Statistical Hypothesis Testing Based on Machine Learning: Large Deviations Analysis
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We study the performance – and specifically the rate at which the error probability converges to zero – 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 ∼exp(-n I + o(n) ), where n is the number of informative observations available for testing (or another relevant parameter, such as the size of the target in an image) and I 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, consequently, the related error rate I, depend on the given training set, which is assumed of finite size. Interestingly, these conditions can be verified and tested numerically exploiting the available dataset, or a synthetic dataset, generated according to the available information on the underlying statistical model. In other words, the classification error probability convergence to zero and its rate can be computed on a portion of the dataset available for training. Coherently with the large deviations theory, we can also establish the convergence, for n large enough, of the normalized D3F statistic to a Gaussian distribution. This property is exploited to set a desired asymptotic false alarm probability, which empirically turns out to be accurate even for quite realistic values of n. Furthermore, approximate error probability curves ∼ζ_n exp(-n I ) are provided, thanks to the refined asymptotic derivation (often referred to as exact asymptotics), where ζ_n represents the most representative sub-exponential terms of the error probabilities.
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