1,503 research outputs found
Successive approximations for Markov decision processes and Markov games with unbounded rewards
The aim of this paper is to give an overview of recent developments in the area of successive approximations for Markov decision processes and Markov games. We will emphasize two aspects, viz. the conditions under which successive approximations converge in some strong sense and variations of these methods which diminish the amount of computational work to be executed. With respect to the first aspect it will be shown how much unboundedness of the rewards may be allowed without violation of the convergence. With respect to the second aspect we will present four ideas, that can be applied in conjunction, which may diminish the amount of work to be done. These ideas are: I. the use of the actual convergence of the iterates for the construction of upper and lower bounds (McQueen bounds), 2. the use of alternative policy improvement procedures (based on stopping times), 3. a better evaluation of the values of actual policies in each iteration step by a value oriented approach, 4. the elimination of suboptimal actions not only permanently, but also temporarily. The general presentation ~s given for Markov decision processes with a final section devoted to the possibilities of extension to Markov games
Robust Markov Decision Processes
Markov decision processes (MDPs) are powerful tools for decision making in uncertain dynamic environments. However, the solutions of MDPs are of limited practical use due to their sensitivity to distributional model parameters, which are typically unknown and have to be estimated by the decision maker. To counter the detrimental effects of estimation errors, we consider robust MDPs that offer probabilistic guarantees in view of the unknown parameters. To this end, we assume that an observation history of the MDP is available. Based on this history, we derive a confidence region that contains the unknown parameters with a pre-specified probability 1-ß. Afterwards, we determine a policy that attains the highest worst-case performance over this confidence region. By construction, this policy achieves or exceeds its worst-case performance with a confidence of at least 1 - ß. Our method involves the solution of tractable conic programs of moderate size.
Play selection in football : a case study in neuro-dynamic programming
Includes bibliographical references (p. 34-35).Supported by the US Army Research Office. AASERT-DAAH04-93-GD169Stephen D. Patek, Dimitri P. Bertsekas
Multigrid methods for two-player zero-sum stochastic games
We present a fast numerical algorithm for large scale zero-sum stochastic
games with perfect information, which combines policy iteration and algebraic
multigrid methods. This algorithm can be applied either to a true finite state
space zero-sum two player game or to the discretization of an Isaacs equation.
We present numerical tests on discretizations of Isaacs equations or
variational inequalities. We also present a full multi-level policy iteration,
similar to FMG, which allows to improve substantially the computation time for
solving some variational inequalities.Comment: 31 page
On Markov games
In the paper it is demonstrated, how a dynamic programming approach may be useful for the analysis of Markov games. Markov games with finitely many stages are dealt with extensively. The existence of optimal Markov strategies is proven for finite stage Markov games using a shortcut of a proof by Derman for the analogous result for Markov decision processes. For Markov games with a countably infinite number of stages some results are summarized. Here again the results and the methods of prove have much in common with results and proofs for Markov decision processes. Actually the theory of Markov games is a generalisation. The paper contains short introductions into the theories of matrix games and tree games
Certified Reinforcement Learning with Logic Guidance
This paper proposes the first model-free Reinforcement Learning (RL)
framework to synthesise policies for unknown, and continuous-state Markov
Decision Processes (MDPs), such that a given linear temporal property is
satisfied. We convert the given property into a Limit Deterministic Buchi
Automaton (LDBA), namely a finite-state machine expressing the property.
Exploiting the structure of the LDBA, we shape a synchronous reward function
on-the-fly, so that an RL algorithm can synthesise a policy resulting in traces
that probabilistically satisfy the linear temporal property. This probability
(certificate) is also calculated in parallel with policy learning when the
state space of the MDP is finite: as such, the RL algorithm produces a policy
that is certified with respect to the property. Under the assumption of finite
state space, theoretical guarantees are provided on the convergence of the RL
algorithm to an optimal policy, maximising the above probability. We also show
that our method produces ''best available'' control policies when the logical
property cannot be satisfied. In the general case of a continuous state space,
we propose a neural network architecture for RL and we empirically show that
the algorithm finds satisfying policies, if there exist such policies. The
performance of the proposed framework is evaluated via a set of numerical
examples and benchmarks, where we observe an improvement of one order of
magnitude in the number of iterations required for the policy synthesis,
compared to existing approaches whenever available.Comment: This article draws from arXiv:1801.08099, arXiv:1809.0782
Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning
Many problems in sequential decision making and stochastic control often have
natural multiscale structure: sub-tasks are assembled together to accomplish
complex goals. Systematically inferring and leveraging hierarchical structure,
particularly beyond a single level of abstraction, has remained a longstanding
challenge. We describe a fast multiscale procedure for repeatedly compressing,
or homogenizing, Markov decision processes (MDPs), wherein a hierarchy of
sub-problems at different scales is automatically determined. Coarsened MDPs
are themselves independent, deterministic MDPs, and may be solved using
existing algorithms. The multiscale representation delivered by this procedure
decouples sub-tasks from each other and can lead to substantial improvements in
convergence rates both locally within sub-problems and globally across
sub-problems, yielding significant computational savings. A second fundamental
aspect of this work is that these multiscale decompositions yield new transfer
opportunities across different problems, where solutions of sub-tasks at
different levels of the hierarchy may be amenable to transfer to new problems.
Localized transfer of policies and potential operators at arbitrary scales is
emphasized. Finally, we demonstrate compression and transfer in a collection of
illustrative domains, including examples involving discrete and continuous
statespaces.Comment: 86 pages, 15 figure
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