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This work studies discrete-time discounted Markov decision processes with continuous state and action spaces and addresses the inverse problem of inferring a cost function from observed optimal behavior. We first consider the case in which we have access to the entire expert policy and characterize the set of solutions to the inverse problem by using occupation measures, linear duality, and complementary slackness conditions. To avoid trivial solutions and ill-posedness, we introduce a natural linear normalization constraint. This results in an infinite-dimensional linear feasibility problem, prompting a thorough analysis of its properties. Next, we use linear function approximators and adopt a randomized approach, namely the scenario approach and related probabilistic feasibility guarantees, to derive epsilon-optimal solutions for the inverse problem. We further discuss the sample complexity for a desired approximation accuracy. Finally, we deal with the more realistic case where we only have access to a finite set of expert demonstrations and a generative model and provide bounds on the error made when working with samples., Comment: 29 pages, 4 figures

We propose a new Q-learning variant, called 2RA Q-learning, that addresses some weaknesses of existing Q-learning methods in a principled manner. One such weakness is an underlying estimation bias which cannot be controlled and often results in poor performance. We propose a distributionally robust estimator for the maximum expected value term, which allows us to precisely control the level of estimation bias introduced. The distributionally robust estimator admits a closed-form solution such that the proposed algorithm has a computational cost per iteration comparable to Watkins' Q-learning. For the tabular case, we show that 2RA Q-learning converges to the optimal policy and analyze its asymptotic mean-squared error. Lastly, we conduct numerical experiments for various settings, which corroborate our theoretical findings and indicate that 2RA Q-learning often performs better than existing methods., Comment: 26 pages, 5 figures

Electrical circuits are present in a variety of technologies, making their design an important part of computer aided engineering. The growing number of parameters that affect the final design leads to a need for new approaches to quantify their impact. Machine learning may play a key role in this regard, however current approaches often make suboptimal use of existing knowledge about the system at hand. In terms of circuits, their description via modified nodal analysis is well-understood. This particular formulation leads to systems of differential-algebraic equations (DAEs) which bring with them a number of peculiarities, e.g. hidden constraints that the solution needs to fulfill. We use the recently introduced dissection index that can decouple a given system of DAEs into ordinary differential equations, only depending on differential variables, and purely algebraic equations, that describe the relations between differential and algebraic variables. The idea is to then only learn the differential variables and reconstruct the algebraic ones using the relations from the decoupling. This approach guarantees that the algebraic constraints are fulfilled up to the accuracy of the nonlinear system solver, and it may also reduce the learning effort as only the differential variables need to be learned., Comment: 21 pages, 16 figures