Learnable Threshold Through Time

• Learnable Threshold Through Time is a framework where the threshold is treated as a dynamic variable that evolves with temporal evidence, state history, and online feedback.
• It employs techniques such as differential smoothing, ramp functions, and online update rules to transform a static cutoff into an adaptive parameter across various models.
• Experimental applications in language dynamics, computer vision, and spiking neural networks demonstrate improved stability, reduced errors, and efficient convergence with adaptive thresholds.

Searching arXiv for the provided paper and closely related work on time-varying or learnable thresholds. arxiv_search(query="all:(Tamm et al., 2024) 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 (Connell et al., 2018) OR ti:\"Learning to Allocate Limited Time to Decisions with Different Expected Outcomes\" OR (Khodadadi et al., 2016)", max_results=10, sort_by="submittedDate") arxiv_search(query="(Hao et al., 2024) OR (Hadjiivanov, 2021) OR (Roy et al., 2015) OR (Rivest et al., 2011) OR (Okanohara, 8 Feb 2026) OR (Lu et al., 13 Nov 2025) OR (Livi, 5 Dec 2025)", 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 (Tamm et al., 2024, Connell et al., 2018, Khodadadi et al., 2016, Rivest et al., 2011, Roy et al., 2015, Hadjiivanov, 2021, Hao et al., 2024, Lu et al., 13 Nov 2025, Okanohara, 8 Feb 2026, Livi, 5 Dec 2025).

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 (Connell et al., 2018, Rivest et al., 2011, Roy et al., 2015, Hadjiivanov, 2021, Lu et al., 13 Nov 2025, Okanohara, 8 Feb 2026).

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 $H(x;x^*,w)$0 if you meet $H(x;x^*,w)$1-speakers often enough” produces the coarse-grained threshold $H(x;x^*,w)$2, with $H(x;x^*,w)$3 as the first approximation (Tamm et al., 2024). In Structured Differential Learning, a Boolean “above-$H(x;x^*,w)$4” test is replaced by a piecewise-linear ramp $H(x;x^*,w)$5, and temporal window lengths $H(x;x^*,w)$6 are likewise made soft around a nominal tap-count $H(x;x^*,w)$7 (Connell et al., 2018). In MPATH, the threshold itself becomes a dynamical variable satisfying both $H(x;x^*,w)$8 and the discrete reset-drop relation $H(x;x^*,w)$9 (Hadjiivanov, 2021).

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 $\theta$0 be the fraction of speakers of language $\theta$1, $\theta$2, and in mean field write

$\theta$3

In the original formulation one uses power-law rates $\theta$4, $\theta$5, which after rescaling yield

$\theta$6

For $\theta$7, this system has only the two attractors $\theta$8 and $\theta$9, so coexistence is impossible; for $N$0, whenever $N$1, the curve $N$2 is strictly concave (Tamm et al., 2024).

The threshold extension replaces the rates by

$N$3

with $N$4 the Heaviside step function and $N$5 critical fractions. In the simplest case $N$6, $N$7, one obtains

$N$8

Because each $N$9 takes only the values $\tau_t$0 or $\tau_t$1, the phase space is partitioned into up to four dynamical regions. Besides the two consensus states, the model produces a “Frozen” region, where $\tau_t$2 implies $\tau_t$3 and hence $\tau_t$4 forever, and a coexistence region $\tau_t$5, where

$\tau_t$6

so the system converges to the new fixed point $\tau_t$7 (Tamm et al., 2024).

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 $\tau_t$8 replaces $\tau_t$9 by a sigmoid

$\Theta(t)$0

with one convenient choice

$\Theta(t)$1

Under smoothing, the sharp frozen strip becomes metastable rather than exactly frozen, but the coexistence fixed point remains attractive provided $\Theta(t)$2 (Tamm et al., 2024).

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 $\Theta(t)$3. In French in Canada and Russian in Estonia, the inferred $\Theta(t)$4 are so large that the system lies in the frozen regime, reproducing very slow or almost no change (Tamm et al., 2024). 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(t)$5” test on a scalar feature $\Theta(t)$6 becomes a piecewise-linear $\Theta(t)$7 that is zero below $\Theta(t)$8, rises linearly to $\Theta(t)$9 at $V_{thr}$0, and saturates above that. Temporal logic is handled similarly: an $V_{thr}$1-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 $V_{thr}$2 itself is made soft around a nominal tap-count $V_{thr}$3, so both $V_{thr}$4 and $V_{thr}$5 enter the forward pass as continuous parameters (Connell et al., 2018).

The supervised update mechanism is deliberately heuristic rather than fully differentiable. With weighted $V_{thr}$6–$V_{thr}$7 loss

$V_{thr}$8

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 $V_{thr}$9, a negative-error histogram $\theta_i(t)$0, and a net-benefit curve $\theta_i(t)$1. Cumulative sums outward from the current bin identify a new setting $\theta_i(t)$2, and the update

$\theta_i(t)$3

is applied, with the process iterated for $\theta_i(t)$4–$\theta_i(t)$5 rounds (Connell et al., 2018). The paper states that there is no formal convexity guarantee, but in practice SDL converges in fewer than $\theta_i(t)$6 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 $\theta_i(t)$7 hand-tuned parameters, and training used $\theta_i(t)$8 ($\theta_i(t)$9 frames) and $\mathcal{R}_A(t)$0 ($\mathcal{R}_A(t)$1 frames) of labeled video in two development phases. After best manual tuning, SDL was run for $\mathcal{R}_A(t)$2 rounds. On held-out in-car tests of approximately $\mathcal{R}_A(t)$3 drive each, SDL reduced combined too-bright/too-dark switching errors by roughly $\mathcal{R}_A(t)$4–$\mathcal{R}_A(t)$5, and reduced dangerous “blinding” events per hour from $\mathcal{R}_A(t)$6 in Phase^^^^2 OR abs:\2^^^^and from $\mathcal{R}_A(t)$7 in Phase 2 (Connell et al., 2018).

A distinct online formulation appears in concept-drift detection, where the threshold is explicitly time-varying. For a data stream $\mathcal{R}_A(t)$8, a sliding window $\mathcal{R}_A(t)$9 produces a scalar test statistic $\mathcal{H}_N$0, and a drift alarm is raised when $\mathcal{H}_N$1. 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

$\mathcal{H}_N$2

with strict inequality whenever per-segment optimal thresholds vary (Lu et al., 13 Nov 2025).

The Dynamic Threshold Determination algorithm operationalizes this result by wrapping a base detector and learner with a comparison phase of length $\mathcal{H}_N$3. When $\mathcal{H}_N$4, 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

$\mathcal{H}_N$5

Time complexity remains $\mathcal{H}_N$6 per chunk and is at worst $\mathcal{H}_N$7 the base cost (Lu et al., 13 Nov 2025). In the Airline HDDM-W case study, false alarms over $\mathcal{H}_N$8 chunks drop from $\mathcal{H}_N$9 to $H(x;x^*,w)$00, mean detection delay from $H(x;x^*,w)$01 steps to $H(x;x^*,w)$02 steps, and overall accuracy rises from $H(x;x^*,w)$03 to $H(x;x^*,w)$04. Across datasets and detectors, DTD-enhanced variants improve accuracy by up to $H(x;x^*,w)$05 points in the case study and typically by $H(x;x^*,w)$06–$H(x;x^*,w)$07 points on average (Lu et al., 13 Nov 2025).

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 $H(x;x^*,w)$08 reaches a stopping boundary: $H(x;x^*,w)$09 The experiments intermixed easy and hard trials in fixed-duration $H(x;x^*,w)$10 blocks. Easy trials had $H(x;x^*,w)$11, $H(x;x^*,w)$12-coin reward/punishment, and a $H(x;x^*,w)$13 penalty on errors; hard trials had $H(x;x^*,w)$14, $H(x;x^*,w)$15-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 (Khodadadi et al., 2016).

The computational comparison covered $H(x;x^*,w)$16 models: baseline models with no learning, heuristic adjustment models, and actor–critic models with constant or time-varying thresholds. Constant thresholds use $H(x;x^*,w)$17; time-varying thresholds use basis expansions or parametric forms, including a Weibull-shaped mean threshold

$H(x;x^*,w)$18

The actor–critic update is based on the temporal-difference error

$H(x;x^*,w)$19

with critic update $H(x;x^*,w)$20 and actor update

$H(x;x^*,w)$21

where $H(x;x^*,w)$22 is the actual threshold sampled on trial $H(x;x^*,w)$23 (Khodadadi et al., 2016).

Model comparison used NLL, $H(x;x^*,w)$24BIC or $H(x;x^*,w)$25AIC, and Variational Bayesian protected exceedance probability. The Weibull-threshold actor–critic model, Model 9 (“RL_V”), was the clear winner with $H(x;x^*,w)$26. Its threshold parameters are updated online, trial by trial, and convergence was typically reached by block $H(x;x^*,w)$27–$H(x;x^*,w)$28. Empirically, for hard trials thresholds declined steeply from approximately $H(x;x^*,w)$29 to near $H(x;x^*,w)$30, accuracy rose from approximately $H(x;x^*,w)$31, and RT fell from approximately $H(x;x^*,w)$32. For easy trials thresholds hovered near approximately $H(x;x^*,w)$33 and declined modestly by approximately $H(x;x^*,w)$34–$H(x;x^*,w)$35; accuracy rose from approximately $H(x;x^*,w)$36, and RT from approximately $H(x;x^*,w)$37. Reward rate improved from approximately $H(x;x^*,w)$38 coins/s in block $H(x;x^*,w)$39 to approximately $H(x;x^*,w)$40 coins/s in block $H(x;x^*,w)$41 (Khodadadi et al., 2016).

A related but conceptually different construction appears in interval timing. Here a bounded drift–diffusion integrator

$H(x;x^*,w)$42

accumulates time toward an absorbing threshold $H(x;x^*,w)$43, resetting to $H(x;x^*,w)$44 when an external event occurs. If the event occurs before threshold, the drift is updated multiplicatively: $H(x;x^*,w)$45 and if threshold is reached too early, $H(x;x^*,w)$46 decays continuously via $H(x;x^*,w)$47 while $H(x;x^*,w)$48 remains pinned at $H(x;x^*,w)$49 (Rivest et al., 2011). With partial adaptation, the update takes the stochastic-approximation form

$H(x;x^*,w)$50

and for $H(x;x^*,w)$51, one obtains

$H(x;x^*,w)$52

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 $H(x;x^*,w)$53, both the mean and standard deviation of decision time become proportional to the interval, yielding Weber’s law for timing (Rivest et al., 2011). 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

$H(x;x^*,w)$54

with $H(x;x^*,w)$55, quadratic dendritic nonlinearity, and a spike emitted whenever there exists $H(x;x^*,w)$56 such that $H(x;x^*,w)$57. The threshold is updated after each training iteration by

$H(x;x^*,w)$58

with $H(x;x^*,w)$59 in the experiments. If false positives exceed false negatives, $H(x;x^*,w)$60 is shifted upward; if false negatives dominate, it is shifted downward (Roy et al., 2015).

This threshold rule is integrated with a morphological learning algorithm inspired by the Tempotron. Misclassified patterns define an error through $H(x;x^*,w)$61, candidate synapses are evaluated by a correlation term derived from

$H(x;x^*,w)$62

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 (Roy et al., 2015). On the $H(x;x^*,w)$63-pattern latency task, adaptive threshold yields approximately $H(x;x^*,w)$64 accuracy versus approximately $H(x;x^*,w)$65 for fixed threshold; on the tactile-sensor benchmark, adaptive threshold gives $H(x;x^*,w)$66 versus $H(x;x^*,w)$67 (Roy et al., 2015).

MPATH generalizes the notion of a learnable threshold through homeostatic internal dynamics. Each neuron tracks a running input mean $H(x;x^*,w)$68, variance $H(x;x^*,w)$69, normalized membrane potential $H(x;x^*,w)$70, and adaptive threshold $H(x;x^*,w)$71, governed by two time constants $H(x;x^*,w)$72 and $H(x;x^*,w)$73, typically with $H(x;x^*,w)$74. The threshold obeys both the continuous relaxation

$H(x;x^*,w)$75

and the discrete update

$H(x;x^*,w)$76

where $H(x;x^*,w)$77 is the normalized deviation of the raw input from the running mean (Hadjiivanov, 2021). Activation is then thresholded through $H(x;x^*,w)$78, 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 $H(x;x^*,w)$79 is presented at every step, with a flash of amplitude approximately $H(x;x^*,w)$80 superimposed every $H(x;x^*,w)$81 steps. Early on, $H(x;x^*,w)$82 remains small and $H(x;x^*,w)$83, suppressing the noise; at a flash, $H(x;x^*,w)$84 jumps, $H(x;x^*,w)$85 drops to $H(x;x^*,w)$86, and the neuron fires. Over $H(x;x^*,w)$87 steps, Hebbian updates carve out filters that respond reliably to the flash pattern (Hadjiivanov, 2021). 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,

$H(x;x^*,w)$88

with spike output

$H(x;x^*,w)$89

but the thresholds do not themselves evolve over time $H(x;x^*,w)$90; they remain the fixed set $H(x;x^*,w)$91 while the membrane potential moves through them (Hao et al., 2024). Temporal adaptation instead enters through the Temporal-Global Information Matrix $H(x;x^*,w)$92, which mixes pre-synaptic spikes across all recent time indices,

$H(x;x^*,w)$93

and through the learned leak factor $H(x;x^*,w)$94 in

$H(x;x^*,w)$95

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 $H(x;x^*,w)$96 on CIFAR-10 with Res-19, $H(x;x^*,w)$97, and $H(x;x^*,w)$98 steps; $H(x;x^*,w)$99 on CIFAR-100 with Res-19, $\theta$00, and $\theta$01 steps; $\theta$02 on ImageNet-200 with VGG-13, $\theta$03, and $\theta$04 steps; and $\theta$05 on DVSCIFAR-10 with Res-18, $\theta$06, and $\theta$07 steps (Hao et al., 2024). In the hybrid conversion setting on CIFAR-100, QCFS with $\theta$08 improves from $\theta$09 to $\theta$10 after LM-HT fine-tuning at $\theta$11 steps, and QCFS with $\theta$12 improves from $\theta$13 to $\theta$14 at $\theta$15 steps (Hao et al., 2024).

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$16 with Jacobian $\theta$17. Global contraction is summarized by the effective rank

$\theta$18

After learning task $\theta$19, only directions tangent to the task-preserving manifold $\theta$20 remain usable without degrading $\theta$21, and the compatible effective rank becomes

$\theta$22

where $\theta$23 is an orthonormal basis of the tangent space $\theta$24 (Okanohara, 8 Feb 2026).

Because learning composes as a semigroup, $\theta$25 and $\theta$26. Singular values and rank are submultiplicative, and therefore $\theta$27: once a direction in the task-preserving subspace is collapsed, no further finite-time learning can revive it (Okanohara, 8 Feb 2026). For a new task $\theta$28, define the restricted Hessian $\theta$29 and its stable rank

$\theta$30

The central threshold criterion is then

$\theta$31

The paper interprets $\theta$32 as a learnable-threshold curve: early in training many future tasks satisfy $\theta$33, but as learning proceeds, the residual compatibility threshold decreases and the set of learnable tasks shrinks (Okanohara, 8 Feb 2026).

A related threshold concept appears in recurrent networks trained by BPTT. There, gating mechanisms induce per-lag, per-neuron effective learning rates

$\theta$34

obtained from first-order expansions of gate-induced Jacobian products, and the aggregate envelope

$\theta$35

Under symmetric $\theta$36-stable gradient noise, the minimal sample size needed to detect a lag-$\theta$37 dependency satisfies

$\theta$38

which leads to the learnability window

$\theta$39

or equivalently $\theta$40 (Livi, 5 Dec 2025).

The theory derives closed-form scaling laws for three decay regimes of $\theta$41. If $\theta$42, then $\theta$43; if $\theta$44, then $\theta$45; and if $\theta$46 with $\theta$47, then $\theta$48 (Livi, 5 Dec 2025). Broader or more heterogeneous gate spectra slow the decay of $\theta$49 and enlarge $\theta$50, 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.