Task Space Decomposition Approach

• Task Space Decomposition is a formal method that partitions complex tasks into manageable subtasks using hierarchical planning and resource-rational optimization.
• It employs both exact enumeration and gradient-based optimization to compute optimal subgoal sets, reducing search costs and improving planning efficiency.
• Practical applications include gridworlds and human planning tasks, where the approach predicts subgoal selection that aligns with both graph properties and cognitive strategies.

Task Space Decomposition Approach

Task space decomposition refers to a set of formal methodologies for representing, analyzing, and solving complex tasks by partitioning a monolithic task into smaller, more tractable subtasks or segments. These approaches are foundational in hierarchical planning, multi-agent systems, reinforcement learning, robotics, automated reasoning, and human cognitive modeling. Central to modern computational perspectives is the idea of resource-rationality: decompositions are selected to optimize the use of computational or cognitive resources, often formulated as minimizing planning costs, coordination overhead, or error (Correa et al., 2020).

1. Formal Principles of Task Space Decomposition

Given a finite state space $S$, deterministic transitions $T \subseteq S \times S$, a distribution over start–goal pairs $p(s,g)$, and a planning algorithm $\text{Alg}$ with per-pair search cost $C_{\text{Alg}}(s, z)$ and distance $D(s, z)$, the decomposition problem is to select a fixed set of subgoals $Z \subseteq S$ (with $g \in Z$ by convention) for reuse across all tasks. An agent utilizes a two-level hierarchical planner:

• At the subtask-level, it chooses a sequence of subgoals $z_1, ..., z_k$ (with $z_k = g$) to maximize cumulative reward minus search cost.
• At the action-level, $\text{Alg}$ plans from one subgoal to the next.

The subtask-level value function $V_Z^g(s)$ is uniquely characterized by the Bellman equation: $$V_Z^g(s) = \max_{z \in Z} \left[ R(s, z) - C_{\text{Alg}}(s, z) + V_Z^g(z) \right],$$ where $R(s, z) = -D(s, z)$. The resource-rational task decomposition is the choice of $Z$ that maximizes the expected subtask-level value over $p(s,g)$: $$Z^* = \arg\max_{Z \subseteq S} \mathbb{E}_{(s,g) \sim p}[V_Z^g(s)].$$ This selection may be regularized or size-constrained for complexity control (Correa et al., 2020).

2. Computational Algorithms for Decomposition

Two principal computational methods are used:

A) Exact Enumeration (small $|S|$, small $K$):

1. Precompute all $D(s, z)$ and $C_{\text{Alg}}(s, z)$.
2. For each candidate $Z \subseteq S$ of size $K$ (with $g \in Z$), solve for $V_Z^g(s)$ via value iteration until convergence.
3. Compute $\mathbb{E}_{(s,g)}[V_Z^g(s)]$ and select $Z$ maximizing expected value.

B) Gradient-Based Optimization (large $|S|$):

1. Introduce a continuous parametrization over subgoal weights.
2. Temper the Bellman “max” with a softmax for differentiability.
3. Compute gradients of the cost function via VI unrolling/backprop or implicit differentiation.
4. Run gradient-based optimization.

These algorithms efficiently handle both exhaustive search and scalable soft selection, leveraging value iteration and policy gradients (Correa et al., 2020).

3. Theoretical Characterizations and Key Results

The decomposition framework exhibits three-level nested optimization: selection of a global subgoal set $Z$, subtask-level scheduling over these subgoals, and action-level plans between subgoals. The Bellman formulation guarantees policy optimality with uniqueness arguments relying on standard contraction properties in finite state spaces.

The choice of action-level planner critically influences optimal decompositions:

• BFS (uninformed search) leads to bottleneck subgoals that strategically partition the state graph, minimizing breadth expansions.
• Heuristic planners (e.g., A* with admissible heuristics) prioritize subgoals that disperse to bound heuristic error and simultaneously minimize search effort (Correa et al., 2020).

Existence and uniqueness of optimal decompositions are guaranteed under finite $S$ and positive regularization or cardinality constraints.

4. Practical Applications and Empirical Validation

Resource-rational decomposition robustly predicts empirical findings in both synthetic and human planning domains:

• Gridworlds: Identifies central, door-like, or corridor cells as bottleneck subgoals, reducing action-level search expansions by 40–50%.
• Community-structured graphs: Recovers subgoals at community boundaries or bottleneck nodes, aligning with human subgoal identification in network navigation tasks.
• The Tower of Hanoi: The method recovers decompositions matching patterns of human hierarchical subtask planning depending on the planner’s available heuristics (Correa et al., 2020).

Metrics include node expansion counts, reaction-time simulation (as a proxy for search cost), and alignment with empirical subgoal and path choices.

5. Normative Explanations for Human and Machine Subgoal Selection

The framework explains empirical regularities in human planning by characterizing subgoal selection as resource-rational: bottleneck states optimally reduce overall computational effort across typical start–goal queries. When heuristic information is available, the decomposition adapts to maintain both low search costs and bounded error, bridging AI “option discovery” and psychological studies of hierarchical planning. These results clarify why subgoals reflect both graph-theoretic structure (e.g., bottlenecks) and the computational properties of the planning algorithm in use (Correa et al., 2020).

6. Implications, Extensions, and Contemporary Directions

Resource-rational task space decomposition serves as a foundation for broad classes of hierarchical and modular planning strategies, including hierarchical reinforcement learning, symbolic-planning decomposition, and graph-based option discovery. Its theoretical clarity provides a basis for further integration with large-scale function approximation, differentiable planners, and adaptive amortization of subgoal sets in nonstationary environments.

Extensions include:

• Adaptive regularization strategies for complexity-aware planning.
• Online updating of subgoal sets in dynamic architectures.
• Application to explainability in learned hierarchical policies.
• Integration with symbolic abstraction or skill-discovery modules (Correa et al., 2020).

By delivering a principled normative objective and scalable computational tools, resource-rational task space decomposition is central to understanding and engineering efficient plans and representations in both human and artificial agents.