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SUN-DSBO: A Structured Unified Framework for Nonconvex Decentralized Stochastic Bilevel Optimization

Yaoshuai Ma, Xiao Wang, Wei Yao, Jin Zhang

TL;DR

This work introduces SUN-DSBO, a Structured Unified framework for nonconvex decentralized stochastic bilevel optimization, enabling both upper- and lower-level objectives to be nonconvex and supporting decentralized schemes with gradient-tracking to mitigate data heterogeneity. It builds on a Moreau-envelope penalty formulation to convert the bilevel problem into a smooth min–max objective, facilitating single-loop, first-order stochastic updates across agents. The authors provide finite-time convergence guarantees for two instantiations, SUN-DSBO-SE and SUN-DSBO-GT, with SUN-DSBO-GT achieving linear speedup in the number of agents without gradient-heterogeneity assumptions, and validate the approach with extensive experiments on data hyper-cleaning and hyper-representation tasks. The results demonstrate practical effectiveness and scalability in decentralized learning settings, with clear guidance on hyperparameter schedules and topology effects, suggesting broad applicability to multi-agent hyperparameter tuning and meta-learning under nonconvex regimes.

Abstract

Decentralized stochastic bilevel optimization (DSBO) is a powerful tool for various machine learning tasks, including decentralized meta-learning and hyperparameter tuning. Existing DSBO methods primarily address problems with strongly convex lower-level objective functions. However, nonconvex objective functions are increasingly prevalent in modern deep learning. In this work, we introduce SUN-DSBO, a Structured Unified framework for Nonconvex DSBO, in which both the upper- and lower-level objective functions may be nonconvex. Notably, SUN-DSBO offers the flexibility to incorporate decentralized stochastic gradient descent or various techniques for mitigating data heterogeneity, such as gradient tracking (GT). We demonstrate that SUN-DSBO-GT, an adaptation of the GT technique within our framework, achieves a linear speedup with respect to the number of agents. This is accomplished without relying on restrictive assumptions, such as gradient boundedness or any specific assumptions regarding gradient heterogeneity. Numerical experiments validate the effectiveness of our method.

SUN-DSBO: A Structured Unified Framework for Nonconvex Decentralized Stochastic Bilevel Optimization

TL;DR

This work introduces SUN-DSBO, a Structured Unified framework for nonconvex decentralized stochastic bilevel optimization, enabling both upper- and lower-level objectives to be nonconvex and supporting decentralized schemes with gradient-tracking to mitigate data heterogeneity. It builds on a Moreau-envelope penalty formulation to convert the bilevel problem into a smooth min–max objective, facilitating single-loop, first-order stochastic updates across agents. The authors provide finite-time convergence guarantees for two instantiations, SUN-DSBO-SE and SUN-DSBO-GT, with SUN-DSBO-GT achieving linear speedup in the number of agents without gradient-heterogeneity assumptions, and validate the approach with extensive experiments on data hyper-cleaning and hyper-representation tasks. The results demonstrate practical effectiveness and scalability in decentralized learning settings, with clear guidance on hyperparameter schedules and topology effects, suggesting broad applicability to multi-agent hyperparameter tuning and meta-learning under nonconvex regimes.

Abstract

Decentralized stochastic bilevel optimization (DSBO) is a powerful tool for various machine learning tasks, including decentralized meta-learning and hyperparameter tuning. Existing DSBO methods primarily address problems with strongly convex lower-level objective functions. However, nonconvex objective functions are increasingly prevalent in modern deep learning. In this work, we introduce SUN-DSBO, a Structured Unified framework for Nonconvex DSBO, in which both the upper- and lower-level objective functions may be nonconvex. Notably, SUN-DSBO offers the flexibility to incorporate decentralized stochastic gradient descent or various techniques for mitigating data heterogeneity, such as gradient tracking (GT). We demonstrate that SUN-DSBO-GT, an adaptation of the GT technique within our framework, achieves a linear speedup with respect to the number of agents. This is accomplished without relying on restrictive assumptions, such as gradient boundedness or any specific assumptions regarding gradient heterogeneity. Numerical experiments validate the effectiveness of our method.
Paper Structure (69 sections, 27 theorems, 165 equations, 17 figures, 14 tables, 2 algorithms)

This paper contains 69 sections, 27 theorems, 165 equations, 17 figures, 14 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. First attempt: regularization
  3. Main contributions
  4. Related works.
  5. Proposed method
  6. Preliminaries
  7. A unified framework for decentralized stochastic bilevel optimization
  8. Other possible algorithmic designs.
  9. Theoretical analysis
  10. General assumptions
  11. Theoretical Results
  12. Results for SUN-DSBO-SE.
  13. Results for SUN-DSBO-GT.
  14. Experiments
  15. Data hyper-cleaning.
  16. ...and 54 more sections

Key Result

Theorem 3.5

Under Assumptions ass1–ass4, let $\mu_k = \mu_0(k+1)^{-p}$, where $\mu_0 > 0$, $p \in (0,1/4)$, and $\gamma \in (0, \frac{1}{2L_{2}})$. Choose learning rates as follows: $\lambda_\theta^k = c_{\theta} n^{1/2}K^{-1/2}$, $\lambda_x^k = \lambda_y^k = c_\lambda \lambda_\theta^k$, where $c_{\theta}, c_{\ where $\mathcal{C}_{1}$ is a positive constant depending on $\delta_f^2$, $\delta_g^2$, $\sigma_f^{

Figures (17)

  • Figure 1: Comparison of the algorithms on hyper-cleaning with different regularization parameters.
  • Figure 2: Comparison of algorithms for data hyper-cleaning under different values of the heterogeneity parameter $\mathcal{H}$ with a corruption rate of $\texttt{cr} = 0.3$.
  • Figure 3: Comparison of algorithms for data hyper-cleaning across different corruption rates (cr) with a heterogeneity parameter of $\mathcal{H} = 0.1$.
  • Figure 4: Comparison of algorithms for hyper-representation under varying values of the heterogeneity parameter $\mathcal{H}$.
  • Figure 5: Data hyper-cleaning is performed with varying values of $\rho$, which quantifies the connectivity of the communication network. A smaller $\rho$ indicates stronger connectivity.
  • ...and 12 more figures

Theorems & Definitions (40)

  • Theorem 3.5: Convergence rate of SUN-DSBO-SE
  • Corollary 3.6
  • Theorem 3.7
  • Corollary 3.8
  • Remark D.1
  • Proposition D.2
  • Proposition D.3
  • Corollary D.4
  • proof
  • Corollary D.5
  • ...and 30 more