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Achieving Better Local Regret Bound for Online Non-Convex Bilevel Optimization

Tingkai Jia, Haiguang Wang, Cheng Chen

TL;DR

This work tackles online non-convex bilevel optimization with time-varying objectives by developing hypergradient-based online algorithms. It introduces AOBO with adaptive inner iterations to achieve the optimal standard regret Reg(T) = $O(1+V_T)$ and an efficient fully single-loop variant FSOBO with Reg(T) = $O(1+V_T+H_{2,T}+E_{2,T})$, while also addressing windowed non-stationarity through WOBO, which attains Reg_w(T) = $O(T/W^2)$ with a matching lower bound $Ω(T/W^2)$. The paper provides sharp upper and lower bounds, including a lower bound demonstrating optimality, and validates the methods with online data cleaning and loss-tuning experiments. By removing reliance on sublinear environmental variation assumptions for windowed regret and reducing inner-loop costs, the work significantly advances provable efficiency in dynamic OBO settings. The results offer both theoretically tight guarantees and practical algorithms for dynamic bilevel problems in machine learning applications.

Abstract

Online bilevel optimization (OBO) has emerged as a powerful framework for many machine learning problems. Prior works have developed several algorithms that minimize the standard bilevel local regret or the window-averaged bilevel local regret of the OBO problem, but the optimality of existing regret bounds remains unclear. In this work, we establish optimal regret bounds for both settings. For standard bilevel local regret, we propose an algorithm that achieves the optimal regret $Ω(1+V_T)$ with at most $O(T\log T)$ total inner-level gradient evaluations. We further develop a fully single-loop algorithm whose regret bound includes an additional gradient-variation terms. For the window-averaged bilevel local regret, we design an algorithm that captures sublinear environmental variation through a window-based analysis and achieves the optimal regret $Ω(T/W^2)$. Experiments validate our theoretical findings and demonstrate the practical effectiveness of the proposed methods.

Achieving Better Local Regret Bound for Online Non-Convex Bilevel Optimization

TL;DR

This work tackles online non-convex bilevel optimization with time-varying objectives by developing hypergradient-based online algorithms. It introduces AOBO with adaptive inner iterations to achieve the optimal standard regret Reg(T) = and an efficient fully single-loop variant FSOBO with Reg(T) = , while also addressing windowed non-stationarity through WOBO, which attains Reg_w(T) = with a matching lower bound . The paper provides sharp upper and lower bounds, including a lower bound demonstrating optimality, and validates the methods with online data cleaning and loss-tuning experiments. By removing reliance on sublinear environmental variation assumptions for windowed regret and reducing inner-loop costs, the work significantly advances provable efficiency in dynamic OBO settings. The results offer both theoretically tight guarantees and practical algorithms for dynamic bilevel problems in machine learning applications.

Abstract

Online bilevel optimization (OBO) has emerged as a powerful framework for many machine learning problems. Prior works have developed several algorithms that minimize the standard bilevel local regret or the window-averaged bilevel local regret of the OBO problem, but the optimality of existing regret bounds remains unclear. In this work, we establish optimal regret bounds for both settings. For standard bilevel local regret, we propose an algorithm that achieves the optimal regret with at most total inner-level gradient evaluations. We further develop a fully single-loop algorithm whose regret bound includes an additional gradient-variation terms. For the window-averaged bilevel local regret, we design an algorithm that captures sublinear environmental variation through a window-based analysis and achieves the optimal regret . Experiments validate our theoretical findings and demonstrate the practical effectiveness of the proposed methods.
Paper Structure (34 sections, 33 theorems, 231 equations, 3 figures, 2 tables, 5 algorithms)

This paper contains 34 sections, 33 theorems, 231 equations, 3 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Related Work
  4. Preliminaries
  5. Notations and Assumptions
  6. Online Bilevel Optimization
  7. Hypergradient-Based Algorithm
  8. Online Non-Convex Bilevel Optimization under Standard Bilevel Local Regret
  9. Adaptive Inner-loop Iteration Method Achieves the Optimal Regret Bound
  10. Adaptive inner-loop Algorithm design
  11. Bilevel Local Regret Analysis
  12. Fully Single-Loop Method with Regret Analysis
  13. Online Non-Convex Bilevel Optimization with Window-Averaged Bilevel Local Regret
  14. Regret Definition and Preliminaries
  15. Window-averaged Online Bilevel Optimizer
  16. ...and 19 more sections

Key Result

Lemma 3.1

Under Assumption asm1, asm2, let $\alpha\leq\frac{1}{L_{g,1}}$ and $\beta\leq\frac{1}{L_{g,1}}$, we can obtain where $c_\mathbf{y}$ and $c_\mathbf{v}$ are some constants and

Figures (3)

  • Figure 1: Performance of different algorithms on the online hyper-cleaning.
  • Figure 2: Performance of WOBO with different window sizes $w$ with $\eta=0.9$ and weight parameters $\eta$ with $w=5$ in subfigures (a) and (b), respectively. The running time of different algorithms in Figure \ref{['fig:figure1']} is shown in subfigure (c).
  • Figure 3: Performance of different algorithms on the parametric loss tuning for imbalanced data.

Theorems & Definitions (57)

  • Definition 2.4: Online Hypergradient-Based Algorithm Class
  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 3.4
  • Lemma 3.5
  • Theorem 3.6
  • Definition 4.1: Window-Averaged Bilevel Local Regret
  • Theorem 4.3
  • Theorem 4.4
  • ...and 47 more