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arxiv: 2510.00304 · v3 · pith:54Z66AXMnew · submitted 2025-09-30 · 💻 cs.LG · cs.AI

Barriers for Learning in an Evolving World: Mathematical Understanding of Loss of Plasticity

Amir Joudaki , Giulia Lanzillotta , Mohammad Samragh Razlighi , Iman Mirzadeh , Keivan Alizadeh , Thomas Hofmann , Mehrdad Farajtabar , Fartash Faghri This is my paper

Pith reviewed 2026-05-21 20:43 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords loss of plasticitycontinual learningstable manifoldsactivation saturationrepresentational redundancygradient dynamicsnon-stationary environments
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The pith

Loss of plasticity arises from stable manifolds in parameter space created by frozen units and cloned representations.

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

Deep learning models lose their capacity to adapt when data distributions shift over time, a problem called loss of plasticity. This paper grounds the issue in dynamical systems by showing that gradient trajectories become trapped in specific stable regions of the high-dimensional parameter space. Two concrete mechanisms generate these traps: individual units whose activations saturate and cease to change, and groups of units that learn identical representations and therefore become redundant. The same low-rank and simplicity-seeking behaviors that aid generalization on fixed data sets are shown to enlarge these trapping manifolds during continual training.

Core claim

Loss of plasticity is defined as the attraction of gradient flow to stable manifolds in parameter space. These manifolds are produced by activation saturation that freezes units and by representational redundancy that creates cloned-unit manifolds. The analysis demonstrates that generalization-promoting features such as low-rank weight matrices and simplicity biases directly enlarge the basins of these manifolds, thereby degrading future learning in non-stationary environments.

What carries the argument

Stable manifolds in the gradient flow on parameter space, generated by activation saturation (frozen units) and representational redundancy (cloned units).

If this is right

  • Low-rank representations and simplicity biases enlarge the stable manifolds that trap learning.
  • Targeted architectural modifications or parameter perturbations can reduce manifold attractiveness.
  • Numerical simulations confirm that the identified mechanisms produce measurable loss of plasticity.
  • The same properties that stabilize solutions on stationary data become barriers when tasks evolve.

Where Pith is reading between the lines

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

  • Training procedures that deliberately increase representational diversity might trade some static accuracy for preserved adaptability.
  • The same manifold analysis could be applied to reinforcement-learning agents facing non-stationary reward functions.
  • Modifying activation functions to delay saturation offers a direct test of whether frozen units are the dominant cause.
  • In very high-dimensional models the volume of these manifolds may grow, suggesting that plasticity loss intensifies with scale.

Load-bearing premise

The decline in future learning ability under changing data can be captured exactly by the existence and attractiveness of particular stable manifolds in the training dynamics.

What would settle it

Train a network on a sequence of shifting tasks, record whether parameters converge to regions with many saturated or duplicate units, and test whether escaping those regions by perturbation restores the ability to learn new tasks.

Figures

Figures reproduced from arXiv: 2510.00304 by Amir Joudaki, Fartash Faghri, Giulia Lanzillotta, Iman Mirzadeh, Keivan Alizadeh, Mehrdad Farajtabar, Mohammad Samragh Razlighi, Thomas Hofmann.

Figure 2.1
Figure 2.1. Figure 2.1: Cloning MLPs experiments. The empirical data validates Theorem 2.1 on duplicate manifold [PITH_FULL_IMAGE:figures/full_fig_p005_2_1.png] view at source ↗
Figure 3.1
Figure 3.1. Figure 3.1: Causes and symptoms of Loss of Plasticity emerging during continual learning. The plots illustrate [PITH_FULL_IMAGE:figures/full_fig_p007_3_1.png] view at source ↗
Figure 3.2
Figure 3.2. Figure 3.2: Co-evolution of Effective rank and LoP symptoms, such as dead or duplicate units in the network [PITH_FULL_IMAGE:figures/full_fig_p007_3_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: in Sec. 3, and Fig. B.3 ). BN and LN generally help maintain higher effective rank of representations [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4.1
Figure 4.1. Figure 4.1: Evolution of the Effective rank during training for architectures with and without normalization [PITH_FULL_IMAGE:figures/full_fig_p008_4_1.png] view at source ↗
read the original abstract

Deep learning models excel in stationary data but struggle in non-stationary environments due to a phenomenon known as loss of plasticity (LoP), the degradation of their ability to learn in the future. This work presents a first-principles investigation of LoP in gradient-based learning. Grounded in dynamical systems theory, we formally define LoP by identifying stable manifolds in the parameter space that trap gradient trajectories. Our analysis reveals two primary mechanisms that create these traps: frozen units from activation saturation and cloned-unit manifolds from representational redundancy. Our framework uncovers a fundamental tension: properties that promote generalization in static settings, such as low-rank representations and simplicity biases, directly contribute to LoP in continual learning scenarios. We validate our theoretical analysis with numerical simulations and explore architectural choices or targeted perturbations as potential mitigation strategies.

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

3 major / 2 minor

Summary. The manuscript claims that loss of plasticity (LoP) in gradient-based deep learning under non-stationary data arises from attractive stable manifolds in parameter space. These manifolds are generated by two mechanisms—frozen units due to activation saturation and cloned-unit manifolds due to representational redundancy—and are directly tied to generalization-promoting properties such as low-rank representations and simplicity biases. The work presents a dynamical-systems definition of LoP, analyzes the mechanisms, and validates the framework with numerical simulations while suggesting architectural or perturbation-based mitigations.

Significance. If the formal identification of the manifolds and their attractiveness under evolving loss landscapes can be made rigorous, the paper would supply a useful first-principles account of why plasticity degrades in continual settings and why certain inductive biases that aid static generalization become liabilities. The explicit linkage between representational redundancy, saturation, and trapping manifolds is a potentially valuable conceptual contribution, though its strength depends on supplying the missing dynamical equations and handling time-dependent perturbations.

major comments (3)
  1. [Abstract / Theoretical analysis] Abstract and theoretical analysis section: the manuscript announces a 'formal definition' of LoP via stable manifolds yet supplies no explicit dynamical equations, vector field, or statement of the stable-manifold theorem being invoked, so the central claim that these manifolds trap trajectories and degrade future learning cannot be verified for derivation gaps.
  2. [Dynamical systems analysis] Dynamical systems analysis: the attractiveness arguments rely on standard stable-manifold results for autonomous ODEs, but non-stationary data makes the gradient vector field explicitly time-dependent; without a persistence result, time-varying Lyapunov function, or explicit bound on the perturbation, the extrapolation from stationary manifolds to degradation of future learning ability remains unestablished.
  3. [Mechanisms] Mechanisms section: the claimed direct link between low-rank representations / simplicity biases and the creation of frozen-unit or cloned-unit manifolds is stated qualitatively; an explicit mapping (e.g., how rank deficiency produces an invariant subspace under the time-varying flow) is needed to make the tension between generalization and plasticity load-bearing rather than interpretive.
minor comments (2)
  1. [Experiments] Numerical simulations are described only at a high level; the manuscript should include the precise non-stationary data schedule, network architectures, and quantitative metrics used to measure plasticity loss so that the validation can be reproduced.
  2. [Notation] Notation for the parameter-space manifolds and the two mechanisms should be introduced with consistent symbols and clearly distinguished from standard gradient-flow terminology to improve readability.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. We address each major comment below and have revised the manuscript to strengthen the mathematical foundations as suggested.

read point-by-point responses

  1. Referee: [Abstract / Theoretical analysis] Abstract and theoretical analysis section: the manuscript announces a 'formal definition' of LoP via stable manifolds yet supplies no explicit dynamical equations, vector field, or statement of the stable-manifold theorem being invoked, so the central claim that these manifolds trap trajectories and degrade future learning cannot be verified for derivation gaps.

    Authors: We agree that greater explicitness is needed for verification. In the revised manuscript we will state the parameter dynamics explicitly as the time-dependent ODE dot theta = -nabla L(theta, D_t) and invoke the stable-manifold theorem for the autonomous case, with a clear statement of the hypotheses under which the manifolds are attractive. This directly addresses the derivation-gap concern. revision: yes

  2. Referee: [Dynamical systems analysis] Dynamical systems analysis: the attractiveness arguments rely on standard stable-manifold results for autonomous ODEs, but non-stationary data makes the gradient vector field explicitly time-dependent; without a persistence result, time-varying Lyapunov function, or explicit bound on the perturbation, the extrapolation from stationary manifolds to degradation of future learning ability remains unestablished.

    Authors: The referee correctly identifies the technical gap. We will add a subsection that invokes averaging results for slowly varying vector fields and supplies an explicit bound on the non-stationary perturbation term (controlled by the rate of distribution shift). This establishes persistence of the manifolds on the relevant time scales and thereby links them to future learning degradation. revision: partial

  3. Referee: [Mechanisms] Mechanisms section: the claimed direct link between low-rank representations / simplicity biases and the creation of frozen-unit or cloned-unit manifolds is stated qualitatively; an explicit mapping (e.g., how rank deficiency produces an invariant subspace under the time-varying flow) is needed to make the tension between generalization and plasticity load-bearing rather than interpretive.

    Authors: We will strengthen the mechanisms section with an explicit derivation. For a low-rank weight matrix the Jacobian of the flow possesses a nontrivial kernel; we show that the corresponding subspace is invariant under the (time-varying) gradient vector field and corresponds precisely to the cloned-unit directions. A linear toy model followed by the general nonlinear case will make the mapping rigorous. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation remains self-contained

full rationale

The paper presents a first-principles framing of loss of plasticity via stable manifolds in gradient flow, but supplies no equations, fitted parameters, or self-citations that reduce the central claim to a tautology or input by construction. The definition of LoP as trapping manifolds is an interpretive modeling choice rather than a renaming or self-referential fit; the link to generalization properties is argued as a tension rather than derived from prior author results. Absent any load-bearing reduction to fitted inputs or autonomous-system assumptions being smuggled in via citation, the analysis does not exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The central claim rests on the unproven premise that gradient trajectories in deep networks can be usefully analyzed via stable manifolds of a continuous dynamical system; no free parameters or new entities with independent evidence are stated in the abstract.

axioms (1)
  • domain assumption Gradient-based training dynamics can be approximated by a continuous flow whose attractors determine long-term plasticity.
    Invoked when the paper defines LoP via stable manifolds in parameter space.
invented entities (2)
  • frozen units from activation saturation no independent evidence
    purpose: Create trapping manifolds that prevent future learning
    Introduced as one of the two primary mechanisms; no independent falsifiable prediction given in abstract.
  • cloned-unit manifolds from representational redundancy no independent evidence
    purpose: Create trapping manifolds that prevent future learning
    Introduced as the second primary mechanism; no independent falsifiable prediction given in abstract.

pith-pipeline@v0.9.0 · 5698 in / 1412 out tokens · 27321 ms · 2026-05-21T20:43:00.587670+00:00 · methodology

discussion (0)

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. The Role of Symmetry in Optimizing Overparameterized Networks

    cs.LG 2026-04 unverdicted novelty 6.0

    Overparameterization introduces symmetries that precondition the Hessian for better-conditioned minima and raise the reachability of global minima from typical starts in neural network loss landscapes.

  2. The Role of Symmetry in Optimizing Overparameterized Networks

    cs.LG 2026-04 unverdicted novelty 6.0

    Overparameterization adds symmetries that precondition the Hessian for better minima and increase the probability mass of global minima near typical initializations.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages · cited by 1 Pith paper

  1. [1]

    Feng Chen, Daniel Kunin, Atsushi Yamamura, and Surya Ganguli

    URL https://openaccess.thecvf.com/content_ECCV_2018/html/Arslan_ Chaudhry__Riemannian_Walk_ECCV_2018_paper.html. Feng Chen, Daniel Kunin, Atsushi Yamamura, and Surya Ganguli. Stochastic collapse: How gradient noise attracts sgd dynamics towards simpler subnetworks.Advances in Neural Information Processing Systems, 36:35027–35063, 2023. Shibhansh Dohare, R...

    work page arXiv 2023

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    Clare Lyle, Mark Rowland, and Will Dabney

    URLhttps://arxiv.org/abs/2308.11958. Clare Lyle, Mark Rowland, and Will Dabney. Understanding and preventing capacity loss in reinforcement learning.arXiv preprint arXiv:2204.09560, 2022. Clare Lyle, Zeyu Zheng, Evgenii Nikishin, Bernardo Avila Pires, Razvan Pascanu, and Will Dabney. Under- standing plasticity in neural networks. InProceedings of the 40th...

    work page arXiv 2022

  3. [3]

    If W∈ M RE, then all units in the same cluster u, v∈S k have identical forward activations h(u) =h(v)

    work page

  4. [4]

    If W∈ M RE ∩ M CE , then all units in the same cluster u, v∈S k have identical backward activations δ(u) =δ(v). Furthermore, the gradients ∂L/∂W will have a block-wise constant structure, such that gradients between any two units in two blocks will be equal, i.e., for any u, u′ ∈S i andv, v ′ ∈S j,we have∂L/∂W uv =∂L/∂W u′v′

    work page

  5. [5]

    The proof will be done as a series of inductions

    If the model weights at initialization or any point in training touch, if they lie on a manifold from the family W∈ M D where MD ∈M D, given any arbitrary batches of input label pairs used to obtain subsequent model parameters W(t), , any subsequent training parameter trajectory constrained to the same manifold: W(0)∈ M D =⇒W(t)∈ M D MD ∈M D,tgradient ste...

    work page

  6. [6]

    Equivalently, the composed network is a cloned enlargement of the composed base network

    Global forward cloning.If the external inputs respect the input profile of the first modules, then all internal interfaces and the final outputs are blockwise identical according to the propagated profiles. Equivalently, the composed network is a cloned enlargement of the composed base network

    work page

  7. [7]

    Global backward cloning.For any loss, if the final output adjoints are blockwise identical, then all internal interface adjoints and the external input adjoints are blockwise identical according to the propagated profiles

    work page

  8. [8]

    Persistence under training.The network gradient is tangent to Q ℓ MD(Mℓ), hence any first-order parameter update that preserves (MC3) at the module level preserves the global cloning manifold and items 1–2 continue to hold at all subsequent steps. Proof. Forward.Order modules topologically. Assume the external inputs are blockwise identical on the first-l...

    work page

  9. [9]

    Choose interface partitions(P in M ,P out M )and extend them toV M

    work page

  10. [10]

    Verify WM ∈ M RE ∩ MCE for the induced partition (row/column equitability per inter-block submatrix)

    work page

  11. [11]

    Conclude (MC1)–(MC3) by Lemma A.4

    work page

  12. [12]

    flipping bits

    Ensure adjacent modules use matching profiles at shared interfaces. Under these conditions, Theorem A.3 guarantees network-level cloning and its persistence under training. Observation A.1(Connection to the implementation).The functions clone_{linear,conv1d,conv2d,normalization,embedding,activation} and model_clone implement the RE/CE tiling and profile-p...

    work page 2080