  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 $\sim \exp\left(-n\,I + o(n) \right)$, 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 $\sim \zeta_n
\exp\left(-n\,I \right)$ are provided, thanks to the refined asymptotic
derivation (often referred to as exact asymptotics), where $\zeta_n$ represents
the most representative sub-exponential terms of the error probabilities.
