Subgraph Bellman Operators in RL

• Subgraph Bellman Operators are specialized formulations that restrict Bellman updates to selected state subsets, blending TD and MC methods for localized evaluation.
• They yield rigorous error analysis by offering sharp probabilistic error bounds and minimax lower bounds that improve over classical global operators.
• They are applied in RL policy evaluation, distributed decision making, and verification tasks, and extend to function approximation and spectral methods for scalable decision processes.

Subgraph Bellman Operators are a class of Bellman operator formulations arising in dynamic programming, reinforcement learning (RL), and operator theory, defined by restricting the operator’s action to a subset (“subgraph”) of the state space or, more generally, to substructures within an operator framework. This notion unifies several distinct technical approaches—ranging from RL estimators that interpolate between temporal difference (TD) and Monte Carlo (MC) methods on specific state subsets, to operator-theoretic inequalities for functional aggregation over subgraphs, and reachability/verification problems for piecewise affine maps restricted to induced subgraphs of Markov decision processes (MDPs). Subgraph Bellman Operators provide mechanisms for localized estimation, sharper error analysis, and adaptive policy evaluation, enabling rigorous treatment of partitioned or structured state spaces in both theoretical and computational regimes.

1. Formal Definition and Operator Construction

A Subgraph Bellman Operator is formulated by selecting a subset $G \subset S$ of the state space (or nodes in a network) and defining an operator whose fixed-point equation uses bootstrapping on transitions within $G$ and Monte Carlo (rollout-style) evaluation on transitions that exit $G$. Formally, for a Markov reward process (MRP) or MDP, the subgraph operator $\mathcal{T}_G$ acts as: $$(\mathcal{T}_G V)(s) = \begin{cases} r_G(s) + P_G V(s) + O_G(s), & s \in G \end{cases}$$ with

• $r_G(s)$: empirical reward accumulated for $s \in G$
• $P_G$: transition operator restricted to $G$
• $O_G(s)$: Monte Carlo correction term aggregating rewards from trajectories that exit $G$.

This local split allows explicit interpolation: TD-style updates when the trajectory remains in $G$, and MC rollouts upon leaving $G$ (Mou et al., 14 Nov 2024). The fixed point $V_{\text{est}}$ solves

$$V_{\text{est}} = r_G + P_G V_{\text{est}} + \mathcal{O}_G$$

and can be computed via stochastic approximation techniques adapted to the occupancy measure on $G$.

2. Mathematical Analysis: Error Bounds and Lower Bounds

The operator’s design admits sharp probabilistic error bounds. For large sample sizes $n$, the estimation satisfies asymptotic normality: $$\sqrt{n}(V_{\text{est}} - V^*) \to \mathcal{N}(0, (I - P_G)^{-1} \Sigma_G^* (I - P_G)^{-T})$$ where $\Sigma_G^*$ is a state-dependent conditional covariance comprising TD variance and an additional term scaling with the exit probability from $G$. Explicit non-asymptotic bounds hold: $$\|V_{\text{est}} - V^*\|_{L^2(\nu)} \leq C \left[ \sum_{s \in G} \nu(s) \big((I - P_G)^{-1} \Sigma_G^* (I - P_G)^{-T}\big)_{s,s} \right]^{1/2} \sqrt{ \frac{\log(1/\delta)}{n} + \frac{h^3}{n\sqrt{\nu_{\min}}} }$$ where $h$ is the effective planning horizon and $\nu_{\min}$ the minimal occupancy in $G$ (Mou et al., 14 Nov 2024).

Additionally, a minimax lower bound establishes that the variance increment due to MC rollouts at exits from $G$ is information-theoretically unavoidable (scaling as $q/\nu_0$ for exit probability $q$ and occupancy $\nu_0$), unless $n$ becomes large.

3. Methodology: Comparison with Classical and Alternative Operators

Unlike global Bellman operators (classical TD), subgraph operators locally pool data and adapt error analysis to visitation patterns, sidestepping the bias that affects TD methods when sample sizes are insufficient relative to the state space. This also distinguishes them from pure MC estimators, which do not exploit trajectory sharing and thus incur higher variance.

Practically, algorithms such as ROOT–SA (Algorithm 1/2 in (Mou et al., 14 Nov 2024)) efficiently solve the subgraph fixed-point equations using data-dependent weighting (e.g., $w(s) := 1/(2\hat{\nu}(s))$ computed from auxiliary samples), and a greedy algorithm can learn $G$ to minimize variance using a hold-out set.

4. Applications and Generalizations

Subgraph Bellman Operators have wide applicability:

• RL policy evaluation focusing on frequently visited regions, reducing sample complexity for these states.
• Distributed decision making and networked control, where only partial state and transition data are accessible.
• Modular verification in MDPs, by applying Bellman-type analysis to subgraphs and ensuring fixed-point reachability and decidability.
• Extension to function approximation, off-policy evaluation, and online adaptive selection of $G$.

Set-based Bellman operators (Li et al., 2020) further generalize this principle by mapping compact sets of value functions, incorporating parameter uncertainty via Hausdorff contracting mappings in complete metric spaces.

5. Operator Inequalities and Operator-Theoretic Connections

Operator-theoretic Bellman inequalities provide additional tools for analyzing subgraph aggregation. The reverse operator Bellman inequality (Bakherad et al., 2015) states: $$\delta I_{\mathscr K} + \sum_{j=1}^n \omega_j \Phi_j\left( (I_{\mathscr H} - A_j)^p \right) \geq \left( \sum_{j=1}^n \omega_j \Phi_j(I_{\mathscr H} - A_j) \right)^p$$ where $A_j$ are self-adjoint contractions (potentially associated with subgraph segments), $\Phi_j$ unital positive linear maps, and $\omega_j$ positive weights. The Mond–Pečarić method enables reversals and refinements of such inequalities, providing sharper bounds on functional calculus over localized operator blocks.

6. Reachability, Verification, and Decidability

Piecewise affine Bellman operators—arising from MDPs—admit reachability analysis restricted to subgraphs, with decidability guaranteed in arbitrary dimension when the target vector is not the fixed point, or the initial and target vectors are componentwise comparable. In dimension two, the reachability question is decidable for all cases, contrasting the undecidability in general piecewise affine maps (Varonka et al., 27 Feb 2025). Techniques employed include contraction arguments, sign-abstraction, and reduction to matrix semigroups.

7. Future Directions and Open Problems

Key open research directions include:

• Adaptive online selection and resizing of $G$ in response to changing data streams or visitation patterns.
• Extension of subgraph operators to policy optimization, integrating maximization steps (as in Q-learning), and function approximation.
• Analysis of planning and value propagation using spectral methods (as in the Spectral Bellman Method (Nabati et al., 17 Jul 2025)), where feature representations are aligned with Bellman dynamics over subgraphs or multi-step operators.
• Investigating subgraph operator inequalities using alternative operator monotone or concave functions.
• Application to distributed optimization, quantum networks, and modular spectral graph theory.

A plausible implication is that deeper integration of spectral analysis, operator inequalities, and subgraph locality will yield scalable algorithms capable of rigorous performance certification on large-scale or non-homogeneous decision processes.

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In summary, Subgraph Bellman Operators constitute a mathematically rigorous and practically flexible framework for localized dynamic programming and reinforcement learning. They interpolate TD and MC estimation, enable sharp error bounds and lower bounds governed by subgraph occupancy and exit probabilities, and admit generalizations incorporating parameter uncertainty, operator-theoretic inequalities, and reachability analysis for modular verification tasks. Their continued paper is poised to inform the development of robust, adaptive, and scalable decision-making systems in engineering, computer science, and mathematics.