Truncated Rollout Backward-Forward ADP Framework

• The paper introduces a two-phase ADP framework that separates offline Q-factor computation from online 1-step rollout to manage intractable MDPs with high-dimensional information states.
• It employs state quantization and Lagrange duality to incorporate directed information penalties, significantly reducing computational complexity.
• Empirical results on binary Markov chain examples validate improved cost performance and scalability, making it ideal for networked control and IoT systems.

A truncated rollout-based backward-forward Approximate Dynamic Programming (ADP) framework is an algorithmic approach for large-scale and information-constrained Markov decision processes (MDPs), designed to address intractability arising from continuous or high-dimensional information states by decomposing the dynamic program into an efficient two-phase structure. This framework integrates directed information and stage-wise cost constraints directly into the optimization, advances solution methods by separating offline base-policy approximation from online rollout-based lookahead, and leverages provable convergence guarantees to enable robust control in settings where communication limitations or complexity of latent states dominate.

1. Problem Setting and Information-Theoretic Constraints

The central focus is a finite-horizon MDP in which each decision-maker aims to minimize information transfer (as measured by directed information) from the controlled source process $X^N$ to the control process $U^N$ under stage-wise cost constraints. Here, the cost at each stage is not only the conventional state-action cost $\rho_t(x_t, u_t)$ but also includes an explicit penalty for the flow of information, quantified as

$$I(X_t; U_t \mid U^{t-1}) = \mathbb{E}\left\{ \log \frac{\mu_t}{\nu_t} \right\}$$

where $\mu_t$ is the conditional control policy and $\nu_t$ is the output distribution under a reference policy. This yields a constrained optimization where the Lagrangian is constructed as

$$g_t(b_t, \mu_t) = \log \frac{\mu_t}{\nu_t} - s_t (\rho_t(x_t, u_t) - D_t)$$

with $s_t$ denoting the dual Lagrange multipliers and $D_t$ the fidelity thresholds. The information state $b_t$ is a filter-based sufficient statistic encapsulating the causal conditional distribution of source states, representing the dynamical evolution of information as actions are taken and new measurements are incorporated.

2. Q-Factor Recursion and Unconstrained Reformulation

The information-theoretic MDP is reformulated—via Lagrange duality—into an unconstrained dynamic program over a continuous information state, where the principal object is the Q-factor, recursively defined by

$$Q_t^*(b_t, \mu_t) = \sum_{x_t, u_t} \left\{ \log \frac{\mu_t}{\nu_t} - s_t \rho_t(x_t, u_t) + \min_{\mu_{t+1}} Q_{t+1}^*(b_{t+1}, \mu_{t+1}) \right\} w_t \mu_t b_t + s_t D_t$$

with $w_t$ representing an exogenous weighting. The minimizing policy at each stage is obtained by simultaneously optimizing over $(\mu_t, \nu_t)$, where $\nu_t$ is itself implicitly determined through a structural relationship (see Equation (3) in the original formulation). Direct solution is intractable due to the continuous nature and high dimension of the information state space $b_t$, as well as the need to recompute optimal Q-factors for each possible realization on the fly.

3. Truncated Rollout-Based Backward-Forward Decomposition

To address the computational intractability, the framework divides the decision process into two primary phases:

a. Offline Base-Policy Approximation (Backward Phase)

• Short-Horizon Truncation: Rather than solving the Q-factor recursion backward over the entire horizon $N$ and full continuous information state space, the offline phase uses a shortened horizon $N_s \ll N$.
• State-Discretization: The information state space is discretized (forming $\overline{\mathcal{B}}_t$), reducing the problem to a finite but tractable set of representative belief points.
• Iterative Policy Computation: The Q-factor and policy are iteratively updated according to (paraphrased from Eq. (5)):

$$\mu_t^{(k+1)} = \frac{\nu_t^{(k)} \, \mathcal{A}_t[b_t](x_t, u_t, s_t)} { \sum_{u_t} \nu_t^{(k)} \, \mathcal{A}_t[b_t](x_t, u_t, s_t) }$$

with $\mathcal{A}_t$ denoting a softmin over the cost-to-go and dual variables, until convergence is reached or a tolerance $\epsilon$ is satisfied.

• Result: The output is a base policy $\bar{\pi}$ with approximate Q-factors $\tilde{Q}_t^{(\bar{\pi})}$ defined over the quantized information states.

b. Online Rollout Lookahead (Forward Phase)

• Rollout-Based Policy Improvement: At each time step, the current information state is used to perform a 1-step rollout (lookahead minimization) using the precomputed $\tilde{Q}_t^{(\bar{\pi})}$ values:

$$\tilde{\mu}_t = \arg\min_{\mu_t \in \bar{\pi}} \sum_{u^{t-1}} \tilde{Q}_t^{(\bar{\pi})}(b_t, \mu_t) P_t(u^{t-1})$$

• Information-State Update: Upon selection of control $\tilde{\mu}_t$ and observation of the next source realization, the information state $b_{t+1}$ is updated in accordance with the belief update (filtering) equations.
• Iterative Forward Simulation: This process continues for $t = 1, \ldots, N$, progressively building the control sequence and preserving stage-wise improvement guarantees.

4. Theoretical Properties and Convergence Guarantees

Rigorous guarantees are established for both phases:

• Offline Double Minimization: The base policy computation (offline phase) admits precise double minimization, guaranteeing convergence to the optimal Q-factor for the truncated horizon as iteration index $k \to \infty$.
• Rollout Cost Improvement: Backward induction is employed to show that the online rollout ensures nonincreasing cost-to-go relative to the base policy:

$$\tilde{J}_t^{(\hat{\pi})}(b_t) \leq \tilde{J}_t^{(\bar{\pi})}(b_t), \quad \forall b_t$$

• Complexity Analysis: The overall computational complexity scales with the size of the discretized information state space and truncation horizon as $O(N_s n^2 / \epsilon)$ in the offline phase (for quantized state space of size $n$ and tolerance $\epsilon$), which is substantially less than a full-horizon, full-state-space backward sweep.

5. Empirical Illustration and Numerical Example

A canonical example is constructed using binary source and control alphabets $(X_t, U_t)\in\{0,1\}$, with Hamming distance cost:

$$\rho(x_t,u_t) = \begin{cases} 0 & \text{if } x_t = u_t\ 1 & \text{if } x_t \neq u_t \end{cases}$$

A binary symmetric controlled Markov chain serves as the process model. The information state is quantized with a finite number of grid points (e.g., for belief over $X_t$) and a memory-1 (Markov) assumption is used, ensuring tractable state evolution. Simulation results demonstrate:

• Improved Cost: The rollout-enhanced policy achieves strictly lower (or equal) stage-wise and cumulative cost compared to baseline policies previously used in similar information-theoretic MDPs.
• Reduced Offline Complexity: Offline policy computation for the truncated horizon requires significantly less time and memory, supporting larger grids or longer horizons with practical resource use.
• Scalability of Online Phase: The main online burden is isolated to 1-step forward minimization and belief update, which is substantially more efficient than full backward DP.

6. Implications for Communication-Control Systems and Limitations

The truncated rollout-based backward-forward ADP framework is particularly well-suited for networked control, cyber-physical systems, and Internet of Things (IoT) scenarios with stringent communication or information constraints, as the explicit minimization of directed information promotes control policies that are frugal in channel usage without sacrificing control performance. The decoupling of offline and online computations allows system designers to precompute base policies for a wide range of likely information states or cost regimes.

However, practical deployment must account for:

• Online Computation: Although the online phase is minimized, for systems with very rapid dynamics or tight latency limits, further acceleration or parallelization may be needed.
• State Space Quantization: The selection and adaptation of the quantized information state space affects both the tractability and the tightness of the performance guarantees.
• Adaptation to Time-Varying or Unmodeled Dynamics: The offline phase can be periodically repeated or quantization grids can be adaptively refined in response to observed empirical information states.

7. Relation to Broader ADP and Rollout Literature

This framework synthesizes and extends multiple active research directions in ADP:

• It generalizes rollout algorithms—traditionally employed for tractable approximations of value-to-go functions—by embedding them in information-constrained settings with general, possibly high-dimensional continuous state statistics.
• It incorporates and improves upon theoretical advances in guaranteeing policy improvement and convergence through backward induction and double minimization arguments.
• It enables explicit tradeoffs between computational complexity (via offline rollout truncation and information state quantization) and solution optimality, grounded in concrete empirical performance metrics.

The methodological architecture is broadly compatible with other structural extensions of ADP, such as distributionally robust ADP, constraint approximation via linear programs, and function approximation via kernel or subsemimodule projections, provided the formulation permits backward-forward decomposition and rollout-based improvement steps.

Summary Table: Framework Components and Roles

Component	Role in Framework	Complexity Impact
Offline Base Policy	Approximates Q-factors over truncated, discretized horizon	$O(N_s n^2 / \epsilon)$
Online Rollout	1-step lookahead minimization using precomputed Q-factors	Dominant per-stage cost, scalable
Directed Info Constraint	Penalizes excessive information flow via mutual information term	Enforces communication frugality
State Quantization	Discretizes continuous info states for offline optimization	Governs memory/runtime tradeoff

This truncated rollout-based backward-forward ADP framework provides a scalable, convergence-guaranteed method for MDPs with complex information-theoretic requirements, offering provably improved performance over base policies while maintaining computational practicality for application-scale problems (He et al., 2 Sep 2025).