Two-Stage Reinforcement Learning

• Two-Stage Reinforcement Learning is a structured framework that divides tasks into high-level planning and low-level execution, enabling effective decision decomposition.
• It leverages hierarchical architectures and feedback loops to improve credit assignment and sample efficiency in adversarial and complex environments.
• Practical implementations like HGFormer demonstrate its advantages in dynamic resource allocation through enhanced global context integration and refined tactical adjustments.

Two-stage reinforcement learning refers to the structured decomposition of a sequential decision-making problem into two interdependent learning subproblems, with each handled by a specialized agent, module, or optimization process. Rather than treating the decision process as a monolithic end-to-end task, the two-stage framework introduces an explicit hierarchy (or cascade) of decisions, typically corresponding to distinct phases, temporal scales, or abstraction levels. This architecture is leveraged not only to address modeling or computational challenges intrinsic to high-dimensional, adversarial, or combinatorial domains, but also to facilitate credit assignment, sample efficiency, and targeted learning of complementary capabilities.

1. Formal Structure and Hierarchical Decomposition

The canonical two-stage reinforcement learning setup partitions policy generation or control into:

1. Stage 1 – Strategic/Planning Stage: An agent generates high-level actions or coarse allocations of resources (e.g., initial deployment, task assignment, trajectory planning), often with access to global structural context and under long-horizon objectives.
2. Stage 2 – Tactical/Execution Stage: Conditioned on the first stage's output and current state, a separate agent (or an augmented policy) makes finer-grained or dynamic decisions (e.g., reallocation, local adjustments, fine motion control) typically under stricter constraints and shorter timescales.

In the context of adversarial resource allocation, such as the two-stage Colonel Blotto game on graphs, the initial allocation (Stage 1) determines the distribution of resources on a network, and the dynamic transfer (Stage 2) handles sequential, constrained redistribution across graph edges (Lv et al., 10 Jun 2025).

This staged approach generally recognizes strong interdependencies between stages: optimality in the second stage (execution/tactics) can be severely constrained by suboptimal high-level plans, while strategies that ignore second-stage adaptability may underperform in adversarial or uncertain environments.

2. Core Algorithmic Components

Hierarchical Architectures

Two-stage RL frameworks often employ modular, hierarchical agent architectures. In "HGFormer: A Hierarchical Graph Transformer Framework for Two-Stage Colonel Blotto Games via Reinforcement Learning" (Lv et al., 10 Jun 2025), the hierarchy consists of:

• Planner Agent (Stage 1): Sequentially assigns resources to nodes, leveraging global state embeddings from an enhanced graph Transformer encoder (EGTE) with structural bias.
• Transfer Agent (Stage 2): At each round, determines dynamic local resource transfers, combining global EGTE features and fine-grained local signals via a GATv2 module.

This separation enables efficient combinatorial optimization while maintaining end-to-end differentiability and credit assignment across both stages.

Layer-by-Layer Feedback Reinforcement Learning

To ensure synergy between stages, HGFormer introduces layer-by-layer feedback RL (LFRT):

• Independent pre-training: The Planner is first trained (REINFORCE) to maximize an immediate proxy objective (e.g., nodes initially controlled).
• Downstream training: The Transfer agent (PPO) is then optimized, taking the initial allocation as input, to maximize overall dynamic utility (e.g., maximizing final node control after transfer, minus transfer costs).
• Feedback loop: The long-term cumulative reward of the Transfer agent feeds back as an augmentation to the Planner’s optimization target:

$$J'(\theta^P) = \mathbb{E}\left[R_0^P + \lambda U_r(\hat{\theta}^T)\right]$$

This feedback ensures that Stage 1 learns a policy not merely for myopic immediate payoff but for enabling downstream adaptability and global performance in Stage 2.

Attention and Structural Bias for Representation

In the HGFormer framework, state encoding is performed via a graph Transformer with shortest-path distance-based structural bias in self-attention:

$$e_{ij}^{(l,h)} = \frac{\langle q_{i}^{(l,h)}, k_{j}^{(l,h)} \rangle}{\sqrt{d/H} + \mathscr{b}_{\delta(i,j)}}$$

This formulation enables capturing of long-range dependencies critical for planning on graphs and ensures representation is sensitive to both node features and topological context.

3. Mathematical Coordination Across Stages

The two-stage coordination is formalized by linking the upper-level (Stage 1) policy's optimization objective to the long-term outcome as realized by the lower-level (Stage 2) agent. For the Colonel Blotto scenario:

$$U_r = \sum_{i} I(\mathscr{s}_i^r(\mathcal{T}) \geq \mathscr{s}_i^b(\mathcal{T})) v_i - \sum_{t=0}^{\mathcal{T}} C^r(t)$$

Here, $U_r$ captures the final strategic payoff (controlled nodes minus transfer cost). The Planner’s learning objective is augmented to account for $U_r$ resulting from the Transfer agent’s rollout, ensuring that global coordination across both stages is explicitly enforced.

In contrast to previous hierarchical or decomposed RL algorithms that either decouple stages or use surrogate objectives, the explicit feedback loop mechanism ensures full credit assignment and learning of policies that trade off short- and long-term gains.

4. Performance Benefits and Comparative Analysis

Empirical and numerical results obtained with HGFormer demonstrate:

• Superior performance on large, adversarial, graph-structured resource allocation benchmarks, measured in final utility, transfer cost minimization, and inference speed.
• EGTE’s ability to capture global and long-range dependencies robustly outperforms local aggregation approaches typical of GNN-based baselines in both policy quality and scalability.
• The layer-by-layer feedback mechanism is the critical differentiator. Methods without feedback, or that do not coordinate policies across stages, underperform both in dynamic adaptability and strategic optimality.

A summary table of comparative results illustrates these trends:

Method	Final Utility	Transfer Cost	Inference Time
HGFormer (+LFRT)	Highest	Lowest	Fastest
Hierarchical GNN RL	Lower	Higher	Slower
Separable RL	Lower	Higher	Variable

(As reported in (Lv et al., 10 Jun 2025), see Table 1/2 and figures for specific numbers.)

5. Extensions, Limitations, and Generalization

The two-stage RL framework instantiated by HGFormer generalizes to other sequential resource allocation and multi-stage decision problems exhibiting:

• Strong coupling between high-level strategic planning and low-level execution
• Adversarial or uncertain, dynamic environments
• Operational constraints reflecting underlying topological or combinatorial structure

Key limitations include: the added complexity of feedback-based joint training, the requirement for differentiable or at least backpropagatable reward functions across hierarchical agents, and potential sample inefficiency if coordination is not effectively scheduled or if reward signals are too sparse or delayed.

In summary, two-stage reinforcement learning as deployed in hierarchical and feedback-coordinated architectures (such as HGFormer (Lv et al., 10 Jun 2025)) offers a principled and empirically validated approach to solving large-scale, adversarial, dynamic resource allocation problems, by decomposing policy learning into strategically aligned stages, embedding global-structural context, and ensuring each stage is optimized in light of its downstream effect on overall system utility.