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The paper concerns the stochastic approximation recursion, \[ \theta_{n+1}= \theta_n + \alpha_{n + 1} f(\theta_n, \Phi_{n+1}) \,,\quad n\ge 0, \] where the {\em estimates} $$\{ \theta_n\} $$ evolve on $$\Re^d$$, and $$\bfPhi \eqdef \{ \Phi_n \}$$ is a stochastic process on a general state space, satisfying a conditional Markov property that allows for parameter-dependent noise. In addition to standard Lipschitz assumptions and conditions on the vanishing step-size sequence, it is assumed that the associated \textit{mean flow} $$ \ddt \odestate_t = \barf(\odestate_t)$$ is globally asymptotically stable, with stationary point denoted $$\theta^*$$. The main results are established under additional conditions on the mean flow and an extension of the Donsker-Varadhan Lyapunov drift condition known as~(DV3): (i) A Lyapunov function is constructed for the joint process $$\{\theta_n,\Phi_n\}$$ that implies convergence of the estimates in $$L_4$$. (ii) A functional central limit theorem (CLT) is established, as well as the usual one-dimensional CLT for the normalized error. Moment bounds combined with the CLT imply convergence of the normalized covariance $$\Expect [ z_n z_n^\transpose ]$$ to the asymptotic covariance $$\SigmaTheta$$ in the CLT, where $$z_n\eqdef (\theta_n-\theta^*)/\sqrt{\alpha_n}$$. (iii) The CLT holds for the normalized averaged parameters $$\zPR_n\eqdef \sqrt{n} (\thetaPR_n -\theta^*)$$, with $$\thetaPR_n \eqdef n^{-1} \sum_{k=1}^n\theta_k$$, subject to standard assumptions on the step-size. Moreover, the covariance of $$\zPR_n$$ converges to $$\SigmaPR$$, the minimal covariance of Polyak and Ruppert. (iv) An example is given where $$f$$ and $$\barf$$ are linear in $$\theta$$, and $$\bfPhi$$ is a geometrically ergodic Markov chain but does not satisfy~(DV3). While the algorithm is convergent, the second moment of $$\theta_n$$ is unbounded and in fact diverges. 

Sample complexity bounds are a common performance metric in the Reinforcement Learning literature. In the discounted cost, infinite horizon setting, all of the known bounds can be arbitrarily large, as the discount factor approaches unity. These results seem to imply that a very large number of samples is required to achieve an epsilon-optimal policy. The objective of the present work is to introduce a new class of algorithms that have sample complexity uniformly bounded over all discount factors. One may argue that this is impossible, due to a recent min-max lower bound. The explanation is that these prior bounds concern value function approximation and not policy approximation. We show that the asymptotic covariance of the tabular Q-learning algorithm with an optimized step-size sequence is a quadratic function of a factor that goes to infinity, as discount factor approaches 1; an essentially known result. The new relative Q-learning algorithm proposed here is shown to have asymptotic covariance that is uniformly bounded over all discount factors. 

This paper examines the problem of real-time optimization of networked systems and develops online algorithms that steer the system towards the optimal trajectory without explicit knowledge of the system model. The problem is modeled as a dynamic optimization problem with time-varying performance objectives and engineering constraints. The design of the algorithms leverages the online zero-order primal-dual projected-gradient method. In particular, the primal step that involves the gradient of the objective function (and hence requires a networked systems model) is replaced by its zero-order approximation with two function evaluations using a deterministic perturbation signal. The evaluations are performed using the measurements of the system output, hence giving rise to a feedback interconnection, with the optimization algorithm serving as a feedback controller. The paper provides some insights on the stability and tracking properties of this interconnection. Finally, the paper applies this methodology to a real-time optimal power flow problem in power systems, and shows its efficacy on the IEEE 37-node distribution test feeder for reference power tracking and voltage regulation. 

Stochastic approximation (SA) algorithms are recursive techniques used to obtain the roots of functions that can be expressed as expectations of a noisy parameterized family of functions. In this paper two new SA algorithms are introduced: 1) PolSA, an extension of Polyak's momentum technique with a specially designed matrix momentum, and 2) NeSA, which can either be regarded as a variant of Nesterov's acceleration method, or a simplification of PolSA. The rates of convergence of SA algorithms is well understood. Under special conditions, the mean square error of the parameter estimates is bounded by σ 2 /n+o(1/n), where σ 2 ≥ 0 is an identifiable constant. If these conditions fail, the rate is typically sub-linear. There are two well known SA algorithms that ensure a linear rate, with minimal value of variance, σ 2 : the Ruppert-Polyak averaging technique, and the stochastic Newton-Raphson (SNR) algorithm. It is demonstrated here that under mild technical assumptions, the PolSA algorithm also achieves this optimality criteria. This result is established via novel coupling arguments: It is shown that the parameter estimates obtained from the PolSA algorithm couple with those of the optimal variance (but computationally more expensive) SNR algorithm, at a rate O(1/n 2 ). The newly proposed algorithms are extended to a reinforcement learning setting to obtain new Q-learning algorithms, and numerical results confirm the coupling of PolSA and SNR. 

The authors develop a theory characterizing optimal stopping times for discrete-time ergodic Markov processes with discounted rewards. The theory differs from prior work by its view of per-stage and terminal reward functions as elements of a certain Hilbert space. In addition to a streamlined analysis establishing existence and uniqueness of a solution to Bellman's equation, this approach provides an elegant framework for the study of approximate solutions. In particular, the authors propose a stochastic approximation algorithm that tunes weights of a linear combination of basis functions in order to approximate a value function. They prove that this algorithm converges (almost surely) and that the limit of convergence has some desirable properties. The utility of the approximation method is illustrated via a computational case study involving the pricing of a path dependent financial derivative security that gives rise to an optimal stopping problem with a 100-dimensional state space