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Efficient Curvature-Aware Hypergradient Approximation for Bilevel Optimization

Youran Dong, Junfeng Yang, Wei Yao, Jin Zhang

TL;DR

The work addresses efficient hypergradient estimation in bilevel optimization by exploiting curvature information through an inexact Newton-based framework (NBO). It introduces two instantiations, NBO-GD for deterministic settings and NSBO-SGD for stochastic settings, and provides convergence guarantees with improved deterministic complexity over prior gradient-based methods. A unified hypergradient approximation ties the lower-level solution with Hessian inverse-vector computations via a shared curvature matrix, enabling quadratic improvements in error decay with Newton steps. Empirically, curvature-aware updates yield significant gains across synthetic problems, hyperparameter optimization, and data cleaning, with extensions to variance reduction, momentum, and non-strongly convex settings discussed as future directions.

Abstract

Bilevel optimization is a powerful tool for many machine learning problems, such as hyperparameter optimization and meta-learning. Estimating hypergradients (also known as implicit gradients) is crucial for developing gradient-based methods for bilevel optimization. In this work, we propose a computationally efficient technique for incorporating curvature information into the approximation of hypergradients and present a novel algorithmic framework based on the resulting enhanced hypergradient computation. We provide convergence rate guarantees for the proposed framework in both deterministic and stochastic scenarios, particularly showing improved computational complexity over popular gradient-based methods in the deterministic setting. This improvement in complexity arises from a careful exploitation of the hypergradient structure and the inexact Newton method. In addition to the theoretical speedup, numerical experiments demonstrate the significant practical performance benefits of incorporating curvature information.

Efficient Curvature-Aware Hypergradient Approximation for Bilevel Optimization

TL;DR

The work addresses efficient hypergradient estimation in bilevel optimization by exploiting curvature information through an inexact Newton-based framework (NBO). It introduces two instantiations, NBO-GD for deterministic settings and NSBO-SGD for stochastic settings, and provides convergence guarantees with improved deterministic complexity over prior gradient-based methods. A unified hypergradient approximation ties the lower-level solution with Hessian inverse-vector computations via a shared curvature matrix, enabling quadratic improvements in error decay with Newton steps. Empirically, curvature-aware updates yield significant gains across synthetic problems, hyperparameter optimization, and data cleaning, with extensions to variance reduction, momentum, and non-strongly convex settings discussed as future directions.

Abstract

Bilevel optimization is a powerful tool for many machine learning problems, such as hyperparameter optimization and meta-learning. Estimating hypergradients (also known as implicit gradients) is crucial for developing gradient-based methods for bilevel optimization. In this work, we propose a computationally efficient technique for incorporating curvature information into the approximation of hypergradients and present a novel algorithmic framework based on the resulting enhanced hypergradient computation. We provide convergence rate guarantees for the proposed framework in both deterministic and stochastic scenarios, particularly showing improved computational complexity over popular gradient-based methods in the deterministic setting. This improvement in complexity arises from a careful exploitation of the hypergradient structure and the inexact Newton method. In addition to the theoretical speedup, numerical experiments demonstrate the significant practical performance benefits of incorporating curvature information.
Paper Structure (35 sections, 14 theorems, 104 equations, 9 figures, 2 tables, 5 algorithms)

This paper contains 35 sections, 14 theorems, 104 equations, 9 figures, 2 tables, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Notation
  4. Curvature-Aware Bilevel Optimization Framework
  5. Hypergradient Approximation
  6. Description of the Algorithmic Framework
  7. Examples
  8. Deterministic Setting: the NBO-GD algorithm.
  9. Stochastic Setting: the NSBO-SGD algorithm.
  10. Convergence Analysis
  11. Assumptions
  12. Convergence Analysis for NBO-GD
  13. Convergence Analysis for NSBO-SGD
  14. Proof Sketch
  15. Experiments
  16. ...and 20 more sections

Key Result

Theorem 3.2

Under Assumption asp:base, choose an initial iterate $(y^0, u^0, x^0)$ in BOX 1. Then, for any constant step size $\gamma_k=\gamma\leq 1/L_{g,1}$, there exists a proper constant step size $\alpha_k = \alpha = \Theta(\kappa^{-3})$ and $T \geq \Theta(\kappa)$ such that NBO-GD has the following proper

Figures (9)

  • Figure 1: Experimental results on synthetic data with $r'=1$.
  • Figure 2: Comparison between NSBO-SGD and other algorithms on hyperparameter optimization. Left: IJCNN1 dataset; Right: Covtype dataset.
  • Figure 3: Comparison between NSBO-SGD and other algorithms on data hyper-cleaning. Left: MNIST dataset; Right: FashionMNIST dataset.
  • Figure 4: Comparison between the NBO framework and other algorithms incorporating variance reduction and the moving average technique. Left: Hyperparameter optimization on IJCNN1; Right: Data hyper-cleaning on MNIST.
  • Figure 5: Performance of the NBO framework on a toy example with a non-strongly convex lower-level problem.
  • ...and 4 more figures

Theorems & Definitions (32)

  • Remark 2.1
  • Remark 2.2
  • Theorem 3.2
  • Remark 3.3
  • Remark 3.4
  • Theorem 3.7
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 22 more