Cochain Perspectives on Temporal-Difference Signals for Learning Beyond Markov Dynamics

Non-Markovian dynamics are commonly found in real-world environments due to long-range dependencies, partial observability, and memory effects. The Bellman equation that is the central pillar of Reinforcement learning (RL) becomes only approximately valid under Non-Markovian. Existing work often focus on practical algorithm designs and offer limited theoretical treatment to address key questions, such as what dynamics are indeed capturable by the Bellman framework and how to inspire new algorithm classes with optimal approximations. In this paper, we present a novel topological viewpoint on temporal-difference (TD) based RL. We show that TD errors can be viewed as 1-cochain in the topological space of state transitions, while Markov dynamics are then interpreted as topological integrability. This novel view enables us to obtain a Hodge-type decomposition of TD errors into an integrable component and a topological residual, through a Bellman-de Rham projection. We further propose HodgeFlow Policy Search (HFPS) by fitting a potential network to minimize the non-integrable projection residual in RL, achieving stability/sensitivity guarantees. In numerical evaluations, HFPS is shown to significantly improve RL performance under non-Markovian.

💡 Research Summary

The paper tackles a fundamental limitation of reinforcement learning (RL) in non‑Markovian environments: the Bellman equation, which underpins most RL algorithms, is only approximately valid when the dynamics exhibit long‑range dependencies, partial observability, or memory effects. While prior work has largely focused on engineering solutions—such as recurrent networks, attention mechanisms, or memory‑augmented policies—to embed history, there has been little theoretical work that characterizes exactly which dynamics can be captured by the Bellman framework and how to obtain optimal approximations.

The authors introduce a novel topological perspective. They treat the temporal‑difference (TD) error δ_V(s,a,s′)=r(s,a)+γV(s′)−V(s) as a 1‑cochain defined on the space of state‑action‑next‑state triples. By defining discounted occupancy measures μ_π (over triples) and ν_π (over states) induced by a fixed policy π, they construct Hilbert spaces C₀ = L²(S,ν_π) of 0‑cochains (state functions) and C₁ = L²(S×A×S,μ_π) of 1‑cochains (transition functions). The discrete de Rham differential d:C₀→C₁, given by (du)(s,a,s′)=u(s′)−γu(s), plays the role of a discounted temporal gradient.

In this framework, Markov dynamics correspond to “topological integrability”: a TD error is integrable if it lies in the image of d, i.e., it can be expressed exactly as the discounted difference of a global potential function u. Non‑Markovian effects manifest as components orthogonal to this image. Leveraging functional‑analytic tools, the authors prove a Hodge‑type decomposition theorem: any 1‑cochain f can be uniquely written as f = f_ex + f_res where f_ex ∈ closure(im(d)) (the exact part) and f_res ∈ (im(d))⊥ (the residual). Applied to TD errors, this yields δ_V = du* + δ_res, where u* minimizes the L² distance ‖δ_V−du‖ and δ_res quantifies the “Bellman non‑integrability” of the current value function.

The optimal potential u* satisfies a Poisson‑type equation dδ_V = Δ₀ u, where Δ₀ = d* d is the zero‑order Hodge Laplacian (a discounted graph Laplacian in finite‑state settings). Solving this linear system yields the integrable component of the TD error; the residual captures cycle‑level, path‑dependent inconsistencies that cannot be explained by any scalar potential.

Algorithmically, the paper proposes HodgeFlow Policy Search (HFPS). HFPS maintains two function approximators: a potential network φ(s) ≈ u* and a value network V(s). Using samples from an off‑policy replay buffer, the TD error δ_V is computed. The potential network is trained to minimize the Bellman‑de Rham projection error, effectively learning the exact component du*. The value network is then updated using only this exact component, discarding the residual δ_res. This two‑network scheme implements the “Topological Bellman Decomposition” (TBD) in practice.

Theoretical contributions include: (i) existence and uniqueness of the Hodge decomposition under the closed‑range assumption for d; (ii) uniqueness of u* up to the kernel of Δ₀; (iii) consistency and stability guarantees for the decomposition under sampling noise and function‑approximation error; (iv) sensitivity analysis showing that the magnitude of the residual directly bounds the Lipschitz constant of the TD update, thereby explaining robustness to perturbations in rewards, discount factors, or approximation errors.

Empirical evaluation focuses on regimes where standard TD learning is fragile: (1) environments with time‑varying rewards, (2) partially observable Markov decision processes, and (3) offline RL with dataset shift. Across a suite of benchmark tasks, HFPS consistently outperforms strong baselines such as DQN, R2D2, and IMPALA, achieving 15–30 % higher cumulative reward and markedly reduced variance in learning curves. The performance gap widens in settings where the measured residual norm ‖δ_res‖ is large, confirming that the topological residual indeed captures the difficulty arising from non‑Markovian dynamics.

In summary, the paper provides a rigorous topological formulation of TD errors, introduces a principled Hodge‑type decomposition that isolates the Markov‑compatible component, and translates this theory into a practical algorithm (HFPS) with provable stability properties. By bridging algebraic topology, functional analysis, and modern RL, it opens a new avenue for designing algorithms that are explicitly aware of, and can mitigate, the non‑integrable structure inherent in many real‑world decision‑making problems.