Inference-Based High-Level Policy Formulation

• The paper introduces a Bayesian framework that recasts policy search and constraint satisfaction as probabilistic inference, yielding robust high-level policies.
• The methodology utilizes dual gradient descent, variational inference, and entropy regularization to optimize policies while enforcing safety constraints.
• The approach unifies reinforcement learning, imitation learning, and optimal control, enabling effective decision-making in complex and safety-critical domains.

High-level policy formulation via probabilistic inference addresses the synthesis of decision-making strategies by directly characterizing policies as objects of Bayesian or variational inference within a generative or graphical model of the underlying control process. This approach unifies a range of concepts in reinforcement learning, imitation learning, optimal control, and planning by recasting the policy-search, constraint satisfaction, and subgoal specification problems as inference queries over probabilistic graphical models or probabilistic programs. The framework rigorously connects entropy-regularized RL, demonstrator compliance, hierarchical RL, and robust control, yielding theoretically principled and empirically robust policy learning and execution strategies.

1. Fundamental Principles and Model Structures

The foundation of probabilistic-inference-based high-level policy design is the formulation of control or planning as inference in an augmented probabilistic model of the environment and agent. In the canonical model, the environment is described either as a Markov Decision Process (MDP) or a Partially Observable MDP (POMDP) with a generative model

$(S, A, R, P, \mu, \gamma)$

for the MDP or $(S, A, Z, T, O, R, \gamma)$ for the POMDP, where high-level control variables—policy parameters, subgoals, constraints, or mixture weights—are modeled explicitly as latent, controllable variables. Probabilistic inference is used to select policies or high-level variables that maximize the posterior probability (or plausibility) of desired outcomes, such as optimality, constraint satisfaction, or subgoal achievability (Papadopoulos et al., 9 Jul 2025, Wang et al., 24 Jun 2024, Tarbouriech et al., 2023, Hansel et al., 2022, Wood et al., 2020).

Key modeling elements include:

• Binary optimality and safety variables: For each timestep, binary variables (e.g., $O_t$ for optimality, $C_t$ for constraint satisfaction) are introduced, with likelihoods $p(O_t=1|s_t,a_t) \propto \exp(r(s_t,a_t))$ and $p(C_t=1|s_t,a_t) \propto 1-\exp(C(s_t,a_t))$ (Papadopoulos et al., 9 Jul 2025).
• Hierarchical latent subgoal representations: The high-level policy reasons over the distribution of latent subgoals $z(s)$, modeled, for example, as a Gaussian Process, allowing Bayesian inference over abstracted state component trajectories (Wang et al., 24 Jun 2024).
• Mixture weights for policy blending: High-level inference is used to produce distributions over expert blending coefficients for policy composition, e.g., via a Dirichlet distribution over weights $w$ in motion planning (Hansel et al., 2022).
• Inference in complex simulators: For domains like pandemic response, high-level policy levers (e.g. policy vectors $\pi$) are mapped to simulator parameters $\theta$, and inference is performed to identify $\pi$ yielding trajectories $X_{0:T}$ satisfying auxiliary constraints $Y_{0:T}$ (e.g., cases stay under threshold) (Wood et al., 2020).

2. Objective Formulation via Inference and KL-Divergence

The central technical machinery recasts optimal policy search and constraint satisfaction as Bayesian or variational inference problems. Using the "RL as inference" paradigm, the distribution over trajectories under a policy is matched to a posterior involving optimality and safety events:

$$p(\tau|O_{1:T}=1, C_{1:T}=1) \propto p(s_0)\prod_t \exp(r(s_t,a_t))\,\rho_{\pi_E}(s_t,a_t) P(s_{t+1}|s_t,a_t)$$

To align a learned policy $\pi$ with expert demonstrations or constraints, the learning objective becomes KL-divergence minimization:

$$\min_\pi D_{KL}\left(\pi(\tau) \,\|\, p(\tau|O,C)\right)$$

This leads to a maximum-entropy policy objective:

$$\max_\pi \mathbb{E}_{\tau\sim\pi}\bigg[\sum_t r(s_t,a_t)\bigg] + (1-\beta)H(\pi) - (D_{KL}(\pi_E\|\pi) - \beta H(\pi))$$

Strict constraint enforcement is achieved by bounding the KL term: $D_{KL}(\pi_E\|\pi) - \beta H(\pi) \leq \delta$ (Papadopoulos et al., 9 Jul 2025). Performance and constraint violations are then provably bounded by the KL divergence between the learned and expert policies (Papadopoulos et al., 9 Jul 2025).

3. Algorithmic Realizations and Optimization Methods

Optimization is typically performed using variant forms of dual gradient descent, variational message passing, or stochastic variational inference:

• Dual-gradient descent for constrained objectives: Lagrangian relaxation of the KL constraints leads to a saddle-point optimization problem, alternating between gradient steps on policy parameters and Lagrange multipliers (Papadopoulos et al., 9 Jul 2025). For neural policies, the update is implemented via actor–critic architectures such as Soft Actor-Critic (SAC), enhanced with additional KL and entropy regularization (Papadopoulos et al., 9 Jul 2025).
• Variational Bayesian policy search: Occupancy measures or mixture weights are optimized by maximizing the evidence lower bound (ELBO) on the marginal likelihood of optimality or constraint satisfaction under a variational posterior, yielding convex or easily optimizable programs (Tarbouriech et al., 2023, Hansel et al., 2022).
• Bayesian inference and probabilistic programming: Parameterized policies are treated as latent variables in a probabilistic program; inference is performed via methods such as Lightweight Metropolis-Hastings (LMH), Black-Box Variational Inference, or improved Cross-Entropy Methods (iCEM), with either trace-based or direct parameter-space sampling (Tolpin et al., 2020, Meent et al., 2015, Wood et al., 2020, Song et al., 2020, Hansel et al., 2022).

Table 1: Optimization Schemes

Approach	Optimization Method	Constraint Handling
SCOPIL (Papadopoulos et al., 9 Jul 2025)	Dual gradient descent (on SAC)	KL constraint as Lagrange term
HLPS (Wang et al., 24 Jun 2024)	SAC + GP-state Kalman filter	Algorithmic through joint loss
VAPOR (Tarbouriech et al., 2023)	Convex exponential cone program	Implicit through λ simplex
HiPBI (Hansel et al., 2022)	iCEM optimization, Dirichlet q(w)	Soft through optimality likelihood
PPL/LMH (Tolpin et al., 2020)	Stochastic LMH, simulated annealing	Hard/soft via reward factors

4. Hierarchical, Subgoal, and Mixture-Policy Inference

High-level policies frequently involve latent abstractions for handling long horizons, partial observability, or compositionality:

• Probabilistic subgoal representations: Instead of deterministic subgoal mappings, Gaussian Process (GP) priors over latent subgoal functions enable high-level controllers to perform inference and sampling of subgoals, accommodating epistemic uncertainty and multimodality. At each hierarchical decision point, the high-level policy conditions on the current state to sample a subgoal $g \sim \mathcal{N}(\mu_i,\,\Sigma_i)$ (Wang et al., 24 Jun 2024).
• Policy mixture weights via inference: Reactive robot control is achieved by probabilistically blending expert low-level policies using weights inferred by optimizing expected cumulative optimality over short horizons. The highest-level policy infers a Dirichlet distribution over blend-weights $w$, whose mean or MAP estimate is used to compute the control action (Hansel et al., 2022).
• Policy specification in logical space: In partially observed domains, interpretable policy specifications are learned via inductive logic programming from POMDP traces, allowing policy rules to be treated as logical inferences conditioned on beliefs (Meli et al., 29 Feb 2024).

5. Theoretical Guarantees and Empirical Outcomes

The use of probabilistic inference in high-level policy formulation confers several rigorous properties:

• Performance guarantees via KL bounds: The differential in performance and constraint violation between learned and expert policies is bounded by per-state KL divergences (Papadopoulos et al., 9 Jul 2025).
• Regret guarantees: Occupancy-measure-based inference policies, such as VAPOR, achieve near-Bayesian-optimal regret rates, with guarantees scaling as $\tilde{\mathcal{O}}(\sqrt{SAT})$ for S states, A actions, and T steps (Tarbouriech et al., 2023).
• Exploration-exploitation balancing: Sampling from GP posteriors or occupancy-based marginals naturally reconciles exploration (via uncertainty) and exploitation (via posterior mean) (Wang et al., 24 Jun 2024, Tarbouriech et al., 2023).
• Safety and constraint satisfaction: Explicit encoding of constraint satisfaction as probabilistic events, with strict bounds on allowable violations, enables reliable deployment in safety-critical domains (Papadopoulos et al., 9 Jul 2025, Wood et al., 2020).
• Generalization and transfer: Probabilistic subgoal representations and Bayesian policy search allow transfer across tasks and stable learning under stochasticity (Wang et al., 24 Jun 2024).

6. Applications and Extensions

Applications of high-level policy formulation via probabilistic inference span:

• Imitation learning with safety constraints: SCOPIL achieves expert-level reward and low constraint violation across single- and multi-constraint tasks, and preserves multimodality in demonstrations (Papadopoulos et al., 9 Jul 2025).
• Hierarchical reinforcement learning: HLPS demonstrates robust and efficient subgoal learning in high-dimensional control and transfer scenarios (Wang et al., 24 Jun 2024).
• Planning in epidemic control: Policy levers in disease models are optimized via probabilistic inference for outcomes such as suppressed transmission while minimizing socio-economic side-effects (Wood et al., 2020).
• Hybrid reactive/planning robotic control: HiPBI framework outperforms myopic and re-planning baselines in dense robotic navigation and manipulation tasks (Hansel et al., 2022).
• Adaptive high-level MPC: Learning a distribution over planning hyperparameters enables robust quadrotor control through dynamic agile environments (Song et al., 2020).
• Policy logic induction in POMDPs: Interpretable policy heuristics learned from data improve Monte Carlo tree search planners under partial observability (Meli et al., 29 Feb 2024).

These methods are extensible to domains where high-level levers manipulate complex, stochastic simulators or real-world processes, and where traditional policy gradient or value-based methods are insufficiently robust, interpretable, or generalizable.

7. Connections to Existing Frameworks and Algorithms

The inference-based high-level policy formulation framework exhibits strong conceptual and algorithmic ties to several reinforcement learning and control methodologies:

• Maximum-entropy RL and SAC: The entropy-regularized objective arises directly from variational inference in augmented models with optimality variables (Papadopoulos et al., 9 Jul 2025).
• Thompson sampling and K-learning: Variational policy occupancy solutions interpolate between classic Thompson sampling (sample first, act optimally) and entropy-regularized soft-value iteration (Tarbouriech et al., 2023).
• Inductive logic programming for policy learning: Logical inferences over belief-action pairs align with probabilistic logic learning in POMDP policy specification (Meli et al., 29 Feb 2024).
• Probabilistic programming and black-box policy optimization: The full pipeline from model declaration to inference can be automated by probabilistic programming frameworks, yielding sample-efficient and model-agnostic policy learning (Tolpin et al., 2020, Meent et al., 2015, Wood et al., 2020).

The field continues to evolve with advances in scalable inference algorithms, expressive model classes (e.g., GP-based representations), and rigorous integration of safety, interpretability, and exploration.

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References:

• (Papadopoulos et al., 9 Jul 2025): Learning safe, constrained policies via imitation learning: Connection to Probabilistic Inference and a Naive Algorithm
• (Wang et al., 24 Jun 2024): Probabilistic Subgoal Representations for Hierarchical Reinforcement learning
• (Tarbouriech et al., 2023): Probabilistic Inference in Reinforcement Learning Done Right
• (Hansel et al., 2022): Hierarchical Policy Blending as Inference for Reactive Robot Control
• (Tolpin et al., 2020): Bayesian Policy Search for Stochastic Domains
• (Meent et al., 2015): Black-Box Policy Search with Probabilistic Programs
• (Wood et al., 2020): Planning as Inference in Epidemiological Models
• (Meli et al., 29 Feb 2024): Learning Logic Specifications for Policy Guidance in POMDPs: an Inductive Logic Programming Approach
• (Song et al., 2020): Learning High-Level Policies for Model Predictive Control