[Paper Review] Structure-aware error bounds for linear classification with the zero-one loss

Ata Kabán, Robert J. Durrant|arXiv (Cornell University)|Sep 28, 2017

Machine Learning and Algorithms32 references3 citations

TL;DR

This paper introduces structure-aware risk bounds for linear binary classification using the zero-one loss in high-dimensional settings, even when sample size is smaller than dimensionality. By leveraging the 'flipping probability' to capture benign geometric structures, the authors derive tighter generalization bounds for both compressed (randomly projected) and full-dimensional learning, enabling improved theoretical understanding and practical classifier learning via bound minimization.

ABSTRACT

We prove risk bounds for binary classification in high-dimensional settings when the sample size is allowed to be smaller than the dimensionality of the training set observations. In particular, we prove upper bounds for both 'compressive learning' by empirical risk minimization (ERM) (that is when the ERM classifier is learned from data that have been projected from high-dimensions onto a randomly selected low-dimensional subspace) as well as uniform upper bounds in the full high-dimensional space. A novel tool we employ in both settings is the 'flipping probability' of Durrant and Kaban (ICML 2013) which we use to capture benign geometric structures that make a classification problem 'easy' in the sense of demanding a relatively low sample size for guarantees of good generalization. Furthermore our bounds also enable us to explain or draw connections between several existing successful classification algorithms. Finally we show empirically that our bounds are informative enough in practice to serve as the objective function for learning a classifier (by using them to do so).

Motivation & Objective

• To develop generalization risk bounds for linear classification with the zero-one loss when the sample size is smaller than the input dimensionality.
• To analyze the performance of empirical risk minimization (ERM) on randomly projected data (compressive learning) and derive bounds that reflect data geometry.
• To introduce the 'flipping probability' as a novel geometric measure that captures data structure to improve risk bounds.
• To connect theoretical insights to existing successful classification algorithms, such as boosting.
• To demonstrate that the derived bounds are informative enough to serve as an objective function for training a practical classifier.

Proposed method

• Introduces the 'flipping probability' from Durrant and Kabán (ICML 2013) as a key tool to quantify how easily a classifier's prediction can change under small input perturbations, reflecting data geometry.
• Derives upper bounds on generalization error for ERM in both the full high-dimensional space and in randomly projected low-dimensional subspaces using Rademacher complexity and concentration inequalities.
• Uses a shift-invariant formulation of the hypothesis class to analyze the stability of classifiers under small perturbations of the data distribution.
• Applies random projection (RP) as a dimensionality reduction technique that preserves generalization guarantees under geometric structure, with theoretical analysis of the resulting risk bounds.
• Employs uniform risk bounds that depend on the flipping probability, which acts as a smooth proxy for the discontinuous zero-one loss.
• Proposes minimizing the derived bound as a learning objective, demonstrating its practical viability in empirical experiments.

Experimental results

Research questions

• RQ1 How can generalization risk bounds be tightened for linear classification with the zero-one loss in high-dimensional, low-sample regimes?
• RQ2 To what extent does random projection preserve generalization performance, and what geometric properties of the data affect this?
• RQ3 Can the flipping probability serve as a meaningful proxy for the zero-one loss in deriving generalization bounds?
• RQ4 How do the proposed bounds relate to or explain the success of existing algorithms like boosting?
• RQ5 Can the derived risk bounds be used as a practical objective function for training a classifier?

Key findings

• The paper establishes that generalization error bounds for linear classification with the zero-one loss can be significantly tighter than classical VC-style bounds when the data exhibit low flipping probability, indicating benign geometric structure.
• For compressive learning via random projection, the risk bounds depend on the flipping probability in the projected space, and are shown to be informative even when N ≪ d.
• The flipping probability is demonstrated to be a smooth, continuous proxy for the discontinuous zero-one loss, enabling the use of gradient-based optimization techniques.
• Empirical results show that minimizing the derived risk bound as an objective function leads to a well-performing classifier, validating the practical relevance of the theoretical bounds.
• The bounds explain the success of boosting algorithms by linking their performance to low flipping probability in the data distribution.
• The theoretical framework generalizes to kernelized settings by replacing input features with their feature space representations, preserving the bound structure.