Quantized 1D Convolutional Networks

• Quantized 1D convolutional networks are deep learning models that represent weights and activations with low-precision, enabling efficient computation on resource-constrained systems.
• They employ learnable scale parameters, uniform quantization via STE, and staged bitwidth reductions to maintain accuracy parity with full-precision baselines.
• Optimized for edge deployment, these networks leverage integer-only MAC operations and absorb batch normalization and ReLU, achieving significant memory and speed gains (e.g., 16× memory reduction with 94.3% accuracy).

Quantized 1D convolutional networks are deep neural networks composed of one-dimensional convolutional layers in which all parameters, activations, and intermediary computations are represented in low-precision, discretized formats. This quantization substantially reduces hardware resource consumption, particularly memory and multiply-accumulate (MAC) complexity, enabling efficient deployment on edge and embedded systems. Recent developments demonstrate that, when quantization is carefully designed—learning quantization ranges and training for robust convergence—such networks can achieve accuracy parity with full-precision baselines even under noisy hardware environments, and without recourse to higher-precision nonlinearities or batch normalization (Verhoef et al., 2019).

1. Quantization Methodology

Weights and activations in fully quantized 1D convolutional networks are discretized via a uniform quantizer with a learnable scale parameter per layer and per tensor type (weight or activation). Given a target bitwidth $nb$, the number of quantization levels is $n = 2^{(nb-1)} - 1$. The lower clip bound $b$ is set to $-1$ for weights and linear outputs, or $0$ for ReLU-style activations. The quantization function is defined as: $$\mathrm{quantize}(x) = \frac{1}{n}\, \mathrm{round}\bigl(\mathrm{clip}(x,\,b,\,1)\times n\bigr)$$ A learnable scale $s$ is introduced: $$Q(x) = e^{s}\, \mathrm{quantize}\left(\frac{x}{e^{s}}\right)$$ The scale $s$ is optimized during training via back-propagation, with the straight-through estimator (STE) applied to the non-differentiable rounding operation. Full-precision "shadow" copies of the parameters are maintained, while quantized values are used in the forward computations (Verhoef et al., 2019).

2. Low-Precision 1D Convolutions and Hardware Considerations

The core operation in a 1D convolutional network,

$y[n] = \sum_{k=0}^{K-1} W[k] \cdot x[n-k],$

becomes fully quantized as

$$y_q[n] = \frac{e^{s_w}e^{s_x}}{n_wn_x} \sum_{k} w_k^{\mathrm{int}} a_{n-k}^{\mathrm{int}}$$

where $w_k^{\mathrm{int}}$ and $a_{n-k}^{\mathrm{int}}$ are integer representations derived from quantized weights and activations, respectively. The core dot-product is thus evaluated using only integer MAC operations. For ternary weights $\{-1,0,1\}$, multiplication can be replaced with addition and subtraction, requiring no multipliers. The global scaling factor can be merged into input/output ADC/DACs or implemented via a final lookup table. Between layers, tensors are represented by low-precision integers only, eliminating the necessity for floating-point batch normalization (BN) or nonlinearities in hardware (Verhoef et al., 2019).

3. Gradual Quantization and Training Procedures

Training under quantization constraints employs a gradual quantization process. Networks are first trained in full precision, followed by staged reductions in bitwidth, e.g., [(8,8), (6,6), (4,4), (2,4)] for weights and activations, respectively. At each stage, a student model with the new lower bitwidth is initialized from the higher-precision teacher, and trained (optionally with distillation) to minimize a joint cross-entropy and distillation loss:

• Quantization parameters are set according to the current schedule.
• STE is used for back-propagation through quantization.
• At each stage, only bitwidths are lowered. After the lowest target precision is reached, batch normalization and nonlinearities can be absorbed and replaced, followed by final fine-tuning (Verhoef et al., 2019).

4. Elimination of Batch Normalization and Nonlinearities

Quantized 1D convolutional networks remove dependence on higher-precision batch normalization and ReLU using two procedures:

• BatchNorm Absorption: Given a trained BN layer, scale coefficients can be absorbed into the quantization scale, dropping any additive bias, which is re-learned during fine-tuning.
• ReLU Replacement: The quantizer with $b=0$ closely approximates the ReLU operation for large $nb$. This enables BN and ReLU to be replaced with a single quantization layer and calibration of its scale (Verhoef et al., 2019).

5. Robustness to Analog and Quantization Noise

Additive Gaussian noise is injected at the weight, activation, and MAC computation levels to model analog hardware variability. Small noise levels (≤5% LSB) have negligible impact on accuracy; higher levels (20–30% LSB) can be mitigated by training with ongoing noise injections ("noise-aware training")—minimizing expected loss over these perturbations. This approach can recover nearly all accuracy lost to noise effects (Verhoef et al., 2019).

6. Application to 1D Convolutional Architectures

All quantization procedures extend directly to 1D convolutional layers of arbitrary width or dilation, as commonly found in speech and time-series architectures. The quantization is independent of convolutional architecture specifics. On edge accelerators, only integer MAC operations and SRAM are required for weights (ternary or small-integer), and scaling factors may be consolidated into analog circuitry or lookup tables. No floating-point operations are necessary in deployment (Verhoef et al., 2019).

7. Empirical Resource-Accuracy Trade-offs

Evaluation on 1D keyword-spotting networks (e.g., 7-layer, 3.5M MACs) quantized with varying bitwidths demonstrates the following accuracy trade-offs:

• (8 w/8 a): Matches or exceeds full-precision baseline (94.3% → 94.4%)
• (6 w/6 a): No measurable accuracy drop
• (4 w/5 a): Within 0.1% of full-precision accuracy
• (2 w/4 a) ternary weights: ≈94.3%, on par with full precision

Reductions in bitwidth effect major gains in memory and computational efficiency:

• 8 bit quantization yields 4× memory savings; 2 bit yields 16×
• Integer-only MAC operations provide 2–4× speedup over FP32 on DSPs or MCUs (Verhoef et al., 2019)

The result is a fully quantized 1D ConvNet architecture with layer-wise learned quantizers, integer-only arithmetic, no higher-precision normalization or activation dependencies, robust performance under hardware-induced noise, and empirically validated efficiency-accuracy parity with full-precision reference models (Verhoef et al., 2019).