Remez-type inequalities bound the suprema of low-degree polynomials over some domain K by their suprema over a subset S of K. Existing multi-dimensional Remez inequalities bear constants with strong dependence on dimension. In this talk we will prove a dimension-free Remez-type estimate when K is the polydisc D^n and S is from a certain class of discrete subsets. As a direct consequence we also obtain a Bohnenblust-Hille-type inequality for products of cyclic groups, which in turn has consequences for learning algorithms. Based on joint work with Lars Becker, Ohad Klein, Alexander Volberg, and Haonan Zhang.