Activation Variability Score Overview

• Activation Variability Score is a scalar metric that quantifies the spread or stability of neural activations across different layers and experimental runs.
• It utilizes methods such as exponential smoothing, variance normalization, and covariance analysis to inform learning rate adaptation and model pruning.
• AVS plays a crucial role in improving model interpretability, reproducibility in fMRI studies, and neural architecture search by guiding effective model tuning.

An Activation Variability Score (AVS) quantifies the dispersion or stability of neural activations over samples, layers, or experimental runs. AVS has emerged as a critical metric in modern deep learning optimization, interpretability, fMRI reproducibility analysis, model pruning, concept-based explainability, and neural architecture search. This article presents a detailed account of AVS definitions, computational methodologies, domain-specific applications, empirical impacts, and limitations, referencing major frameworks where the concept plays a central role.

1. Formal Definitions and Core Variants

The term "Activation Variability Score" encapsulates several related, but context-specific, formulations. All share the goal of quantifying, in scalar form, the spread or inconsistency of activations:

Neural Optimization and AdaAct: The core AVS in AdaAct is the per-neuron, exponentially-smoothed second moment of the (augmented) activation vector across a mini-batch:

• Let $\tilde a^{(\ell-1)}_{i,j}$ be the $j$-th activation of layer $\ell-1$ for sample $i$.
• The score is defined as

$$\widehat v_{t,j} = \frac{V_t^{(\ell-1)}}{1 - \beta_2^t}$$

where $$V_t^{(\ell-1)} = \beta_2 V_{t-1}^{(\ell-1)} + (1-\beta_2)\tilde A_t^{(\ell-1)}$$ and $$(\tilde A_t^{(\ell-1)})_{jj} = \frac{1}{|\mathcal{B}_t|}\sum_{i \in \mathcal{B}_t}(\tilde a^{(\ell-1)}_{i,j})^2$$ (Seung et al., 10 Jun 2025).

Layer Importance and AVSS in LLMs: For a model with $M$ layers, AVSS for each layer $L_i$ combines normalized variance and normalized sparsity:

• $\mathrm{AVSS}(L_i) = \frac{\sigma^2(L_i)}{S(L_i)}$, where $\sigma^2(L_i)$ is the variance of activations and $S(L_i)$ is the fraction below a threshold $\epsilon$ (Song et al., 15 Nov 2024).

Covariance-Based AVS for Architecture Search: For a set of post-ReLU activations $A(x_i)$, AVS can be defined via the log-determinant of the sample covariance matrix:

• $$\mathrm{AVS}_\lambda = \log\det(\Sigma) + \lambda \log(N_A)$$, where $\Sigma$ is the empirical activation covariance (Liu et al., 2022).

Reproducibility in fMRI: In multi-study fMRI, AVS is the complement of the largest-eigenvalue-based Jaccard overlap $\mathrm{AVS} = 1 - R$, with $R$ the reliability index over $M$ activation maps (Maitra, 2022).

Concept Activation Vectors (CAVs): The variability of CAVs is the total variance of the CAV coefficients across random draws of reference examples:

• $$\mathrm{Var}(\beta_N) = \sum_{k=1}^d \mathbb{E}[(\beta_{N,k} - \mathbb{E}[\beta_{N,k}])^2]$$ (Wenkmann et al., 28 Sep 2025).

2. Computational Methodologies

The computation of AVS depends on context, but follows a few general principles:

• Batch Aggregation: Activations are aggregated over a mini-batch or multiple runs to estimate second moments, mean, and sparsity.
• Exponential Smoothing: In AdaAct, a running exponential moving average with bias correction produces the smoothed AVS, aligning with Adam's second-moment estimator.
• Normalization: For model-wide comparison (e.g., AVSS), layer-wise statistics are normalized to allow cross-layer ranking and network pruning (Song et al., 15 Nov 2024).
• Covariance/Log-Determinant: In architecture search and continuous analogues, AVS is computed as the log-determinant of the covariance matrix of activations, quantifying the "spread" in activation space (Liu et al., 2022).
• Mixture Models: In fMRI, a two-component mixture model is fit for the distribution of $p$-values (null: uniform, alternative: noncentral-$t$), and AVS is derived from posterior probabilities of classifying a voxel as activated or not (Maitra, 2022).
• Empirical Variance Estimation: For CAVs, AVS is empirically estimated via repeated resampling, fitting, and covariance computation (Wenkmann et al., 28 Sep 2025).

3. Applications Across Domains

Deep Learning Optimization: AdaAct

AdaAct leverages AVS to adapt per-neuron learning rates. High activation variability leads to reduced step size, mitigating instability and promoting generalization. By integrating the per-neuron AVS into the gradient update, AdaAct achieves a balance between fast early-stage convergence and late-stage generalization, outperforming Adam in final accuracy and matching or exceeding SGD in many scenarios (Seung et al., 10 Jun 2025).

Model Pruning and Layer Selection in LLMs

AVSS provides a straightforward ranking of layer importance by quantifying both activity (variance) and inactivity (sparsity). Layers with low AVSS are often redundant and can be pruned with minimal loss in performance. The method enables pruning up to 25% of layers in LLMs while retaining over 90% of task-specific accuracy, attesting to over-parameterization in current architectures (Song et al., 15 Nov 2024).

Hallucination and Robustness Mitigation

The enhanced AVSS (EAVSS) combines standard AVSS with hallucination-specific activation metrics, identifying and regularizing hallucination-prone layers. Contrastive learning is then selectively applied to these layers, effectively reducing hallucination propensity (e.g., ∼12% improvement in certain QA benchmarks), and improving calibration metrics such as ECE and Brier score (Song et al., 15 Nov 2024).

Functional Brain Imaging and Reproducibility

In fMRI, AVS measures variability (or, equivalently, unreliability) across activation maps from repeated experiments. The global AVS quantifies overall reproducibility, while jackknife-based outlier detection can flag anomalous sessions or runs, aiding the identification of outlier activation maps (Maitra, 2022).

Neural Architecture Search and Early Evaluation

AVS, as an extension of the Distinguishing Activation Score, enables rapid, zero-cost architectural pre-selection by summarizing how richly a network's early activations partition a mini-batch. Both the "spread" (determinant of the activation covariance) and total number of units are considered, yielding strong correlation to downstream accuracy when calibrated correctly (Liu et al., 2022).

Concept-Based Model Interpretability

For concept activation methods (CAVs), the AVS offers a principled means to quantify and control the randomness-induced uncertainty in interpretation vectors. Theoretical analysis demonstrates the $O(1/N)$ decay of AVS with the number of reference samples, offering recipe-like guidance on required sample sizes for prescribed precision in concept-based explainability studies (Wenkmann et al., 28 Sep 2025).

4. Empirical Impact and Best Practices

Implementing and tuning AVS-based metrics requires careful methodological choices:

• Exponent Selection: In AdaAct, using the square-root of AVS ($p=0.5$) yields superior balance between convergence and stability, compared to full variance scaling (Seung et al., 10 Jun 2025).
• Normalization for Ranking: For pruning, normalization across layers/batches prevents scale artifacts and ensures robust importance rankings (Song et al., 15 Nov 2024).
• Batch Size and Reference Sampling: In CAVs, achieving low AVS (stable concept vectors) depends crucially on the number of random references; empirical fits of $\mathrm{Var}(\beta_N)\approx a/N + b$ guide dataset and experimental design (Wenkmann et al., 28 Sep 2025).
• Regularization and Stability: Strong regularization or activation clipping is generally not necessary when the AVS or EAVSS metric is inherently stabilized by design (e.g., AdaAct's EMA loop) (Seung et al., 10 Jun 2025).
• Computational Efficiency: Restricting to diagonal covariances suffices in most optimization contexts and matches the memory footprint of Adam; covariance determinants are used only in architecture evaluation (Seung et al., 10 Jun 2025, Liu et al., 2022).

5. Limitations, Interpretational Caveats, and Extensions

Several caveats affect the utility and generalizability of AVS:

• The exact statistical efficiency of AVS-based metrics can depend on model architecture, activation regularity, and batch size.
• In fMRI, the AVS is sensitive to thresholding protocols and spatial smoothing, and highly sparse activations may artifactually inflate variability estimates (Maitra, 2022).
• Asymptotic properties (e.g., $O(1/N)$ scaling for CAVs) may not hold in small-sample or highly non-Gaussian settings (Wenkmann et al., 28 Sep 2025).
• Kernel-based AVS formulations in architecture search omit deeper task-specific dependencies, and decoupling measures must be carefully tuned to avoid misleading rankings (Liu et al., 2022).
• For concept-based explanations, downstream interpretability scores (such as average sensitivity) may retain $\Theta(1)$ variance due to borderline inputs, requiring tailored variance reduction beyond naive sample averaging (Wenkmann et al., 28 Sep 2025).

6. Comparative Summary Table

Context	AVS Definition/Formula	Function
Deep nets (AdaAct)	$\widehat v_{t,j}$: neuron EMA of squared activations	Learning-rate adaptation
LLMs (AVSS/EAVSS)	$\frac{\sigma^2(L_i)}{S(L_i)}$, optionally hallucination-aware	Layer ranking/pruning
fMRI reproducibility	$1 - (λ_{(1)}-1)/(M-1)$ (Jaccard eigenvalue overlap)	Between-map variability
NAS/arch search	$\log\det(\Sigma) + \lambda \log(N_A)$	Early architecture screening
Concept Activation Vectors	$\mathrm{Var}(\beta_N)$, total CAV variance over random references	Interpretability reliability

These AVS definitions are tailored to each application, but all share the objective of quantifying and leveraging neural activation spread for model selection, stability, reliability, and interpretability in modern AI systems.