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Exploring the Generalization Capabilities of AID-based Bi-level Optimization

Congliang Chen, Li Shen, Zhiqiang Xu, Wei Liu, Zhi-Quan Luo, Peilin Zhao

TL;DR

The uniform stability of AID-based methods is ascertain, which achieves similar results to a single-level nonconvex problem, and the generalization ability of AID-based bi-level optimization methods is given.

Abstract

Bi-level optimization has achieved considerable success in contemporary machine learning applications, especially for given proper hyperparameters. However, due to the two-level optimization structure, commonly, researchers focus on two types of bi-level optimization methods: approximate implicit differentiation (AID)-based and iterative differentiation (ITD)-based approaches. ITD-based methods can be readily transformed into single-level optimization problems, facilitating the study of their generalization capabilities. In contrast, AID-based methods cannot be easily transformed similarly but must stay in the two-level structure, leaving their generalization properties enigmatic. In this paper, although the outer-level function is nonconvex, we ascertain the uniform stability of AID-based methods, which achieves similar results to a single-level nonconvex problem. We conduct a convergence analysis for a carefully chosen step size to maintain stability. Combining the convergence and stability results, we give the generalization ability of AID-based bi-level optimization methods. Furthermore, we carry out an ablation study of the parameters and assess the performance of these methods on real-world tasks. Our experimental results corroborate the theoretical findings, demonstrating the effectiveness and potential applications of these methods.

Exploring the Generalization Capabilities of AID-based Bi-level Optimization

TL;DR

The uniform stability of AID-based methods is ascertain, which achieves similar results to a single-level nonconvex problem, and the generalization ability of AID-based bi-level optimization methods is given.

Abstract

Bi-level optimization has achieved considerable success in contemporary machine learning applications, especially for given proper hyperparameters. However, due to the two-level optimization structure, commonly, researchers focus on two types of bi-level optimization methods: approximate implicit differentiation (AID)-based and iterative differentiation (ITD)-based approaches. ITD-based methods can be readily transformed into single-level optimization problems, facilitating the study of their generalization capabilities. In contrast, AID-based methods cannot be easily transformed similarly but must stay in the two-level structure, leaving their generalization properties enigmatic. In this paper, although the outer-level function is nonconvex, we ascertain the uniform stability of AID-based methods, which achieves similar results to a single-level nonconvex problem. We conduct a convergence analysis for a carefully chosen step size to maintain stability. Combining the convergence and stability results, we give the generalization ability of AID-based bi-level optimization methods. Furthermore, we carry out an ablation study of the parameters and assess the performance of these methods on real-world tasks. Our experimental results corroborate the theoretical findings, demonstrating the effectiveness and potential applications of these methods.

Paper Structure

This paper contains 23 sections, 28 theorems, 91 equations, 3 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminary
  4. Bi-Level Optimization Algorithms
  5. Generalization Decomposition
  6. Theoretical Analysis
  7. Basic Assumptions and Definitions
  8. Stability of Algorithm \ref{['algo-aid']}
  9. Convergence Analysis
  10. Trade-off in Generalization Ability
  11. Proof Sketch
  12. Proof Sketch for Theorem \ref{['thm_sta']}
  13. Proof Sketch for Theorem \ref{['converge']}
  14. Experiments
  15. Toy Example
  16. ...and 8 more sections

Key Result

Proposition 3.1

The gradient of the function $\Phi(x)$ can be given as where $\nabla_{xy}^2 G(x,y^*(x)) = \frac{1}{q} \sum_{j=1}^q \nabla_{xy}^2 g(x,y^*(x),\zeta_j)$, $\nabla_{yy}^2 G(x,y^*(x)) = \frac{1}{q} \sum_{j=1}^q \nabla_{yy}^2 g(x,y^*(x),\zeta_j)$, and $\nabla_y F(x,y^*(x)) = \frac{1}{n}\sum_{i=1}^n \nabla_y f(x,y^*(x),\xi_i)$.

Figures (3)

  • Figure 1: Results for Toy Example. The left figure shows the results when learning rates are constant, the middle figure shows the results when we use diminishing learning rates, and the right figure compares the results for constant and diminishing learning rates.
  • Figure 2: Results for Data selection on MNIST. The first figure shows the result with constant learning rates. The second figure shows the results with diminishing learning rates. The third figure and fourth figure compare the results between constant learning rates and diminishing learning rates with 100 samples in the validation set and 200 samples in the validation set, respectively.
  • Figure 3: Results for Dataset Mixture on Fashion MNIST. The figure shows the text accuracy of using constant learning rates and diminishing learning rates with 100 samples in the validation set.

Theorems & Definitions (59)

  • Proposition 3.1: Lemma 2.1 in ghadimi2018approximation
  • Proposition 3.2: Theorem 2.2 in hardt2016train
  • Remark 3.3
  • Definition 4.3
  • Definition 4.4: Uniform stability in bao2021stability
  • Proposition 4.5
  • Remark 4.6
  • Theorem 4.7
  • Corollary 4.8
  • Remark 4.9
  • ...and 49 more