Power-Law Decaying Feedback Kernel

• Power-Law Decaying Feedback Kernel is a causal memory function whose influence decays algebraically over time, capturing persistent non-Markovian effects.
• It appears in diverse models including generalized Langevin equations, predictive spiking systems, and reinforced stochastic processes, bridging physics, neuroscience, and machine learning.
• The kernel’s algebraic decay enables phase transitions and complex temporal dynamics by integrating infinitely many slow modes, informing innovations in learning and response optimization.

Searching arXiv for the cited papers and closely related terminology. A power-law decaying feedback kernel is a causal or history-dependent weighting function whose contribution from past states decays algebraically with lag rather than on a single exponential timescale. In the cited literature, this object appears in several mathematically distinct forms: as the memory kernel in generalized Langevin equations (GLEs), as a spike-triggered refractory or prediction kernel in predictive coding, as a memory kernel in reinforced stochastic processes, and as a forgetting or response kernel in learning dynamics (Chu et al., 2017, Bohte et al., 2010, Mori et al., 2021, Li et al., 6 Feb 2026). A separate but closely related usage concerns kernels whose spectra decay by a power law, where the decaying object is the eigenvalue sequence of a kernel integral operator rather than a temporal kernel value itself (Li et al., 2023).

1. Conceptual scope and terminology

In its literal dynamical sense, a feedback kernel enters an evolution equation through a history integral or an equivalent temporal weighting rule. In the GLE derived from a coarse-grained lattice, the feedback term is

$\int_0^t \Theta(t-\tau)p(\tau)\,d\tau,$

so $\Theta$ is the memory kernel that feeds past coarse-grained velocities back into the present dynamics (Chu et al., 2017). In predictive spiking models, each spike inserts a delayed causal kernel into the internal reconstruction,

$\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$

so the same kernel acts as a refractory response and as a predictive feedback signal (Bohte et al., 2010). In the generalized Pólya urn, the kernel $d(t)$ weights the influence of past draws in the current draw probability, which is again a direct feedback mechanism (Mori et al., 2021). In the functional scaling law for SGD, past noise contributes to final loss through

$\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$

which is the closest analogue of a temporal feedback or forgetting kernel in that setting (Li et al., 6 Feb 2026).

A common source of ambiguity is that not every “power-law kernel” result concerns a literal temporal kernel value. In kernel ridge regression, the central assumption is instead

$$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$

where $\lambda_i$ are the eigenvalues of the kernel integral operator. Here the relevant power law is spectral, not temporal, and the paper does not use “feedback kernel” in a technical sense (Li et al., 2023). Likewise, in random-feature regression dynamics, the power law is imposed on the mode variances $\eta_i=\eta_1 i^{-1-\gamma}$, while the temporal memory kernel $K$ is generated self-consistently from the dynamics rather than assumed to decay as a power law in time (Kramp et al., 26 Feb 2026).

2. Generalized Langevin equations and coarse-grained lattice dynamics

A mathematically explicit instance of a power-law decaying feedback kernel is obtained by coarse-graining an infinite one-dimensional harmonic lattice with nearest- and second-nearest-neighbor interactions. After linearization,

$\ddot{u}=-\mathcal A u,$

with

$\Theta$0

where $\Theta$1, $\Theta$2, and $\Theta$3. Under the stability assumptions $\Theta$4 and $\Theta$5, the Mori–Zwanzig formalism yields the exact coarse-grained GLE

$\Theta$6

with random force $\Theta$7 satisfying

$\Theta$8

through the second fluctuation-dissipation theorem (Chu et al., 2017).

The kernel itself is matrix-valued: $\Theta$9 Because the lattice and coarse-graining are translationally invariant, the entries $\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$0 admit a block-Fourier representation. For the diagonal temporal kernel, the central asymptotic result is that, for piecewise constant coarse-graining,

$\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$1

which the authors describe as decay “with rate at least equal to $\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$2” (Chu et al., 2017). The result is an upper-order asymptotic estimate obtained by stationary phase; the paper does not provide a full explicit constant $\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$3 in an asymptotic formula $\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$4, nor a matching lower bound.

The same analysis also distinguishes temporal and spatial behavior sharply. For piecewise constant averaging,

$\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$5

so spatial decay is faster than any algebraic power, while the temporal tail is slow (Chu et al., 2017). Numerically, the paper reports exponential-looking spatial decay, which accentuates the contrast between fast spatial localization and persistent temporal memory. At $\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$6, the kernel magnitude depends on the coarse-graining level $\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$7: with piecewise constant averaging,

$\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$8

hence $\hat{x}(t)=\sum_{t_j<t}\kappa(t-(t_j+\Delta)),$9, whereas for piecewise linear averaging the estimate is $d(t)$0 (Chu et al., 2017). The rigorously established $d(t)$1 tail, however, is only proved for the piecewise constant case.

3. Power-law memory and phase transition in reinforced stochastic processes

In the generalized Pólya urn with memory kernel, the feedback variable is a weighted empirical average of past binary states $d(t)$2. With

$d(t)$3

the draw probability becomes

$d(t)$4

The kernel $d(t)$5 therefore controls how strongly lag-$d(t)$6 observations feed back into present reinforcement (Mori et al., 2021).

For the power-law case,

$d(t)$7

the asymptotics of the cumulative memory weight are

$d(t)$8

This leads to a phase transition at $d(t)$9. The autocorrelation function

$\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$0

has order parameter $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$1, and the paper reports $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$2 for $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$3, $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$4 for $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$5, and critical algebraic decay at $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$6 (Mori et al., 2021). The long-time exponents are summarized as

$\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$7

Thus the power-law decay exponent changes discontinuously at the critical point.

The paper also contrasts this behavior with exponentially decaying memory. For $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$8, the total memory weight converges, the effective state variable obeys an Ornstein–Uhlenbeck equation, and the autocorrelation decays exponentially: $\mathcal K(t-\tau)=(1+t-\tau)^{-(2-1/\beta)},$9 By contrast, the power-law kernel is represented as a continuum mixture of exponentials,

$$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$0

which produces a Markovian embedding in an infinite-dimensional auxiliary field $$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$1 driven by a common Wiener process (Mori et al., 2021). This suggests that the nontrivial long-memory behavior is not merely a consequence of slow decay, but of the collective effect of infinitely many slow modes coupled through shared noise.

4. Predictive spiking neurons, refractory kernels, and fractional derivatives

In predictive spiking models, the power-law decaying feedback kernel is a spike-triggered refractory or prediction kernel. The idealized form is a shifted power law,

$$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$2

and the practical regularized version is

$$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$3

The kernel is causal, zero before onset, and characterized by exponent $$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$4, onset parameter $$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$5, and scale $$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$6 (Bohte et al., 2010). With positive and negative spikes, the reconstruction is

$$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$7

Each emitted spike therefore adds a long-tailed kernel to the internally reconstructed signal and suppresses future spikes over many timescales.

The same model admits a fractional-calculus interpretation. If

$$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$8

then the paper argues that when a signal is approximated by a sum of power-law kernels with exponent $$c_\beta i^{-\beta}\le \lambda_i\le C_\beta i^{-\beta},$$9, the spike train behaves as the fractional derivative of order

$\lambda_i$0

The key Fourier-domain statement is that the transform of $\lambda_i$1 is proportional to $\lambda_i$2; thus applying fractional differentiation of order $\lambda_i$3 maps the kernel to a constant spectrum, corresponding to a Dirac spike (Bohte et al., 2010). The paper explicitly notes that “a spike-train is the $\lambda_i$4 fractional derivative of a signal approximated by a sum of power-law kernels with exponent $\lambda_i$5.”

The power-law kernel is contrasted directly with exponentially decaying kernels in online coding. For synthesized fractional Brownian motion signals with $\lambda_i$6, the paper reports that power-law kernels required less than half the number of spikes for similar SNR compared to similar exponentially decaying kernels; one figure caption gives 1398 spikes for exponential-kernel encoding versus 618 spikes for power-law-kernel encoding at the same SNR (Bohte et al., 2010). The paper adds an important caveat: without negative spikes, the advantage can disappear because a slow power-law tail cannot retract quickly on descending signals. For implementation, the paper states that the normalized power-law kernel can be approximated very accurately over multiple orders of magnitude by a sum of just 11 $\lambda_i$7-function exponentials, and that this decomposition supports downstream temporal filtering by selectively suppressing slow or fast exponential components (Bohte et al., 2010).

5. Spectral power laws, collective response kernels, and optimization dynamics

In learning theory, “power-law decaying kernel” frequently refers to spectral decay. For kernel ridge regression, the kernel integral operator

$\lambda_i$8

has eigenvalues satisfying

$\lambda_i$9

Under this assumption and a source condition

$\eta_i=\eta_1 i^{-1-\gamma}$0

the excess risk follows the asymptotic form

$\eta_i=\eta_1 i^{-1-\gamma}$1

while in the nearly interpolating regime $\eta_i=\eta_1 i^{-1-\gamma}$2 one has

$\eta_i=\eta_1 i^{-1-\gamma}$3

in the noisy case (Li et al., 2023). The effective dimension satisfies $\eta_i=\eta_1 i^{-1-\gamma}$4, so the power law controls variance through the number of active spectral directions.

A related but dynamically richer construction appears in random-feature regression with power-law-distributed kernel eigenvalues

$\eta_i=\eta_1 i^{-1-\gamma}$5

After disorder averaging and dynamical mean-field reduction, the mode discrepancies $\eta_i=\eta_1 i^{-1-\gamma}$6 satisfy

$\eta_i=\eta_1 i^{-1-\gamma}$7

where $\eta_i=\eta_1 i^{-1-\gamma}$8 is a self-consistent collective memory kernel defined by

$\eta_i=\eta_1 i^{-1-\gamma}$9

The paper explicitly interprets this as a nonlocal self-coupling and as an Onsager reaction term (Kramp et al., 26 Feb 2026). It does not claim that $K$0 itself decays as a power law in time; rather, the power-law spectrum $K$1 generates a broad hierarchy of mode time scales that shapes the time dependence of the feedback kernel and the resulting bias–variance dynamics.

The functional scaling law for one-pass SGD makes the temporal-memory interpretation fully explicit. Under

$K$2

the expected excess risk is modeled as

$K$3

Here $K$4 is a power-law forgetting kernel for injected noise (Li et al., 6 Feb 2026). Solving the finite-horizon schedule optimization yields the easy-task optimal learning-rate schedule

$K$5

up to the stated asymptotic correction, and a warmup-stable-decay structure in the hard-task regime $K$6 (Li et al., 6 Feb 2026). A common misconception is therefore to identify every power-law kernel in learning theory with a temporal convolution kernel; in some cases the power law is spectral, in some it is a forgetting kernel, and in some it shapes a self-consistent response operator.

6. Approximation theory, perturbation bounds, and limitations

For stochastic Volterra formulations of GLEs, power-law decaying feedback kernels are treated through weighted Banach spaces and weighted Schur norms. The first-order model

$K$7

and its perturbed analogue with kernel $K$8 are compared under synchronized noise coupling $K$9. The central kernel norm is

$\ddot{u}=-\mathcal A u,$0

with $\ddot{u}=-\mathcal A u,$1, $\ddot{u}=-\mathcal A u,$2, and the paper states explicitly that examples of $\ddot{u}=-\mathcal A u,$3 include

$\ddot{u}=-\mathcal A u,$4

Thus integrable power-law tails are directly admissible (Lang et al., 11 Dec 2025).

For translation-invariant power-law kernels

$\ddot{u}=-\mathcal A u,$5

the framework applies naturally when $\ddot{u}=-\mathcal A u,$6. If $\ddot{u}=-\mathcal A u,$7 with $\ddot{u}=-\mathcal A u,$8, then the weighted Schur norms are finite, and the trajectory discrepancy satisfies a bound of the form

$\ddot{u}=-\mathcal A u,$9

with analogous estimates for the second-order GLE in a hypocoercive Lyapunov-type distance (Lang et al., 11 Dec 2025). The paper’s explicit scalar example

$\Theta$00

shows that if $\Theta$01, then the Volterra comparison theorem yields $\Theta$02 when $\Theta$03. This means that, in the integrable power-law regime, the decay rate of the perturbation bound is directly controlled by the chosen kernel envelope.

The numerical evidence in that work uses

$\Theta$04

and reports that the observed trajectory-error decay is close to exponent $\Theta$05, near the decay of $\Theta$06, even though the proven envelope uses exponent $\Theta$07 (Lang et al., 11 Dec 2025). This suggests that the weighted-space theory can be conservative. The same paper also sets a clear limit on its scope: the subexponential framework does not directly cover nonintegrable power laws with $\Theta$08 (Lang et al., 11 Dec 2025). Comparable scope limitations appear elsewhere: the lattice GLE proof of $\Theta$09 is confined to a one-dimensional harmonic chain with piecewise constant coarse-graining (Chu et al., 2017); the KRR learning-curve results concern spectral power laws rather than literal temporal feedback kernels (Li et al., 2023); and the random-feature DMFT derives a temporal response kernel from a power-law mode spectrum without proving that the response kernel itself has a power-law tail in time (Kramp et al., 26 Feb 2026). Taken together, these results support a narrow but technically clear conclusion: power-law decay in a feedback kernel can produce persistent non-Markovian memory, phase transitions, and polynomial stability or learning laws, but the precise object that decays by a power law must be identified case by case.