Analyzing Approximate Value Iteration Algorithms

In this paper, we consider the stochastic iterative counterpart of the value iteration scheme wherein only noisy and possibly biased approximations of the Bellman operator are available. We call this counterpart the approximate value iteration (AVI) scheme. Neural networks are often used as function approximators, in order to counter Bellman’s curse of dimensionality. In this paper, they are used to approximate the Bellman operator. Because neural networks are typically trained using sample data, errors and biases may be introduced. The design of AVI accounts for implementations with biased approximations of the Bellman operator and sampling errors. We present verifiable sufficient conditions under which AVI is stable (almost surely bounded) and converges to a fixed point of the approximate Bellman operator. To ensure the stability of AVI, we present three different yet related sets of sufficient conditions that are based on the existence of an appropriate Lyapunov function. These Lyapunov function–based conditions are easily verifiable and new to the literature. The verifiability is enhanced by the fact that a recipe for the construction of the necessary Lyapunov function is also provided. We also show that the stability analysis of AVI can be readily extended to the general case of set-valued stochastic approximations. Finally, we show that AVI can also be used in more general circumstances, that is, for finding fixed points of contractive set-valued maps.

A stochastic Prékopa–Leindler inequality for log-concave functions

The Brunn–Minkowski and Prékopa–Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no special signficance. On the other hand, it was recently shown that the Brunn–Minkowski inequality, specialized to convex sets, follows from a local stochastic dominance for naturally associated random polytopes. We show that for the subclass of log-concave functions and associated stochastic approximations, a similar stochastic dominance underlies the Prékopa–Leindler inequality.