arXiv subjects

Serdar Yuksel

Publications and source records attributed to Serdar Yuksel.

At least 19 recordsLinked to original sources

arXiv2026

Approximations and Learning for Decentralized Stochastic Control and Near Optimal Finite Window Policies

Decentralized stochastic control problems are difficult to study due to information structure dependent subtleties, which prevent many classical methods in stochastic control from being applicable. In this paper we consider such problems with general standard Borel spaces under two related information structures. (a) the one-step delayed information sharing pattern (OSDISP) where agents share their information with one-step delay, and (b) the $K$-step periodic information sharing pattern (KSPISP), where information is shared periodically. It is known that OSDISP and KSPISP problems admit a centralized reduction where the agents view the problem from the perspective of a centralized controller that uses the common information to prescribe function valued actions (local policies) which map each agent's private information to an optimal action in the original problem. We provide rigorous approximation results and performance bounds for the KSPISP and OSDISP problems, which results from replacing the full common information by a finite sliding window of information and we establish near optimality of such policies. The latter depends on a predictor stability condition in expected total variation. As a further contribution, we show that under the information structures provided, corresponding Q-learning algorithms (in quantized or finite memory forms) converge asymptotically to near optimal solutions. While restrictive and hypothetical conditions have been presented in the literature, our contributions are thus to provide, to our knowledge, the first explicit conditions and rigorous approximation and learning results for such decentralized problems with general spaces.

math.OC↗

arXiv2026

Reinforcement Learning for Discounted and Ergodic Control of Diffusion Processes

This paper develops a quantized Q-learning algorithm for the optimal control of controlled diffusion processes on $\mathbb{R}^d$ under both discounted and ergodic (average) cost criteria. We first establish near-optimality of finite-state MDP approximations to discrete-time discretizations of the diffusion, then introduce a quantized Q-learning scheme and prove its almost-sure convergence to near-optimal policies for the finite MDP. These policies, when interpolated to continuous time, are shown to be near-optimal for the original diffusion model under discounted costs and -- via a vanishing-discount argument -- also under ergodic costs for sufficiently small discount factors. The analysis applies under mild conditions (Lipschitz dynamics, non-degeneracy, bounded continuous costs, and Lyapunov stability for ergodic case) without requiring prior knowledge of the system dynamics or restrictions on control policies (beyond admissibility). Our results complement recent work on continuous-time reinforcement learning for diffusions by providing explicit near-optimality rates and extending rigorous guarantees both for discounted cost and ergodic cost criteria for diffusions with unbounded state space.

math.OC↗

arXiv2025

Kernel Mean Embedding Topology: Weak and Strong Forms for Stochastic Kernels and Implications for Model Learning

We introduce a novel topology, called Kernel Mean Embedding Topology, for stochastic kernels, in a weak and strong form. This topology, defined on the spaces of Bochner integrable functions from a signal space to a space of probability measures endowed with a Hilbert space structure, allows for a versatile formulation. This construction allows one to obtain both a strong and weak formulation. (i) For its weak formulation, we highlight the utility on relaxed policy spaces, and investigate connections with the Young narrow topology and Borkar (or $ w^* $)-topology, and establish equivalence properties. We report that, while both the $ w^* $-topology and kernel mean embedding topology are relatively compact, they are not closed. Conversely, while the Young narrow topology is closed, it lacks relative compactness. (ii) We show that the strong form provides an appropriate formulation for placing topologies on spaces of models characterized by stochastic kernels with explicit robustness and learning theoretic implications on optimal stochastic control under discounted or average cost criteria. (iii) We thus show that this topology possesses several properties making it ideal to study optimality and approximations (under the weak formulation) and robustness (under the strong formulation) for many applications.

eess.SY↗

arXiv2025

Near Optimality of Discrete-Time Approximations for Controlled McKean-Vlasov and Large Interacting Particle Diffusions

We study stochastic optimal control problems for (possibly degenerate) McKean-Vlasov controlled diffusions and obtain discrete-time as well as finite interacting particle approximations. (i) Via continuity of the expected cost in control policy by endowing the space of relaxed policies with a compact weak topology, we prove near-optimality of piecewise-constant policies which leads to a discrete-time model. We show that the discrete-time value functions (for finite-horizon and discounted infinite-horizon) converge to their continuous-time counterparts as the timestep converges to zero. In particular, we establish that optimal policies for the discrete-time model exists and they are near-optimal for the original continuous-time problem. (ii) We then extend these approximation and near-optimality results to $N$-particle interacting systems under centralized or decentralized mean-field sharing information structure, proving that the discrete-time McKean-Vlasov policy is asymptotically optimal as $N\to \infty$ and the time step goes to zero. Using discrete-time approximation as an intermediate step leads to complementary conditions compared to those in the literature. (iii) We thus develop a unified approximation framework for McKean-Vlasov optimal control problems via discrete-time McKean-Vlasov control problems (and associated numerical methods such as finite model approximations), and we also show near optimality of such approximate policy solutions for the $N$-agent interacting models under centralized and decentralized control.

math.OC↗

arXiv2025

Quantizer Design for Finite Model Approximations, Model Learning, and Quantized Q-Learning for MDPs with Unbounded Spaces

In this paper, for Markov decision processes (MDPs) with unbounded state spaces we present refined upper bounds presented in [Kara et. al. JMLR'23] on finite model approximation errors via optimizing the quantizers used for finite model approximations. We also consider implications on quantizer design for quantized Q-learning and empirical model learning, and the performance of policies obtained via Q-learning where the quantized state is treated as the state itself. We highlight the distinctions between planning, where approximating MDPs can be independently designed, and learning (either via Q-learning or empirical model learning), where approximating MDPs are restricted to be defined by invariant measures of Markov chains under exploration policies, leading to significant subtleties on quantizer design performance, even though asymptotic near optimality can be established under both setups. In particular, under Lyapunov growth conditions, we obtain explicit upper bounds which decay to zero as the number of bins approaches infinity

math.OC↗

arXiv2025

Sliding Window Codes: Near-Optimality and Q-Learning for Zero-Delay Coding

We study the problem of zero-delay coding for the transmission of a Markov source over a noisy channel with feedback and present a reinforcement learning solution which is guaranteed to achieve near-optimality. To this end, we formulate the problem as a Markov decision process (MDP) where the state is a probability-measure valued predictor/belief and the actions are quantizer maps. This MDP formulation has been used to show the optimality of certain classes of encoder policies in prior work, but their computation is prohibitively complex due to the uncountable nature of the constructed state space and the lack of minorization or strong ergodicity results. These challenges invite rigorous reinforcement learning methods, which entail several open questions: can we approximate this MDP with a finite-state one with some performance guarantee? Can we ensure convergence of a reinforcement learning algorithm for this approximate MDP? What regularity assumptions are required for the above to hold? We address these questions as follows: we present an approximation of the belief MDP using a sliding finite window of channel outputs and quantizers. Under an appropriate notion of predictor stability, we show that policies based on this finite window are near-optimal, in the sense that the lowest distortion achievable by such a policy approaches the true lowest distortion as the window length increases. We give sufficient conditions for predictor stability to hold. Finally, we propose a Q-learning algorithm which provably converges to a near-optimal policy and provide a detailed comparison of~the sliding finite window scheme with another approximation scheme which quantizes the belief MDP in a nearest neighbor fashion.

math.OC↗

arXiv2025

Existence of $ε$-Nash Equilibria in Nonzero-Sum and Zero-Sum Markov Games with Standard Borel Spaces via Finite Model Approximations

Establishing the existence of exact or near Markov or stationary perfect Nash equilibria in nonzero-sum Markov games over Borel spaces is a challenging problem with limited positive results. Motivated by problems in multi-agent and Bayesian learning, this paper demonstrates the existence of approximate Markov and stationary Nash equilibria for such games under mild regularity conditions. Our approach is constructive: For both compact and non-compact state spaces, we approximate the Borel model with finite state-action models and show that their equilibria correspond to $ε$-equilibria for the original game. Compared with previous results in the literature, which we comprehensively review, we provide more general and complementary conditions, along with explicit approximation models whose equilibria are $ε$-equilibria for the original model. For completeness, we also study the approximation of zero-sum Markov games and Markov teams to highlight the key differences between zero-sum and nonzero-sum settings. In particular, while for zero-sum and team games, joint weak (Feller) continuity of the transition kernel is sufficient (as the value function is continuous), this is not the case for general nonzero-sum games.

eess.SY↗

arXiv2025

Quantum Markov Decision Processes: Dynamic and Semi-Definite Programs for Optimal Solutions

In this paper, building on the formulation of quantum Markov decision processes (q-MDPs) presented in our previous work [{\sc N.~Saldi, S.~Sanjari, and S.~Yüksel}, {\em Quantum Markov Decision Processes: General Theory, Approximations, and Classes of Policies}, SIAM Journal on Control and Optimization, 2024], our focus shifts to the development of semi-definite programming approaches for optimal policies and value functions of both open-loop and classical-state-preserving closed-loop policies. First, by using the duality between the dynamic programming and the semi-definite programming formulations of any q-MDP with open-loop policies, we establish that the optimal value function is linear and there exists a stationary optimal policy among open-loop policies. Then, using these results, we establish a method for computing an approximately optimal value function and formulate computation of optimal stationary open-loop policy as a bi-linear program. Next, we turn our attention to classical-state-preserving closed-loop policies. Dynamic programming and semi-definite programming formulations for classical-state-preserving closed-loop policies are established, where duality of these two formulations similarly enables us to prove that the optimal policy is linear and there exists an optimal stationary classical-state-preserving closed-loop policy. Then, similar to the open-loop case, we establish a method for computing the optimal value function and pose computation of optimal stationary classical-state-preserving closed-loop policies as a bi-linear program.

quant-ph↗

arXiv2025

Discrete-Time Approximations of Controlled Diffusions with Infinite Horizon Discounted and Average Cost

We present discrete-time approximation of optimal control policies for infinite horizon discounted/ergodic control problems for controlled diffusions in $\Rd$\,. In particular, our objective is to show near optimality of optimal policies designed from the approximating discrete-time controlled Markov chain model, for the discounted/ergodic optimal control problems, in the true controlled diffusion model (as the sampling period approaches zero). To this end, we first construct suitable discrete-time controlled Markov chain models for which one can compute optimal policies and optimal values via several methods (such as value iteration, convex analytic method, reinforcement learning etc.). Then using a weak convergence technique, we show that the optimal policy designed for the discrete-time Markov chain model is near-optimal for the controlled diffusion model as the discrete-time model approaches the continuous-time model. This provides a practical approach for finding near-optimal control policies for controlled diffusions. Our conditions complement existing results in the literature, which have been arrived at via either probabilistic or PDE based methods.

math.OC↗

arXiv2025

Near Optimal Approximations and Finite Memory Policies for POMPDs with Continuous Spaces

We study an approximation method for partially observed Markov decision processes (POMDPs) with continuous spaces. Belief MDP reduction, which has been the standard approach to study POMDPs requires rigorous approximation methods for practical applications, due to the state space being lifted to the space of probability measures. Generalizing recent work, in this paper we present rigorous approximation methods via discretizing the observation space and constructing a fully observed finite MDP model using a finite length history of the discrete observations and control actions. We show that the resulting policy is near-optimal under some regularity assumptions on the channel, and under certain controlled filter stability requirements for the hidden state process. Furthermore, by quantizing the measurements, we are able to utilize refined filter stability conditions. We also provide a Q learning algorithm that uses a finite memory of discretized information variables, and prove its convergence to the optimality equation of the finite fully observed MDP constructed using the approximation method.

math.OC↗

arXiv2024

Partially Observed Optimal Stochastic Control: Regularity, Optimality, Approximations, and Learning

In this review/tutorial article, we present recent progress on optimal control of partially observed Markov Decision Processes (POMDPs). We first present regularity and continuity conditions for POMDPs and their belief-MDP reductions, where these constitute weak Feller and Wasserstein regularity and controlled filter stability. These are then utilized to arrive at existence results on optimal policies for both discounted and average cost problems, and regularity of value functions. Then, we study rigorous approximation results involving quantization based finite model approximations as well as finite window approximations under controlled filter stability. Finally, we present several recent reinforcement learning theoretic results which rigorously establish convergence to near optimality under both criteria.

math.OC↗

arXiv2024

Quantum Markov Decision Processes: General Theory, Approximations, and Classes of Policies

In this paper, the aim is to develop a quantum counterpart to classical Markov decision processes (MDPs). Firstly, we provide a very general formulation of quantum MDPs with state and action spaces in the quantum domain, quantum transitions, and cost functions. Once we formulate the quantum MDP (q-MDP), our focus shifts to establishing the verification theorem that proves the sufficiency of Markovian quantum control policies and provides a dynamic programming principle. Subsequently, a comparison is drawn between our q-MDP model and previously established quantum MDP models (referred to as QOMDPs) found in the literature. Furthermore, approximations of q-MDPs are obtained via finite-action models, which can be formulated as QOMDPs. Finally, classes of open-loop and classical-state-preserving closed-loop policies for q-MDPs are introduced, along with structural results for these policies. In summary, we present a novel quantum MDP model aiming to introduce a new framework, algorithms, and future research avenues. We hope that our approach will pave the way for a new research direction in discrete-time quantum control.

quant-ph↗

arXiv2024

Reinforcement Learning for Near-Optimal Design of Zero-Delay Codes for Markov Sources

In the classical lossy source coding problem, one encodes long blocks of source symbols that enables the distortion to approach the ultimate Shannon limit. Such a block-coding approach introduces large delays, which is undesirable in many delay-sensitive applications. We consider the zero-delay case, where the goal is to encode and decode a finite-alphabet Markov source without any delay. It has been shown that this problem lends itself to stochastic control techniques, which lead to existence, structural, and general structural approximation results. However, these techniques so far have resulted only in computationally prohibitive algorithmic implementations for code design. To address this problem, we present a reinforcement learning design algorithm and rigorously prove its asymptotic optimality. In particular, we show that a quantized Q-learning algorithm can be used to obtain a near-optimal coding policy for this problem. The proof builds on recent results on quantized Q-learning for weakly Feller controlled Markov chains whose application necessitates the development of supporting technical results on regularity and stability properties, and relating the optimal solutions for discounted and average cost infinite horizon criteria problems. These theoretical results are supported by simulations.

cs.IT↗

arXiv2024

Near Optimality of Lipschitz and Smooth Policies in Controlled Diffusions

For optimal control of diffusions under several criteria, due to computational or analytical reasons, many studies have a apriori assumed control policies to be Lipschitz or smooth, often with no rigorous analysis on whether this restriction entails loss. While optimality of Markov/stationary Markov policies for expected finite horizon/infinite horizon (discounted/ergodic) cost and cost-up-to-exit time optimal control problems can be established under certain technical conditions, an optimal solution is typically only measurable in the state (and time, if the horizon is finite) with no apriori additional structural properties. In this paper, building on our recent work [S. Pradhan and S. Yüksel, Continuity of cost in Borkar control topology and implications on discrete space and time approximations for controlled diffusions under several criteria, Electronic Journal of Probability (2024)] establishing the regularity of optimal cost on the space of control policies under the Borkar control topology for a general class of diffusions, we establish near optimality of smooth/Lipschitz continuous policies for optimal control under expected finite horizon, infinite horizon discounted/average, and up-to-exit time cost criteria. Under mild assumptions, we first show that smooth/Lipschitz continuous policies are dense in the space of Markov/stationary Markov policies under the Borkar topology. Then utilizing the continuity of optimal costs as a function of policies on the space of Markov/stationary policies under the Borkar topology, we establish that optimal policies can be approximated by smooth/Lipschitz continuous policies with arbitrary precision. While our results are extensions of our recent work, the practical significance of an explicit statement and accessible presentation dedicated to Lipschitz and smooth policies, given their prominence in the literature, motivates our current paper.

math.OC↗

arXiv2024

Q-Learning for Stochastic Control under General Information Structures and Non-Markovian Environments

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.

math.OC↗

arXiv2023

Robustness of Stochastic Optimal Control to Approximate Diffusion Models under Several Cost Evaluation Criteria

In control theory, typically a nominal model is assumed based on which an optimal control is designed and then applied to an actual (true) system. This gives rise to the problem of performance loss due to the mismatch between the true model and the assumed model. A robustness problem in this context is to show that the error due to the mismatch between a true model and an assumed model decreases to zero as the assumed model approaches the true model. We study this problem when the state dynamics of the system are governed by controlled diffusion processes. In particular, we will discuss continuity and robustness properties of finite horizon and infinite-horizon $α$-discounted/ergodic optimal control problems for a general class of non-degenerate controlled diffusion processes, as well as for optimal control up to an exit time. Under a general set of assumptions and a convergence criterion on the models, we first establish that the optimal value of the approximate model converges to the optimal value of the true model. We then establish that the error due to mismatch that occurs by application of a control policy, designed for an incorrectly estimated model, to a true model decreases to zero as the incorrect model approaches the true model. We will see that, compared to related results in the discrete-time setup, the continuous-time theory will let us utilize the strong regularity properties of solutions to optimality (HJB) equations, via the theory of uniformly elliptic PDEs, to arrive at strong continuity and robustness properties.

math.OC↗

arXiv2023

Approximations and Learning for Continuous State and Action MDPs under Average Cost Criteria

In this paper, for Markov Decision Processes (MDPs) with standard Borel spaces, (i) we first provide a discretization based approximation method for MDPs with continuous spaces under average cost criteria, and provide error bounds for approximations when the dynamics are only weakly continuous (for asymptotic convergence of errors as the grid sizes vanish) or Wasserstein continuous (with a rate in approximation as the grid sizes vanish) under certain ergodicity assumptions. In particular, we relax the total variation condition given in prior work to weak continuity or Wasserstein continuity. (ii) We provide synchronous and asynchronous (quantized) Q-learning algorithms for continuous spaces via quantization (where the quantized state is taken to be the actual state in corresponding Q-learning algorithms presented in the paper), and establish their convergence. (iii) We finally show that the convergence is to the optimal Q values of a finite approximate model constructed via quantization, which implies near optimality of the arrived solution.

math.OC↗

arXiv2022

Stochastic Observability and Filter Stability under Several Criteria

Despite being a foundational concept of modern systems theory, there have been few studies on observability of non-linear stochastic systems under partial observations. In this paper, we introduce a definition of observability for stochastic non-linear dynamical systems which involves an explicit functional characterization. To justify its operational use, we establish that this definition implies filter stability under mild continuity conditions: an incorrectly initialized non-linear filter is said to be stable if the filter eventually corrects itself with the arrival of new measurement information. Numerous examples are presented and a detailed comparison with the literature is reported. We also establish implications for various criteria for filter stability under several notions of convergence such as weak convergence, total variation, and relative entropy. These findings are connected to robustness and approximations in partially observed stochastic control.

math.PR↗