Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards Differentiable Multilevel Optimization: A Gradient-Based Approach

Yuntian Gu, Xuzheng Chen

TL;DR

A novel gradient-based approach for multilevel optimization that overcomes limitations by leveraging a hierarchically structured decomposition of the full gradient and employing advanced propagation techniques, significantly reduces computational complexity while improving both solution accuracy and convergence speed.

Abstract

Multilevel optimization has gained renewed interest in machine learning due to its promise in applications such as hyperparameter tuning and continual learning. However, existing methods struggle with the inherent difficulty of efficiently handling the nested structure. This paper introduces a novel gradient-based approach for multilevel optimization that overcomes these limitations by leveraging a hierarchically structured decomposition of the full gradient and employing advanced propagation techniques. Extending to n-level scenarios, our method significantly reduces computational complexity while improving both solution accuracy and convergence speed. We demonstrate the effectiveness of our approach through numerical experiments, comparing it with existing methods across several benchmarks. The results show a notable improvement in solution accuracy. To the best of our knowledge, this is one of the first algorithms to provide a general version of implicit differentiation with both theoretical guarantees and superior empirical performance.

Towards Differentiable Multilevel Optimization: A Gradient-Based Approach

TL;DR

A novel gradient-based approach for multilevel optimization that overcomes limitations by leveraging a hierarchically structured decomposition of the full gradient and employing advanced propagation techniques, significantly reduces computational complexity while improving both solution accuracy and convergence speed.

Abstract

Multilevel optimization has gained renewed interest in machine learning due to its promise in applications such as hyperparameter tuning and continual learning. However, existing methods struggle with the inherent difficulty of efficiently handling the nested structure. This paper introduces a novel gradient-based approach for multilevel optimization that overcomes these limitations by leveraging a hierarchically structured decomposition of the full gradient and employing advanced propagation techniques. Extending to n-level scenarios, our method significantly reduces computational complexity while improving both solution accuracy and convergence speed. We demonstrate the effectiveness of our approach through numerical experiments, comparing it with existing methods across several benchmarks. The results show a notable improvement in solution accuracy. To the best of our knowledge, this is one of the first algorithms to provide a general version of implicit differentiation with both theoretical guarantees and superior empirical performance.

Paper Structure

This paper contains 30 sections, 10 theorems, 62 equations, 2 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Bilevel Optimization and Solution Approaches
  4. Implicit Differentiation Method
  5. Procedures and Mathematical Formulation of Multilevel Optimization
  6. Gradient-based Optimization Algorithm for Multilevel Optimization
  7. The gradient of Trilevel Optimization
  8. The gradient of Multilevel Optimization
  9. High-level idea
  10. Algorithm
  11. Theoretical Analysis
  12. Complexity of Calculating Gradients
  13. Complexity Analysis of Algorithm \ref{['alg']}
  14. Complexity Analysis of Chain Rule Calculation
  15. Convergence Analysis
  16. ...and 15 more sections

Key Result

Lemma 2.1

Let $f:\mathbb R^{m+n}\to \mathbb R$ be a continuous function with second derivatives and $\det (\frac{\partial ^2 f}{\partial \bf{y} ^2} ) \neq 0$. Let $\bf{g}(\bf{x}) = \mathop{\mathrm{arg\,min}}\limits _{\bf y} f(\bf x, \bf y)$. Then the derivative of $\bf g$ with respect to $\bf x$ is

Figures (2)

  • Figure 1: The MSE in Generalized Stackelberg's model. Our method converges significantly faster than all the alternatives, and is the only method that do not fall into local minima.
  • Figure 2: In the task of hyperparameter optimization, both the FD method and the ITD approach tend to converge to local minima.

Theorems & Definitions (18)

  • Lemma 2.1
  • Lemma 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Theorem 4.1
  • Theorem 4.2
  • proof
  • proof
  • proof
  • proof
  • ...and 8 more