Provable Robust Classification via Learned Smoothed Densities

Smoothing classifiers and probability density functions with Gaussian kernels appear unrelated, but in this work, they are unified for the problem of robust classification. The key building block is approximating the $\textit{energy function}$ of the random variable $Y=X+N(0,\sigma^2 I_d)$ with a neural network which we use to formulate the problem of robust classification in terms of $\widehat{x}(Y)$, the $\textit{Bayes estimator}$ of $X$ given the noisy measurements $Y$. We introduce $\textit{empirical Bayes smoothed classifiers}$ within the framework of $\textit{randomized smoothing}$ and study it theoretically for the two-class linear classifier, where we show one can improve their robustness above $\textit{the margin}$. We test the theory on MNIST and we show that with a learned smoothed energy function and a linear classifier we can achieve provable $\ell_2$ robust accuracies that are competitive with empirical defenses. This setup can be significantly improved by $\textit{learning}$ empirical Bayes smoothed classifiers with adversarial training and on MNIST we show that we can achieve provable robust accuracies higher than the state-of-the-art empirical defenses in a range of radii. We discuss some fundamental challenges of randomized smoothing based on a geometric interpretation due to concentration of Gaussians in high dimensions, and we finish the paper with a proposal for using walk-jump sampling, itself based on learned smoothed densities, for robust classification.