A New Smooth Approximation to the Zero One Loss with a Probabilistic Interpretation

TL;DR

This paper introduces a smooth, probabilistic approximation to the zero-one loss using a generalized Beta-Bernoulli formulation, improving robustness, scalability, and probabilistic outputs in classification tasks.

Contribution

It proposes a novel generalized logistic function based on a Beta-Bernoulli model, enhancing zero-one loss approximation with probabilistic interpretation and broad applicability.

Findings

Improved robustness to outliers compared to logistic and hinge losses.

Outperforms logistic and max margin models on large benchmark datasets.

Yields sparser solutions with Gaussian-Laplacian priors and improves structured prediction probabilities.

Abstract

We examine a new form of smooth approximation to the zero one loss in which learning is performed using a reformulation of the widely used logistic function. Our approach is based on using the posterior mean of a novel generalized Beta-Bernoulli formulation. This leads to a generalized logistic function that approximates the zero one loss, but retains a probabilistic formulation conferring a number of useful properties. The approach is easily generalized to kernel logistic regression and easily integrated into methods for structured prediction. We present experiments in which we learn such models using an optimization method consisting of a combination of gradient descent and coordinate descent using localized grid search so as to escape from local minima. Our experiments indicate that optimization quality is improved when learning meta-parameters are themselves optimized using a validation set. Our experiments show improved performance relative to widely used logistic and hinge loss methods on a wide variety of problems ranging from standard UC Irvine and libSVM evaluation datasets to product review predictions and a visual information extraction task. We observe that the approach: 1) is more robust to outliers compared to the logistic and hinge losses; 2) outperforms comparable logistic and max margin models on larger scale benchmark problems; 3) when combined with Gaussian- Laplacian mixture prior on parameters the kernelized version of our formulation yields sparser solutions than Support Vector Machine classifiers; and 4) when integrated into a probabilistic structured prediction technique our approach provides more accurate probabilities yielding improved inference and increasing information extraction performance.