Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis

We study the performance of Machine Learning (ML) classification techniques. Leveraging the theory of large deviations, we provide the mathematical conditions for a ML classifier to exhibit error probabilities that vanish exponentially, say <inline-formula><tex-math notation="LaTeX">$\exp (-n\,I)$</tex-math></inline-formula>, where <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula> is the number of informative observations available for testing (or another relevant parameter, such as the size of the target in an image) and <inline-formula><tex-math notation="LaTeX">$I$</tex-math></inline-formula> is the error rate. Such conditions depend on the Fenchel-Legendre transform of the cumulant-generating function of the Data-Driven Decision Function (D3F, i.e., what is thresholded before the final binary decision is made) learned in the training phase. As such, the D3F and the related error rate <inline-formula><tex-math notation="LaTeX">$I$</tex-math></inline-formula> depend on the given training set. The conditions for the exponential convergence can be verified and tested numerically exploiting the available dataset or a synthetic dataset generated according to the underlying statistical model. Coherently with the large deviations theory, we can also establish the convergence of the normalized D3F statistic to a Gaussian distribution. Furthermore, approximate error probability curves <inline-formula><tex-math notation="LaTeX">$\zeta _{n} \exp (-n\,I)$</tex-math></inline-formula> are provided, thanks to the refined asymptotic derivation, where <inline-formula><tex-math notation="LaTeX">$\zeta _{n}$</tex-math></inline-formula> represents the most representative sub-exponential terms of the error probabilities. Leveraging the refined asymptotic, we are able to compute an accurate analytical approximation of the classification performance for both the regimes of small and large values of <inline-formula><tex-math notation="LaTeX">$n$</tex-math></inline-formula>. Theoretical findings are corroborated by extensive numerical simulations and by the use of real-world data, acquired by an X-band maritime radar system for surveillance.