On the Computational and Statistical Efficiency of the Empirical Maximum Entropy on the Mean Method

The Maximum Entropy on the Mean (MEM) method provides a flexible computational framework for solving inverse problems by combining data fidelity with entropy-based regularization. In practice, however, the prior distribution is typically unknown but can be estimated from data, giving rise to the empirical MEM method. We establish a parametric convergence rate of $O(n^{-1/2})$ in expectation for empirical MEM, improving upon the previously established $O(n^{-1/4})$ guarantee by King-Roskamp et al. (2026). Our proof is based on a novel stability analysis of the primal and dual optimization problems under perturbations of the underlying probability measure, relying only on foundational tools from convex analysis and probability. We further show that the MEM dual problem admits a reformulation as an expected risk minimization problem, thereby placing MEM within the modern framework of stochastic optimization and enabling scalable stochastic gradient algorithms for large-scale inverse problems. Together, these results place empirical MEM as a statistically and computationally efficient methodology for data-driven inverse problems.

Where Entropy Is Measured Matters: Policy Geometry in Bounded Continuous-Control PPO

Many continuous-control policies are optimized as unbounded Gaussians and then mapped into bounded actions. We show that where entropy is measured changes the policy geometry learned by proximal policy optimization (PPO). In an 80-muscle MyoLeg task, a clipped Gaussian executes 89.07% of actions within 5% of a bound. A same-state decomposition shows that this is not due to variance alone: setting variance to zero still leaves 83.83% of actions near a bound, while 82.12% of state-conditioned means lie outside the executable interval. Replacing clipping with a tanh map does not remove the high-variance regime. For latent Gaussian entropy H(u), the entropy loss has zero gradient with respect to the mean and a constant variance-increasing gradient. For executed-action entropy H(a), the transform Jacobian adds an inward gradient on the mean. Across three matched MyoLeg seeds, near-boundary occupancy is 71.42%, 29.76%, and 18.83% under latent entropy, no entropy, and executed-action entropy. A 38-dimensional Dog-Stand replication with an independent CleanRL-based PPO implementation reproduces the ordering in mean geometry, which also survives shared-state evaluation and boundary margins from 1% to 10%. Direct mean penalties can match or exceed the centering produced by H(a), showing that interior means are not unique to executed entropy. However, matched mean geometry can coexist with substantially different variance and return. Entropy measurement space is therefore a coupled mean-variance design choice, and task return alone does not characterize bounded-policy geometry.

Reinforcement Learning for Continuous-Time Jump Markov Decision Processes with Applications to Network Dynamic Pricing

We study reinforcement learning (RL) in Continuous-Time Jump Markov Decision Processes (CTJMDPs) featuring general discrete state spaces (which need not possess a vector space structure) and continuous/discrete action spaces. The setup covers many well-known applications in operations such as multi-product dynamic pricing with capacitated resources (Gallego and van Ryzin 1997). To model the exploration-exploitation tradeoff, we formulate an entropy-regularized continuous-time control problem with stochastic policies. Recent continuous-time RL techniques such as $q$-learning for controlled diffusions in (Jia and Zhou 2023) focus on continuous state spaces $\mathbb{R}^d$ and rely heavily on semimartingale theory in $\mathbb{R}^d$ for their theoretical analysis. Consequently, their methods cannot be directly applied to CTJMDPs with general discrete state spaces, which may lack the algebraic addition and subtraction structures inherent to Euclidean spaces. To bridge this gap, we establish the theoretical foundations of $q$-learning for CTJMDPs and develop model-free $q$-learning algorithms. Compared to naïve time discretization and approximating CTJMDPs using discrete-time MDPs, our approach has several conceptual and empirical benefits. Numerical experiments in network dynamic pricing (Gallego and van Ryzin 1997) show that our proposed RL algorithm reliably learns near-optimal policies and consistently outperforms standard benchmark methods, demonstrating superior solution quality and effective scalability to large-scale network instances.

Unregularized Convergence of Single-Loop, Entropy-Regularized Natural Actor-Critic

While entropy regularization is widely used to stabilize and accelerate Natural Policy Gradient methods, its ability to yield faster convergence rates for the unregularized objective remains underexplored. Existing analyses often rely on double-loop architectures and invoke a linear entropy penalty. To bridge the gap between theory and practice, we analyze a single-loop, entropy-regularized Natural Actor-Critic algorithm under compatible linear function approximation. By training an uncentered critic, our critic tracking can remain stable even as the training policy approaches determinism and the Fisher information matrix degenerates. We focus on two primary regimes for the optimization landscape: a Stochastic Regime, where we fuse coupled actor-critic updates into a joint Lyapunov recurrence, and a Deterministic Regime, where we pivot to a Policy Mirror Descent framework to circumvent the collapse of Euclidean geometry. By exploiting a positive Minimal Action Gap in the unregularized Markov decision process, we introduce an Exponential Translation mechanism that maps the regularized gap to the unregularized one up to an exponentially decaying tail. By tuning the fixed temperature, our algorithm achieves accelerated unregularized convergence rates, up to approximation-error terms: $\tilde{\mathcal{O}}(T_{total}^{-1})$ in the Stochastic Regime, and $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ for the average iterate alongside $\tilde{\mathcal{O}}(T_{total}^{-1/3})$ for the last iterate in the Deterministic Regime. Here, $T_{total}$ denotes the total number of stochastic critic updates (or Monte Carlo rollouts). Furthermore, in the tabular setting, our positive-action-gap analysis yields a $\tilde{\mathcal{O}}(T_{total}^{-2/3})$ average-iterate rate, surpassing the $\mathcal{O}(T_{total}^{-1/2})$ worst-case statistical barrier that applies without a positive action margin.

Wasserstein Policy Gradient for Entropy-Regularized Linear-Quadratic Control

Wasserstein policy gradient (WPG) updates state-conditional action laws by transport in the action space. We study entropy-regularized discounted linear-quadratic (LQ) control. A Bellman verification argument shows that the unrestricted problem has a linear-Gaussian optimal policy, and the discounted-occupancy-weighted statewise Wasserstein gradient is tangent to this policy class. WPG therefore reduces exactly to a finite-dimensional ODE for the feedback gain and action covariance. We prove that this ODE is globally well posed and converges exponentially from every admissible initialization. For each fixed LQ problem, the exponent has a positive limit as the entropy temperature tends to zero and contains no perturbative factor of the form $\exp(-c/τ)$, while retaining the usual dependence on the conditioning of the control problem.

Soft-Constrained Optimization of Latent Space in Variational Autoencoders

The usefulness of a variational autoencoder (VAE) depends on two properties of its latent space that are hard to obtain together: high encoding capacity in the individual latent variables, and a low-dimensional, disentangled organization of those variables. Weakening the Kullback-Leibler regularization raises capacity but degrades disentanglement, while strengthening it prunes latent variables away entirely. We formulate VAE training as a soft-constrained optimization problem that addresses both. First, we impose an entropy-based constraint (EC) on individual latent variables, showing that the entropy of a latent code upper-bounds the mutual information it carries about the generative factors of the data. Second, we propose a weight-filter method that exploits the slack of the soft constraint to prune low-entropy dimensions during downstream training. On dSprites, the EC raises the aggregate latent-variable activation score by 43-62% over a vanilla VAE, attains the highest FactorVAE score among the \b{eta} \b{eta}-VAE variants (0.891 vs 0.847), and lowers reconstruction error by up to 38%. On MNIST, the weight filter reduces the latent dimensionality supplied to a downstream classifier from ten to two while holding accuracy above 90%, converging in 37% fewer epochs than the same procedure without the EC. We also find that low-entropy discrete factors tend to merge into a single latent variable, whereas high-entropy continuous factors are distributed across several.

Self-Balancing Sequential Sampling: Fast Convergence with Controlled Predictability

Many instances of sequential sampling, including audit and inspection scheduling, representative sampling, and treatment assignment, require selections to be distributed evenly without becoming easy to anticipate or exploit. We study a family of sequential sampling rules that adaptively bias sampling probabilities in order to achieve faster convergence of the empirical distribution to a desired target law, while keeping the resulting samples as unpredictable as possible. The resulting self-balancing sampler is simple to implement, arises naturally among a class of Markovian samplers sharing a certain invariance property, and admits a stochastic mirror-descent interpretation. Our main results show that (i) this self-balancing sampler converges at the fastest possible $O(n^{-1})$ rate with explicit dependence on biasing parameters, beating the standard $O(n^{-1/2})$ rate of IID sampling, (ii) it is the unique solution to a natural entropy-regularized optimization problem which balances the convergence rate of the empirical law and the unpredictability of the samples, and (iii) in the weak-biasing regime, the properly centered counts process converges to an Ornstein-Uhlenbeck process in the diffusive limit. Together, these results support a practical framework for reducing repeated selections and long gaps in coverage without making future selections overly predictable.

The Curvature Shadow: An Apparent Failure of Maximum-Entropy Equilibrium Selection is a Removable Artifact

In two-player zero-sum games whose Nash equilibria form a convex set, regularized solvers such as Regularized Nash Dynamics (R-NaD) empirically select the maximum-entropy member: the information projection (I-projection) of a uniform reference onto the Nash set. On a panel of small games this match is exact, with one apparent exception: in Kuhn poker R-NaD lands at bluff coordinate 0.180 while the maximum-entropy member sits at 0.201, a coordinate gap of about 0.021, even though R-NaD attains 99.7 percent of the maximum entropy. We ask whether this gap is a genuine selection bias or an artifact, and answer it quantitatively. We show that for selection on a one-dimensional Nash manifold the coordinate gap factorizes as $\mathrm{gap} \approx \sqrt{2δ/κ}$, where $δ$ is the entropy shortfall of the solver and $κ$ is the curvature of the entropy landscape at its peak. Across five games this relation holds to within $2 \times 10^{-4}$ (under 1 percent relative error). The four matrix games have $δ\approx 0$ (R-NaD reaches the maximum-entropy member exactly) and therefore no gap regardless of curvature; only the sequential game (Kuhn) has $δ> 0$. A causal sweep of the magnet strength drives $δ\to 0$ and the gap toward zero along the predicted curve (fitted scaling exponent 0.50, $R^2 > 0.999999$, against the exact prediction of 1/2), until the dynamics destabilize at a stability floor: behavior consistent with a removable shortfall and inconsistent with a fixed bias. We quantify the curvature half of the law from measured curvatures and flag a moving-target pitfall in the natural Tsallis-entropy experiment. The Kuhn gap is thus the curvature shadow of a small, removable entropy shortfall on an unusually flat peak; the I-projection account is upheld up to a flatness-limited residual.

Rationalizing Boltzmann Rationality: An Axiomatic Characterization of Entropy-Regularized Policies

The softmax policy $π(a \mid s) \propto \exp(βQ(s,a))$ is the default model of stochastic choice in reinforcement learning (RL). Various justifications based on robustness, exploration, and optimization have been offered in the RL literature, but none uniquely derives the softmax form from first principles. This leaves a basic tension unresolved: the entropy bonus in the soft Bellman equation violates the Independence axiom that underwrites the Markov decision process (MDP) reward structure. We dissolve this tension by distinguishing two kinds of randomness: chance and choice. By restricting von Neumann-Morgenstern (VNM) Independence to environmental lotteries over base prospects, we show that imposing independence of irrelevant alternatives (IIA) and monotonicity on the policy and value functions at choice nodes uniquely determines the Boltzmann policy, the entropy-regularized representation, and the soft Bellman equation. The choice between the soft and hard Bellman equations thus reduces to a design decision: whether the agent values its own ability to choose. We develop RL-specific consequences, including return monotonicity and convergence under generalized discounting, and synthesize the independent lines from economics and information theory that arrive at the same structure, offering a normative assessment of when IIA is appropriate for agent design.

Entropy Regularization Improves Policy Robustness in Continuous-Time Reinforcement Learning

Entropy regularization is widely used in continuous-time reinforcement learning (RL) to reduce sensitivity to environmental perturbations, yet its robustness benefits lack a rigorous theoretical foundation. This paper establishes the first robustness guarantees for entropy-regularized continuous-time Markov decision processes. We show that maximizing an entropy-regularized objective yields a lower bound on a worst-case robust RL problem with joint reward and transition perturbations. We analytically characterize the induced robust sets and prove that they expand monotonically with the regularization strength, justifying the empirical observation that stronger entropy improves robustness. In contrast to prior discrete-time analyses, our results remove the intractable state-distribution entropy term and provide guarantees invariant to action frequency. Experiments on queueing network control and market making confirm our theory, showing that entropy-regularized policies outperform greedy and $ε$-greedy baselines under dynamics perturbations.

Entropy-Regularized Probabilistic Gates for Sparse Model Discovery in Scarce-Data Federated Learning

Federated Learning (FL) is a distributed machine learning (ML) paradigm with collaboration among multiple clients without sharing data. FL is challenging under data heterogeneity and partial client participation. Learning sparse models is useful for communication and computational efficiency in FL, but it is especially difficult in the small-sample high-dimensional regime (d >> N) where optimization can yield parameter configurations that fail to generalize to unseen test data. While magnitude-based pruning doesn't account for uncertainty exploration in the parameter space, a formulation with probabilistic gates and an L0 constraint allows sampling from competing sparse configurations during training. In this work, we study entropy regularization of gate distributions as a mechanism to maintain uncertainty in sparse federated optimization by preventing early commitment to sparse support. We examine its impact under data heterogeneity, client participation heterogeneity, and sparsity. Experiments on synthetic and real-world benchmarks show consistent improvements over federated iterative hard thresholding (Fed-IHT) and pruning after dense federated averaging (FedAvg) training, both in statistical performance on test data and in sparsity recovery accuracy.

ZEBRA: Zero-Shot Entropy-Regularized Prompt Learning for Base-to-Novel Generalization in Audio-Language Models

Audio-Language Models (ALMs) achieve strong zero-shot performance by aligning audio with textual class descriptions. Although prompt learning improves accuracy on base classes through few-shot supervised adaptation, we observe a critical trade-off: it often degrades performance on novel classes, sometimes falling below zero-shot accuracy. This exposes a base-to-novel generalization gap in prompt learning for ALMs. To address this issue, we propose \textbf{ZEBRA} (Zero-shot Entropy-Regularized Prompt Learning for Base-to-Novel Generalization), a plug-and-play framework that fuses zero-shot logits with prompt-learning logits, and employs self-entropy regularization to reduce overfitting to base classes. Experiments across multiple audio classification datasets show that ZEBRA consistently improves novel-class performance while maintaining strong base accuracy, significantly reducing the base-to-novel gap compared to standard prompt learning. The code is available at: https://github.com/asif-hanif/zebra.

Entropy Regularized Reinforcement Learning for Zero-Sum Stochastic Differential Games in a Regime-Switching Jump-Diffusion Process

To address parameter misspecification and sudden structural environmental changes in conventional stochastic differential game (SDG) frameworks, this paper introduces a distributional control approach that characterizes optimal strategies as probability distributions over actions, conditioned on the continuous state, the discrete regime state, and parameters. This forms a reinforcement learning framework for entropy-regularized zero-sum stochastic differential games (ERRL-ZSSDGs) in a regime-switching jump-diffusion process. Using the dynamic programming principle (DPP), we derive the associated coupled systems of Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations, from which equilibrium strategies are expressed via gradients of the value function. For linear-quadratic problems, semi-analytical solutions for both value function and equilibrium strategies are obtained by solving a system of coupled ordinary differential equations (ODEs). In more general settings, an Actor-Critic policy improvement algorithm is developed to approximate the value functions and equilibrium policies across different regimes. The method is applied to an investment game, and numerical examples illustrate the effect of the temperature parameter and regime transitions on optimal policies and values.

Information Gap and Feasibility-Aware Inference in Binomial Logistic Mixtures

This paper studies the information gap between mixture detection and label recovery in binomial logistic mixtures. Standard likelihood-based criteria such as the Bayesian information criterion (BIC) can detect the presence of two components, but this does not guarantee that the corresponding labels are recoverable. We show that this gap is intrinsic to binomial logistic mixtures with a fixed number of trials: observed-data evidence for mixture structure and per-observation information for label recovery have different local orders in the component separation, and only the former accumulates with the sample size. As a result, there exists a detectable-but-unrecoverable regime in which BIC selects two components while the posterior labels remain essentially uninformative. To address this issue, we propose two feasibility-aware inference procedures: a recoverability-aware BIC with a posterior-entropy penalty and an entropy-regularized estimator that mitigates the tendency of the maximum likelihood estimator to produce overly separated components and overly concentrated posterior responsibilities. Numerical experiments confirm the predicted gap and demonstrate that the proposed methods avoid misleading component selections and improve the calibration of posterior label probabilities.

PAEC: Position-Aware Entropy Calibration for LLM Reasoning in RLVR

Reinforcement learning with verifiable rewards (RLVR) improves large language model reasoning but often suffers from rapid policy-entropy collapse, where the policy prematurely concentrates on narrow high-probability reasoning paths. While global entropy regularization can encourage exploration, uniformly increasing entropy across all token positions is inefficient for long reasoning trajectories, where many tokens are not decision-relevant. We propose Position-Aware Entropy Calibration (PAEC), a token-level entropy-management framework that constructs a soft mask from local top-p entropy and top-two candidate competition, and applies an anchor-based lower-bound penalty to prevent selected-position entropy collapse. Experiments on five mathematical reasoning benchmarks show that PAEC improves macro-average majority-vote performance over strong RLVR baselines, with clear gains on AIME-style tasks. Our results suggest that entropy management in reasoning RL should be formulated as selective exploration allocation over decision-sensitive positions rather than uniform randomness injection.

DICE: Entropy-Regularized Equilibrium Selection for Stable Multi-Agent LLM Coordination

Multi-agent large language model (LLM) systems often fail to reliably outperform a single strong model equipped with best-of-N sampling. We argue that a core source of this instability is ill-posed equilibrium selection: current systems specify what information agents share, but not which coordination convention should be selected. We formalize a broad class of such systems as discounted incomplete-information Markov games and show that two common pathologies, oscillation between competing conventions and drift across them, can both induce unstable learning and linear Bayesian regret. To obtain a well-posed target, we introduce the Heterogeneous Quantal Response Equilibrium (HQRE), an entropy-regularized equilibrium concept with agent- and state-dependent temperatures. Under a monotonicity condition, HQRE is unique, admits linearly convergent mirror updates, and yields bounded Bayesian regret; the same condition yields rollout-measurable stability diagnostics. We instantiate this objective in two algorithms: DICE-PC, which coordinates frozen models through prompt-control actions, and DICE-FT, which performs parameter-efficient mirror fine-tuning. Across eleven benchmarks in four domains, DICE improves accuracy-cost trade-offs over strong within-class baselines; on reasoning and planning tasks, DICE-PC improves by 4.3 percentage points on average and DICE-FT by 8.5 points.

\textit{Stochastic} MeanFlow Policies: One-Step Generative Control with Entropic Mirror Descent

Online off-policy reinforcement learning (RL) is shaped by two coupled choices: the policy class and the update rule. Gaussian policies are fast and have tractable entropy, but struggle with multimodal action distributions. Generative policies are more expressive, but often require iterative sampling or lack tractable entropy estimates. On the optimisation side, SAC-style soft policy improvement and mirror descent (MD) can be viewed as minimising different KL divergences: the former moves the policy towards a value-induced Boltzmann distribution, while the latter regularises each update against the previous policy. Combining entropy regularisation with an MD constraint is therefore attractive, as it supports exploration while stabilising policy improvement; however, the resulting target can be multimodal and is poorly matched by unimodal Gaussian policies. We propose Stochastic MeanFlow Policies (SMFP), a one-step generative policy class that maps Gaussian noise to actions through a MeanFlow transformation. This stochastic reparameterisation yields a tractable entropy surrogate and allows MeanFlow policies to be trained within off-policy mirror descent under a unified objective for exploratory yet stable improvement. Across seven MuJoCo benchmarks, SMFP improves over Gaussian and generative baselines while retaining single-step inference efficiency.

Continuous-time q-learning for mean-field control with common noise, part-I: Theoretical foundations

This paper investigates the continuous-time counterpart of the Q-function for entropy-regularized mean-field control (MFC) with controlled common noise, coined as q-function by Jia and Zhou (2023) in the single agent's model. We first show that, under discretely sampled actions, the value function in the exploratory formulation converges to the one in the relaxed control formulation as the time grid refines. Leveraging the relaxed control formulation, we derive the exploratory Hamilton-Jacobi-Bellman (HJB) equation, in which the controlled common noise gives rise to an additional nonlinear functional of policy, rendering the policy iteration intricate. Under certain concavity condition, we establish the existence and uniqueness of the optimal one-step policy iteration via a first-order condition using the partial linear functional derivative with respect to policy. The policy improvement at each iteration is verified by relating to an entropy-regularized optimization problem over the space of policies. In the mean-field setting, we introduce the integrated q-function (Iq-function) defined on the state distribution and the policy, and it is shown that an optimal policy is identified as a two-layer fixed point to the argmax operator of the Iq-function. Finally, we provide the explicit characterization of an optimal policy as a Gaussian distribution in the general linear-quadratic (LQ) setting.