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Unlock the Power of Algorithm Features: A Generalization Analysis for Algorithm Selection

Xingyu Wu, Yan Zhong, Jibin Wu, Yuxiao Huang, Sheng-hao Wu, Kay Chen Tan

TL;DR

The paper addresses when and how algorithm features improve per-instance algorithm selection by deriving provable generalization guarantees. It distinctly analyzes adaptive versus predefined features under transductive and inductive learning using Rademacher complexity, showing how training size, number of candidate algorithms, and distribution shift (captured by $χ^2$-divergence) shape generalization bounds. Key contributions include tight upper bounds on generalization error, insights into how feature design interacts with learning paradigms, and corollaries linking theory to practice under distribution shifts. The empirical simulations on simulated continuous-optimization tasks validate the theoretical insights and offer practical guidelines: predefined features generalize better under shift, adaptive features excel with many known algorithms, and model complexity should be tuned to the extent of distribution differences.

Abstract

In the algorithm selection research, the discussion surrounding algorithm features has been significantly overshadowed by the emphasis on problem features. Although a few empirical studies have yielded evidence regarding the effectiveness of algorithm features, the potential benefits of incorporating algorithm features into algorithm selection models and their suitability for different scenarios remain unclear. In this paper, we address this gap by proposing the first provable guarantee for algorithm selection based on algorithm features, taking a generalization perspective. We analyze the benefits and costs associated with algorithm features and investigate how the generalization error is affected by different factors. Specifically, we examine adaptive and predefined algorithm features under transductive and inductive learning paradigms, respectively, and derive upper bounds for the generalization error based on their model's Rademacher complexity. Our theoretical findings not only provide tight upper bounds, but also offer analytical insights into the impact of various factors, such as the training scale of problem instances and candidate algorithms, model parameters, feature values, and distributional differences between the training and test data. Notably, we demonstrate how models will benefit from algorithm features in complex scenarios involving many algorithms, and proves the positive correlation between generalization error bound and $χ^2$-divergence of distributions.

Unlock the Power of Algorithm Features: A Generalization Analysis for Algorithm Selection

TL;DR

The paper addresses when and how algorithm features improve per-instance algorithm selection by deriving provable generalization guarantees. It distinctly analyzes adaptive versus predefined features under transductive and inductive learning using Rademacher complexity, showing how training size, number of candidate algorithms, and distribution shift (captured by -divergence) shape generalization bounds. Key contributions include tight upper bounds on generalization error, insights into how feature design interacts with learning paradigms, and corollaries linking theory to practice under distribution shifts. The empirical simulations on simulated continuous-optimization tasks validate the theoretical insights and offer practical guidelines: predefined features generalize better under shift, adaptive features excel with many known algorithms, and model complexity should be tuned to the extent of distribution differences.

Abstract

In the algorithm selection research, the discussion surrounding algorithm features has been significantly overshadowed by the emphasis on problem features. Although a few empirical studies have yielded evidence regarding the effectiveness of algorithm features, the potential benefits of incorporating algorithm features into algorithm selection models and their suitability for different scenarios remain unclear. In this paper, we address this gap by proposing the first provable guarantee for algorithm selection based on algorithm features, taking a generalization perspective. We analyze the benefits and costs associated with algorithm features and investigate how the generalization error is affected by different factors. Specifically, we examine adaptive and predefined algorithm features under transductive and inductive learning paradigms, respectively, and derive upper bounds for the generalization error based on their model's Rademacher complexity. Our theoretical findings not only provide tight upper bounds, but also offer analytical insights into the impact of various factors, such as the training scale of problem instances and candidate algorithms, model parameters, feature values, and distributional differences between the training and test data. Notably, we demonstrate how models will benefit from algorithm features in complex scenarios involving many algorithms, and proves the positive correlation between generalization error bound and -divergence of distributions.
Paper Structure (14 sections, 13 theorems, 49 equations, 6 figures)

This paper contains 14 sections, 13 theorems, 49 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Algorithm Feature-based Models and Learning Paradigms
  3. Model Definition and Basic Notations
  4. Transductive and Inductive Generalization in Algorithm Selection
  5. Generalization Error of the Adaptive Feature-based Model
  6. Generalization Error of the Predefined Feature-based Model
  7. Experiment
  8. Data Simulation
  9. Impact of the Number of Problems
  10. Impact of the Number of Algorithms
  11. Impact of the Distribution Shift
  12. Impact of the Training Scale under Distribution Shift
  13. Impact of the Model Complexity
  14. Conclusion

Key Result

Lemma 1

(Contraction of Rademacher Complexity, following from el2009transductive) Let $\mathcal{V}\in \mathbb{R}^{m+u}$ be a set of vectors. Let $f$ and $g$ be real-valued functions. Let $\boldsymbol{\sigma}=\{\sigma_i\}_{i=1}^{m+u}$ be Rademacher variables as defined in Definition tran_R_comp. If for all $

Figures (6)

  • Figure 1: Comparison of the problem feature-based framework and the algorithm feature-based framework.
  • Figure 2: The impact of the number of problem instances on model performance.
  • Figure 3: The impact of the number of candidate algorithms on model performance.
  • Figure 4: The impact of the distribution shift on model performance.
  • Figure 5: The impact of the number of problem instances on model performance under distribution shift.
  • ...and 1 more figures

Theorems & Definitions (23)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Lemma 2
  • Theorem 2
  • proof
  • ...and 13 more