Estimating optimal PAC-Bayes bounds with Hamiltonian Monte Carlo
This addresses a specific issue in PAC-Bayes theory for machine learning researchers, but it is incremental as it focuses on quantifying an existing limitation.
The paper tackled the problem of tightness loss in PAC-Bayes bounds when using factorized Gaussian posteriors, by estimating optimal bounds with Hamiltonian Monte Carlo and comparing them to MFVI, revealing gaps of up to 5-6% on MNIST.
An important yet underexplored question in the PAC-Bayes literature is how much tightness we lose by restricting the posterior family to factorized Gaussian distributions when optimizing a PAC-Bayes bound. We investigate this issue by estimating data-independent PAC-Bayes bounds using the optimal posteriors, comparing them to bounds obtained using MFVI. Concretely, we (1) sample from the optimal Gibbs posterior using Hamiltonian Monte Carlo, (2) estimate its KL divergence from the prior with thermodynamic integration, and (3) propose three methods to obtain high-probability bounds under different assumptions. Our experiments on the MNIST dataset reveal significant tightness gaps, as much as 5-6\% in some cases.