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## Section: New Results

### Optimal Control of Markov Processes

Participants: B. Scherrer

Finite-horizon lookahead policies are abundantly used in Reinforcement Learning and demonstrate impressive empirical success. Usually, the lookahead policies are implemented with specific planning methods such as Monte Carlo Tree Search (e.g. in AlphaZero). Referring to the planning problem as tree search, a reasonable practice in these implementations is to back up the value only at the leaves while the information obtained at the root is not leveraged other than for updating the policy. Here, we question the potency of this approach. Namely, the latter procedure is non-contractive in general, and its convergence is not guaranteed. Our proposed enhancement, in [9], published in AAAI'2019, is straightforward and simple: use the return from the optimal tree path to back up the values at the descendants of the root. This leads to a ${\gamma }^{h}$-contracting procedure, where $\gamma$ is the discount factor and $h$ is the tree depth. To establish our results, we first introduce a notion called multiple-step greedy consistency. We then provide convergence rates for two algorithmic instantiations of the above enhancement in the presence of noise injected to both the tree search stage and value estimation stage.

Value iteration is a method to generate optimal control inputs for generic nonlinear systems and cost functions. Its implementation typically leads to approximation errors, which may have a major impact on the closed-loop system performance. We talk in this case of approximate value iteration (AVI). In [24], published in CDC'2019, we investigate the stability of systems for which the inputs are obtained by AVI. We consider deterministic discrete-time nonlinear plants and a class of general, possibly discounted, costs. We model the closed-loop system as a family of systems parameterized by tunable parameters, which are used for the approximation of the value function at different iterations, the discount factor and the iteration step at which we stop running the algorithm. It is shown, under natural stabilizability and detectability properties as well as mild conditions on the approximation errors, that the family of closed-loop systems exhibit local practical stability properties. The analysis is based on the construction of a Lyapunov function given by the sum of the approximate value function and the Lyapunov-like function that characterizes the detectability of the system. By strengthening our conditions, asymptotic and exponential stability properties are guaranteed.

Many recent successful (deep) reinforcement learning algorithms make use of regularization, generally based on entropy or Kullback-Leibler divergence. In [10], published in ICML'2019, we propose a general theory of regularized Markov Decision Processes that generalizes these approaches in two directions: we consider a larger class of regularizers, and we consider the general modified policy iteration approach, encompassing both policy iteration and value iteration. The core building blocks of this theory are a notion of regularized Bellman operator and the Legendre-Fenchel transform, a classical tool of convex optimization. This approach allows for error propagation analyses of general algorithmic schemes of which (possibly variants of) classical algorithms such as Trust Region Policy Optimization, Soft Q-learning, Stochastic Actor Critic or Dynamic Policy Programming are special cases. This also draws connections to proximal convex optimization, especially to Mirror Descent.