Layer Importance Scores in Deep Networks

• Layer Importance Scores are scalar metrics that quantify the contribution of individual layers to overall model performance and interpretability.
• They integrate diverse methodologies—such as attribution, sensitivity analysis, and game-theoretic sampling—to assess layer-specific impacts in various neural architectures.
• These scores drive practical applications including model pruning, compression, adaptation, and enhanced interpretability by guiding resource allocation and optimizing performance.

Layer Importance Scores quantify the contribution of individual layers or submodules within deep neural networks, capturing their roles in overall model performance, interpretability, and computational efficiency. These scores are critical in diverse settings, including biological networks, LLMs, neural compression, pruning, task adaptation, and interpretability. Their computation and aggregation span attribution methods, reward-based learning, clustering, sensitivity analysis, and cooperative game theory. The following sections detail key formalisms, methodologies, empirical findings, applications, and implications.

1. Formal Definitions and Core Methodologies

Layer importance scoring methodologies are context-dependent but share the goal of condensing rich, layer-specific signals into scalar values reflecting performance impact or interpretive relevance.

• Data-driven centrality aggregation: In the context of biological networks, multi-layer refined models (MR-PIN) compute protein node importance as the geometric mean across layer-specific normalized centrality scores (e.g., degree, PageRank, clustering) (Wang et al., 2023). Given $K$ refined layers and scores $s_i^{(k)}$, overall importance is $S_i = [\prod_{k=1}^K s_i^{(k)}]^{1/K}$.
• Influence-function-based estimation: LayerIF computes the sensitivity of validation loss to training data via layer-restricted influence functions, isolating gradients and Hessians for each layer (Askari et al., 27 May 2025). Layer importance $S^{(\ell)}$ for layer $\ell$ aggregates positive influences $I^{(\ell)}(z_i)$ from training examples, highlighting layers undertrained for the task.
• Game-theoretic attribution: Shapley-value-based approaches treat each layer as a player in a cooperative game; the value $\phi_\ell$ measures the marginal contribution of layer $\ell$ by evaluating model performance under all possible layer subsets (Zhang et al., 2024). Efficient sampling (early truncation, neighborhood sampling) makes computation tractable.
• Attention-generated scores: LAYA computes layer contribution via input-dependent attention over projected layer features, with softmax-normalized weights $\alpha_i(x)$ providing per-sample attribution (Vessio, 16 Nov 2025).
• Bandit-style reward learning: IST updates a trainable importance vector $I = (I_1, ..., I_{N_L})$ by measuring reward signals when subsets of layers are activated during fine-tuning. This non-gradient-based method accumulates utility over time (Yao et al., 2024).
• Activation statistics: The AVSS metric combines normalized activation variance $\sigma^2(\ell)$ and sparsity $S(\ell)$, scoring each layer via $\mathrm{AVSS}(\ell) = \sigma^2(\ell)/S(\ell)$; non-essential layers (with low AVSS) can be pruned with minimal performance impact (Song et al., 2024, Song et al., 2024).
• Sensitivity-based ranking: SAfER measures the expected accuracy drop ($\Delta A_\ell$) upon layerwise perturbation (noise, dropout), providing a direct estimate of performance sensitivity and guiding resource allocation (Yvinec et al., 2023).

2. Aggregation, Normalization, and Sampling Techniques

Layer importance scores often require normalization across layers and aggregation methods sensitive to context or downstream application:

• Geometric mean penalizes inconsistent performance across layers and raises reliability over arithmetic mean for multi-view biological PIN scoring (Wang et al., 2023).
• Normalization and standardization: Scores are min-max normalized, standardized (zero mean, unit variance), or $\ell_1$-normalized to enable fair comparison across varying parameter or activation scales (Zhang et al., 30 Sep 2025, Ikeda et al., 25 Aug 2025).
• Weighted chunking in model merging: Expert Merging++ combines per-layer learned coefficients, task vector magnitudes, and parameter counts into a unified normalized importance metric. Chunk allocation for merging is proportional to $I_\ell$ (Zhang et al., 30 Sep 2025).
• Sampling for tractability: Shapley-value estimation leverages early truncation and neighborhood sampling, substantially reducing exponential complexity by ignoring negligible contributions and focusing on local layer interactions (Zhang et al., 2024).
• Voting and ranking aggregation: In influence estimation, positional voting and aggregated ranks are superior to mean averaging for filtering noisy training data, providing robust layerwise discrimination without domination by outlier samples (Vitel et al., 6 Nov 2025).

3. Experimental Validation and Quantitative Findings

Rigorous empirical evaluation is central to layer importance research:

• Biological PINs: MR-PIN achieves a recall boost of 8.94%–31.61% and ROCAUC/PRAUC increases up to 21.9%/65% compared to earlier refinements. Orthology-based layers contributed most, gene expression the least (Wang et al., 2023).
• LLM specialization: LayerIF shows low-level layers peak for syntax (CoLA), middle layers for NLI/sentence-pair tasks, upper layers for commonsense reasoning. Layerwise allocation in LoRA-MoE yields 1.61% accuracy gain over uniform baselines (Askari et al., 27 May 2025).
• Cornerstone layers in LLMs: Ablation guided by Shapley values reveals that removing top-ranked layers collapses accuracy to random-guess levels (~29–37% drop), whereas non-cornerstone removal has marginal effect. Cornerstone layers are typically early, especially as model size increases (Zhang et al., 2024).
• Pre-training stability: Layerwise ablation for math reasoning in Qwen-2.5-7B and Llama-3.1-8B shows critical layers that are invariant under instruction tuning, RL, or distillation—removal drops math accuracy by 60–85% (Nepal et al., 27 Jun 2025).
• Adaptive attention aggregation: LAYA consistently matches or slightly improves baseline accuracy (+1 pp), and learned $\alpha$ profiles accurately reflect class- and sample-dependent abstraction needs (Vessio, 16 Nov 2025).
• Compression, quantization, and pruning: AVSS and related activation-based metrics allow removal of ~25% lowest-scoring layers while retaining >90% performance in multitask LLMs. In quantization, learned scale factors directly map sensitivity to bit-width assignments, achieving state-of-the-art efficiency on ResNet/ImageNet with ILP-driven search (Tang et al., 2022, Song et al., 2024, Song et al., 2024).

4. Applications: Compression, Adaptation, Interpretability

Layer importance scores drive diverse downstream applications:

• Model pruning/platform adaptation: Non-essential layers identified by AVSS or Shapley analysis can be directly pruned, reducing inference FLOPs and memory (15–25%) with negligible accuracy cost. NISP propagates neuron importance backward to enable closed-form channel selection in CNNs (Yu et al., 2017, Song et al., 2024).
• Sparse fine-tuning/PEFT: IST updates only 20–30% most valuable layers, cutting memory by 10–15 GB across 7B-parameter LLMs and raising accuracy by 1–2 points, with convergence supported by VC-dimension and Taylor-expansion arguments (Yao et al., 2024).
• Expert merging: Importance-guided chunk allocation via $I_\ell$ in Expert Merging++ offers parameter-efficient multi-domain capability, outperforming both layerwise and uniform chunking strategies in LLM and multimodal MLLM merging (Zhang et al., 30 Sep 2025).
• Hallucination mitigation: Enhanced variance–sparsity scores (EAVSS) identify hallucination-prone layers; targeted contrastive learning on these submodules reduces calibration error and Brier score by up to 12% (Song et al., 2024).
• Interpretability: Aggregated layerwise attention (LAYA) and bias-aware, multi-layer fusion (Integrative CAM) yield interpretable attribution heatmaps and reveal sample-specific depth usage, outperforming conventional CAM methods in visual recognition IoU (Vessio, 16 Nov 2025, Singh et al., 2024).
• Data influence and auditing: Influence aggregation across layers is most reliable in early–middle attention blocks (for large LLMs) with positional voting, outperforming cancellation-based metrics for noisy data detection (Vitel et al., 6 Nov 2025).

5. Comparative Analysis and Theoretical Insights

Comparative studies elucidate methodology efficacy and limitations:

• Aggregated vs. single-layer analysis: Geometric mean fusion for node importance penalizes missingness in any layer, while arithmetic mean may obscure unreliable (spurious) edges, particularly in biological networks (Wang et al., 2023). Influence estimation via mean averaging is often suboptimal relative to rank or vote-based aggregation in LLM data auditing (Vitel et al., 6 Nov 2025).
• Location of critical layers: Multiple studies independently confirm the elevated functional importance of early and middle layers across architectures, especially for foundational knowledge, reasoning, or representation transformation (Zhang et al., 2024, Nepal et al., 27 Jun 2025, Song et al., 2024, Ikeda et al., 25 Aug 2025).
• Sampling and estimation bias: Shapley value approximations based on early truncation sampling may overestimate early-layer contributions, while neighborhood windows mitigate distant interaction underestimation (Zhang et al., 2024).
• Attribution stability: Layerwise normalized importance (e.g., $L_1/L_2$ rescaling) mitigates layer collapse and preserves diversity in sparse solution masks (as in Lottery Ticket Hypothesis) (Vandersmissen et al., 2023).
• Competition and redundancy: Activation variance–sparsity frameworks reveal that deep stacks in LLMs and CNNs contain substantial redundancy, with the lowest-scoring quartile removable without performance loss (Song et al., 2024, Song et al., 2024, Tang et al., 2022).

6. Future Directions, Limitations, and Research Trends

Current limitations and opportunities for development include:

• Interaction effects: Most importance measurement protocols ignore higher-order synergies or antagonisms between layer groups; extensions to multi-layer Shapley analysis or influence co-aggregation are warranted (Zhang et al., 2024).
• Generalization across tasks: Task-dependent variability in layer importance (e.g., seen in LayerIF) suggests a need for adaptive, context-sensitive scoring pipelines, especially for zero-shot or domain-shifted applications (Askari et al., 27 May 2025).
• Mechanistic understanding: Despite advances, mechanistic links between importance scores and functional subcircuits (e.g., logic gates, compositional patterns) remain underexplored. Attribution analyses focused on “cornerstone” or high-HCS layers may elucidate specific operations (Nepal et al., 27 Jun 2025, Song et al., 2024).
• Calibration and hallucination: Detailed hallucination-propensity scoring and targeted loss design offer promising routes for robust model deployment in safety-critical settings (Song et al., 2024).
• Computational scalability: Efficient sampling, optimization (ILP, bandit updates), and reduction techniques mitigate traditional prohibitive costs, pointing toward scalable layerwise adaptation in ever-larger models (Tang et al., 2022, Yao et al., 2024, Zhang et al., 2024).

Layer importance scores thus constitute a foundational tool for model optimization, resource allocation, interpretability, adaptation, and error mitigation, with validated methodologies and significant impact across biological, language, and vision domains.