Transitive Reinforcement Learning (TRL)

• Transitive Reinforcement Learning (TRL) is a reinforcement learning paradigm that utilizes compositional subgoal structures to improve value function estimation and policy learning.
• It employs a divide-and-conquer Bellman backup using the triangle inequality, enabling O(log T) recursions to reduce bias accumulation compared to classical TD methods.
• TRL has shown robust performance in long-horizon tasks, facilitating transfer learning and hierarchical planning in robotics, navigation, and multi-step decision problems.

Transitive Reinforcement Learning (TRL) refers to a paradigm in reinforcement learning that leverages transitive or compositional structures of state transitions to improve value function estimation, policy learning, and transfer across tasks. Unlike classical approaches that focus on immediate state-action pairs, TRL algorithms exploit the triangle inequality inherent in shortest-path goal-conditioned reinforcement learning, enabling divide-and-conquer strategies for learning in long-horizon or large-action problems.

1. Mathematical Foundations of Transitive RL

TRL is founded on the principle that in goal-conditioned reinforcement learning, the optimal value of reaching a goal from an initial state satisfies a triangle inequality structure. In settings such as shortest-path problems, this is formalized as: $d^*(s,g) \leq d^*(s,w) + d^*(w,g)$ where $d^*(s,g)$ is the optimal distance (or expected steps) from state $s$ to goal $g$, and $w$ is an intermediate “subgoal”. For $\gamma$-discounted environments, the corresponding value function relation is: $$V^*(s,g) = \gamma^{d^*(s,g)} \geq V^*(s,w) \cdot V^*(w,g)$$ This compositional property allows TRL algorithms to update long-horizon values via combinations of shorter, already-learned segments, rather than propagating estimates from immediate transitions alone.

2. Divide-and-Conquer Value Update Rule

The central algorithmic innovation of TRL is the transitive Bellman backup for offline value learning. In the context of deterministic goal-conditioned RL, the value function update is: $$V(s,g) \leftarrow \begin{cases} \gamma^0 & \text{if } s = g \ \gamma^1 & \text{if direct edge } (s,g) \ \max_{w \in \mathcal{S}} V(s,w)V(w,g) & \text{otherwise} \end{cases}$$ For Q-functions with actions: $$Q(s,a,g) \leftarrow \begin{cases} \gamma^0 & s = g \ \gamma^1 & g = p(s,a),\ s \neq g \ \max_{w\in\mathcal{S}, a'\in \mathcal{A}} Q(s,a,w) Q(w,a',g) & \text{otherwise} \end{cases}$$ This divide-and-conquer update requires only $O(\log T)$ recursions for a length-$T$ trajectory, drastically reducing the bias accumulation compared to the $O(T)$ steps in classical temporal-difference algorithms.

3. Practical Algorithm and Implementation

Direct maximization over all potential subgoals $w$ introduces the risk of overestimation in function-approximation settings. To address this, practical TRL implementations restrict intermediate subgoals to those encountered within sampled trajectories and use soft maximization schemes (e.g., expectile regression with parameter $\kappa$) instead of hard maxima.

The TRL value learning loss is structured as: $$L^{\rm TRL}(Q) = \mathbb{E}_{\tau \sim \mathcal{D}}\left[w(s_i,s_j)\; D_\kappa({Q}(s_i,a_i,s_j),\, \bar{Q}(s_i,a_i,s_k)\; \bar{Q}(s_k,a_k,s_j)) \right]$$ where $i < k < j$ are indices sampled along a trajectory $\tau$, $w(s_i,s_j)$ is a distance-based weighting emphasizing shorter segments, and $D_\kappa$ is an expectile loss promoting soft maximization.

A typical TRL learning loop (see Algorithm 1 in (Park et al., 26 Oct 2025)) samples trajectory chunks, computes targets from recursively composed Q-values, and performs updates with weighted losses.

4. Algorithmic Advantages Over Classical Methods

TRL’s divide-and-conquer recursion achieves logarithmic ($O(\log T)$) dependency on trajectory horizon for value update propagation. This sharply reduces bias accumulation compared to temporal-difference (TD) learning and mitigates the variance blow-up typical of Monte Carlo approaches. Theoretical analysis in (Park et al., 26 Oct 2025) confirms that the expected number of recursions $B(n)$ for a trajectory of length $n$ satisfies: $B(n) \leq \frac{\log n}{\log(4/3)}$ leading to robust value estimation on long-horizon tasks.

Method	Recursions per Trajectory	Bias	Variance
TD	$O(T)$	High	Low
MC	$O(1)$	Low	High
TRL	$O(\log T)$	Low	Low

TRL also decouples the estimation of valuable transitions from skill acquisition—a key conceptual advantage first discussed in the context of T-Learning (Graziano et al., 2011), where transitions rather than state-action pairs are valued policy-independently, enabling efficient learning in environments with few relevant states and large action spaces.

5. Empirical Performance and Applications

TRL has demonstrated superior or matching performance compared to best-tuned TD-n and MC methods, especially in offline goal-conditioned RL tasks characterized by long horizons (e.g., humanoid navigation, extended multi-step robotics benchmarks with up to $3{,}000$ steps). Unlike previous triangle-inequality backup approaches, TRL successfully handles realistic continuous control scenarios.

Key ablations confirm that in-trajectory behavioral subgoals, expectile regression with $\kappa > 0.5$, and log-distance reweighting are all crucial for accurate, efficient learning. TRL adapts automatically to the effective horizon length, without requiring manual tuning of $n$-step parameters or explicit hierarchical planning.

6. Contextualization Within RL Theory and Transfer

TRL exemplifies the exploitation of transitive structures for compositional planning and transfer, naturally extending to settings where task or environment structure admits subgoal decompositions. The triangle inequality update underpins a general principle for efficient value learning in RL and suggests new directions for algorithm design, including integration with hierarchical and transfer learning frameworks (Graziano et al., 2011), as well as benchmarking and generalization analysis (Müller-Brockhausen et al., 2021).

TRL’s structure is especially relevant for transfer RL (also abbreviated TRL in the literature), where rapid adaptation across similar tasks relies on the ability to estimate transition values or composite policies. The algorithmic innovations discussed above provide mechanisms for scalable, horizon-adaptive, and robust transfer in both offline and online reinforcement learning.

7. Summary Table: TRL Loss Structure

Component	Description
$Q(s,a,g)$	Goal-conditioned Q-value function
$i < k < j$	Sampled indices along trajectory chunk
$w(s_i,s_j)$	Inverse log-distance weighting factor
$D_\kappa(\cdot)$	Expectile regression loss, soft maximization
Target	$Q(s_i,a_i,s_k) \times Q(s_k,a_k,s_j)$ (compositional target via subgoal $k$)

The TRL update combines compositional targets, soft maximization over behavioral subgoals, and distance reweighting to efficiently propagate value estimates across long trajectories.

8. Implications and Future Directions

TRL offers a mathematically principled solution to long-horizon value learning, opening avenues for robust offline RL, efficient transfer, and compositional policy construction. Applications extend to robotics, navigation, and multi-step planning. The framework admits generalization beyond deterministic goals with appropriate modifications and can catalyze future research into transitive, hierarchical, and divide-and-conquer RL algorithms. Continued exploration may yield further improvements in generalization, transfer, and scaling to high-dimensional state-action spaces.

For algorithmic details and code, refer to the published source at (Park et al., 26 Oct 2025).