In particular, we leverage the concept of stability in the literature of partial differential equation tostudy the asymptotic behavior ofthe learned solution asthe loss approaches zero. Withthis concept, we study animportant class of high-dimensional non-linear PDEs in optimal control, the Hamilton-JacobiBellman (HJB) Equation, and provethat for generalLp Physics-Informed Loss, a wide class of HJB equation is stable only ifp is sufficiently large.

The optimal control problem of stochastic systems is commonly solved via robust [2, 21] or scenario-based [7, 19, 17] optimization methods, which are both challenging to scale to long optimization horizons due to their open-loop nature. Dynamic programming formulations [4], while applicable to stochastic systems, typically involve nonconvex optimization problems and do not support specifying the terminal distribution. Polynomial optimization has been proposed for deterministic nonlinear [11] and hybrid systems [16]. We extend the method to stochastic systems using a weak formulation of the Fokker-Planck equation. As a cost function, we propose to use the Christoffel polynomial, which can be estimated from data.

We consider the problem of designing a control policy for an infinite-horizon discounted cost Markov decision process $\mathcal{M}$ when we only have access to an approximate model $\hat{\mathcal{M}}$. How well does an optimal policy $\hat{\pi}^{\star}$ of the approximate model perform when used in the original model $\mathcal{M}$? We answer this question by bounding a weighted norm of the difference between the value function of $\hat{\pi}^\star $ when used in $\mathcal{M}$ and the optimal value function of $\mathcal{M}$. We then extend our results and obtain potentially tighter upper bounds by considering affine transformations of the per-step cost. We further provide upper bounds that explicitly depend on the weighted distance between cost functions and weighted distance between transition kernels of the original and approximate models. We present examples to illustrate our results.

As a primary contribution, we present a convergence theorem for stochastic iterations, and in particular, Q-learning iterates, under a general, possibly non-Markovian, stochastic environment. Our conditions for convergence involve an ergodicity and a positivity criterion. We provide a precise characterization on the limit of the iterates and conditions on the environment and initializations for convergence. As our second contribution, we discuss the implications and applications of this theorem to a variety of stochastic control problems with non-Markovian environments involving (i) quantized approximations of fully observed Markov Decision Processes (MDPs) with continuous spaces (where quantization break down the Markovian structure), (ii) quantized approximations of belief-MDP reduced partially observable MDPS (POMDPs) with weak Feller continuity and a mild version of filter stability (which requires the knowledge of the model by the controller), (iii) finite window approximations of POMDPs under a uniform controlled filter stability (which does not require the knowledge of the model), and (iv) for multi-agent models where convergence of learning dynamics to a new class of equilibria, subjective Q-learning equilibria, will be studied. In addition to the convergence theorem, some implications of the theorem above are new to the literature and others are interpreted as applications of the convergence theorem. Some open problems are noted.

Decentralized stochastic control (DSC) considers the optimal control problem of a multi-agent system. However, DSC cannot be solved except in the special cases because the estimation among the agents is generally intractable. In this work, we propose memory-limited DSC (ML-DSC), in which each agent compresses the observation history into the finite-dimensional memory. Because this compression simplifies the estimation among the agents, ML-DSC can be solved in more general cases based on the mean-field control theory. We demonstrate ML-DSC in the general LQG problem. Because estimation and control are not clearly separated in the general LQG problem, the Riccati equation is modified to the decentralized Riccati equation, which improves estimation as well as control. Our numerical experiment shows that the decentralized Riccati equation is superior to the conventional Riccati equation.

There has been much recent interest in screening populations for an infectious disease. Here, we present a stochastic-control model, wherein the optimum screening policy is provably difficult to find, but wherein Thompson sampling has provably optimal performance guarantees in the form of Bayesian regret. Thompson sampling seems applicable especially to diseases, for which we do not understand the dynamics well, such as to the super-spreading COVID-19.

Bidding in real-time auctions can be a difficult stochastic control task; especially if underdelivery incurs strong penalties and the market is very uncertain. Most current works and implementations focus on optimally delivering a campaign given a reasonable forecast of the market. Practical implementations have a feedback loop to adjust and be robust to forecasting errors, but no implementation, to the best of our knowledge, uses a model of market risk and actively anticipates market shifts. Solving such stochastic control problems in practice is actually very challenging. This paper proposes an approximate solution based on a Recurrent Neural Network (RNN) architecture that is both effective and practical for implementation in a production environment. The RNN bidder provisions everything it needs to avoid missing its goal. It also deliberately falls short of its goal when buying the missing impressions would cost more than the penalty for not reaching it.