Maximum State-Visitation Entropy (MSVE)

• MSVE is a reinforcement learning exploration method that maximizes the entropy of state visitation distributions through optimized policy search and constraint enforcement.
• It establishes a rigorous entropic framework by leveraging path-dependent formulations, Markov decision processes, and geometry-aware techniques to ensure uniform state coverage.
• The approach is applied in sparse-reward and partially observable environments, enhancing sample efficiency and scalability through methods like off-policy intrinsic rewards and belief-space exploration.

Maximum State-Visitation Entropy (MSVE) characterizes a class of exploration objectives in stochastic processes and reinforcement learning that seek to maximize the entropy of the distribution of state visitations induced by an agent’s behavior. Rather than incentivizing exploration with extrinsic rewards or transient novelty bonuses, MSVE formulates exploration as a principled optimization problem aiming for maximal dispersion across the state space under relevant dynamical and structural constraints. This paradigm finds rigorous expression in random walks, Markov decision processes (MDPs), and their extensions, and is distinguished by its treatment of system topology, path dependencies, and constraint interactions.

1. Entropic Frameworks: Path-Dependent and State-Dependent Formulations

In stochastic dynamical systems, the stationary properties of MSVE emerge from maximizing the entropy of transition dynamics subject to normalization and consistency constraints. The central functional is the path entropy: $\mathcal{E} = -\sum_{a,b} p_a k_{ab} \log k_{ab}$ where $p_a$ and $k_{ab}$ represent stationary probabilities and transition kernel entries, respectively (Dixit, 2015). When state- and path-dependent constraints (e.g., energies, currents) are enforced via Lagrange multipliers, the optimal transition probabilities are derived as: $$k_{ab} = \frac{1}{\eta} \frac{\phi_b}{\phi_a} W_{ab}$$ with $$W_{ab} = \exp\left( -\sum_{i} \gamma_i r_{ab}^{(i)} \right)$$ encoding the impact of constraints and $\phi$ the positive right eigenvector according to the Perron–Frobenius theorem.

The stationary distribution then takes the form: $p_a = \psi_a \phi_a$ where $\psi$ is the left eigenvector, capturing both connectivity and imposed restrictions. This structure illustrates the direct competition between dynamical path multiplicity and constraint-based enthalpy.

2. Markov Decision Processes: Optimization and Oracle-Based Algorithms

In reward-free MDPs, the MSVE objective is to find a policy $\pi$ such that the state-visitation distribution $d_\pi$ is made as uniform as possible, formalized by maximizing the entropy: $$H(d_\pi) = -\mathbb{E}_{s \sim d_\pi}[\log d_\pi(s)]$$ The canonical algorithm uses conditional (Frank–Wolfe) gradient methods in the space of achievable state distributions, iteratively computing gradient rewards and adding near-optimal policies via a black box planning oracle (Hazan et al., 2018). Policy mixtures are updated as: $$d_{\text{mix}, t+1} = (1-\eta) d_{\text{mix}, t} + \eta d_{\pi_{t+1}}$$ Ensuring sample and computational efficiency, this procedure leverages the decoupling of intrinsic exploration objectives from planning machinery, permitting the adoption of deep RL methods as robust oracles.

3. Geometry-Aware and Nonparametric Entropy Maximization

Classical MSVE methods are most effective in discrete state spaces; in continuous domains, the lack of geometry-awareness presents significant challenges. Geometric Entropy Maximisation (GEM) extends MSVE by replacing indicator functions with smooth similarity kernels $k(x, x')$, yielding the generalized entropy

$$H_k(p^\pi) = -\mathbb{E}_{x \sim p^\pi}[\ln p_k^\pi(x)]$$

where $p_k^\pi(x) = \mathbb{E}_{x' \sim p^\pi}[k(x, x')]$ (Guo et al., 2021). The tractability of optimizing this objective is achieved via a noise-contrastive formulation, with unbiased gradient estimation enabled by joint optimization over policy and similarity function parameters.

A complementary approach in high-dimensional continuous spaces involves non-parametric density estimation via balanced k-means clustering (Nedergaard et al., 2022). Local density is approximated as $p(x) \approx 1/(k \cdot m(c_i))$, where $m(c_i)$ is the measure ("size") of cluster $i$. The differential entropy lower bound is tied to inter-cluster distances: $$H(p) \gtrsim \frac{d}{k} \sum_i \log(\min_{j \neq i} \|\mu_i - \mu_j\| + (w_i - w_j)) + C$$ The intrinsic reward is defined by changes in this objective upon new state visits, offering computational tractability and scalability in on-policy RL contexts.

4. Temporal and Policy Structure: Markovianity, Episodic, and Lifelong MSVE

In finite-sample (single-trial) regimes, maximizing the entropy of empirical visitation is fundamentally enhanced by non-Markovian, history-dependent policies (Mutti et al., 2022). While Markovian stochastic policies suffice for asymptotic entropy maximization, non-Markovian deterministic policies eliminate regret in finite trials by steering towards not-yet-visited states. However, finding such policies is NP-hard due to exponential growth in the history-conditioned policy class, prompting the investigation of recurrent function approximation and windowed-history relaxations.

Recognition of episodic versus lifelong entropy scales motivates multiscale frameworks such as ELEMENT (Li et al., 5 Dec 2024). Episodic entropy rewards states contributing to diverse short-term trajectories, formally via the "average episodic state entropy": $$r_{\text{ep}}(s) := \mathbb{E}_{\text{ep}}(H_{\{s \in \tau\}}(s) \cdot \mathbb{1}[s \in \tau])$$ Lifelong entropy maintains state diversity over extended runs, with efficient estimation via k-nearest neighbor (kNN) graph constructions.

5. Sample Complexity, Oracle Efficiency, and Game-Theoretic Perspectives

Sample and computational complexity improvements for MSVE objectives are characterized in minimax game-theoretic analyses (Tiapkin et al., 2023). The visitation entropy maximization problem is recast as a two-player prediction game: $\max_{d \in \mathcal{K}_p} \VE(d) = \min_{\mu \in \mathcal{K}} \max_{d \in \mathcal{K}_p} \sum_{(h,s,a)} d_h(s,a) \log \frac{1}{\mu_h(s,a)}$ Here, a forecaster predicts visitation distributions, while a sampler "surprises" it via exploration policy choice. Algorithms derived from this dual formulation (e.g., EntGame and its regularized variants) exhibit sample complexity rates of $\widetilde{O}(H^2SA/\epsilon^2)$, a notable improvement over prior methods. Regularization of planning steps via entropy bonuses further enhances statistical efficiency.

6. Extensions to Partial Observability and Belief-Space Exploration

MSVE objectives and algorithms are generalized to POMDPs, where agents receive only partial observations (Zamboni et al., 4 Jun 2024). The main challenge is the disparity between entropy over observations and true state visitation entropy. Proxy objectives such as Maximum Observation Entropy (MOE) and Maximum Believed Entropy (MBE) use belief states as surrogates, with explicit regularization to combat the "hallucination problem" (i.e., agents increasing belief entropy rather than true coverage). The policy class is compactly parametrized around beliefs, and first-order policy gradients for entropy objectives are derived, supporting deployable learning in realistic scenarios.

7. Off-Policy Approaches: Intrinsic Reward via Future Visitation Measures

Intrinsic rewards based on the KL divergence between future visitation distributions and a uniform reference can be incorporated into off-policy RL frameworks (Bolland et al., 9 Dec 2024). For state–action pair $(s, a)$: $$R^{(\text{int})}(s, a) = -\mathrm{KL}_z\left[q^\pi(z|s,a) \ \| \ q^*(z)\right]$$ where $q^\pi(z|s,a)$ aggregates features from future discounted visitation and $q^*(z)$ is a target distribution. The future visitation distribution is shown to be the fixed point of a $\gamma$-contractive operator, enabling estimation via bootstrapping. Practical adaptation involves supplementing existing critic/policy updates with visitation network learning; empirical results demonstrate superior entropy and coverage in sparse-reward control domains.

Summary Table: Core Formulations in MSVE

Formulation	Key Formula	Domain
Path Entropy (stationary)	$k_{ab} = \frac{1}{\eta} (\phi_b/\phi_a) W_{ab}$	Random Walks
MDP State Entropy	$$H(d_\pi) = -\mathbb{E}_{s \sim d_\pi}[\log d_\pi(s)]$$	RL / MDP
Geometric Entropy (GEM)	$$H_k(p^\pi) = -\mathbb{E}_{x \sim p^\pi}[\ln p_k^\pi(x)]$$	RL, Continuous States
kNN Graph Entropy (Elem.)	$$r_\ell(s) = \log(\| s - \text{GNNS}(\mathcal{G}, s) \|_2 + 1)$$	Multiscale RL
POMDP Belief Entropy	$J^S(\pi) = -\sum_s d^\pi(s) \log d^\pi(s)$, proxies via belief/observation entropy	Partial Observability
Off-Policy KL Intrinsic	$$R^{(\text{int})}(s, a) = \mathbb{E}_{z}[\, \log q^*(z) - \log q^\pi(z|s,a)]$$	Off-Policy RL

Concluding Perspective

MSVE principles provide a rigorous foundation for exploration-based learning and modeling across discrete, continuous, and partially observable environments. By optimizing entropy over visitation distributions, agents approximate uniform coverage critical for efficient data collection, robust skill acquisition, and downstream task transfer. Key research advances include conditional gradient methods, geometry-aware entropy objectives, kNN-based estimators, game-theoretic formulations, and belief-space extensions, all converging to address practical issues of sample efficiency, computational tractability, and policy expressiveness in modern reinforcement learning and stochastic process analysis.