Finite time bounds for fitted value iteration
R Munos and Csaba Szepesvari
Journal of Machine Learning Research
In this paper we develop a theoretical analysis of the performance of sampling-based fitted value iteration
(FVI) to solve infinite state-space, discounted-reward Markovian decision processes (MDPs)
under the assumption that a generative model of the environment is available. Our main results
come in the form of finite-time bounds on the performance of two versions of sampling-based FVI.
The convergence rate results obtained allow us to show that both versions of FVI are well behaving
in the sense that by using a sufficiently large number of samples for a large class of MDPs,
arbitrary good performance can be achieved with high probability. An important feature of our
proof technique is that it permits the study of weighted Lp-norm performance bounds. As a result,
our technique applies to a large class of function-approximation methods (e.g., neural networks,
adaptive regression trees, kernel machines, locally weighted learning), and our bounds scale well
with the effective horizon of the MDP. The bounds show a dependence on the stochastic stability
properties of the MDP: they scale with the discounted-average concentrability of the future-state
distributions. They also depend on a new measure of the approximation power of the function
space, the inherent Bellman residual, which reflects how well the function space is “aligned” with
the dynamics and rewards of the MDP. The conditions of the main result, as well as the concepts
introduced in the analysis, are extensively discussed and compared to previous theoretical results.
Numerical experiments are used to substantiate the theoretical findings.