Reinforcement learning (RL) algorithms are typically based on optimizing a Markov Decision Process (MDP) using the optimal Bellman equation. Recent studies have revealed that focusing the optimization of Bellman equations solely on in-sample actions tends to result in more stable optimization, especially in the presence of function approximation. Upon on these findings, in this paper, we propose an Empirical MDP Iteration (EMIT) framework. For each of these empirical MDPs, it learns an estimated Q-function denoted as \widehat{Q} . The key strength is that by restricting the Bellman update to in-sample bootstrapping, each empirical MDP converges to a unique optimal \widehat{Q} function.

This study investigates the computational speed and accuracy of two numerical integration methods, cubature and sampling-based, for integrating an integrand over a 2D polygon. Using a group of rovers searching the Martian surface with a limited sensor footprint as a test bed, the relative error and computational time are compared as the area was subdivided to improve accuracy in the sampling-based approach. The results show that the sampling-based approach exhibits a $14.75\%$ deviation in relative error compared to cubature when it matches the computational performance at $100\%$. Furthermore, achieving a relative error below $1\%$ necessitates a $10000\%$ increase in relative time to calculate due to the $\mathcal{O}(N^2)$ complexity of the sampling-based method. It is concluded that for enhancing reinforcement learning capabilities and other high iteration algorithms, the cubature method is preferred over the sampling-based method.

Even though most interfaces in the real world are discrete, no efficient way exists to train neural networks to make use of them, yet. We enhance an Interaction Network (a Reinforcement Learning architecture) with discrete interfaces and train it on the generalized Dyck language. This task requires an understanding of hierarchical structures to solve, and has long proven difficult for neural networks. We provide the first solution based on learning to use discrete data structures. We encountered unexpected anomalous behavior during training, and utilized pre-training based on execution traces to overcome them. The resulting model is very small and fast, and generalizes to sequences that are an entire order of magnitude longer than the training data.