Action-Constrained Markov Decision Processes With Kullback-Leibler Cost

This paper concerns computation of optimal policies in which the one-step reward function contains a cost term that models Kullback-Leibler divergence with respect to nominal dynamics. This technique was introduced by Todorov in 2007, where it was shown under general conditions that the solution to the average-reward optimality equations reduce to a simple eigenvector problem. Since then many authors have sought to apply this technique to control problems and models of bounded rationality in economics. A crucial assumption is that the input process is essentially unconstrained. For example, if the nominal dynamics include randomness from nature (e.g., the impact of wind on a moving vehicle), then the optimal control solution does not respect the exogenous nature of this disturbance. This paper introduces a technique to solve a more general class of action-constrained MDPs. The main idea is to solve an entire parameterized family of MDPs, in which the parameter is a scalar weighting the one-step reward function. The approach is new and practical even in the original unconstrained formulation

Kullback-Leibler-Quadratic Optimal Control of Flexible Power Demand

International audienceA new stochastic control methodology is introduced for distributed control, motivated by the goal of creating virtual energy storage from flexible electric loads, i.e. Demand Dispatch. In recent work, the authors have introduced Kullback-Leibler-Quadratic (KLQ) optimal control as a stochastic control methodology for Markovian models. This paper develops KLQ theory and demonstrates its applicability to demand dispatch. In one formulation of the design, the grid balancing authority simply broadcasts the desired tracking signal, and the heterogeneous population of loads ramps power consumption up and down to accurately track the signal. Analysis of the Lagrangian dual of the KLQ optimization problem leads to a menu of solution options, and expressions of the gradient and Hessian suitable for Monte-Carlo-based optimization. Numerical results illustrate these theoretical results

Sparse Randomized Shortest Paths Routing with Tsallis Divergence Regularization

This work elaborates on the important problem of (1) designing optimal randomized routing policies for reaching a target node t from a source note s on a weighted directed graph G and (2) defining distance measures between nodes interpolating between the least cost (based on optimal movements) and the commute-cost (based on a random walk on G), depending on a temperature parameter T. To this end, the randomized shortest path formalism (RSP, [2,99,124]) is rephrased in terms of Tsallis divergence regularization, instead of Kullback-Leibler divergence. The main consequence of this change is that the resulting routing policy (local transition probabilities) becomes sparser when T decreases, therefore inducing a sparse random walk on G converging to the least-cost directed acyclic graph when T tends to 0. Experimental comparisons on node clustering and semi-supervised classification tasks show that the derived dissimilarity measures based on expected routing costs provide state-of-the-art results. The sparse RSP is therefore a promising model of movements on a graph, balancing sparse exploitation and exploration in an optimal way

Action-Constrained Markov Decision Processes With Kullback-Leibler Cost

International audienceThis paper concerns computation of optimal policies in which the one-step reward function contains a cost term that models Kullback-Leibler divergence with respect to nominal dynamics. This technique was introduced by Todorov in 2007, where it was shown under general conditions that the solution to the average-reward optimality equations reduce to a simple eigenvector problem. Since then many authors have sought to apply this technique to control problems and models of bounded rationality in economics. A crucial assumption is that the input process is essentially unconstrained. For example, if the nominal dynamics include randomness from nature (e.g., the impact of wind on a moving vehicle), then the optimal control solution does not respect the exogenous nature of this disturbance. This paper introduces a technique to solve a more general class of action-constrained MDPs. The main idea is to solve an entire parameterized family of MDPs, in which the parameter is a scalar weighting the one-step reward function. The approach is new and practical even in the original unconstrained formulation

How to Rank Answers in Text Mining

In this thesis, we mainly focus on case studies about answers. We present the methodology CEW-DTW and assess its performance about ranking quality. Based on the CEW-DTW, we improve this methodology by combining Kullback-Leibler divergence with CEW-DTW, since Kullback-Leibler divergence can check the difference of probability distributions in two sequences. However, CEW-DTW and KL-CEW-DTW do not care about the effect of noise and keywords from the viewpoint of probability distribution. Therefore, we develop a new methodology, the General Entropy, to see how probabilities of noise and keywords affect answer qualities. We firstly analyze some properties of the General Entropy, such as the value range of the General Entropy. Especially, we try to find an objective goal, which can be regarded as a standard to assess answers. Therefore, we introduce the maximum general entropy. We try to use the general entropy methodology to find an imaginary answer with the maximum entropy from the mathematical viewpoint (though this answer may not exist). This answer can also be regarded as an “ideal” answer. By comparing maximum entropy probabilities and global probabilities of noise and keywords respectively, the maximum entropy probability of noise is smaller than the global probability of noise, maximum entropy probabilities of chosen keywords are larger than global probabilities of keywords in some conditions. This allows us to determinably select the max number of keywords. We also use Amazon dataset and a small group of survey to assess the general entropy. Though these developed methodologies can analyze answer qualities, they do not incorporate the inner connections among keywords and noise. Based on the Markov transition matrix, we develop the Jump Probability Entropy. We still adapt Amazon dataset to compare maximum jump entropy probabilities and global jump probabilities of noise and keywords respectively. Finally, we give steps about how to get answers from Amazon dataset, including obtaining original answers from Amazon dataset, removing stopping words and collinearity. We compare our developed methodologies to see if these methodologies are consistent. Also, we introduce Wald–Wolfowitz runs test and compare it with developed methodologies to verify their relationships. Depending on results of comparison, we get conclusions about consistence of these methodologies and illustrate future plans