Error analysis for approximate CVaR-optimal control with a maximum cost

We consider a risk-aware optimal control problem, where the objective is the conditional value-at-risk of a maximum of stagewise and terminal costs along a finite time-horizon. Previous techniques for this problem rely on dynamic programming (DP), which is notorious for scalability issues. Since approximate DP (ADP) in risk-neutral settings can alleviate such issues, we study an ADP method for the aforementioned risk-aware setting that relies on empirical sampling and function approximation in a reproducing kernel Hilbert space. Our contribution is the derivation of sup-norm approximation error bounds that are pointwise functions of the sampling process, using techniques from functional analysis, probability theory and a relaxed Lipschitz condition. The performance of the method is evaluated using a single-stage reservoir management problem, and the effect of different algorithm parameters on the error is illustrated.