition 1: $\epsilon$-stationary point
  • Definition 2: Node heterogeneity
  • Remark 2: Weaker assumption on heterogeneity
  • Remark 3: Loopless structure
  • Remark 4: Time-scale separation
  • Theorem 1
  • proof
  • Remark 5: Effect of the penalty parameter
  • Corollary 1
  • ...and 20 more