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arxiv: 2606.00243 · v1 · pith:KR4PFOMFnew · submitted 2026-05-29 · 💻 cs.NE · q-bio.NC· stat.ML

Dynamics and Representation Structure of Local Approximations to Gradient-Based Learning in Linear Recurrent Neural Networks

Ezekiel Williams , Alexandre Payeur , Guillaume Lajoie This is my paper

Pith reviewed 2026-06-28 19:16 UTC · model grok-4.3

classification 💻 cs.NE q-bio.NCstat.ML
keywords RFLOlinear recurrent neural networkslocal learning ruleslow-rank perturbationsBPTTtBPTTlearning dynamicsdata-aligned RNNs
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The pith

RFLO learning in linear RNNs restricts solutions to low-rank perturbations of initial parameters

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

The paper applies dynamical systems theory to data-aligned linear recurrent neural networks to analyze local approximations to gradient descent such as RFLO and truncated BPTT. These methods display qualitatively different stability properties and convergence rates compared to full BPTT. The central observation is that RFLO reaches only solutions that are low-rank perturbations around the starting parameters, and this holds outside the data-aligned setting as well. The work clarifies how requirements for spatial and temporal locality during learning constrain the parameter space that can be reached.

Core claim

In data-aligned linear RNNs the learning dynamics under RFLO, BPTT and one-step tBPTT exhibit qualitatively distinct behaviour. The solutions learned by RFLO are restricted to low-rank perturbations of the initial parameters, a result which holds beyond the data-aligned setting.

What carries the argument

Separation of the RNN into orthogonal modes in the data-aligned linear case, which permits mode-by-mode analysis of the learning dynamics and reveals the low-rank restriction on RFLO solutions

If this is right

  • RFLO exhibits different stability properties and convergence rates than BPTT and one-step tBPTT
  • RFLO solutions cannot reach arbitrary parameters but remain confined to low-rank perturbations around initialization
  • The low-rank restriction applies more generally and is not limited to the data-aligned linear setting
  • Locality constraints therefore shape both the dynamics and the final representation structure of the learned network

Where Pith is reading between the lines

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

  • The low-rank restriction may limit the tasks that purely local rules can solve without additional mechanisms such as specific initializations
  • Hybrid rules that occasionally allow non-local information could be tested to see whether they escape the low-rank constraint
  • The result offers a possible explanation for why some biological learning models succeed only on restricted classes of problems

Load-bearing premise

The learning dynamics of these algorithms can be usefully analyzed by separating the RNN into orthogonal modes in the data-aligned linear case

What would settle it

A numerical simulation of RFLO on a data-aligned linear RNN that produces full-rank changes from the initial parameters would falsify the restriction claim

Figures

Figures reproduced from arXiv: 2606.00243 by Alexandre Payeur, Ezekiel Williams, Guillaume Lajoie.

Figure 1
Figure 1. Figure 1: Fixed-point curves for the three learning algorithms. Optima (ab = a⋆b⋆, w = w⋆) are in cyan and non-optima (a = b = 0, w free) in red. BPTT and tBPTT share the same fixed-point structures, whereas RFLO lacks the non-optimal line. Parameters: w⋆ = 0.7, a⋆ = 0.4, b⋆ = 0.25, aˆ = 0.2, wˆ = 0.3. unlike BPTT and tBPTT, RFLO admits only optimal fixed points. 3.2. Stability For each algorithm, each eigenmode obe… view at source ↗
Figure 2
Figure 2. Figure 2: Vector fields for all learning rules (left: BPTT; center: tBPTT (τ = 1); right: RFLO). Top row: slice through 3D space at w = w⋆. Bottom row: slice at a = b. Same parameters as in [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Real part of the largest (λ+, top) and smallest (λ−, bottom) eigenvalues on the optimal manifold ab = a⋆b⋆, w = w⋆. (A) a⋆, b⋆ and w⋆ are the same as in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Parameter space trajectories. (A) Trajectories. All algorithms start at the same point (a, b, w) = (1, 1, 0.6). The termination points are indicated by star symbols. (B) Parameters as a function of time for the trajectories in A. Parameters: w⋆ = 0.7, a 2 ⋆b 2 ⋆ = 0.25, wˆ = 0.3, aˆ = −0.2. − P k,t,s ηwˆ sE k t . Lastly, o is the output dimension. This means the rank of the weight matrix changes learned by… view at source ↗
Figure 5
Figure 5. Figure 5: Comparison of data-aligned theory with non-data-aligned numerical experiments for RFLO. (A) Theory error for each mode as a function of initialization epoch (see Methods). (B) Alignment as a function of training epoch. In panels A and B, curves show mean ± standard error (shaded region) over 6 seeds. (C) Comparison of experimental dynamics (dotted lines) and theory (solid) for the student modes correspondi… view at source ↗
Figure 6
Figure 6. Figure 6: (A) Spectrum of the change in W relative to initialization (i.e., spectrum of WK −W0, where K is the final training iteration) learned by BPTT, tBPTT, RFLO, or e-prop, on a linear student￾teacher task with a single output dimension. Y -axis: absolute value of the eigenvalues; x-axis: eigenvalue index, ordered from largest to smallest magnitude. All spectra are normalized by the absolute value of the larges… view at source ↗
Figure 7
Figure 7. Figure 7: Same as [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: As in [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: As in [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Negative log error of the four algorithms on the task from Section 3.6. RFLO and tBPTT both find low rank solutions and have lower performance; BPTT finds the highest rank solution and sees best performance, and e-prop is intermediate on both counts. 24 [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
read the original abstract

Biological and neuromorphic recurrent neural networks (RNNs) are subject to spatial and temporal locality constraints on the information that can plausibly be used during learning. A common strategy to satisfy these constraints is to modify gradient descent by neglecting non-local terms to varying degrees, as in random feedback local online (RFLO) learning and truncated backpropagation through time (tBPTT). However, the learning dynamics of these algorithms, and how they compare with BPTT, remain poorly understood. We apply dynamical systems theory to data-aligned linear RNNs -- whose dynamics can be separated into orthogonal modes -- to compare stationary solutions, stability properties, and convergence rates, finding qualitatively distinct behaviour for RFLO versus BPTT and one-step tBPTT. We further observe that the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting. Our work provides analytical insight into how locality constraints shape learning dynamics, with implications for neuroscientific models of learning and alternative optimization approaches for RNNs.

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.

Referee Report

1 major / 0 minor

Summary. The manuscript applies dynamical systems theory to data-aligned linear RNNs (whose dynamics separate into orthogonal modes) to analyze and compare the stationary solutions, stability properties, and convergence rates of RFLO, one-step tBPTT, and BPTT. It further claims that RFLO solutions are restricted to low-rank perturbations of the initial parameters, with this low-rank result asserted to hold beyond the data-aligned setting.

Significance. If the dynamical analysis and the low-rank result are rigorously established, the work would provide useful analytical insight into how locality constraints in learning rules shape RNN dynamics, with implications for neuromorphic hardware and neuroscientific models of learning. The separation into orthogonal modes and comparison of qualitative behaviors across algorithms is a constructive application of dynamical systems tools.

major comments (1)
  1. [Abstract] Abstract: The claim that 'the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting' is load-bearing for the paper's contribution yet rests on an unshown extension. The mode-separation analysis is valid only when the input covariance is diagonal in the eigenbasis of the recurrent weights; no separate argument, relaxed derivation, numerical counter-example check, or explicit statement of the relaxed assumptions is supplied for non-aligned covariances or nonlinear networks.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback. We address the single major comment below and will revise the manuscript to clarify the low-rank result.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that 'the solutions learned by RFLO are restricted to low-rank perturbations of initial parameters, a result which holds beyond the data-aligned setting' is load-bearing for the paper's contribution yet rests on an unshown extension. The mode-separation analysis is valid only when the input covariance is diagonal in the eigenbasis of the recurrent weights; no separate argument, relaxed derivation, numerical counter-example check, or explicit statement of the relaxed assumptions is supplied for non-aligned covariances or nonlinear networks.

    Authors: The low-rank property follows directly from the structure of the RFLO learning rule itself: each update is a rank-1 perturbation (outer product of the local eligibility trace and the feedback signal) applied to the recurrent weights. This algebraic property of the update does not depend on the input covariance matrix or on the data-aligned assumption used for the orthogonal-mode decomposition. We will add an explicit, short derivation of this fact (valid for arbitrary input covariances in the linear case) to the revised manuscript. The mode-separation analysis is indeed limited to the data-aligned setting, which is already stated in the text; it is used only to obtain closed-form expressions for stability and convergence rates. The manuscript makes no claims about nonlinear networks. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard dynamical systems analysis to learning rules without reduction to inputs by construction

full rationale

The paper applies dynamical systems theory to the equations governing RFLO, tBPTT, and BPTT in data-aligned linear RNNs, separating dynamics into orthogonal modes to derive stationary solutions, stability, and convergence rates. The low-rank perturbation observation for RFLO solutions is presented as a direct consequence of those dynamics in the aligned case and asserted to extend more generally, but the provided text contains no self-definitional loops, fitted parameters renamed as predictions, load-bearing self-citations, uniqueness theorems imported from prior author work, smuggled ansatzes, or renamings of known results. The central claims rest on explicit mode decomposition and comparison of the learning rules themselves rather than on quantities defined in terms of the target outputs. This is the most common honest finding for papers whose analysis is self-contained against external dynamical-systems benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the applicability of linear dynamical systems analysis to the learning rules and the separation of dynamics into orthogonal modes; no free parameters or invented entities are introduced in the abstract.

axioms (2)
  • domain assumption Dynamical systems theory can be applied to derive stationary solutions, stability properties, and convergence rates of the learning algorithms in linear RNNs.
    Invoked to compare RFLO, tBPTT, and BPTT.
  • domain assumption Data-aligned linear RNNs allow separation of dynamics into orthogonal modes.
    Used as the setting for the main analysis.

pith-pipeline@v0.9.1-grok · 5724 in / 1324 out tokens · 32600 ms · 2026-06-28T19:16:38.104738+00:00 · methodology

discussion (0)

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Reference graph

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