Bias-Variance Trade-Off in Hierarchical Probabilistic Models Using Higher-Order Feature Interactions

TL;DR

This paper investigates the bias-variance trade-off in hierarchical probabilistic models, specifically higher-order Boltzmann machines, using a new inference algorithm and bias-variance decomposition.

Contribution

It introduces an efficient inference method for higher-order Boltzmann machines and analyzes the bias-variance trade-off between hidden layers and higher-order interactions.

Findings

Higher-order interactions produce less variance with small sample sizes.

Hidden layers and higher-order interactions have comparable errors.

The study provides insights into model complexity and generalization.

Abstract

Hierarchical probabilistic models are able to use a large number of parameters to create a model with a high representation power. However, it is well known that increasing the number of parameters also increases the complexity of the model which leads to a bias-variance trade-off. Although it is a classical problem, the bias-variance trade-off between hidden layers and higher-order interactions have not been well studied. In our study, we propose an efficient inference algorithm for the log-linear formulation of the higher-order Boltzmann machine using a combination of Gibbs sampling and annealed importance sampling. We then perform a bias-variance decomposition to study the differences in hidden layers and higher-order interactions. Our results have shown that using hidden layers and higher-order interactions have a comparable error with a similar order of magnitude and using higher-order interactions produce less variance for smaller sample size.