Bayesian Perspective on Generalization (Sam Smith & Quoc Le, 2018)

Bayesian model comparisons were first made on Neural Networks by Mackay. Consider a classification model \(M\) with a single parameter \(w\), training inputs \(x\) and training labels \(y\). We can infer a posterior probability distribution over the parameter by applying Bayes theorem:

\begin{equation} P(w|y,x;M) = \frac{P(y|w,x;M)P(w;M)}{P(y|x;M)} \end{equation}

The assumption of a Gaussian prior for \(P(w;M)\) leads to a posterior density of the parameter given the new training data \(P(w|y;x;M) \propto \sqrt{\lambda/2\pi}e^{-C(w;M)}\), where \(C(w;M) = H(w;M) + \lambda w^2 / 2\), which is the L2 regularized cross-entropy.

We can evaluate the normalizing constant, \(P(y|x;M) = \sqrt{\frac{\lambda}{2\pi}} \int dw e^{-C(w;M)}\). Assuming that the integral is dominated by the region near the minimum \(w_0\), we can estimate the evidence by Taylor expanding \(C(w;M) \approx C(w_0) + C”(w_0)(w-w_0)^2\).

\begin{equation} P(y|x;M) = \mathrm{exp} \left\{ -\left( C(w_0) + \frac{1}{2}ln(C”(w_0)/\lambda) \right) \right\} \end{equation}

In models with many parameters, \(P(y|x;M) \approx \frac{\lambda^{p/2}f^{-C(w_0)}} {| \nabla \nabla C(w) |_{w_0}^{1 / 2}}\), where the denominator can be thought of as an “Occam factor”, causing the network to prefer broad minima.

Bibliography

Smith, S., & Le, Q. V., A bayesian perspective on generalization and stochastic gradient descent, In , (pp. ) (2018). : .