We propose RUDDER, a novel reinforcement learning approach for delayed rewards in finite Markov decision processes (MDPs). In MDPs the Q-values are equal to the expected immediate reward plus the expected future rewards. The latter are related to bias problems in temporal difference (TD) learning and to high variance problems in Monte Carlo (MC) learning. Both problems are even more severe when rewards are delayed. RUDDER aims at making the expected future rewards zero, which simplifies Q-value estimation to computing the mean of the immediate reward. We propose the following two new concepts to push the expected future rewards toward zero. (i) Reward redistribution that leads to return-equivalent decision processes with the same optimal policies and, when optimal, zero expected future rewards. (ii) Return decomposition via contribution analysis which transforms the reinforcement learning task into a regression task at which deep learning excels. On artificial tasks with delayed rewards, RUDDER is significantly faster than MC and exponentially faster than Monte Carlo Tree Search (MCTS), TD(λ), and reward shaping approaches. At Atari games, RUDDER on top of a Proximal Policy Optimization (PPO) baseline improves the scores, which is most prominent at games with delayed rewards.

BACKGROUND: Cluster analysis is widely used to discover patterns in multi-dimensional data. Clustered heatmaps are the standard technique for visualizing one-way and two-way clustering results. In clustered heatmaps, rows and/or columns are reordered, resulting in a representation that shows the clusters as contiguous blocks. However, for biclustering results, where clusters can overlap, it is not possible to reorder the matrix in this way without duplicating rows and/or columns. RESULTS: We present Furby, an interactive visualization technique for analyzing biclustering results. Our contribution is twofold. First, the technique provides an overview of a biclustering result, showing the actual data that forms the individual clusters together with the information which rows and columns they share. Second, for fuzzy clustering results, the proposed technique additionally enables analysts to interactively set the thresholds that transform the fuzzy (soft) clustering into hard clusters that can then be investigated using heatmaps or bar charts. Changes in the membership value thresholds are immediately reflected in the visualization. We demonstrate the value of Furby by loading biclustering results applied to a multi-tissue dataset into the visualization. CONCLUSIONS: The proposed tool allows analysts to assess the overall quality of a biclustering result. Based on this high-level overview, analysts can then interactively explore the individual biclusters in detail. This novel way of handling fuzzy clustering results also supports analysts in finding the optimal thresholds that lead to the best clusters.