---
title: Learnable Threshold Through Time
url: https://www.emergentmind.com/topics/learnable-threshold-through-time
type: topic
---

# Learnable Threshold Through Time

Searching arXiv for the provided paper and closely related work on time-varying or learnable thresholds.
arxiv_search(query="all:2406.14522 OR ti:\"Learning thresholds lead to stable language coexistence\" OR abs:\"learning thresholds\" ", max_results=5, sort_by="submittedDate")
arxiv_search(query="ti:\"Structured Differential Learning for Automatic Threshold Setting\" OR 1808.00361 OR ti:\"Learning to Allocate Limited Time to Decisions with Different Expected Outcomes\" OR 1607.05334", max_results=10, sort_by="submittedDate")
arxiv_search(query="2402.00411 OR 2104.10851 OR 1506.05212 OR 1103.2382 OR 2602.07950 OR 2511.09953 OR 2512.05790", max_results=10, sort_by="submittedDate")
Learnable Threshold Through Time denotes a family of models in which a threshold is not treated as a fixed hyperparameter, but as a quantity that changes with temporal evidence, state history, online feedback, or the compositional geometry of learning dynamics. In the surveyed literature, the threshold may be an explicit decision boundary, a firing threshold, a temporal window length, a concept-drift alarm level, or an effective capacity boundary that separates learnable from non-learnable future tasks. The common feature is that thresholding is embedded in a dynamical process rather than applied once and held constant [2406.14522] [1808.00361] [1607.05334] [1103.2382] [1506.05212] [2104.10851] [2402.00411] [2511.09953] [2602.07950] [2512.05790].

## 1. Conceptual scope and recurring mathematical structure

Across these works, threshold learning appears in several mathematically distinct forms. Some models modify a scalar cutoff directly, as in adaptive firing thresholds or online drift-detection thresholds. Others soften a discrete threshold into a ramp or sigmoid so that approximate descent becomes possible. Still others hold the nominal threshold fixed while learning an associated temporal parameter, such as a drift rate or effective learning-rate envelope, which changes when the threshold is reached or what remains learnable at a given time [1808.00361] [1103.2382] [1506.05212] [2104.10851] [2511.09953] [2602.07950].

| Framework | Threshold object | Temporal mechanism |
|---|---|---|
| Language competition | \(x^*, y^*\), or \(H(x;x^*,w)\) | Memory of encounters and smoothed threshold heterogeneity |
| Structured rule systems | \(\theta\), \(N\), \(\tau_t\) | Ramps, histogrammed blame, or comparison-phase updates |
| Sequential decision and spiking models | \(\Theta(t)\), \(V_{thr}\), \(\theta_i(t)\) | TD updates, FP/FN balancing, or homeostatic dynamics |
| Learning-capacity theories | \(\mathcal{R}_A(t)\), \(\mathcal{H}_N\) | Jacobian contraction or gate-filtered sample-complexity bounds |

A recurring pattern is the replacement of a hard discontinuity by either a coarse-grained threshold derived from temporal aggregation or a smooth surrogate that is easier to optimize. In the language-competition model, the condition “you only learn language \(X\) if you meet \(X\)-speakers often enough” produces the coarse-grained threshold \(x^* = K_x^*/(\mathrm{int}\,T^*)\), with \(\int_{t-T^*}^t x(s)\,ds \approx T^*x(t)\) as the first approximation [2406.14522]. In Structured Differential Learning, a Boolean “above-\(\theta\)” test is replaced by a piecewise-linear ramp \(\sigma(s;\theta,\delta^-,\delta^+)\), and temporal window lengths \(N\) are likewise made soft around a nominal tap-count \(V\) [1808.00361]. In MPATH, the threshold itself becomes a dynamical variable satisfying both \(\tau_\theta\,d\theta_i/dt = 1-\theta_i(t)\) and the discrete reset-drop relation \(\theta_i(t_k)=\exp(-\Delta\tilde V_i(t_k)/\tau_\theta)\) [2104.10851].

This suggests that “learnable threshold through time” is less a single algorithm than a design principle: thresholding is made temporally adaptive either by explicit online updates, by differentiable relaxation, or by embedding the threshold in a broader dynamical state space.

## 2. Coarse-grained learning thresholds in language dynamics

A particularly clear analytical realization appears in an extension of the Abrams–Strogatz model for language competition. Let \(x(t)\in[0,1]\) be the fraction of speakers of language \(X\), \(y(t)=1-x(t)\), and in mean field write
\[
\frac{dx}{dt} = (1-x)r_x(x) - x r_y(1-x).
\]
In the original formulation one uses power-law rates \(r_x(x)=j_x x^a\), \(r_y(y)=j_y y^a\), which after rescaling yield
\[
\frac{dx}{dt} = \gamma(1-x)x^a - (1-\gamma)x(1-x)^a.
\]
For \(a>1\), this system has only the two attractors \(x=0\) and \(x=1\), so coexistence is impossible; for \(a\ge 1\), whenever \(x(t)\to 0\), the curve \(\ln x(t)\) is strictly concave [2406.14522].

The threshold extension replaces the rates by
\[
r_x(x)\to r_x(x)\Theta(x-x^*), \qquad r_y(y)\to r_y(y)\Theta(y-y^*),
\]
with \(\Theta\) the Heaviside step function and \(x^*,y^*\) critical fractions. In the simplest case \(r_x=j_x\), \(r_y=j_y\), one obtains
\[
\frac{dx}{dt}=\gamma\,\Theta(x-x^*)(1-x) - (1-\gamma)\,\Theta(1-x-y^*)x.
\]
Because each \(\Theta\) takes only the values \(0\) or \(1\), the phase space is partitioned into up to four dynamical regions. Besides the two consensus states, the model produces a “Frozen” region, where \(1-y^*<x(t)<x^*\) implies \(dx/dt\equiv 0\) and hence \(x(t)=x(0)\) forever, and a coexistence region \(x\in[x^*,1-y^*]\), where
\[
\frac{dx}{dt}=\gamma-x,\qquad x(t)=\gamma-[\gamma-x_0]e^{-t},
\]
so the system converges to the new fixed point \(x_{III}=\gamma\) [2406.14522].

The exact solvability is significant because it isolates two regimes absent from the Abrams–Strogatz model: a stable dynamical coexistence of the two languages and a frozen state coinciding with the initial state. Smoothing the thresholds by averaging over a distribution of personal thresholds \(\xi\) replaces \(\Theta\) by a sigmoid
\[
H(x;x^*,w)\equiv \int \Theta(x-\xi)\,f((\xi-x^*)/w)\,d\xi,
\]
with one convenient choice
\[
H(x;x^*,w)=\frac{S(x;x^*,w)-S(0;x^*,w)}{S(1;x^*,w)-S(0;x^*,w)},\qquad
S(x;x^*,w)=\frac12[1+\tanh((x-x^*)/w)].
\]
Under smoothing, the sharp frozen strip becomes metastable rather than exactly frozen, but the coexistence fixed point remains attractive provided \(\gamma\in(x^*,1-y^*)\) [2406.14522].

The comparison with historical datasets is central to the interpretation. In Welsh-in-Wales and Swedish-in-Finland data, the threshold model captures a slowing-down of decline and approach to a nonzero plateau, whereas an Abrams–Strogatz fit always underestimates this plateau because it lacks \(x_{III}\). In French in Canada and Russian in Estonia, the inferred \(x^*,y^*\) are so large that the system lies in the frozen regime, reproducing very slow or almost no change [2406.14522]. A plausible implication is that temporally aggregated exposure thresholds can qualitatively alter the phase portrait of a social-dynamical system even when the underlying state variable remains one-dimensional.

## 3. Differentiable, heuristic, and online threshold adaptation in engineered systems

In rule-based computer vision, learnable thresholds through time are realized by converting a structured threshold-and-logic pipeline into an almost-differentiable system. Structured Differential Learning replaces every Boolean or counting decision by a small linear ramp around a hand-picked setting. A one-sided “above-\(\theta\)” test on a scalar feature \(s\) becomes a piecewise-linear \(\sigma(s;\theta,\delta^-,\delta^+)\) that is zero below \(\theta-\delta^-\), rises linearly to \(1\) at \(\theta+\delta^+\), and saturates above that. Temporal logic is handled similarly: an \(N\)-frame smooth-AND is approximated by a soft minimum over recent thresholded values, while a retriggerable monostable is approximated by a soft maximum. The time constant \(N\) itself is made soft around a nominal tap-count \(V\), so both \(\theta\) and \(N\) enter the forward pass as continuous parameters [1808.00361].

The supervised update mechanism is deliberately heuristic rather than fully differentiable. With weighted \(0\)–\(1\) loss
\[
L=\sum_t\bigl[w_{FN}\mathbf{1}\{H(x(t))=0,g(t)=1\}+w_{FP}\mathbf{1}\{H(x(t))=1,g(t)=0\}\bigr],
\]
SDL assigns blame for each frame’s error to exactly one suspect parameter and one alternate value that would have changed the hard decision on that frame. Each parameter builds a positive-error histogram \(H^+(k)\), a negative-error histogram \(H^-(k)\), and a net-benefit curve \(B(k)=H^+(k)-H^-(k)\). Cumulative sums outward from the current bin identify a new setting \(\theta_i'\), and the update
\[
\theta_i \leftarrow \theta_i + \eta_i\,\mathrm{sign}(\Delta\theta)\,|\Delta\theta|
\]
is applied, with the process iterated for \(5\)–\(10\) rounds [1808.00361]. The paper states that there is no formal convexity guarantee, but in practice SDL converges in fewer than \(10\) iterations when initialized from a reasonable hand-tuned starting point.

The automotive headlight-controller experiments exemplify threshold learning under temporal supervision. The system had on the order of \(200\) hand-tuned parameters, and training used \(1.9\,\mathrm{h}\) (\(200\mathrm{K}\) frames) and \(12.6\,\mathrm{h}\) (\(1.3\mathrm{M}\) frames) of labeled video in two development phases. After best manual tuning, SDL was run for \(8\) rounds. On held-out in-car tests of approximately \(1\,\mathrm{h}\) drive each, SDL reduced combined too-bright/too-dark switching errors by roughly \(30\)–\(80\%\), and reduced dangerous “blinding” events per hour from \(3\to 0\) in Phase 1 and from \(10.3\to 1.0\) in Phase 2 [1808.00361].

A distinct online formulation appears in concept-drift detection, where the threshold is explicitly time-varying. For a data stream \(D=\{(x_t,y_t)\}_{t=1}^T\), a sliding window \(\Omega_t\) produces a scalar test statistic \(S_t=f(\Omega_t)\), and a drift alarm is raised when \(S_t>\tau\). The paper proves three theorems: perfect detection may not maximize online predictive accuracy; no constant threshold is universally optimal; and dynamic thresholds outperform stationary thresholds in the sense that
\[
\max_{\{\tau_t\}\in\Theta_{\mathrm{dyn}}} A(\{\tau_t\};D) \ge \max_{\tau\in\Theta_{\mathrm{const}}} A(\tau;D),
\]
with strict inequality whenever per-segment optimal thresholds vary [2511.09953].

The Dynamic Threshold Determination algorithm operationalizes this result by wrapping a base detector and learner with a comparison phase of length \(K\). When \(S_t>\tau\), it instantiates three candidates: an Early Drift Model (EDM), a Reactive Drift Model (RDM), and a Previous Model (PM). After parallel evaluation, the winning candidate sets the new threshold by
\[
\tau \leftarrow
\begin{cases}
S_{t-1}, & \text{if EDM wins},\\
\tau, & \text{if RDM wins},\\
S_t+\eta, & \text{if PM wins}.
\end{cases}
\]
Time complexity remains \(O(n)\) per chunk and is at worst \(3\times\) the base cost [2511.09953]. In the Airline HDDM-W case study, false alarms over \(58\) chunks drop from \(36\) to \(3\), mean detection delay from \(4.2\) steps to \(1.1\) steps, and overall accuracy rises from \(48.66\%\) to \(58.31\%\). Across datasets and detectors, DTD-enhanced variants improve accuracy by up to \(10\) points in the case study and typically by \(1\)–\(3\) points on average [2511.09953].

These two lines of work illustrate two non-equivalent interpretations of learnable thresholding. SDL softens thresholds so they can be tuned inside a structured pipeline; DTD keeps the detector statistic fixed and instead learns the threshold schedule that best balances false alarms and delayed adaptation.

## 4. Sequential decision thresholds and interval timing

In sequential decision-making, a time-varying threshold defines when accumulated evidence is sufficient for action. In the “canoe” task, a participant observes a noisy left-or-right random walk and decides when the displacement \(X(t)\) reaches a stopping boundary:
\[
|X(t)| \ge \Theta(t).
\]
The experiments intermixed easy and hard trials in fixed-duration \(1\,\mathrm{min}\) blocks. Easy trials had \(P_0=0.65\), \(\pm 20\)-coin reward/punishment, and a \(3\,\mathrm{s}\) penalty on errors; hard trials had \(P_0=0.51\), \(\pm 1\)-coin reward/punishment, and no delay penalty. The optimal strategy was to adopt a small decision threshold for hard trials, but several participants did not learn this simple strategy [1607.05334].

The computational comparison covered \(10\) models: baseline models with no learning, heuristic adjustment models, and actor–critic models with constant or time-varying thresholds. Constant thresholds use \(\Theta(t)=\theta_0\); time-varying thresholds use basis expansions or parametric forms, including a Weibull-shaped mean threshold
\[
m(t)=\psi - [1-\exp\{-(t/\lambda)^\phi\}]\,[\tfrac12\psi-\psi'].
\]
The actor–critic update is based on the temporal-difference error
\[
\delta_s(k)=r_k-\rho d_k+V_k(s')-V_k(s),
\]
with critic update \(V_{k+1}(s)=V_k(s)+\alpha_c\delta_s(k)\) and actor update
\[
m_{s,k+1}=m_{s,k}+\alpha_m\,\delta_s(k)\,(a_k-m_{s,k}),
\]
where \(a_k\) is the actual threshold sampled on trial \(k\) [1607.05334].

Model comparison used NLL, \(-\tfrac12\)BIC or \(-\tfrac12\)AIC, and Variational Bayesian protected exceedance probability. The Weibull-threshold actor–critic model, Model 9 (“RL_V”), was the clear winner with \(\mathrm{PE}\approx 1\). Its threshold parameters are updated online, trial by trial, and convergence was typically reached by block \(25\)–\(30\). Empirically, for hard trials thresholds declined steeply from approximately \(120\,\mathrm{px}\) to near \(0\,\mathrm{px}\), accuracy rose from approximately \(60\%\to 75\%\), and RT fell from approximately \(8\,\mathrm{s}\to 4\,\mathrm{s}\). For easy trials thresholds hovered near approximately \(140\,\mathrm{px}\) and declined modestly by approximately \(10\)–\(20\,\mathrm{px}\); accuracy rose from approximately \(80\%\to 90\%\), and RT from approximately \(6\,\mathrm{s}\to 5\,\mathrm{s}\). Reward rate improved from approximately \(0.7\) coins/s in block \(1\) to approximately \(0.9\) coins/s in block \(40\) [1607.05334].

A related but conceptually different construction appears in interval timing. Here a bounded drift–diffusion integrator
\[
d\phi(t)=w\,dt+c\,dW_t,\qquad \phi\in[0,1],
\]
accumulates time toward an absorbing threshold \(\theta=1\), resetting to \(0\) when an external event occurs. If the event occurs before threshold, the drift is updated multiplicatively:
\[
\Delta w = w\Bigl(\frac{\theta}{\phi(t_i)}-1\Bigr),\qquad \theta=1,
\]
and if threshold is reached too early, \(w\) decays continuously via \(dw/dt=-w^2\) while \(\phi\) remains pinned at \(1\) [1103.2382]. With partial adaptation, the update takes the stochastic-approximation form
\[
w_{n+1}=w_n+a_n(I_n^{-1}-w_n),
\]
and for \(a_n=1/n\), one obtains
\[
w_n=\frac1n\sum_{i=1}^n I_i^{-1}.
\]

The significance is that a nominally fixed threshold process becomes effectively learnable through adaptation of the drift needed to hit that threshold at the correct interval. The model proves that learning requires a number of trials independent of the interval length, and when the diffusion is set by \(c^2=Bw\), both the mean and standard deviation of decision time become proportional to the interval, yielding Weber’s law for timing [1103.2382]. This suggests that “threshold learning” may reside either in the boundary itself or in the dynamical law that determines how the boundary is approached.

## 5. Adaptive thresholds in spiking and neuromorphic systems

In spiking classifiers with binary synapses, threshold adaptation is often coupled to structural plasticity rather than continuous weight adjustment. The neuron with nonlinear dendrites (NNLD) computes
\[
V(t)=\sum_{j=1}^m b(v_j(t)), \qquad
v_j(t)=\sum_{i=1}^d w_{ij}\sum_f K(t-t_f^i),
\]
with \(w_{ij}\in\{0,1\}\), quadratic dendritic nonlinearity, and a spike emitted whenever there exists \(t\) such that \(V(t)\ge V_{thr}\). The threshold is updated after each training iteration by
\[
\Delta V_{thr}=\eta\,(w_{fp}\,FP-w_{fn}\,FN),
\]
with \(w_{fp}=w_{fn}=1\) in the experiments. If false positives exceed false negatives, \(V_{thr}\) is shifted upward; if false negatives dominate, it is shifted downward [1506.05212].

This threshold rule is integrated with a morphological learning algorithm inspired by the Tempotron. Misclassified patterns define an error through \(V_{max}=\max_t V(t)\), candidate synapses are evaluated by a correlation term derived from
\[
\frac{\partial E}{\partial w_{ij}}=-b'(v_j(t_{max}))\sum_f K(t_{max}-t_f^i),
\]
and a weak existing synapse is replaced by a stronger silent candidate. No formal proof of global convergence is given, but the paper states that threshold adaptation co-optimizes the decision boundary so that misclassifications of both types are balanced and empirically stabilizes learning [1506.05212]. On the \(500\)-pattern latency task, adaptive threshold yields approximately \(95.6\%\) accuracy versus approximately \(94.8\%\) for fixed threshold; on the tactile-sensor benchmark, adaptive threshold gives \(96.54\pm0.66\%\) versus \(95.64\pm0.65\%\) [1506.05212].

MPATH generalizes the notion of a learnable threshold through homeostatic internal dynamics. Each neuron tracks a running input mean \(V_{\mu,i}(t)\), variance \(V_{\sigma,i}^2(t)\), normalized membrane potential \(\rho_i(t)\in[-1,1]\), and adaptive threshold \(\theta_i(t)\), governed by two time constants \(\tau_m\) and \(\tau_\theta\), typically with \(\tau_\theta=0.1\,\tau_m\). The threshold obeys both the continuous relaxation
\[
\tau_\theta\frac{d\theta_i}{dt}=1-\theta_i(t)
\]
and the discrete update
\[
\theta_i(t_k)=\exp\!\Bigl(-\frac{\Delta\tilde V_i(t_k)}{\tau_\theta}\Bigr),
\]
where \(\Delta\tilde V_i(t_k)\) is the normalized deviation of the raw input from the running mean [2104.10851]. Activation is then thresholded through \(\rho_{\theta,i}(t_k)=\rho_i(t_k)-\theta_i(t_k)\), with positive net drive producing activation and otherwise exponential decay of the activation trace.

The flash-in-noise experiment clarifies the temporal role of the threshold. White noise \(x_i\sim\mathcal{N}(0,1)\) is presented at every step, with a flash of amplitude approximately \(3\) superimposed every \(10\) steps. Early on, \(\Delta\tilde V\) remains small and \(\theta\approx 1\), suppressing the noise; at a flash, \(\Delta\tilde V\) jumps, \(\theta\) drops to \(\exp(-\text{large}/\tau_\theta)\ll 1\), and the neuron fires. Over \(2000\) steps, Hebbian updates carve out filters that respond reliably to the flash pattern [2104.10851]. Here threshold learning functions as unsupervised contrast-dependent gain control and as a short-term temporal memory mechanism.

LM-HT occupies an intermediate position. Its multi-hierarchical thresholds are explicit and equidistant,
\[
\tau_k^\ell = k\,\theta^\ell,\qquad k=1,\dots,L,
\]
with spike output
\[
s^\ell(t)=\mathrm{clip}(\lfloor m^\ell(t)/\theta^\ell\rfloor,0,L),
\]
but the thresholds do not themselves evolve over time \(t\); they remain the fixed set \(\{\theta^\ell,2\theta^\ell,\dots,L\theta^\ell\}\) while the membrane potential moves through them [2402.00411]. Temporal adaptation instead enters through the Temporal-Global Information Matrix \(\Omega^\ell\), which mixes pre-synaptic spikes across all recent time indices,
\[
I^\ell(t)=\sum_{j=1}^T \omega^\ell_{tj} W^\ell s^{\ell-1}(j)\theta^{\ell-1},
\]
and through the learned leak factor \(\lambda^\ell\) in
\[
m^\ell(t)=\lambda^\ell v^\ell(t-1)+I^\ell(t).
\]

This distinction matters. LM-HT is a time-structured threshold architecture rather than a threshold trajectory in the MPATH or NNLD sense. Nevertheless, it supports direct STBP training and hybrid ANN-SNN conversion plus fine-tuning. Reported results include \(96.9\%\) on CIFAR-10 with Res-19, \(L=2\), and \(2\) steps; \(81.8\%\) on CIFAR-100 with Res-19, \(L=2\), and \(2\) steps; \(61.0\%\) on ImageNet-200 with VGG-13, \(L=2\), and \(2\) steps; and \(81.9\%\) on DVSCIFAR-10 with Res-18, \(L=4\), and \(2\) steps [2402.00411]. In the hybrid conversion setting on CIFAR-100, QCFS with \(T_q=4\) improves from \(63.3\%\) to \(75.97\%\) after LM-HT fine-tuning at \(2\) steps, and QCFS with \(T_q=8\) improves from \(64.9\%\) to \(76.31\%\) at \(2\) steps [2402.00411].

A common misconception is that threshold learning in neural systems must mean direct optimization of a scalar cutoff. These works show three alternatives: direct threshold updates from FP/FN counts, homeostatic threshold dynamics coupled to input statistics, and fixed multi-threshold ladders combined with learned temporal current regulation.

## 6. Thresholds as temporal capacity limits in continual and recurrent learning

In more abstract learning theory, the threshold is no longer a scalar decision boundary but a time-dependent limit on future reconfiguration. In the thermodynamic theory of continual learning, stochastic learning is modeled as a transport map \(\theta_t=\Psi_t(\theta_0;\omega)\) with Jacobian \(J_t=\partial\Psi_t/\partial\theta_0\). Global contraction is summarized by the effective rank
\[
\mathcal{R}(t)=\exp\!\Bigl[\frac1d\,E\{\log\det(J_t^\top J_t)\}\Bigr].
\]
After learning task \(A\), only directions tangent to the task-preserving manifold \(\mathcal{G}_A(\epsilon_A)\) remain usable without degrading \(A\), and the compatible effective rank becomes
\[
\mathcal{R}_A(t)=\exp\!\Bigl[\frac1{k_A}\,E\{\log\det(Q_A^\top J_t^\top J_t Q_A)\}\Bigr],
\]
where \(Q_A\) is an orthonormal basis of the tangent space \(T_\theta\mathcal{G}_A\) [2602.07950].

Because learning composes as a semigroup, \(\Psi_{t+s}=\Psi_s\circ\Psi_t\) and \(J_{t+s}=J_sJ_t\). Singular values and rank are submultiplicative, and therefore \(\mathcal{R}_A(t+s)\le \mathcal{R}_A(t)\): once a direction in the task-preserving subspace is collapsed, no further finite-time learning can revive it [2602.07950]. For a new task \(B\), define the restricted Hessian \(H_{B|A}=Q_A^\top H_B Q_A\) and its stable rank
\[
m_B=\frac{\|H_{B|A}\|_F^2}{\|H_{B|A}\|_2^2}.
\]
The central threshold criterion is then
\[
m_B>\mathcal{R}_A(t) \quad \Longrightarrow \quad \text{any trajectory learning \(B\) must exit }\mathcal{G}_A \text{ and induce forgetting.}
\]
The paper interprets \(t\mapsto \mathcal{R}_A(t)\) as a learnable-threshold curve: early in training many future tasks satisfy \(m_B\le \mathcal{R}_A(t)\), but as learning proceeds, the residual compatibility threshold decreases and the set of learnable tasks shrinks [2602.07950].

A related threshold concept appears in recurrent networks trained by BPTT. There, gating mechanisms induce per-lag, per-neuron effective learning rates
\[
\mu^{(q)}_{t,\ell}=\mu\bigl(\gamma^{(0,q)}_{t,\ell}+\gamma^{(1,q)}_{t,\ell}\bigr),
\]
obtained from first-order expansions of gate-induced Jacobian products, and the aggregate envelope
\[
f(\ell)=\|\mu_{t,\ell}\|_1.
\]
Under symmetric \(\alpha\)-stable gradient noise, the minimal sample size needed to detect a lag-\(\ell\) dependency satisfies
\[
N(\ell)\propto f(\ell)^{-\alpha},
\]
which leads to the learnability window
\[
\mathcal{H}_N=\max\{\ell\ge 1:\text{a dependency at lag }\ell\text{ is statistically detectable}\}
\]
or equivalently \(\mathcal{H}_N\approx f^{\leftarrow}(C\,N^{-1/\alpha})\) [2512.05790].

The theory derives closed-form scaling laws for three decay regimes of \(f(\ell)\). If \(f(\ell)\asymp c/\log(1+\ell)\), then \(\mathcal{H}_N\asymp \exp(\kappa N^{1/\alpha})-1\); if \(f(\ell)\asymp c\,\ell^{-\beta}\), then \(\mathcal{H}_N\asymp N^{1/(\alpha\beta)}\); and if \(f(\ell)\asymp c\,\lambda^\ell\) with \(0<\lambda<1\), then \(\mathcal{H}_N\asymp \log N/[\alpha\log(1/\lambda)]\) [2512.05790]. Broader or more heterogeneous gate spectra slow the decay of \(f(\ell)\) and enlarge \(\mathcal{H}_N\), whereas heavier-tailed noise compresses the learnability window.

Taken together, these results extend learnable-threshold thinking beyond explicit threshold parameters. The threshold may be a dynamically shrinking compatibility rank or a lag-dependent statistical detectability frontier. This suggests that temporal thresholding is a unifying way to describe when a system changes state, when it should act, and when future adaptation ceases to be feasible without loss.

Source: https://www.emergentmind.com/topics/learnable-threshold-through-time