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Learning long range dependencies through time reversal symmetry breaking

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read RHEL computes loss gradients by running three forward trajectories of a Hamiltonian system, and in the small-nudging limit these gradients equal backpropagation through time, with matching performance on sequence benchmarks up to roughly…

desk verdict The discrete-time RHEL-BPTT equivalence is a real contribution, but the training experiments omit the gradient downscaling the authors admit is needed, so the empirical claim is not yet supported. read the letter →

arxiv 2506.05259 v1 pith:AMIQCX2E submitted 2025-06-05 cs.LG

Guillaume Pourcel , Maxence Ernoult This is my paper
classification cs.LG
keywords RecurrentHamiltonianEchoLearningtimereversalsymmetryUnitsStateSpaceModelsBackpropagationThroughforward-onlygradientestimationlong-rangedependencesymplecticintegration
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proposes Recurrent Hamiltonian Echo Learning (RHEL), a training rule for recurrent networks built as symplectic integrators of separable Hamiltonian dynamics. The central claim is that the loss gradient can be obtained from three forward simulations only: a free forward pass, and two 'echo' passes that replay the inputs backwards while nudging the trajectory by opposite amounts; the finite difference of the two echoes encodes the gradient, with no backward pass through the graph and no state Jacobian. In continuous time the method is proved equivalent to the adjoint state method, and in discrete time, on Hamiltonian Recurrent Units (HRUs) and stacks of them (HSSMs), the small-nudging limit of the echo estimators equals Backpropagation Through Time at every timestep for states, parameters, and inputs. The experiments show near-perfect cosine alignment between RHEL and BPTT gradients and statistically matched test performance on six long-range classification tasks and a roughly 50,000-step regression task. If the equivalence holds, RHEL is a forward-only, zero-variance route to training sequence models, with a direct path to implementation on energy-conserving physical hardware.

What carries the argument

The load-bearing object is the echo trajectory $\Phi^e$ of a reversible leapfrog integrator $M_{H,\delta}$ on the phase space $(\phi, \pi)$, started from the momentum-flipped final state, driven by the time-reversed input sequence, and nudged by $\pm\epsilon$ along the loss gradient. The paper defines a half-step averaged Hamiltonian $H^{1/2}$ so that the RHEL gradient estimators are finite differences of that averaged Hamiltonian between the two echoed trajectories. The proof of equivalence works by showing that the $\epsilon$-derivative of the echo trajectory obeys the same discrete recurrence as the BPTT adjoint $\lambda_k$, so the two agree at all times in the limit $\epsilon\to 0$. The scaling to deep models comes from echo chaining: the input-gradient estimate of one HRU becomes the nudging signal for the HRU below it, mirroring the backward pass through a stack.

What would settle it

Implement the implicit echo dynamics of Eq. (13) and the explicit linearized echo of Corollary A.8 on the same HRU, sweep $\epsilon$ toward zero, and compare both against BPTT gradients; if the linearized estimator's cosine similarity does not approach one, or if the two echo versions disagree at leading order, the implemented algorithm is not the proved one.

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Extended reading notes

Core claim

The core discovery is that time-reversal symmetry turns gradient computation into a physical echo effect. Starting from the final state of a Hamiltonian network, flip the momenta, replay the input sequence in reverse, and apply a small nudging $\pm\epsilon$ in the direction of the loss gradient; because the un-nudged system would exactly retrace its own history, the separation between the $+\epsilon$ and $-\epsilon$ echoed trajectories is precisely the adjoint signal $\lambda_k$, and the analogous finite difference of the half-step averaged Hamiltonian yields the parameter and input sensitivities. Theorem 3.2 states this equality for every timestep in the $\epsilon\to 0$ limit for HRUs, and Theorem 3.3 extends it to hierarchical stacks by chaining each layer's input-gradient estimate into the next echo. The paper's empirical claim is that this theoretical equivalence survives finite-$\epsilon$ practice: gradient cosine similarities are near one and RHEL-trained linear and nonlinear HSSMs match BPTT-trained models on long-range classification and regression, including the roughly 50k-step PPG-DaLiA heart-rate prediction task.

Load-bearing premise

The experiments actually run a linearized, finite-strength version of the echo update, and the paper asserts rather than proves that this version preserves the equivalence; on top of that, numerical accuracy relies on a rescaling whose second half was not implemented in the submitted experiments.

Editorial extensions

If this is right

  • RHEL computes gradients with exactly three simulations per training step, regardless of the number of parameters, with no stored activations and no state-Jacobian computation.
  • Because the estimators are finite differences rather than random projections, the gradient estimate is deterministic; the only sources of deviation from BPTT are finite-$\epsilon$ bias and floating-point precision.
  • The equivalence theorem applies to any model expressible as a stack of leapfrog-integrable separable Hamiltonians, so linear and nonlinear HSSMs of the kind tested inherit the guarantee without architectural modification.
  • The algorithm carries over to physical systems that can only evolve forward in time, which is the paper's stated motivation for self-learning machines on photonic, superconducting, or spintronic substrates.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The formal results are proven for the implicit echo update, while the experiments run the explicit linearized version of Corollary A.8; whether that linearization preserves the equivalence is asserted, not demonstrated, so the tested algorithm is technically a finite-$\epsilon$ approximation of the proved one.
  • An adaptive or higher-order scheme for the nudging strength could eliminate the hand-tuned rescaling factor $\gamma$, and the paper's own gradient-matching plots suggest the optimal $\gamma$ differs between linear and nonlinear architectures.
  • The chaining construction extends naturally to arbitrary directed acyclic graphs of reversible units, as the authors note in Remark 7, which would cover residual connections and other graph-structured sequence models beyond linear stacks.
  • If implemented on analog hardware, the echo dynamics compute the learning signal during the physical evolution itself, so the relevant comparison is not FLOPs but energy per gradient; a concrete test would be measuring gradient quality against BPTT on a photonic testbed.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 7 minor

Summary. The paper proposes Recurrent Hamiltonian Echo Learning (RHEL), a forward-only gradient estimation algorithm for Hamiltonian recurrent models. In continuous time RHEL is shown to be equivalent to the continuous adjoint state method in the limit of small nudging strength; in discrete time it is shown to be equivalent to Backpropagation Through Time (BPTT) for a class of symplectic integrators called Hamiltonian Recurrent Units (HRUs), with a chaining procedure for stacks of HRUs (HSSMs). The authors report static gradient comparisons and training experiments on long-sequence classification and regression benchmarks, claiming that RHEL consistently matches BPTT performance.

Significance. If the equivalence theorems are correct and the algorithm is implemented as stated, this is a significant contribution: RHEL would compute exact loss gradients (in the infinitesimal-nudging limit) using only three forward passes, with no explicit state-Jacobian computation and no gradient-estimation variance, and the derivation is explicitly connected to Hamiltonian physics. The paper's theoretical development is detailed and mostly rigorous, and the static gradient comparisons in Fig. 4 are encouraging evidence that the finite-difference construction can reproduce BPTT gradients. However, the current experimental validation is undermined by an internal inconsistency: the training experiments, which are the main empirical support for the headline claim, were run with a scaled-gradient variant for which the equivalence theorems have not been established. The transparency of Appendix A.4.4 in disclosing this issue is a credit to the authors, but the disclosure itself shows that the submitted results do not instantiate the algorithm analyzed in Theorems 3.1-3.3.

major comments (3)
  1. [Appendix A.4.4] The training experiments in Tables 1 and 2 were run without the downscaling half of the gradient rescaling procedure. The text states: 'In our initial implementation, we applied gradient scaling without the corresponding downscaling step, effectively amplifying the gradient magnitude throughout training.' Since the scaling factor gamma was chosen up to 1e4, the RHEL parameter update used in training was gamma times the gradient defined by Theorem 3.2/3.3. This means the trained models do not optimize the objective with the gradient that the paper proves equivalent to BPTT, so the 'consistently matches BPTT' claim is not supported by the submitted experiments. The static gradient comparison in Fig. 4 does not resolve this, because it is a different procedure from the one used for Tables 1-2. The authors should rerun the training experiments with the full rescaling (upscaling followed by downscaling) and report those results, or clearly restrict the empirical claims to the variant actually implemented.
  2. [Appendix A.2.3, Corollary A.8] The proof of Corollary A.8 is stated as 'Identical to the proof of Corollary A.7', but the two dynamical systems are not identical: the implicit echo update in Corollary A.7 contains the term -epsilon J grad_Phi ell[Phi^e_{k+1}] evaluated at the unknown next state, while Corollary A.8 replaces it with the explicit perturbation +epsilon Sigma_x y_{-(k+1)} for an arbitrary sequence y. The claim that the epsilon-to-zero limit is unaffected by this linearization requires a separate argument, for example a Taylor expansion in epsilon showing that the difference between the implicit and linearized trajectories is of order epsilon^2 in the appropriate norm. Since Algorithm 1 and the HSSM chaining result in Theorem 3.3 explicitly use this linearized update, the proof should be written out rather than asserted. This is a genuine gap in the central derivation.
  3. [Section 3.1 and Section 3.2] Theorems 3.1 and 3.2 are limit statements as epsilon goes to 0, while the experiments use a fixed finite nudging strength epsilon = 1e-1. The paper acknowledges in Section 5 that finite-nudging approximation bias exists, but it does not quantify the bias or provide a bound in terms of epsilon, trajectory length K, and the Hamiltonian's curvature. Given that the benchmarks reach sequence lengths of roughly 50k, an uncontrolled finite-epsilon bias is a correctness risk for the claim that RHEL 'provably computes' the same gradients as BPTT. At minimum, the authors should provide an epsilon-sensitivity study on the actual benchmark sequence lengths, or state precisely in the theorem statements that the equivalence holds only in the limit and that finite-epsilon behavior is not guaranteed.
minor comments (7)
  1. [Section 3.2, Eq. (13)] The echo dynamics are introduced using Phi^e_k(epsilon) before the notation is fully defined; please clarify the indexing of the echo trajectory and the role of the fractional time steps in the definition.
  2. [Algorithm 1] The input is described as 'Phi0 (final state of the forward trajectory)', but the echo initial condition in Eq. (13) uses Phi^star_0, the momentum-reversed final state; please align the algorithm's notation with the equation to avoid confusion.
  3. [Theorem 3.1 vs. Appendix A.1.3] The informal main-text statement of Theorem 3.1 omits the loss-gradient term grad_theta ell appearing in the full statement in the appendix; please make the two versions consistent so that the informal statement does not mislead readers about the content of the theorem.
  4. [Appendix A.3.1, Eqs. (24)-(30)] Some signs in the toy-model estimators appear inconsistent with the general definitions in Eq. (7) (for example, Delta_RHEL_phi is written with a minus sign while Eq. (7) has a plus). Please check and harmonize the sign conventions.
  5. [Appendix A.4.4] The grid search for the rescaling factor gamma reports the 'best performing' value on the regression task and on the averaged classification score without stating whether this is a validation split or the test set; please report the tuning procedure and the split used.
  6. [Section 5, Limitations] The claim that the RHEL-BPTT gap is 'statistically significant only for nonlinear HSSMs on regression tasks' is not supported by any significance test; please describe the test used or qualify the statement.
  7. [Appendix A.4.4] The phrase 'will be implemented in the camera-ready version' is inappropriate for a manuscript under review; the submitted version should already contain the correct procedure or should state that the reported results correspond to the variant actually run.

Circularity Check

1 steps flagged · score 2.0 of 10

Core equivalence theorems are genuine derivations; only the empirical 'matches BPTT' claim is mildly circular, because it rests on a per-task grid search over the rescaling factor γ while omitting the promised downscaling.

  1. fitted input called prediction [Appendix A.4.4, Section 4.2 (Tables 1–2)]
    "we did a grid search over the values {100, 104, 108, 1012} for the regression task (PPG-DaLiA) and found that the best performing parameter was 104. ... In our initial implementation, we applied gradient scaling without the corresponding downscaling step, effectively amplifying the gradient magnitude throughout training."

    The paper's empirical claim that RHEL 'consistently matches' BPTT (Tables 1–2) is evaluated on the same tasks used to select γ, and the submitted implementation omits the division by γ promised earlier in A.4.4. With γ=104 on PPG-DaLiA, the RHEL update is γ times the gradient estimator that Theorem 3.2 proves equivalent to BPTT, so the effective learning rate is γ-scaled. Choosing γ per task for best performance and then reporting a 'match' on that same task makes the validation a tuned-parameter demonstration rather than an independent test of the provably equivalent estimator. This fits the fitted-input-called-prediction pattern: the fitted knob (γ) and the reported outcome (matching BPTT) are evaluated on the same benchmark, with no holdout or default-setting prediction.

full rationale

The core derivation is not circular. Theorems 3.1–3.2, detailed in Theorem A.2 and Corollary A.7, define ΔRHEL as finite differences of echo trajectories and then prove, by differentiating the echo ODE/dynamics around ϵ=0, that the limits satisfy the same ODE/recursions and initial conditions as the continuous adjoint quantities or BPTT sensitivities. Equality follows from uniqueness of the solution; the target gradients are not baked into the definitions of ΔRHEL. The chaining theorem (Theorem 3.3 / Theorem A.4) is an induction that invokes Corollary A.8, whose proof is asserted to be identical to Corollary A.7 rather than actually carried out; that is an omitted proof or reproducibility gap, not a circularity. There is no load-bearing self-citation: the Hamiltonian-echo inspiration [10] and the HRU architectures [29,30] are external prior work, and no uniqueness theorem is imported from the present authors to force the ansatz. The one genuinely circular-adjacent element is the empirical validation: Appendix A.4.4 discloses a grid search over γ on the evaluation tasks and admits that during training the gradient was scaled without the corresponding downscaling, so the submitted experiments instantiate γ times the proven gradient and therefore a γ-scaled effective learning rate. That makes the 'RHEL matches BPTT' result partly a tuned-parameter effect, a mild instance of fitted-input-called-prediction. Because the equivalence theorems themselves remain self-contained and the empirical match is not forced by construction, the overall circularity score is low (2), not higher.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

No new physical entities are introduced. The ledger records two algorithmic hyperparameters (epsilon and gamma) and three domain assumptions that constrain the model class.

free parameters (2)
  • epsilon (nudging strength) = 0.1 (fixed, no tuning)
    Used in all echo finite-difference gradient estimators; theory requires epsilon to go to 0, but experiments use a fixed finite value, creating an uncontrolled bias.
  • gamma (gradient rescaling factor) = 10^4 (grid-searched: {10^0,...,10^12} for PPG; {10^0,...,10^4} for classification)
    Multiplicatively scales the nudge loss during echo computation to avoid numerical underflow; selected on each task to make RHEL gradients match BPTT. The paper admits the downscaling half of the procedure was not implemented in the current version.
assumptions (4)
  • domain assumption Time-reversal symmetry of the Hamiltonian: H[Phi, theta, u] = H[Sigma_z * Phi, theta, u] (Def. A.1, Assumption 1)
    Needed for Lemma A.2 and for the echo trajectory to retrace the forward path at epsilon = 0. Restricts models to energy-conserving (non-dissipative) systems.
  • domain assumption Time-reversal symmetry of the loss: ell[Phi, theta, k] = ell[Sigma_z * Phi, theta, k] (Def. A.4, Assumption 1)
    Used to relate gradients at successive times; not satisfied by every practical loss, though it holds for the losses used here (which depend only on position outputs).
  • domain assumption Separable Hamiltonian and reversibility of the leapfrog integrator (Def. A.3)
    RHEL's discrete-time guarantees rely on the integrator M_H,delta being exactly reversible; this holds for separable Hamiltonians integrated with leapfrog, restricting the model class.
  • standard math Smoothness and differentiability assumptions (Assumption A.1: twice continuously differentiable Hamiltonian and loss)
    Standard regularity needed for gradient and adjoint existence; not checked for the GELU and GLU building blocks but these are smooth almost everywhere.

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Pith. "Pith review of Learning long range dependencies through time reversal symmetry breaking." pith.science (2026). https://pith.science/paper/AMIQCX2E

@misc{pith2026250605259,
  author       = {Pith},
  title        = {Pith review of: Learning long range dependencies through time reversal symmetry breaking},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AMIQCX2E}},
  note         = {Machine review of arXiv:2506.05259}
}
abstract

Deep State Space Models (SSMs) reignite physics-grounded compute paradigms, as RNNs could natively be embodied into dynamical systems. This calls for dedicated learning algorithms obeying to core physical principles, with efficient techniques to simulate these systems and guide their design. We propose Recurrent Hamiltonian Echo Learning (RHEL), an algorithm which provably computes loss gradients as finite differences of physical trajectories of non-dissipative, Hamiltonian systems. In ML terms, RHEL only requires three "forward passes" irrespective of model size, without explicit Jacobian computation, nor incurring any variance in the gradient estimation. Motivated by the physical realization of our algorithm, we first introduce RHEL in continuous time and demonstrate its formal equivalence with the continuous adjoint state method. To facilitate the simulation of Hamiltonian systems trained by RHEL, we propose a discrete-time version of RHEL which is equivalent to Backpropagation Through Time (BPTT) when applied to a class of recurrent modules which we call Hamiltonian Recurrent Units (HRUs). This setting allows us to demonstrate the scalability of RHEL by generalizing these results to hierarchies of HRUs, which we call Hamiltonian SSMs (HSSMs). We apply RHEL to train HSSMs with linear and nonlinear dynamics on a variety of time-series tasks ranging from mid-range to long-range classification and regression with sequence length reaching $\sim 50k$. We show that RHEL consistently matches the performance of BPTT across all models and tasks. This work opens new doors for the design of scalable, energy-efficient physical systems endowed with self-learning capabilities for sequence modelling.

Figures

Figures reproduced from arXiv: 2506.05259 by the authors.

Figure 1
Figure 1. HEB core mechanics [10]. Af￾ter following a free forward trajectory (blue curve) and undergoing “rebound”, the neu￾rons exactly travel backward (grey curve). In￾stead, HEB prescribes nudging the "echo" tra￾jectory (red curve) closer to y before rebound, with the resulting position gap encoding the error gradient with respect to momentum (the left green arrow). RTRL fails to satisfy criteria ii) because it explic￾itl… view at source ↗
Figure 2
Figure 2. (A): Learning Hamiltonian SSMs by RHEL. The forward pass of a HSSM (left, Eq. (15)) reads as a composition of Hamiltonian Recurrent Units (HRUs, Eq. (9)). During the backward pass (right), the momenta of the neurons of the top-most HRU are flipped and nudged by the first error signal, yielding a perturbed trajectory (red curve). The contrast with the time-reversed trajectory (which would be obtained without nudging,… view at source ↗
Figure 3
Figure 3. Computational graph of MH,δ. Hamiltonian Recurrent Units (HRUs). We now assume that the neurons Φ obey the following discrete-time dynamics, starting from Φ−K = x and: Φk+1 = MH,δ[Φk, θ,uk] ∀k = −K · · · − 1, (9) where MH,δ denotes the integrator associated with the Hamiltonian function H with time dis￾cretization δ. We classically pick MH,δ as the 5 [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Static comparison between RHEL and BPTT. Given some (ui , yi) ∼ D, we perform BPTT and RHEL on six blocks-deep linear and nonlinear HSSMs. We measure, layer-wise (first layer in purple), the cosine similarity (top panels) and norm ratio (bottom pannels) between RHEL an…
Figure 5
Figure 5. Figure 5: (A): Parameters gradients comparison between RHEL and BPTT for HSSM. Given some (ui , yi) ∼ D, we perform BPTT and RHEL on six blocks-deep linear HSSMs for different scaling factor γ of RHEL (different colors). We measure, per layer (line plot) and when averaged across…
Figure 6
Figure 6. Figure 6: (A): Parameters gradients comparison between RHEL and BPTT for HSSM. Given some (ui , yi) ∼ D, we perform BPTT and RHEL on six blocks-deep nonlinear HSSMs for different scaling factor γ of RHEL (different colors). We measure, per layer (line plot) and when averaged acr…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.