Recent work in reinforcement learning has emphasized the power of L1 regularization to perform feature selection and prevent overfitting. We propose formulating the L1 regularized linear fixed point problem as a linear complementarity problem (LCP). This formulation offers several advantages over the LARS-inspired formulation, LARS-TD. The LCP formulation allows the use of efficient off-the-shelf solvers, leads to a new uniqueness result, and can be initialized with starting points from similar problems (warm starts). We demonstrate that warm starts, as well as the efficiency of LCP solvers, can speed up policy iteration. Moreover, warm starts permit a form of modified policy iteration that can be used to approximate a greedy" homotopy path, a generalization of the LARS-TD homotopy path that combines policy evaluation and optimization."

We show that linear value-function approximation is equivalent to a form of linear model approximation. We then derive a relationship between the model-approximation error and the Bellman error, and show how this relationship can guide feature selection for model improvement and/or value-function improvement. We also show how these results give insight into the behavior of existing feature-selection algorithms.

We analyze a simple, Bellman-error-based approach to generating basis functions for value-function approximation. We show that it generates orthogonal basis functions that provably tighten approximation error bounds. We also illustrate the use of this approach in the presence of noise on some sample problems.