On rates of convergence for sample average approximations without smoothness

Sample average approximation (SAA) replaces an intractable expected objective by an empirical average and is a basic device of modern stochastic optimization. We develop a rate theory for optimal values and empirical varepsilon-minimizers that does not assume continuity, lower semicontinuity, or smooth perturbation structure of the sample objectives. Working on ℓ∞(X) with the Hoffmann--Jorgensen outer-probability formalism, we show that uniform control of the empirical objective process transfers deterministically to convergence rates for optimal values, excess risks of empirical varepsilon-minimizers, and, under a sharp-growth condition, distances to the expected objective solution set. Combined with the directional differentiability of the infimum functional, this yields weak limits for empirical optimal values at the n-1/2 scale. Combined with LILs and maximal inequalities, it yields outer almost-sure and outer-mean rates. The definability, envelope, and VC-subgraph hypotheses are verified for definable discontinuous or non-Lipschitz classes arising in direct 0--1 classification, fixed-architecture neural networks, threshold regression, and non-Lipschitz ℓp-type objectives with rational 0<p<1. Practical sufficient conditions for measurability hypotheses are discussed. Together, the framework extends continuity-based SAA theory to a tame-topological setting.