arxiv: 2605.04368 · v1 · submitted 2026-05-06 · 💻 cs.LG · cs.AI

Recognition: 2 theorem links

· Lean Theorem

Extending Differential Temporal Difference Methods for Episodic Problems

Kris De Asis , Mohamed Elsayed , Jiamin He

Authors on Pith no claims yet

Pith reviewed 2026-05-08 18:15 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords differential temporal differenceepisodic reinforcement learningreward centeringpolicy orderingtemporal difference learningstreaming reinforcement learning

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The pith

A generalization of differential TD maintains correct policy ordering when episodes terminate by adjusting reward centering.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Differential TD methods center each reward by the average reward to keep returns bounded and remove offset in continuing tasks. In episodic problems this centering can change which policy is optimal because termination affects the average. The paper introduces a modified centering step that respects termination states. It proves the modification leaves the ranking of policies unchanged, so higher-value policies remain preferred. Equivalence to linear TD transfers existing convergence results, and extensions to streaming algorithms show faster learning across episodic test environments.

Core claim

The generalization of differential TD maintains the ordering of policies in the presence of termination, thereby extending the method to episodic problems. Equivalence with a form of linear TD is established, inheriting its theoretical guarantees. Several streaming reinforcement learning algorithms receive differential counterparts, and experiments across base algorithms and environments confirm that the adjusted reward centering improves sample efficiency in episodic settings.

What carries the argument

The adjusted reward-centering term in the differential TD update that subtracts the average reward only over non-terminating transitions.

If this is right

Differential TD methods become applicable to episodic reinforcement learning without distorting policy preferences.
Existing convergence and performance guarantees for linear TD transfer directly to the generalized updates.
Multiple streaming reinforcement learning algorithms can be converted to differential versions with the same centering adjustment.
Reward centering yields measurable gains in sample efficiency when applied to episodic tasks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same centering adjustment could be tested in deep RL settings where normalization already helps streaming stability.
Equivalence to linear TD opens the possibility of importing variance-reduction analyses from the linear case into episodic differential methods.
The approach suggests a route for making other average-reward techniques compatible with clear episode boundaries.

Load-bearing premise

Adjusting reward centering for termination states preserves the original policy ordering and satisfies the conditions needed for equivalence to linear TD.

What would settle it

An episodic MDP in which the policy selected by the generalized differential TD differs from the policy that maximizes true expected return under proper termination handling.

Figures

Figures reproduced from arXiv: 2605.04368 by Jiamin He, Kris De Asis, Mohamed Elsayed.

**Figure 1.** Figure 1: Performance of Q-learning when used with reward- and value-centering compared against view at source ↗

**Figure 2.** Figure 2: Performance of differential Stream Q(0.8) compared against a standard uncentered base view at source ↗

**Figure 3.** Figure 3: Performance of differential Stream AC(0.8) compared against a standard uncentered base view at source ↗

**Figure 4.** Figure 4: Performance of differential Stream AC(0.8) compared against a standard uncentered base view at source ↗

read the original abstract

Differential temporal difference (TD) methods are value-based reinforcement learning algorithms that have been proposed for infinite-horizon problems. They rely on reward centering, where each reward is centered by the average reward. This keeps the return bounded and removes a value function's state-independent offset. However, reward centering can alter the optimal policy in episodic problems, limiting its applicability. Motivated by recent works that emphasize the role of normalization in streaming deep reinforcement learning, we study reward centering in episodic problems and propose a generalization of differential TD. We prove that this generalization maintains the ordering of policies in the presence of termination, and thus extends differential TD to episodic problems. We show equivalence with a form of linear TD, thereby inheriting theoretical guarantees that have been shown for those algorithms. We then extend several streaming reinforcement learning algorithms to their differential counterparts. Across a range of base algorithms and environments, we empirically validate that reward centering can improve sample efficiency in episodic problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper extends differential TD to episodic problems by generalizing reward centering, with a proof that policy ordering holds under termination and equivalence to linear TD.

read the letter

The main thing to know is that this work adapts differential TD methods, which center rewards by their average to keep returns bounded, so they apply to episodic tasks without distorting which policies rank higher. They prove the generalized centering preserves policy ordering even with termination and show it matches a form of linear TD, inheriting those guarantees. From there they convert several streaming algorithms to differential versions and test them on multiple environments, reporting sample-efficiency gains from the centering step. The proof and equivalence are the parts that stand out as useful if the derivations are tight. Extending the approach to streaming methods and checking it across base algorithms gives the empirical side some breadth rather than a single narrow test. A soft spot is the precise definition of the new centering operator for termination; the abstract states it works but the full steps would need checking to confirm no edge cases with episode length or reward structure create ordering shifts. The claimed efficiency improvements also need the actual numbers and variance from the results section to judge how reliable they are across settings. This is for RL people already working with value-based methods or online normalization tricks. A reader who knows the differential TD literature would see the direct extension and could use the proof or the adapted algorithms. It deserves a serious referee because the theoretical claims are stated clearly and the empirical scope is reasonable for the claim.

Referee Report

0 major / 3 minor

Summary. The paper proposes a generalization of differential TD methods to episodic problems via a modified reward centering operator that accounts for termination. It proves that this generalization preserves policy ordering and establishes equivalence to a form of linear TD, thereby inheriting existing theoretical guarantees. The authors extend multiple streaming RL algorithms to their differential counterparts and report empirical results indicating that reward centering improves sample efficiency across base algorithms and episodic environments.

Significance. If the central proofs hold, the work meaningfully extends differential TD beyond its original infinite-horizon restriction, addressing a practical limitation in episodic settings common to RL. The explicit equivalence to linear TD is a strength, as it directly transfers known guarantees without introducing new parameters. The empirical component, spanning multiple algorithms and environments, provides concrete support for the utility of the approach in streaming deep RL contexts where normalization matters.

minor comments (3)

The abstract refers to equivalence with 'a form of linear TD' without naming the specific variant or the exact mapping; stating this explicitly in the introduction would improve immediate clarity for readers.
In the empirical validation, the abstract asserts improvement in sample efficiency but does not report quantitative effect sizes, confidence intervals, or statistical tests; including these in the results section would strengthen the presentation.
The definition of the generalized centering operator (likely introduced in the methods section) should be contrasted more explicitly with the standard average-reward centering to highlight the precise adjustment for termination.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and constructive review, including the recognition of the paper's contributions in generalizing differential TD methods to episodic settings, preserving policy ordering, establishing equivalence to linear TD, and showing empirical benefits for sample efficiency in streaming RL. The recommendation for minor revision is noted, and we will incorporate appropriate changes in the revised manuscript.

Circularity Check

0 steps flagged

No significant circularity; derivation relies on independent proofs

full rationale

The paper proposes a generalization of differential TD for episodic settings, proves policy ordering preservation under termination, and establishes equivalence to linear TD. These steps are presented as mathematical results rather than reductions to fitted parameters, self-definitions, or self-citation chains. The abstract and description provide no equations or claims where a prediction or uniqueness result collapses to the input by construction. The central claims rest on external proofs and equivalences that do not reference the method's own fitted values or prior self-citations as load-bearing justifications. This is the expected self-contained case for a theoretical extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard RL assumptions (Markov property, existence of average reward) plus the new generalization; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Standard Markov decision process assumptions including existence of average reward and proper termination in episodic settings.
Implicit foundation for differential TD and the policy-ordering claim.

pith-pipeline@v0.9.0 · 5458 in / 1083 out tokens · 36856 ms · 2026-05-08T18:15:55.779270+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost (J(x)=½(x+x⁻¹)−1) Jcost_unit0 / washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Φ(s) := b/(1−γ) ... F(s,a,s') = γ·b/(1−γ) − b/(1−γ) = −b
IndisputableMonolith.Foundation (RealityFromDistinction, forcing chain) reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Convergence proof: Ã is negative definite via Tsitsiklis–Van Roy non-expansion of P_π in D_π norm; preconditioning K=diag(η,1,…,1) preserves Hurwitz spectrum.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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