[Paper Review] Stability Analysis and Learning Bounds for Transductive Regression Algorithms

Corinna Cortes, Mehryar Mohri+2 more|ArXiv.org|Apr 5, 2009

Machine Learning and Algorithms16 references9 citations

TL;DR

This paper introduces algorithmic stability-based generalization bounds for transductive regression algorithms, deriving tighter, algorithm-specific learning bounds than traditional complexity measures. It analyzes stability across local regression, unconstrained, and constrained regularization methods, showing that many widely used algorithms are unstable, and demonstrates through experiments that stability bounds enable effective model selection—particularly for neighborhood radius in local transduction.

ABSTRACT

This paper uses the notion of algorithmic stability to derive novel generalization bounds for several families of transductive regression algorithms, both by using convexity and closed-form solutions. Our analysis helps compare the stability of these algorithms. It also shows that a number of widely used transductive regression algorithms are in fact unstable. Finally, it reports the results of experiments with local transductive regression demonstrating the benefit of our stability bounds for model selection, for one of the algorithms, in particular for determining the radius of the local neighborhood used by the algorithm.

Motivation & Objective

• To develop algorithm-specific generalization bounds for transductive regression using algorithmic stability, improving upon general complexity measures like VC-dimension.
• To analyze the stability of key transductive regression algorithms, including local, unconstrained, and constrained regularization methods.
• To identify instability in widely used algorithms and provide theoretical justification for improved design.
• To demonstrate the practical utility of stability bounds in model selection, particularly for choosing neighborhood radius in local transduction.

Proposed method

• Derives a concentration bound for sampling without replacement, generalizing McDiarmid's inequality for use in transductive settings.
• Introduces cost and score stability notions to analyze algorithm behavior under data perturbations.
• Applies convexity and closed-form solutions to derive stability coefficients for local transductive regression (LTR) algorithms.
• Uses score-based stability analysis for unconstrained regularization algorithms with closed-form solutions.
• Applies constrained regularization analysis to graph-based algorithms, including the graph Laplacian, to derive improved stability bounds.
• Employs experimental validation on real datasets (Boston Housing, Ailerons, Elevators) to evaluate model selection using derived bounds.

Experimental results

Research questions

• RQ1 How can algorithmic stability be leveraged to derive tighter generalization bounds for transductive regression compared to VC-dimension-based bounds?
• RQ2 Which families of transductive regression algorithms exhibit stable behavior, and which are inherently unstable?
• RQ3 Can stability-based bounds effectively guide model selection, particularly for hyperparameters like neighborhood radius in local regression?
• RQ4 How does the stability of unconstrained regularization algorithms compare to that of constrained regularization algorithms in transductive settings?
• RQ5 What is the impact of data sampling without replacement on concentration and stability in transductive learning?

Key findings

• The paper proves a new concentration bound for sampling without replacement that is simpler and slightly stronger than prior work.
• Many widely used transductive regression algorithms, including several unconstrained regularization methods, are shown to be unstable under the proposed stability criteria.
• For local transductive regression (LTR), the stability coefficient β_loc is bounded by O(1/m(r)), where m(r) is the number of points within radius r, enabling radius-based model selection.
• For Gaussian kernel-based LTR, the stability coefficient is bounded by β_loc ≤ 4M / (m(r) e^{-2r²/σ²}), showing exponential dependence on kernel bandwidth.
• Experiments on Boston Housing, Ailerons, and Elevators datasets show that minimizing the stability-based bound leads to test MSE performance close to the optimal radius, validating the bound’s utility in model selection.
• The constrained graph Laplacian regularization algorithm exhibits better stability properties than unconstrained counterparts, supporting its use in practice.