Saddle-Point Problems with a Low-Dimensional Block Do Not Need Accurate Inner Solves

Many learning problems couple a high-dimensional block of parameters with a handful of adversarial or dual variables: worst-group risk over a few groups, or learning under a few constraints. We consider $\min_{x\in X}\max_{y\in Y} f(x,y)$ with $X\subseteq\mathbb{R}^n$, a compact convex set and stron