Markovian Function Learning

• Markovian Function Learning is the process of inferring value functions, reward mappings, and transition dynamics in systems governed by Markov processes, with applications in reinforcement learning and model checking.
• The methodology leverages Bayesian inference, data augmentation, and parameter-expanded algorithms to achieve accurate uncertainty quantification and efficient prediction in high-dimensional, noisy environments.
• Advanced techniques such as state merging, low-rank optimization, and active learning enable robust model identification and decision-making under stochastic and non-i.i.d. conditions.

Markovian Function Learning refers to the process of inferring (or learning) mappings—such as value functions, reward or transition models, control laws, abstractions, or dynamical patterns—that govern systems modeled via Markovian dynamics. This includes both classical and contemporary statistical, computational, and algorithmic frameworks, covering domains such as Markov decision processes (MDPs), partially observable or jump Markov systems, population processes, federated systems with Markovian data, and stochastic iterative optimization methods. Markovian function learning is foundational to a broad array of fields (reinforcement learning, computational biology, model checking, control, optimization), as it underpins both system modeling and decision making in environments with temporal or state-dependent dependencies.

1. Inverse Reinforcement Learning and Bayesian Markovian Inference

A central paradigm is the Bayesian inference of latent Markovian functions—most notably, the value function driving sequential decision making. In the Bayesian learning framework for noisy MDPs (Singh et al., 2012), observed state-action trajectories are assumed to result from a “noisy” version of an otherwise optimal policy, with the optimal value function $V$ being an unobserved latent vector. The inference task then becomes estimating $V$ (and potentially nuisance parameters) given the data, under Markovian transition operators. To address intractable likelihoods—arising due to multinomial probit-like models—data augmentation is employed: introducing latent utilities $W_t$ so that posterior computation reduces to tractable alternating updates for $\{W_t\}$ and $V$. Further efficiency is enabled via parameter-expanded data augmentation (PX-DA), which exploits scale and translation invariance in the likelihood, markedly improving MCMC convergence by jointly sampling $(V, z)$. PX-DA leverages transformations of the form $\varphi_z(y) = y/z_1 - z_2 \mathbf{1}$ and results in demonstrably lower sampling autocorrelation.

Empirical results (e.g., mimicking human Tetris play) highlight that Bayesian Markovian function learning yields predictive posterior distributions, naturally combines prior knowledge, and achieves uncertainty quantification in high-dimensional, noisy environments. This approach generalizes elegantly to large state spaces via basis function regression on $V$ and can adapt to structured environments and policy constraints.

2. Structure Discovery and Model Identification in Markov Processes

Markovian function learning extends to the problem of model identification for Markov or controlled systems from data in the absence of a known state space or explicit transition dynamics. Techniques such as state-merging algorithms (Mao et al., 2012), and L*-based active automata learning (Tappler et al., 2019), construct MDP or Markov chain models from observed input/output or test sequences.

For example, state merging for deterministic labeled Markov decision processes (DLMDPs) involves constructing a prefix tree acceptor from data and recursively merging nodes according to statistical tests (Hoeffding bounds) for compatibility of empirical transition distributions. This supports the robust inference of both probabilistic and nondeterministic system behaviors. The learned models are amenable to formal analysis (e.g., model checking of temporal logic properties, scheduler synthesis). Challenges include rigorous parameter selection (for the statistical test threshold $\epsilon$), data requirements for bisimulation-level recovery, and limitations in distinguishing highly similar nondeterministic transitions.

In active learning frameworks, such as L*-MDP algorithms (Tappler et al., 2019), the learner iteratively poses queries (membership, equivalence) to a system and constructs an observation table encoding system responses. Adaptations for Markovian output entail probabilistic tests and sampling, with rigorous guarantees of convergence in the limit to the true canonical MDP. These methods have demonstrated improved accuracy (e.g., in bisimilarity distance and temporal property agreement) over passive learning in empirical studies, though at higher computational cost.

3. Statistical and Algorithmic Approaches for High-dimensional and Non-i.i.d. Markov Data

A recurring theme is learning Markovian structures from large, high-dimensional, or non-i.i.d. data samples. For population models (Zechner et al., 2014), the goal is to infer propensities (e.g., reaction rates) in interacting particle systems, while distinguishing between intrinsic (Markovian event timing) and extrinsic (parameter variation across individuals) stochasticity. Hierarchical Bayesian models with sparsity-inducing priors on the squared coefficient of variation (SCV) are employed to determine the presence of significant extrinsic variability. Inference is performed via variational Bayesian EM with Monte Carlo updates, automatically identifying which reaction channels exhibit random variability versus those that are uniform across the population.

For Markov chains with large state spaces and low-rank transition matrices (Zhu et al., 2019), statistical learning exploits the low-rank property, using nuclear-norm penalized convex optimization and (for sharper estimates) nonconvex rank-constrained maximum likelihood solved by difference-of-convex (DC) programming. Theoretical analysis yields nearly minimax-optimal rates in KL divergence and Frobenius norm, and empirical studies (e.g., NYC taxi trip data) show that low-rank learning enables state aggregation and structure discovery that is not possible with naive frequentist estimators.

Federated learning with Markovian data streams (Huynh et al., 24 Mar 2025) addresses the consequences of non-i.i.d. data across distributed clients. Here, Markov mixing times directly govern sample complexity and convergence rates for SGD-style algorithms. While FL can still support collaborative learning, sample complexity is strictly increased relative to the i.i.d. scenario, but retains 1/M scaling with the number of clients; momentum-based variants mitigate “client drift” and amplify robustness.

4. Function Approximation: Linear, Nonlinear, and History-based Representations

Function approximation in Markovian environments spans linear approximators, non-linear function classes (e.g., MNL models, deep networks), and history or memory-based abstractions.

• In control and reinforcement learning, Markovian function learning with linear function approximation may address RL in POMDPs (Kara, 20 May 2025), where value and Q-function learning is performed in a finite-memory, approximately Markovian, state space; convergence and error bounds are rigorously established in terms of projection and filter stability errors.
• Nonlinear methods include multinomial logistic function approximation for transition probabilities in large MDPs (Park et al., 2024), where optimistic value-iteration-based algorithms with tight regret bounds are derived, and neural network-based models (Reeves et al., 2022) that learn transition rates from time series for CTMCs—successfully reconstructing high-dimensional, non-mass-action systems, as measured by MAE/MSE-type metrics.
• History-based or recurrent feature abstractions (Patil et al., 2022) (approximate information state, AIS) allow RL algorithms to bypass Markovian-memory restrictions, with formal guarantees. Theoretical performance is bounded by reward and transition prediction errors (measured in integral probability metrics such as MMD or Wasserstein). Empirical validation on MuJoCo tasks shows such architectures outperform both feedforward and simple RNN baselines.

5. Stochastic Approximation, Stability, and Dynamic Risk Measures

The learning of Markovian functions often requires stochastic approximation techniques operating under Markovian noise. The extension of ODE-based analyses and the Borkar–Meyn theorem to Markovian noise (Liu et al., 2024) enables almost sure stability and convergence analysis for TD and Q-learning algorithms, even with eligibility traces and non-i.i.d. state evolution. The key is controlling the “diminishing asymptotic rate of change” via strong law-of-large-numbers-type conditions.

In risk-aware Markovian learning (Ruszczynski et al., 2023), value functions and policies are learned using feature-based approximations with coherent Markov risk measures, which are non-linear operators that may not be differentiable (e.g., mean-semi-deviation, AVaR, or max). Mini-batch transition risk mappings enable unbiased learning, and structured policy improvement mitigates error sensitivity.

Learning in Markov jump linear systems (MJLSs) (Badfar et al., 2024) generalizes Q-learning via mode-dependent Q-functions, enabling model-free learning of optimal controllers. The algorithm achieves convergence (matching model-based Riccati solutions) and is unbiased with respect to excitation noise, bypassing the need for dynamics identification and coupled Riccati equations.

6. Future Directions, Open Problems, and Theoretical Implications

Markovian function learning encompasses open challenges across theory and application:

• Extensions to non-smooth and highly nonconvex RL settings (particularly with non-differentiable greedy operators) remain open.
• Non-i.i.d. and dependent data streams pose sample complexity and robustness challenges, motivating future work on variance reduction, adaptivity to mixing times, and asynchronous protocols.
• The automation of Lyapunov function construction, solution of Poisson’s equations, and stationary distribution estimation via deep learning (Qu et al., 22 Aug 2025) demonstrates the synergy of numerical approximation and classic Markov analysis, inviting further exploration in high-dimensional or infinite-dimensional chains.
• PAC-Bayesian analysis of stochastic iterative algorithms via Markovian process modeling (Sucker et al., 2024) yields non-asymptotic generalization bounds for convergence time and rate, offering a rigorous handle on empirical optimizer performance in diverse non-i.i.d. optimization settings.

The field thus offers a unified view of learning mechanisms and statistical inference across Markovian dynamical systems—melding approaches from reinforcement learning, probabilistic modeling, control, and deep learning—to address both Markov process identification and the automated learning of value, policy, and risk-mapping functions in high-dimensional and data-dependent environments.