  This paper explores a theory of generalization for learning problems on
product distributions, complementing the existing learning theories in the
sense that it does not rely on any complexity measures of the hypothesis
classes. The main contributions are two general sample complexity bounds: (1)
$\tilde{O} \big( \frac{nk}{\epsilon^2} \big)$ samples are sufficient and
necessary for learning an $\epsilon$-optimal hypothesis in any problem on an
$n$-dimensional product distribution, whose marginals have finite supports of
sizes at most $k$; (2) $\tilde{O} \big( \frac{n}{\epsilon^2} \big)$ samples are
sufficient and necessary for any problem on $n$-dimensional product
distributions if it satisfies a notion of strong monotonicity from the
algorithmic game theory literature. As applications of these theories, we match
the optimal sample complexity for single-parameter revenue maximization (Guo et
al., STOC 2019), improve the state-of-the-art for multi-parameter revenue
maximization (Gonczarowski and Weinberg, FOCS 2018) and prophet inequality
(Correa et al., EC 2019), and provide the first and tight sample complexity
bound for Pandora's problem.
