Markov

We present EdMCGS (Event-driven Markov chain Gaussian Splatting), an end-to-end method for reconstructing dynamic 3D scenes from extreme-low-frame-rate RGB together with an event stream, which can then be rendered at any intermediate timestamp. Methods relying solely on RGB images generate numerous artifacts due to the lack of evidence from between consecutive frames. To supply this missing evidence, we model the scene motion as an event-driven Markov chain, in which the sparse RGB frames anchor the state at their own timestamps while the events recorded within an interval drive the transition across it. Since the transition reads the events of the current interval, it remains active at inference and produces the in-between motion of the 3D Gaussians directly from the events rather than by interpolation, which sets our method apart from prior work that uses events only as training-time supervision. The state is carried by a compact set of control points, each driven by the events sampled in the neighborhood of its own image projection, and a temporal local isometry term keeps the propagated motion locally rigid. Experiments on synthetic and real-world scenes show that EdMCGS outperforms both RGB-based and event-based baselines, while rendering in real time with far fewer Gaussians than the strongest event-based baseline. We release our source code and a new dataset at https://github.com/joseclipse/EdMCGS.

Constant-stepsize temporal-difference (TD) learning is attractive for policy evaluation, but inference from a single Markov trajectory must account for serial dependence and a stepsize-dependent stationary target. For fixed-stepsize linear TD, we establish a functional central limit theorem whose covariance retains the multiplicative component induced by the random TD matrix and the stationary iterate error. We then derive a joint functional limit for parallel Richardson--Romberg (RR) recursions driven by the same trajectory. A Brownian-bridge self-normalizer yields asymptotically pivotal confidence regions for prespecified state-value contrasts without estimating the long-run covariance or selecting a bandwidth or batch length. For such a contrast, the procedure admits a one-pass implementation whose memory does not grow with the trajectory length. At a fixed stepsize, the inferential center is the RR stationary target. We also study horizon-indexed designs in which the stepsize remains constant within each run and decreases across longer horizons. Under an explicit RR-dependent rate window, the residual RR target shift, multiplicative remainder, and initialization effect are negligible at the root-$n$ scale, yielding inference for the projected Bellman solution. Experiments on FrozenLake and Garnet illustrate stationary-target coverage, RR target correction, and the finite-sample behavior of the horizon-indexed design.

We reexamine the problem of verifying Markov chains with respect to step-bounded reachability probabilities. Prevailing approaches rely on encoding the state-transition matrix using either explicit or symbolic representations. While these approaches are effective for sparse transition dynamics, they scale less favorably in the dense regime. Our insight is to cast probabilistic model checking of Markov chains as computations over dense tensors. This methodology enables the use of off-the-shelf compiler toolchains for optimized execution of these tensor computations on hardware accelerators. We prove the soundness of the methodology of mapping probabilistic model checking to tensor computations. We implement our approach in a tool called Tessa . Empirical evaluation shows that Tessa unlocks massive speedups over state-of-theart methods on selected benchmarks from the literature.

An agent acting under partial observability must retain a recursively updateable statistic of history that restores the Markov property, but the smallest such statistic is generally unknown. We characterize this minimal Markov sufficient statistic for holonomy-cover decision processes, a structured POMDP class in which the visible dynamics are Markov and every realized visible transition applies a fixed permutation to a hidden mode. In particular, we construct the stable quotient, the coarsest observation-wise abstraction preserving one-step rewards and quotient successors, and prove that the pair of the current observation and stable class forms an exact finite Markov state. When the current class is correctly initialized, exact class tracking requires exactly the minimal memory symbols, in the sense that under reachability and pairwise decision separation at a maximizing observation, no arbitrary finite-memory controller can use fewer. Under resettable diagnostics, nearest-prototype class inference has exponentially decaying error, and a calibrate-then-restart reduction transfers finite-MDP guarantees to the recovered state. The results enable \emph{Holonomy Memory Reinforcement Learning}. It represents memory by the current stable class, updates it through ordered edge transports, identifies local class coordinates when diagnostics are available, and applies a standard finite-MDP RL backbone after synchronization. Experiments recover an exact compression from raw states to quotient states and achieve perfect paired-order accuracy with three decision-time memory states, matching the quotient oracle and outperforming the non-oracle baselines.

The problem of learning the graphical Markov blanket (MB) of a variable from data has applications in many areas such as structure learning for Bayesian networks and Markov random fields, causal discovery, and feature selection. However, a common assumption most methods make is that the conditional independencies in the distribution imply the same separation in the graphical structure -- also known as the faithfulness assumption. Unfortunately, this assumption can be violated by higher-order dependencies such as XOR and parity-type relations, and -- on finite samples -- by empirical violations that, in extreme cases, even induce spurious dependencies absent from the true distribution. Therefore, in this paper we propose a "k-order" relaxation of the faithfulness assumption that captures parity type relationships between k+2 variables. We then propose a proof of concept algorithm called k-order Markov blanket (kOMB) that uses this relaxation for MB discovery. Finally, we empirically show how kOMB can recover the MB of a variable under both true and empirical violations of faithfulness. Code available at: https://github.com/lklee9/k-order-Markov-blanket

Human cognition does not separate understanding and generation. A teacher at a whiteboard speaks and draws $\textit{together}$, each modality reshapes the other. In this paper, we bring this coupled loop to artificial systems. Masked Diffusion Models (MDMs) are ideally suited to this task, yet existing samplers either decode text and image interleavedly or independently update them in parallel branches that share only previous-step history, but not the other modality's latest decisions $\textit{within}$ the same step; combined with MDMs' inability to remask, cross-modal contradictions are neither detected nor repaired. We introduce $\textbf{Self-Correcting Coupled Markov Jump Processes (SC-CMJP)}$, a framework in which one modality's transition rates are functionals of the other modality's confidence score, as weighted by cross-modal attention. Furthermore, a remasking jump retracts commitments the moment cross-modal evidence turns against them. In conjunction with SC-CMJP, we introduce $\texttt{CO}_\texttt{2}\texttt{Jump}$ (Self-$\underline{\text{CO}}$rrecting $\underline{\text{CO}}$upled $\underline{\text{Jump}}$), a novel training-free single-pass sampler for joint multimodal geneneration. For training and evaluation purposes, we have created and will release three large-scale joint multimodal generation corpora: $\text{JEdit-1M}$, $\text{JMaze-200K}$, $\text{JNono-200K}$, with matching in- and out-of-distribution benchmarks. $\texttt{CO}_\texttt{2}\texttt{Jump}$ achieves best joint performance for image understanding and editing as well as visual reasoning (maze and nonogram solving). The performance of the sampler scales monotonically with the number of denoising steps, evidence that the benefits of cross-modal coupling $\textit{compound}$ across the trajectory. Project page: https://coupled-jump.github.io

Sampling from discrete Markov random fields (MRFs) is a hard problem. We study amplitude-encoded i.i.d. sampling for small MRFs where $2^n$ target probabilities are precomputed classically. This removes quantum exponential speedup but allows a clean comparison against classical MCMC based on independent circuit samples ($τ\approx 1$). Across 60 instances spanning five graph families (1k-step burn-in, 3k retained samples), the mean ESS ratios of Quantum to Single-Site Gibbs, Block Gibbs, Tuned-Block, and Parallel Tempering are $16.35$, $7.29$, $1.82$, and $1.79$, showing modern classical samplers substantially close this gap. Amortizing $O(2^n)$ preprocessing into wall-clock time, exact inverse-CDF sampling yields $17.7\text{M}$ ESS/s versus $488\text{K}$ ESS/s for the quantum sampler ($36\times$ mean rate, $153\times$ per-instance), confirming no wall-clock advantage. We characterize MCMC autocorrelation costs and benchmark amplitude-encoded state preparation at $n \in \{8,10,12\}$. An MPS scaling study ($n \le 40$) shows bond dimension $χ=32$ achieves $F=0.721\pm0.059$ at $n=40$. Finally, a matched-budget VQC vs. MPS comparison at $n \in \{8,10,12\}$ shows VQC fidelities fall far below MPS: $(F_{\mathrm{VQC}}, F_{\mathrm{MPS}}) = (0.31, 0.99), (0.21, 0.96), (0.17, 0.88)$ at compressions $10.7\times$, $34.1\times$, and $113.8\times$.

We propose a framework for the Markov chain (MC) choice model with panel data, including parameter estimation, personalized choice prediction, and personalized assortment optimization. In contrast to the traditional setting, which assumes that each transaction is independently drawn from a random utility model, our framework accounts for dependencies among transactions for the same customer in historical data, captured by partial-ordering preference information. To the best of our knowledge, our framework initiates the study of choice modeling with panel data under MC. As our primary result, we propose novel expectation-maximization (EM) algorithms for MC parameter estimation by incorporating partial-ordering-based customer preference information. On synthetic datasets and the sushi dataset, our EM algorithms outperform the traditional EM algorithm of Simsek and Topaloglu (Operations Research, 66, 2018) and multinomial-logit-based partial-order benchmarks adapted from Jagabathula and Vulcano (Management Science, 64, 2018). As our secondary contribution, we present hardness and computational results for conditional choice prediction and assortment optimization problems. These results complement our estimation framework and clarify the computational landscape of conditional choice and assortment optimization, which may be of independent interest.

While many explainable AI (XAI) methods have been proposed, most are not designed for time-series forecasting models and often rely on the implicit assumption that timestamp features are independent. This assumption ignores the fundamental property of temporal dependence and can lead to explanations that violate the sequential and causal structure of the data. We introduce \textsc{KARMA}, a method for explaining time-series predictors by constructing a Markov surrogate model that captures the temporal dependencies learned by the predictor. Our approach revolves around three main aspects: identifying the minimal history length $K$ that is predictively sufficient for the model, estimating the best-fitting $K$-order Markov transition kernel from the discretized history space, and a five-level global explanation hierarchy that can be derived from the Markov transition kernel, which we illustrate using real-world weather data (Beijing PM 2.5). We also certify using complex synthetic data with known true causal edges that KARMA (i) recovers the data causal structure as learned by the model via a controlled experiment and (ii) identifies temporal dependencies better than established attribution methods such as TimeSHAP.

We investigate how time-varying interactions, modeled via a Markov switching graph (MSG), impact the robustness of noisy multi-agent dynamics in both continuous- and discrete-time settings. Our focus is on the steady-state performance of consensus and leader-follower tracking dynamics subject to stochastic noise. Using the framework of Markov jump linear systems (MJLS), we derive expressions for the steady-state covariance of each agent's deviation from consensus and tracking error, respectively, and use them to quantify individual and group performance as a function of the interaction graphs and the switching dynamics. We extend established notions of robustness, certainty indices, and joint centrality from static graphs to the MSG setting. To gain analytical insight, we specialize our results to systems switching between two topologies and characterize how switching influences performance. Numerical simulations further illustrate how switching topologies affects system robustness in both coordination tasks.

Model Predictive Control (MPC) is widely used in industrial and robotic systems for enforcing constraints and embedding domain knowledge through finite-horizon optimization-based planning. However, despite these strengths, an MPC scheme typically does not yield optimal policies for sequential decision-making problems formulated as Markov Decision Processes (MDPs). Recent combinations of MPC with Reinforcement Learning (RL) alleviate this issue by treating MPC as a parameterized model of the optimal policy of an MDP and adjusting its parameters using data. While these approaches typically consider classical MDPs, many real-world problems include future information--such as forecasts, prices, or reference trajectories--at decision time, which must be included in the MDP state for optimal decision-making. Current MPC-RL approaches do not directly account for this augmented-state structure, raising the question of how to incorporate future information into MPC to obtain an optimal policy. This work establishes the structural requirements under which a parameterized MPC can exactly represent the optimal value functions and policy of an MDP with future information. We further demonstrate that such a parameterized MPC can serve as a structured function approximator, with its parameters learned using RL. The approach is illustrated on a point-mass racing task with future reference information.

We propose Markov Chain from Human Feedback (MCHF), an elementary approach for aligning generative models from pairwise human preferences. Unlike Reinforcement Learning from Human Feedback (RLHF), which reduces comparisons to a scalar reward, and Nash Learning from Human Feedback (NLHF), which preserves pairwise utilities through a KL-regularized minimax optimization, MCHF uses pairwise preferences directly to define a transition mechanism over model outputs. Given a pairwise utility $U(x,y)$, which quantifies human preference for $y$ over $x$, and a reference probability distribution $μ_{\mathsf{ref}}$, we define a Markov kernel $\mathsf{P}(x, dy)\propto \exp(U(x,y))μ_{\mathsf{ref}}(dy)$, and take the Markov chain starting from $μ_{\mathsf{ref}}$ as an iterative alignment procedure. We show that MCHF converges geometrically fast to the stationary distribution, with a convergence rate governed by the seminorm $\|U\|_\oplus=\inf_{g,f\in L^\infty(μ_{\mathsf{ref}})}\|U-g\oplus f\|_\infty$, which quantifies the non-transitive structure of the pairwise utility. We further show that a mirror-descent algorithm for NLHF satisfies an analogous structure-adaptive convergence guarantee. Finally, through a perturbation analysis, we prove that when $\|U\|_\oplus$ is small, MCHF and NLHF agree up to first order around an RLHF solution, which yields a unified view of reward-based, game-theoretic, and Markovian approaches to alignment. In particular, for two natural algorithms that converge to the MCHF/NLHF equilibria, we show that the first step of MCHF and NLHF recovers the RLHF solution based on the column-sum reward $\hat{f}(y)=\int μ_{\mathsf{ref}}(dx) U(x, y)$, and starting from the second iteration, both algorithms incorporate the same linear functional of the residual $U-(-\hat f)\oplus \hat f$, which captures the non-transitive structure of the pairwise utility $U$.

Score-based diffusion models typically use Brownian perturbations, which provide tractable reverse-time dynamics but impose memoryless noising. We introduce Volterra generative models, a continuous-time score-based framework whose forward process injects path-dependent noise through fractional kernels. To handle the non-Markovian and non-semimartingale dynamics, we construct finite-dimensional Markovian lifts using Gaussian quadrature in both regimes and a hybrid finite-difference exponential approximation in the smooth regime. We prove squared error bounds, derive an augmented linear-Gaussian forward process, and show that the learning can remain data-dimensional by considering residual states and analytic auxiliary Gaussian scores. We also identify covariance and reverse-time degeneracies caused by shared Brownian factors and signed smooth-regime weights. The degeneracy motivates stabilized conditioning and, for stiff larger lifts, a Gaussian-bridge reconstruction sampler. Experiments on MNIST and CIFAR-10 show that persistent fractional perturbations with small Markovian lifts can improve score-based generation on MNIST and provide a promising extension to natural images, while the bridge sampler provides a stability mechanism for larger lifts.

This paper considers neural channel state information (CSI) compression for time-varying massive multiple-input multiple-output (MIMO) channels in frequency division duplex (FDD) systems with limited feedback resources. The main challenge lies in obtaining a compact and efficient representation of the CSI given that it exhibits strong temporal correlation across successive snapshots. Existing memoryless compression models do not exploit this property, while simple temporal extensions often incorporate multiple observations without explicitly modeling the latent dynamics. We propose a context-aware compression framework based on a k-memory Markov variational autoencoder (k-MMVAE), which uses a finite temporal window to capture the evolution of CSI in the latent space. The model introduces Markov-structured latent dynamics with finite memory, enabling efficient use of temporal dependencies for compression. Simulation results show that the proposed approach improves target CSI reconstruction performance compared to memoryless and weakly sequential baselines, particularly at low and moderate compression rates. These results suggest that explicit latent temporal modeling can provide an effective mechanism for CSI compression under limited feedback constraints.

Contrastive Reinforcement Learning (CRL) has seen recent success in a wide variety of goal-conditioned robotics tasks by learning structured representations of the dynamics. However, despite its success in locomotion and simpler control domains, CRL often struggles in interaction-rich manipulation. We argue that a key source of this difficulty is object-centric interaction, such as contact or grasping, that induces distinct changes in the underlying dynamic modes. In this work, we formulate manipulation dynamics as a piecewise-smooth Markov process and show that interaction-induced mode changes create piecewise nonlinear reachability structures that are difficult for standard CRL energy functions to represent and plan over. Based on this analysis, we introduce Interaction-weighted Resampling (IWR). IWR performs interaction-aware resampling around phases before, during, and after interactions, encouraging the learned representation to preserve the mode boundaries that determine future reachability to capture multi-modal and piecewise nonlinear reachability. Across interaction-centric environments, including 2D dynamic control, robotic manipulation, and robot air hockey, IWR improves both sample efficiency and overall performance over prior CRL methods, with 19.8% average improvement in simulation. Finally, using a sim-to-real pipeline with policies trained by IWR, we demonstrate the first real-world goal-conditioned robot air hockey agent capable of hitting goals, improving success from 25% to 60%. Project Page: IWR-arxiv.github.io.

Learning a Markov transition model is not merely conditional density estimation: the learned object must be a valid transition kernel before it is iterated in downstream dynamics. This paper introduces a Doeblin-anchored contrastive chart, a statistical-to-dynamical coordinate framework for learning transition kernels from contrastive objectives. Given a restart law and an anchor strength, the chart mixes the target transition with the restart law. The resulting anchored kernel is simultaneously a Doeblin-minorized Markov kernel, the positive conditional law in a binary contrastive experiment, and an explicitly invertible coordinate for the original transition law. We prove that the anchored contrastive risk identifies the anchored transition density and calibrates excess risk to density error. Since inversion of a learned score may produce a signed or unnormalized object, we introduce a measurable Markovization operator that restores kernel validity while preserving integrated $L^1$ accuracy up to a constant factor. Oracle inequalities and Hölder--ReLU approximation bounds yield nonparametric rates for independent transition pairs. For stationary geometrically $β$-mixing trajectories, a conservative thinning-and-coupling extension yields the same reconstruction interface with an effective sample size. Occupancy-weighted perturbation bounds transfer one-step kernel error to finite-horizon marginal, path-law, and occupation-measure errors under explicit coverage.

Time series forecasting often requires learning nonlinear and time-delayed dependencies. A paradigmatic class of forecasting models are nonlinear vector autoregressive processes (NVAR), also known as next-generation reservoir computers (NG-RCs). These models approximate the Koopman operator on the space spanned by their explicit feature library. We consider the identifiability problem for learning Markovian nonlinear dynamical systems and show that the training error as a function of time resolution follows characteristic (pre-)asymptotic scaling laws. These laws depend on whether the feature library can represent the early Lie-series coefficients of the flow map (propagator) exactly or merely approximately. For dynamical systems governed by polynomial vector fields, we demonstrate the mechanism for NVAR/NG-RC models with monomial and Fourier feature libraries. We determine the dependence of the training error on the temporal resolution, the involved nonlinear degree, and the number of delay terms. While delay terms reduce the optimal one-step training error, they improve long-horizon forecasts only when the library provides sufficient nonlinearity. Thus, small training error coexists with weak generalization as the model class is mismatched to the true data-generating process. Numerical experiments on various chaotic dynamical systems confirm the theoretical predictions.

Under standard graphical assumptions, the Markov boundary of a target variable is the smallest set of features that renders every other feature redundant. Once the boundary is observed, the target is conditionally independent of the rest of the table. This is a tempting object for tabular prediction, since it names exactly the columns a model should need. Yet modern regressors are still trained on the full feature set. We ask whether the Markov boundary is genuinely useful for prediction on SCM3K, a 3,450-task synthetic SCM benchmark with feature counts from 40 to 1000 and six SCM families, evaluated with six regressors. The answer is more nuanced than the theory suggests. Restricting a regressor to the oracle boundary often improves prediction substantially, and the improvement grows as the feature space becomes larger and sparser. But the natural pipeline of recovering the boundary with causal discovery and training on the recovered mask does not deliver. Existing estimators exhaust the compute budget before reaching the regime where the boundary helps most, and even where they run they rarely beat the full feature set. We trace this to three causes. Discovery optimizes structural recovery rather than prediction. False negatives and false positives carry sharply asymmetric predictive cost. The exact boundary is only one of many feature sets that beat all features. We then develop what these facts imply for prediction-aligned feature selection and for tabular models that learn to use causal structure.

Heavy-tailed distributions are prevalent in performance evaluation, network traffic, and risk modeling. This behavior poses a fundamental challenge for modern deep generative models. Standard Variational Autoencoders (VAEs) employ Gaussian decoder likelihoods and Lipschitz-constrained neural networks, a combination that is structurally incapable of producing heavy-tailed outputs: the Gaussian tail decays exponentially, and Lipschitz continuity prevents the decoder from amplifying rare events from the latent space input to sufficiently overcome this decay. We provide both a theoretical characterization of this limitation and a controlled empirical demonstration using synthetic Pareto data across a grid of tail indices $α$ $\in$ {2, 3, 5, 30} and dimensions d $\in$ {1, 5, 10}. As a solution, we replace the Gaussian decoder with a Phase-Type (PH) distribution based on Markov chains, while keeping the encoder, latent space, and training procedure identical. PH distributions allow for arbitrarily precise approximations of any positive-valued distributions, including heavy-tailed families. Experiments showed that the PH-based model reduces tail Kolmogorov-Smirnov distance by up to x6 and extreme quantile error by up to x10 compared to the Gaussian baseline for heavy-tailed data. These results demonstrate that integrating Markov chain-based distributions into the decoder of a generative model institutes a principled and practically effective solution to the heavy-tail generation problem.

We propose MATE, a simple yet effective memory architecture for solving Contextual Markov Decision Processes (CMDPs), a family of MDPs parameterized by an unobserved context. In CMDPs, an optimal agent can adapt online by maintaining the posterior belief over contexts. MATE replaces this intractable posterior with a sum-aggregated memory, leveraging the posterior's permutation invariance to retain provably sufficient expressiveness. Compared to prior memory architectures, MATE avoids the growing per-step rollout cost of Transformers and the gradient issues commonly associated with Recurrent Neural Networks (RNNs). Extensive evaluations across diverse benchmarks demonstrate that MATE provides clear computational advantages while achieving performance comparable to standard sequence-model baselines.

MeanFlow enables one-step generation in continuous spaces by learning an average velocity over a time interval rather than the instantaneous velocity field of flow matching. However, discrete state spaces do not have smooth trajectories or spatial derivatives, so the continuous formulation does not directly apply. We introduce Discrete MeanFlow, which replaces the motion of a point with the transport of probability mass over finite states. Our key object is the conditional transition kernel of a continuous-time Markov chain (CTMC), from which we define a mean discrete rate that measures the average change in transition probability over a time interval. We prove a Discrete MeanFlow identity that relates this finite-interval rate to the instantaneous CTMC generator at the endpoint, with the Kolmogorov forward equation replacing the spatial chain rule of continuous MeanFlow. Based on this identity, we parameterize the transition kernel directly using a boundary-by-construction design that guarantees valid probability outputs and exact boundary conditions without auxiliary losses. Since the learned kernel is itself a probability distribution, generation reduces to a single forward pass followed by one categorical draw meaning no iterative denoising, ODE integration, or multi-step refinement is required. We validate the framework on exact finite-state Markov chains, where the learned kernel recovers the analytical ground truth to high precision, and on factorized synthetic sequence generation tasks with varying alphabet sizes and sequence lengths.

We propose and analyze a model-based bootstrap for transition kernels in finite controlled Markov chains (CMCs) with possibly nonstationary or history-dependent control policies, a setting that arises naturally in offline reinforcement learning (RL) when the behavior policy generating the data is unknown. We establish distributional consistency of the bootstrap transition estimator in both a single long-chain regime and the episodic offline RL regime. The key technical tools are a novel bootstrap law of large numbers (LLN) for the visitation counts and a novel use of the martingale central limit theorem (CLT) for the bootstrap transition increments. We extend bootstrap distributional consistency to the downstream targets of offline policy evaluation (OPE) and optimal policy recovery (OPR) via the delta method by verifying Hadamard differentiability of the Bellman operators, yielding asymptotically valid confidence intervals for value and $Q$-functions. Experiments on the RiverSwim problem show that the proposed bootstrap confidence intervals (CIs), especially the percentile CIs, outperform the episodic bootstrap and plug-in CLT CIs, and are often close to nominal ($50\%$, $90\%$, $95\%$) coverage, while the baselines are poorly calibrated at small sample sizes and short episode lengths.

Many reinforcement learning (RL) tasks have discrete action spaces, but most generative policy methods based on diffusion and flow matching are designed for continuous control. Meanwhile, generative policies usually rely heavily on offline datasets and offline-to-online RL is itself challenging, as the policy must improve from new interaction without losing useful behavior learned from static data. To address those challenges, we introduce DRIFT, an online fine-tuning method that updates an offline pretrained continuous-time Markov chain (CTMC) policy with an advantage-weighted discrete flow matching loss. To preserve useful pretrained knowledge, we add a path-space penalty that regularizes the full CTMC trajectory distribution, rather than only the final action distribution. For large discrete action spaces, we introduce a candidate-set approximation that updates the actor over a small subset of actions sampled from reference-policy rollouts and uniform exploration. Our theoretical analysis shows that the candidate-set error is controlled by missing target probability mass, and the induced CTMC generator error decreases as the candidate set covers more high-probability actions. Experiments on prevailing discrete action RL task show that our method provides stable offline-to-online improvement across all tasks, achieving the highest average score on Jericho with a simple GRU encoder while outperforming methods that use pretrained language models. Controlled experiments further confirm that the path-space penalty remains bounded during fine-tuning and that the CTMC generator adapts to shifted rewards faster than deterministic baselines. The candidate-set mechanism is supported by a stability analysis showing that the generator error decreases exponentially with candidate coverage.

We consider anonymous multi-agent path finding (MAPF) where a set of robots is tasked to travel to a set of targets on a finite, connected graph. We show that MAPF can be cast as a special class of multi-marginal optimal transport (MMOT) problems with an underlying Markovian structure, under which the exponentially large MMOT collapses to a linear program (LP) polynomial in size. Focusing on the anonymous setting, we establish conditions under which the corresponding LP is feasible, totally unimodular, and consequently, yields min-cost, integral $(\{0,1\})$ transports that do not overlap in both space and time. To adapt the approach to large-scale problems, we cast the MAPF-MMOT in a probabilistic framework via Schrödinger bridges. Under standard assumptions, we show that the Schrödinger bridge formulation reduces to an entropic regularization of the corresponding MMOT that admits an iterative Sinkhorn-type solution. The Schrödinger bridge, being a probabilistic framework, provides a shadow (fractional) transport that we use as a template to solve a reduced LP and demonstrate that it results in near-optimal, integral transports at a significant reduction in complexity. Extensive experiments highlight the optimality and scalability of the proposed approaches.

We investigate interactive trajectory planning subject to uncertainty in the decisions of surrounding agents. To control the ego-agent, we aim to first learn the decision distribution and solve a Stochastic Model Predictive Control (SMPC) problem. To account for errors in the learned distribution, we show that it is possible to utilize Probably Approximately Correct (PAC) learning in combination with Distributionally Robust (DR) optimization to obtain a solution which accounts for the errors induced by the learning model. The results indicate that our PAC learning-based DR-MPC framework provides a method to interpolate between a robust MPC and an omnipotent SMPC, based on the available number of samples.

Online reinforcement learning (RL) relies on the Markov property for guaranteed performance, but real-world applications often lack well-defined states given raw observed variables. While causal RL has attracted growing interest, existing work typically assumes Markovian states are provided and focuses on using causality to accelerate learning, leaving a fundamental gap: \emph{given a longitudinal causal graph over observed variables, how does one construct MDP states that provably satisfy the Markov property?} We address this by providing a procedure that constructs a provably minimal state representation. In deep RL, we observe that the minimal representation alone empirically fails to improve performance, indicating that neural networks cannot directly exploit Markovian minimality. To address this, we propose \textbf{MOSE} (Multi-Order State Exposure), which feeds multi-order historical state constructions into the same $Q$-function. MOSE consistently outperforms both the minimal state construction and single-window policies on common benchmarks and synthetic datasets. Including the minimal representation alongside MOSE can further improve performance. Our results establish a core principle for causal deep RL: minimal sufficiency is not enough, and \emph{controlled redundancy} is necessary to unlock the benefit of causal state information.

Modeling the dynamics of non-stationary stochastic systems requires balancing the representational power of deep learning with the mathematical transparency of classical models. While classical Markov transition operators provide explicit, theoretically grounded rules for system evolution, their empirical estimation collapses due to severe data sparsity when applied to high-resolution, high-noise environments. We explore this statistical barrier using financial time series as a canonical, real-world testbed. To overcome the degeneracy of empirical counting, we introduce a framework that utilizes neural networks strictly as parameterization engines to generate explicit, time-varying Markov transition matrices. By constraining the neural network to output its predictions as a formal stochastic operator, we maintain complete structural interpretability. We demonstrate that these learned operators successfully capture complex regime shifts: the state-conditioned model achieves mean row heterogeneity $\barρ = 0.0073$ while the state-free ablation collapses to exactly zero, and operator row entropy correlates with realized variance at $r = -0.62$ ($p \approx 10^{-251}$), revealing that high-volatility regimes homogenize transition dynamics rather than diversify them. Furthermore, rather than enforcing the Chapman-Kolmogorov equations as a rigid structural requirement, we repurpose them as a localized diagnostic tool to pinpoint specific temporal windows where first-order memory assumptions break down. Ultimately, this framework demonstrates how neural networks can be constrained to make rigorous, classical operator analysis viable for complex real-world time series.

Continual incorporation of new knowledge is essential for the long-term evolution of large language models (LLMs). Existing approaches typically rely on parameter-update algorithms to mitigate catastrophic forgetting, yet they suffer from fundamental limitations: 1) forgetting is unavoidable as the amount of newly injected knowledge grows; and 2) model updates are often irreversible. As modern LLMs become increasingly expressive, it is natural to question whether large-scale weight updates are necessary for acquiring a small amount of new knowledge. In this work, we propose a principled framework that models autoregressive language generation as a Markov process over tokens, where model memory is represented by a Markov transition matrix. Under this formulation, incorporating new knowledge/tokens corresponds to extending the state space, and preserving existing transitions guarantees retention of previously learned knowledge. We then prove a sample complexity bound for incorporating new tokens via a token-to-dictionary mapping strategy. In particular, for learning the transition behavior of each new token, the required number of samples scales linearly with the number of existing tokens it is mapped to. To realize this mapping, we propose an embedding-tuning algorithm that requires minimal parameter updates and induces zero forgetting. Experimental results further demonstrate the effectiveness of our method and validate our theoretical findings.

Ancestral sequence reconstruction (ASR) aims to infer extinct protein sequences at internal nodes of a phylogenetic tree. Classical ASR methods are typically based on continuous-time Markov substitution models, but they treat sites largely independently and handle insertions and deletions only weakly or not at all. We introduce a tree-conditioned edit-flow model for variable-length ASR. Given two descendant sequences and their branch distances to a shared ancestor, the model reconstructs the ancestor through paired bidirectional edit trajectories constrained to agree on a common ancestral state. On a benchmark of experimentally evolved sequences with only context-independent substitutions, the model does not match the accuracy of the best classical method, yet still achieves reasonable performance despite being trained on natural sequences that include insertions, deletions, and substitutions. On a benchmark of natural homologous sequences with abundant insertions and deletions, the model most accurately localizes inferred evolutionary change.

Offline goal-conditioned RL (GCRL) learns goal-reaching policies from static datasets, but real-world datasets are often partially observable and history-dependent, exhibiting a mix of Markovian and non-Markovian that violate standard RL assumptions. History-aware sequence models such as Decision Transformer (DT) are a natural fit for long-term dependency modeling, yet pure attention is inefficient and brittle when handling local Markovian structure and long-range context simultaneously. Although recent hybrid architectures (e.g., LSDT) introduce local extractors to improve local dependencies modeling, the fixed-window extraction cannot adapt its effective memory to varying dependency lengths in temporally heterogeneous settings, often truncating long-range context rather than compressing its content adaptively. Moreover, sequential offline GCRL faces a key bottleneck: under sparse rewards, return-to-go (RTG) becomes non-discriminative across sub-trajectories, providing little guidance signal for stitching goal-reaching behaviors from diverse demonstrations. To address these, we propose \textbf{QHyer}, which replaces RTG with a flow-parameterized, state-conditioned goal-reaching Q-estimator to support stitching across demonstrations, and introduces a gated Hybrid Attention-Mamba backbone that performs content-adaptive history compression while preserving local dynamics. Extensive experiments demonstrate that \textbf{QHyer} achieves state-of-the-art performance on both non-Markovian and Markovian datasets, validating its effectiveness for diverse scenarios.

Stochastic gradient methods are central to large-scale learning, yet their generalization theory typically relies on independent sampling assumptions. In many practical applications, data are generated by Markov chains and learning is performed in a decentralized manner, which introduces significant analytical challenges. In this work, we investigate the stability and generalization of decentralized stochastic gradient descent (SGD) and stochastic gradient descent ascent (SGDA) under Markov chain sampling. Leveraging a stability-based framework, we characterize how Markovian dependence and decentralized communication jointly influence generalization behavior. Our analysis captures the effects of network topology, Markov chain mixing properties, and primal-dual dynamics. We establish non-asymptotic generalization bounds for both algorithms, extending existing results on Markov stochastic gradient methods to decentralized and minimax settings.

For a risk-averse finite-horizon Markov Decision Problem, we introduce a special class of Markov coherent risk measures, called mini-batch measures. We also define the class of multipattern risk-averse problems that generalizes the class of linear systems. We use both concepts in a feature-based $Q$-learning method with multipattern $Q$-factor approximation and we prove a high-probability regret bound of $\mathcal{O}\big(H^2 N^H \sqrt{ K}\big)$, where $H$ is the horizon, $N$ is the mini-batch size, and $K$ is the number of episodes. We also propose an economical version of the $Q$-learning method that streamlines the policy evaluation (backward) step. The theoretical results are illustrated on a stochastic assignment problem and a short-horizon multi-armed bandit problem.

This study investigates the structural organisation of Dante's Divina Commedia through a symbolic representation based on vowel-consonant (V/C) encoding. Modelling the resulting sequence as a four-state Markov chain yields a parsimonious index of graphemic memory, capturing local persistence and alternation patterns. Across the poem, this index shows a slight but consistent increase from the Inferno to the Paradiso, indicating a directional shift in local dependency structure. Trigram analysis identifies a restricted set of recurrent configurations acting as graphemic probes, linking Markov patterns to lexical environments and orthographic phenomena such as apostrophised forms. A complementary classification analysis identifies cantica-specific lexical anchors, showing that local symbolic dependencies reflect both the separation among the three cantiche and a continuous progression across the poem. The results provide an interpretable framework connecting local symbolic structure with higher-level textual organisation.

This study applies symbolic time series analysis and Markov modeling to explore the phonological structure of Evgenij Onegin-as captured through a graphemic vowel/consonant (V/C) encoding-and one contemporary Italian translation. Using a binary encoding inspired by Markov's original scheme, we construct minimalist probabilistic models that capture both local V/C dependencies and large-scale sequential patterns. A compact four-state Markov chain is shown to be descriptively accurate and generative, reproducing key features of the original sequences such as autocorrelation and memory depth. All findings are exploratory in nature and aim to highlight structural regularities while suggesting hypotheses about underlying narrative dynamics. The analysis reveals a marked asymmetry between the Russian and Italian texts: the original exhibits a gradual decline in memory depth, whereas the translation maintains a more uniform profile. To further investigate this divergence, we introduce phonological probes-short symbolic patterns that link surface structure to narrative-relevant cues. Tracked across the unfolding text, these probes reveal subtle connections between graphemic form and thematic development, particularly in the Russian original. By revisiting Markov's original proposal of applying symbolic analysis to a literary text and pairing it with contemporary tools from computational statistics and data science, this study shows that even minimalist Markov models can support exploratory analysis of complex poetic material. When complemented by a coarse layer of linguistic annotation, such models provide a general framework for comparative poetics and demonstrate that stylized structural patterns remain accessible through simple representations grounded in linguistic form.

Decision makers exploiting photonic chaotic dynamics obtained by semiconductor lasers provide an ultrafast approach to solving multi-armed bandit problems by using a temporal optical signal as the driving source for sequential decisions. In such systems, the sampling interval of the chaotic waveform shapes the temporal correlation of the resulting time series, and experiments have reported that decision accuracy depends strongly on this autocorrelation property. However, it remains unclear whether the benefit of autocorrelation can be explained by a minimal mathematical model. Here, we analyze a stochastic-process model for solving the two-armed bandit problem based on time series, where the threshold and a two-valued Markov signal evolve jointly. Numerical results reveal an environment-dependent structure: negative (positive) autocorrelation is optimal in reward-rich (reward-poor) environments. These findings show that negative autocorrelation of the time series is advantageous when the sum of the winning probabilities is more than one, whereas positive autocorrelation is useful when the sum of the winning probabilities is less than one. Moreover, the performance is independent of autocorrelation if the sum of the winning probabilities equals one, which is mathematically clarified. This study paves the way for solving the two-armed bandit problems for reinforcement learning applications in wireless communications and robotics.

An important question in the field of AI is the extent to which successful behaviour requires an internal representation of the world. In this work, we quantify the amount of information an optimal policy provides about the underlying environment. We consider a Controlled Markov Process (CMP) with $n$ states and $m$ actions, assuming a uniform prior over the space of possible transition dynamics. We prove that observing a deterministic policy that is optimal for any non-constant reward function then conveys exactly $n \log m$ bits of information about the environment. Specifically, we show that the mutual information between the environment and the optimal policy is $n \log m$ bits. This bound holds across a broad class of objectives, including finite-horizon, infinite-horizon discounted, and time-averaged reward maximization. These findings provide a precise information-theoretic lower bound on the ``implicit world model'' necessary for optimality.

While transformer-based language models have driven the AI revolution thus far, their computational complexity has spurred growing interest in viable alternatives, such as structured state space sequence models (SSMs) and Selective SSMs. Among these, Mamba (S6) and its variant Mamba-2 have shown remarkable inference speed-ups over transformers while achieving comparable or superior performance on complex language modeling tasks. However, despite these architectural innovations and empirical successes, the fundamental learning capabilities of Mamba remain poorly understood. In this paper, we address this gap by studying in-context learning (ICL) on Markov chains and uncovering an interesting phenomenon: even a single-layer Mamba efficiently learns the in-context Laplacian smoothing estimator, which is both Bayes and minimax optimal. To explain this, we theoretically characterize the representation capacity of Mamba and reveal the fundamental role of convolution in enabling it to represent the optimal Laplacian smoothing. These theoretical insights align strongly with empirical results and, to the best of our knowledge, represent the first formal connection between Mamba and optimal statistical estimators. Finally, we outline promising research directions inspired by these findings.

Many learning algorithms can be represented as Markov processes, and understanding their generalization error is a central topic in learning theory. For specific continuous-time noisy algorithms, a prominent analysis technique relies on information-theoretic tools and the so-called ``entropy flow'' method. This technique is compatible with a broad range of assumptions and leverages the convergence properties of learning dynamics to produce meaningful generalization bounds, which can also be informative or extend to discrete-time settings. Despite their success, existing entropy flow formulations are limited to specific noise and algorithm structures (\eg, Langevin dynamics). In this work, we exploit new technical tools to extend its applicability to all learning algorithms whose iterative dynamics is governed by a time-homogeneous Markov process. Our approach builds on a principled continuous-time approximation of Markov algorithms and introduces a new, exact entropy flow formula for such processes. Within this unified framework, we establish novel connections to a well-studied family of modified logarithmic Sobolev inequalities, which we use to connect the generalization error to the ergodic properties of Markov processes. Finally, we provide a detailed analysis of all the terms appearing in our theory and demonstrate its effectiveness by deriving new generalization bounds for several concrete algorithms.

We study Nash equilibrium learning in partially observable Markov games (POMGs), a multi-agent reinforcement learning framework in which agents cannot fully observe the underlying state. Prior work in this setting relies on centralization or information sharing, and suffers from sample and computational complexity that scales exponentially in the number of players. We focus on a subclass of POMGs with independent state transitions, where agents remain coupled through their rewards, and assume that the underlying fully observed Markov game is a Markov potential game. For this class, we present an independent learning algorithm in which players, observing only their own actions and observations and without communication, jointly converge to an approximate Nash equilibrium. Due to partial observability, optimal policies may in general depend on the full action-observation history. Under a filter stability assumption, we show that policies based on finite history windows provide sufficient approximation guarantees. This enables us to approximate the POMG by a surrogate Markov game that is near-potential, leading to quasi-polynomial sample and computational complexity for independent Nash equilibrium learning in the underlying POMG.