Penalized Spike Accuracy (PSA) in SNNs

• Penalized Spike Accuracy (PSA) is an evaluation metric that combines spike prediction accuracy with a penalty for deviations in spike rates, ensuring outputs are statistically aligned with empirical targets.
• It employs an exponential penalty factor to enforce rate alignment within a specified tolerance, guiding hyperparameter optimization via Bayesian methods.
• PSA is significant in event-driven applications like high-frequency trading, where balancing precision and rate consistency enhances true positive rates and overall model utility.

Penalized Spike Accuracy (PSA) is an evaluation metric and hyperparameter optimization objective designed for spiking neural networks (SNNs), particularly in scenarios where both event prediction precision and output firing-rate alignment with ground-truth are crucial. PSA addresses the limitations of metrics that solely reward spike prediction accuracy but neglect the necessity for matched temporal activity levels, as illustrated in high-frequency financial forecasting applications and other event-driven domains (Ezinwoke et al., 5 Dec 2025).

1. Formal Definition and Motivation

Penalized Spike Accuracy augments standard spike accuracy (SA)—the fraction of correctly predicted “real” spikes over the total predicted spikes—with an explicit penalty for deviations in the overall spike rate when compared to empirical target rates. This design compels SNNs to produce outputs that are both precise and statistically aligned with observed event frequencies, thereby mitigating degenerate solutions such as persistent underactivity or overactivity.

Mathematically, PSA is defined by

$\text{PSA} = \text{SA} \times \text{PenaltyFactor}$

where

$$\text{PenaltyFactor} = \exp(-\max(|\text{SRD}| - \alpha, 0))$$

and

$\text{SRD} = \frac{\text{SR}}{\text{RSR}} - 1.$

Here, SR is the model’s average spike rate, RSR is the empirical real spike rate, and $\alpha$ is a tolerance hyperparameter (typically $\alpha=0.05$). The penalty remains inactive (PenaltyFactor $=1$) for spike-rate deviations within $\pm\alpha$ and decays exponentially for larger mismatches (Ezinwoke et al., 5 Dec 2025).

2. Context: Spike Accuracy and Its Limitations

Conventional Spike Accuracy (SA) measures the precision of event (spike) predictions: $$\text{SA} = \frac{\text{Number of correctly predicted real spikes}}{\text{Total predicted spikes}}$$ While high SA indicates precise prediction of target events, optimizing solely for SA favors policies with infrequent spiking (conservatism), leading to low recall and suppressed responsivity. This is particularly suboptimal in domains such as high-frequency trading (HFT), where event rates exhibit significant structure and both under- and over-spiking entail economic risk. PSA, by construction, penalizes such policies whose output spike rates substantially deviate from the empirical event rate, enforcing realistic network reactivity (Ezinwoke et al., 5 Dec 2025).

3. Implementation in Hyperparameter Optimization

Within the Bayesian Optimization (BO) framework for SNN hyperparameter selection, PSA serves directly as the scalar reward function to be maximized. For each hyperparameter candidate:

1. The SNN is trained (e.g., using unsupervised STDP or supervised backpropagation).
2. Model predictions are evaluated on a validation batch, computing SA, SR, SRD, PenaltyFactor, and PSA.
3. The BO acquisition function uses the PSA score to guide the search.

This ensures that selected hyperparameters yield models which are not just accurate but whose spike output matches the empirical event temporal statistics within a prespecified tolerance. The use of exponential decay for the penalty provides smooth gradients for BO’s surrogate model and strictly discounts candidates with severely misaligned rates (Ezinwoke et al., 5 Dec 2025).

4. Practical Significance in Spiking Neural Networks

PSA is especially relevant in tasks where the actionability or realism of SNN outputs depends on both the precision and activity rate of spikes. In the context of HFT price-spike forecasting, SNNs tuned with pure SA were observed to fire at rates up to $-69\%$ below the real event rate, missing actionable trading opportunities. In contrast, PSA-driven BO produced models within $\pm$5\% of the target rate, simultaneously achieving high true positive rates and practical utility—yielding greater economic returns and improved risk metrics such as Sharpe Ratio and drawdown (Ezinwoke et al., 5 Dec 2025).

More broadly, PSA exemplifies a class of objective functions for SNNs and event-based models where rate-aligned behavior is essential, complementing related approaches in spike-sparse learning (Allred et al., 2020).

5. Comparison with Related Objectives: Penalized Spike Count

While PSA is a metric and optimization target for rate-matched output in prediction tasks, earlier approaches, such as the explicit “spike penalty” regularization in SNN backpropagation, focused on minimizing the total number of spikes during supervised learning. The penalized-spike loss augments the primary error objective by

$$L_{\rm total} = L_{\rm error} + \lambda \sum_{i=1}^N \sum_{t=1}^T S_i(t)$$

where $S_i(t)$ is the binary output of neuron $i$ at time $t$, and $\lambda$ controls the tradeoff between accuracy and spiking sparsity (Allred et al., 2020). Both PSA and spike-penalty methods encode rate awareness, but PSA enforces alignment with a dynamical reference rate via multiplicative penalization in the optimization objective, whereas spike-penalty approaches encourage overall sparsity, optionally via annealing schedules. A plausible implication is that PSA is suited to contexts where under- or overactivity both have task-specific costs, while penalized-spike objectives target energy efficiency or biological plausibility by minimizing average activity.

Metric/Objectives	Key Feature	Typical Use Case
Spike Accuracy (SA)	Precision only	General SNN event prediction
Penalized Spike Accuracy (PSA)	Precision and rate-alignment	Event prediction with rate targets
Penalized-Spike Loss ($L_{spikes}$)	Activity minimization	Spike-sparse SNN training

6. Empirical Evaluation and Comparative Results

In financial event prediction, PSA optimization consistently yielded models whose spike-rate deviation (SRD) was close to zero (e.g., SRD ≈ $-0.01$ and $+0.078$ for two SNN variants), compared to large negative SRD ($-0.69$, $-0.43$) under SA optimization. PSA-tuned models, while sometimes exhibiting marginally lower raw SA, demonstrated substantially higher true positive rates and network activity levels. In backtesting, these models outperformed not only the SA-tuned alternatives but also supervised baselines, achieving total returns up to 76.8% compared to 42.5% for the supervised control (Ezinwoke et al., 5 Dec 2025).

7. Limitations and Prospects

PSA’s main limitation lies in its specificity: the metric presupposes well-defined empirical spike-rate targets, which may not generalize to all domains or tasks. The penalty magnitude and tolerance parameter $\alpha$ require calibration to application context (e.g., risk-sensitive versus energy-constrained environments). The exponential penalty form is selected for smoothness and strictness beyond the tolerance band, but other forms are conceivable. A systematic theoretical paper of PSA's effect on the optimization landscape, especially for online or adaptive SNNs, remains an open direction.

PSA constitutes an instructive example of augmenting standard precision-based metrics with explicit rate-alignment objectives to yield SNNs that are both practically actionable and statistically plausible in event-driven contexts (Ezinwoke et al., 5 Dec 2025, Allred et al., 2020).