Weight-Activation Gap in Neural Networks

• The Weight-Activation Gap is a discrepancy between weight statistics and activation behavior that impairs signal propagation, convergence, and quantization accuracy in neural networks.
• Techniques like WeightAlign, Linearly Constrained Weights, and learnable scaling are employed to stabilize initialization and mitigate vanishing or exploding gradients.
• Reducing the gap enhances both model performance and hardware efficiency, as validated by improvements in deep network benchmarks and low-bit quantization experiments.

The weight-activation gap refers to the discrepancy or mismatch between the statistical properties, information propagation, robustness to quantization, or computational contributions of neural network weights versus activations. It emerges across domains: from theoretical signal propagation, practical initialization, normalization and centering, to low-bit quantization for efficient deployment, and even in the mechanistic interpretability of learned feature spaces. The weight-activation gap often manifests as instability, vanishing/exploding gradients, information loss, convergence slowdowns, accuracy drops after quantization, or direct impacts on hardware throughput, depending on context. Addressing this gap at the architectural, algorithmic, or hardware level is crucial for the trainability, robustness, and efficient deployment of modern neural networks.

1. Mathematical and Statistical Origins of the Weight–Activation Gap

Sources of the weight-activation gap are fundamentally statistical. In both fully-connected and convolutional layers, the ideal scenario is for linear pre-activations $a = w^\top x + b$ to have zero mean and uniform variance across layers, consistent with stable signal propagation. However, the mean and variance of $w$ are generally neither matched to, nor sufficient to ensure, the corresponding properties for $a$. As a result, $E[a]\neq0$ and $\mathrm{Var}[a]$ varies across layers unless additional normalizations, constraints, or specific initializations are applied.

This gap is exacerbated in deep networks, or where sample-based normalization (e.g., BatchNorm) is unavailable or unreliable—such as in small-batch regimes or in specialized architectures. In those cases, unmatched and drifting activation statistics induce internal covariate shift, exponential vanishing or exploding of signals and gradients, and impede convergence (Shi et al., 2020, Hayou et al., 2018, Kumar, 2017).

In quantized regimes, the statistical gap is more operational: weight tensors $W$ typically have bell-shaped distributions and limited dynamic ranges, while activation tensors $X$ often have channel-wise or token-wise outliers, heavy tails, or dynamic range explosion. Quantizing $X$ introduces much larger errors (and accuracy drops) than quantizing $W$ to similar bit-widths (Li et al., 2023, Lee et al., 2023, Huang et al., 2024, Zhou et al., 29 Aug 2025, Choi et al., 2018).

2. Effects on Initialization, Signal Propagation, and Deep Trainability

Proper initialization critically determines the extent of the weight-activation gap during the first forward pass and throughout training. If weights are not selected carefully with respect to non-linearity and network depth, variance at each layer can exponentially shrink (vanishing signal) or grow (exploding signal). For activations $g$ differentiable at $0$, closing the gap so that $$\mathrm{Var}[x_{m+1}] \approx \mathrm{Var}[x_m] \approx 1$$ requires a specific scaling:

$v^2 = \frac{1}{N (g'(0))^2 (1+g(0)^2)}$

where $v^2$ is the variance for weight initialization, $N$ is fan-in, $g$ is the activation function (Kumar, 2017).

For ReLU networks, the correct initialization (He) is $v^2 = 2/N$, rather than the classic Xavier formula; otherwise signal variance halves at each layer. With improper scaling, deep networks become untrainable as signal or gradient norms collapse or explode (Kumar, 2017, Hayou et al., 2018).

Theoretical developments using infinite-width mean-field theory formalize the weight-activation gap in terms of the variance ($q^\ell$) and correlation ($c_{ab}^\ell$) propagation maps across layers. The critical point (“edge of chaos”) in $(\sigma_w, \sigma_b)$ hyperparameter space, defined by $\chi_1 = 1$, is where this gap closes—allowing deep information propagation without exponential attenuation (Hayou et al., 2018).

3. Normalization, Centering, and Practical Mechanisms to Close the Gap

Techniques for reducing or closing the weight-activation gap fall into several categories:

• WeightAlign: This method centers and scales each filter or output channel’s weights to enforce per-filter $E[w]=0$ and $\mathrm{Var}[w]=1$, using

$\hat w = \gamma \frac{w-\mu_w}{\sigma_w}$

where $\gamma$ is a learnable scale. This ensures pre-activations are zero-mean and uniformly scaled, independent of batch or input statistics. When stacked with BatchNorm, LayerNorm, GroupNorm, or InstanceNorm, this yields improved stability—especially in small-batch regimes—by closing the expected statistical gap between weight and activation domains (Shi et al., 2020).

• Linearly Constrained Weights (LCW): By explicitly constraining $w^\top \mu = 0$ (where $\mu$ is the mean of incoming activations), the systematic activation shift $E[w^\top a]= \|w\|\|\mu\| \cos\theta$ (for angle $\theta$ between $w$ and $\mu$) is removed. This enforces that every neuron's pre-activation mean is zero regardless of weights’ orientation, stabilizing deeper training and aligning forward and backward variance amplification properties—most notably rescuing deep sigmoid networks from vanishing gradients (Kutsuna, 2024).
• Learnable activation or weight scaling: Techniques like PACT (learned activation clipping for optimal dynamic range before quantization) and SAWB (closed-form, statistics-matched per-layer quantizer scaling for weights) target the gap from each side in quantized neural networks (Choi et al., 2018).
• Edge-of-chaos initialization and criticality: By choosing $(\sigma_w, \sigma_b)$ parameters so that the correlation map $f$ is nearly the identity (i.e., $\chi_1=1$), forward-propagated signal variances and correlation depth-scales remain nearly constant, and the gap between weight-implied and activation-realized statistics is minimized (Hayou et al., 2018).

4. Impacts on Quantization and Hardware/Firmware Efficiency

The gap between weight and activation distributions is a central concern for low-bit quantization. In practical LLM and DNN deployment, activation quantization is much more error-prone and causes greater accuracy loss than comparable weight quantization. The main sources are:

• Outlier or heavy-tailed activation channels that dictate a coarse global quantization step, wasting grid points and inflating quantization noise for the bulk of values (Li et al., 2023, Huang et al., 2024).
• Poorly aligned dynamic ranges or scaling between $W$ and $X$, leading to a concentration of quantization error on either side, or both (Lin et al., 2023).

Strategies developed to bridge the gap include:

• Activation-Weight Equalization (AWEQ): Utilizes per-channel diagonal scaling so that weights and activations occupy a matched range after transformation. The transform

$$X \rightarrow X \operatorname{diag}(s)^{-1},\quad W \rightarrow \operatorname{diag}(s)W,\quad s_i = 1/ ( r_i^{(W)} \sqrt{ r_i^{(X)} r_i^{(W)} } )$$

ensures both quantizers allocate bits meaningfully, yielding balanced quantization noise and eliminating systematic bias (Li et al., 2023).

• Activation-Quantization-Aware Scaling (AQAS): Simultaneously optimizes channel-wise scales for both weights and activations to minimize total output distortion, preventing mismatch-induced error amplification (Lee et al., 2023).
• Rotation-based outlier elimination (RoLoRA): Applies random or structured orthonormal rotations to re-distribute activation and weight statistics so that both become outlier-free—permitting robust 4-6 bit quantization of both operands in LoRA and multimodal models, nearly matching weight-only benchmarks (Huang et al., 2024).
• Binary Weight, Multi-Bit Activation Quantization (BWMA): In memory-constrained accelerators (CIM), moment-matching binary weight quantization combined with differentiable multi-bit activation quantizers achieves accuracy nearly equivalent to full-precision, showing that optimal allocation of representational "budget" sharply reduces this gap (Zhou et al., 29 Aug 2025).
• Hardware Exploitation and the Gap: In certain accelerators, such as Loom, convolutional layers benefit from reductions in both weight and activation precision ($T_\text{conv} \propto 1/(P_w P_a)$), while fully connected layers accelerate only with reduced weight precision ($T_\text{fc} \propto 1/P_w$). Thus, there is a throughput gap—i.e., an extra factor of $P_a$ in convolutional acceleration due to this differential sensitivity (Sharify et al., 2017). Per-layer and fine-grained precision allocation narrows this hardware gap.

5. Quantitative Empirical Evidence and Benchmarks

Closing the weight-activation gap confers measurable improvements:

• Normalization in Small Batches: WeightAlign maintains constant error across batch sizes (down to 1) for CIFAR/ImageNet tasks and improves top-1 accuracy on ImageNet by up to 0.8% over BatchNorm alone (Shi et al., 2020).
• Initialization: For ReLU, the correct $v^2=2/N$ initialization avoids exponential variance decay, aligning actual activation statistics with theoretical desiderata and stabilizing very deep nets (Kumar, 2017, Hayou et al., 2018).
• Quantization in LLMs: AWEQ, AQAS, RoLoRA, and related approaches reduce the perplexity/accuracy drop (the quantization gap) of 4-bit weight-activation quantization to within 0.5-1 PPL or <1% accuracy on massive models (e.g., LLaMA, OPT-175B) (Li et al., 2023, Lee et al., 2023, Huang et al., 2024). RoLoRA, in particular, closes accuracy gaps up to 29.5% absolute on LLaMA2-13B commonsense reasoning and up to 18.9% on multimodal LLaVA models (Huang et al., 2024).
• Energy and Latency: In hardware, bit-serial designs like Loom and compute-in-memory accelerators leveraging BWMA realize speedups and energy efficiency improvements of 3.5–4.4x over baselines, directly proportional to reductions in $P_a$ and $P_w$ (Sharify et al., 2017, Zhou et al., 29 Aug 2025).
• Mechanistic Interpretability: The Signed Quadratic Shrink (SQS) activation preserves the spectra of bilinear weight matrices so that features extracted from weights (via eigendecomposition) match those observed in activations, as measured by cosine similarities $\geq 0.95$ for top eigenvectors—effectively closing the interpretability gap between weights and activations (Abohwo et al., 2 Sep 2025).

6. Limitations, Open Questions, and Future Directions

Several critical assumptions and limitations persist across approaches:

• Assumptions of independence between $w$ and $x$, or of well-behaved activation distributions, may not hold in all architectures or data regimes. Highly skewed, correlated, or multimodal activations can undermine weight-based normalization or centering (Shi et al., 2020, Kutsuna, 2024).
• Most practical methods rely on hyperparameters (e.g., the $\epsilon$ in normalization, scaling exponents, rotation size) or model design choices (LoRA rank, block size). Hyperparameter robustness and theoretical guidance are active areas of investigation (Shi et al., 2020, Huang et al., 2024, Abohwo et al., 2 Sep 2025).
• For quantization methods, careful matching of calibration and task sequence lengths (as in SLAC), and dynamic or adaptive correction for non-stationary input statistics, remain important challenges (Lee et al., 2023, Huang et al., 2024).
• For large-scale models and high-dimensional layers (transformers, multimodal architectures), the computational cost of full spectral methods, per-channel statistics collection, or per-layer optimization may motivate approximations or co-design with hardware primitives (Zhou et al., 29 Aug 2025, Sharify et al., 2017, Abohwo et al., 2 Sep 2025).

Compositional approaches that jointly optimize weights, activations, initialization, normalization, and quantization are likely to further close the practical weight-activation gap, both in terms of model performance and hardware efficiency.

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Key References:

• WeightAlign: Normalizing Activations by Weight Alignment (Shi et al., 2020)
• On the Selection of Initialization and Activation Function for Deep Neural Networks (Hayou et al., 2018)
• On weight initialization in deep neural networks (Kumar, 2017)
• Linearly Constrained Weights: Reducing Activation Shift for Faster Training of Neural Networks (Kutsuna, 2024)
• AWEQ: Post-Training Quantization with Activation-Weight Equalization for LLMs (Li et al., 2023)
• Enhancing Computation Efficiency in LLMs through Weight and Activation Quantization (Lee et al., 2023)
• RoLoRA: Fine-tuning Rotated Outlier-free LLMs for Effective Weight-Activation Quantization (Huang et al., 2024)
• Loom: Exploiting Weight and Activation Precisions to Accelerate Convolutional Neural Networks (Sharify et al., 2017)
• Binary Weight Multi-Bit Activation Quantization for Compute-in-Memory CNN Accelerators (Zhou et al., 29 Aug 2025)
• Bridging the Accuracy Gap for 2-bit Quantized Neural Networks (QNN) (Choi et al., 2018)
• Preserving Bilinear Weight Spectra with a Signed and Shrunk Quadratic Activation Function (Abohwo et al., 2 Sep 2025)
• AWQ: Activation-aware Weight Quantization for LLM Compression and Acceleration (Lin et al., 2023)