[Paper Review] Robustness and Regularization of Support Vector Machines

Huan Xu, Constantine Caramanis+1 more|arXiv (Cornell University)|Mar 25, 2008

Sparse and Compressive Sensing TechniquesEngineering46 references309 citations

TL;DR

This paper establishes a precise equivalence between regularized support vector machines (SVMs) and a robust optimization formulation under non-box-type uncertainty sets, demonstrating that regularization inherently provides robustness to adversarial input perturbations. The key contribution is a new theoretical justification for SVM generalization: robustness to local data disturbances explains why regularized SVMs generalize well, enabling a consistency proof without relying on VC-dimension or stability arguments.

ABSTRACT

We consider regularized support vector machines (SVMs) and show that they are precisely equivalent to a new robust optimization formulation. We show that this equivalence of robust optimization and regularization has implications for both algorithms, and analysis. In terms of algorithms, the equivalence suggests more general SVM-like algorithms for classification that explicitly build in protection to noise, and at the same time control overfitting. On the analysis front, the equivalence of robustness and regularization, provides a robust optimization interpretation for the success of regularized SVMs. We use the this new robustness interpretation of SVMs to give a new proof of consistency of (kernelized) SVMs, thus establishing robustness as the reason regularized SVMs generalize well.

Motivation & Objective

• To establish a formal connection between regularization in SVMs and robust optimization under non-i.i.d. data disturbances.
• To provide a new theoretical explanation for the generalization performance of regularized SVMs by framing it as robustness to adversarial perturbations.
• To develop a robust classification framework that offers less conservative bounds than previous robust formulations, especially for chance-constrained and Bayesian setups.
• To demonstrate that robustness to local disturbances is sufficient to prove consistency of standard SVMs in the i.i.d. learning setup.
• To enable principled selection of regularization parameters without cross-validation via Bayesian interpretation.

Proposed method

• Formulates a robust optimization problem where data are subject to adversarial perturbations constrained by aggregate norms across samples, rather than per-sample box constraints.
• Derives an equivalent regularized SVM formulation by showing that the robust optimization problem yields the same dual optimization as standard regularized SVMs.
• Uses reproducing kernel Hilbert space (RKHS) theory to relate robustness in input space to robustness in feature space for RBF kernels.
• Proves that for shift-invariant kernels like RBF, robustness in input space with bounded perturbation size c corresponds to robustness in feature space with a norm constraint proportional to √(2f(0)−2f(c)).
• Applies the robustness framework to chance-constrained and Bayesian classification, showing tighter bounds and principled regularization parameter selection.
• Uses the robustness perspective to prove consistency of kernelized SVMs without relying on VC-dimension or stability-based arguments.

Experimental results

Research questions

• RQ1 Is there a formal equivalence between regularized SVMs and a robust optimization formulation under non-box-type uncertainty sets?
• RQ2 Can the generalization performance of regularized SVMs be explained through robustness to adversarial input perturbations rather than just complexity control?
• RQ3 How does the robustness of SVMs relate to probabilistic formulations such as chance-constrained or Bayesian classifiers?
• RQ4 Can the robustness interpretation be used to prove consistency of standard SVMs without VC-dimension or stability arguments?
• RQ5 Can the robust formulation enable a principled, cross-validation-free selection of the regularization parameter?

Key findings

• Regularized SVMs are mathematically equivalent to a robust optimization formulation under a class of non-box-type uncertainty sets, where perturbations are constrained by aggregate norms.
• The equivalence provides a new interpretation: regularization in SVMs inherently protects against adversarial input perturbations, explaining their generalization ability.
• For RBF kernels, robustness in input space with perturbation size c corresponds to robustness in feature space with a norm constraint of √(2f(0)−2f(c)) on the feature vector perturbation.
• The robust formulation approximates chance-constrained classifiers with significantly less conservatism than previous robust methods.
• The robustness framework allows a new proof of consistency for kernelized SVMs that does not rely on VC-dimension or stability, instead grounding generalization in robustness to local disturbances.
• The Bayesian interpretation of the robust formulation enables a principled, data-driven selection of the regularization parameter without cross-validation.