MS-Q2P: Mixed-Precision Bit Pruning

• MS-Q2P is a mixed-precision quantization method that uses differentiable LSB extraction and Hessian-guided pruning to dynamically adjust per-layer bit-widths.
• The approach leverages round-clamp quantization to extract and regularize least significant bits, transforming continuous shrinkage into discrete bit removal.
• Empirical results demonstrate substantial reductions in trainable parameters and training time while maintaining competitive accuracy across CNNs and vision transformers.

MS-Q2P denotes the quantization-to-pruning pathway instantiated within Memory-Efficient Bit Sparsification Quantization (MSQ), a mixed-precision quantization method for deep neural networks that reduces precision by continuously sparsifying least significant bits and then pruning them when layer sensitivity permits. In this formulation, quantization, bit-level sparsification, and Hessian-guided pruning are coupled during quantization-aware training, so that final per-layer bit-widths are discovered adaptively without explicit bit-level parameter splitting (Han et al., 30 Jul 2025).

1. Problem setting and conceptual role

Mixed-precision quantization is motivated by the fact that layer sensitivities to quantization noise vary: sensitive layers need more bits to sustain accuracy, while less sensitive layers can be aggressively compressed. Search-based mixed precision, including RL or pre-trained sensitivity analysis, is expensive and static, and it fails to track sensitivity changes during quantization-aware training. Bit-level sparsity offers a more direct mechanism because zeros induced in specific bits of the fixed-point representation can reduce effective precision with limited perturbation when the targeted bits are the least significant bits (LSBs) (Han et al., 30 Jul 2025).

Prior bit-level approaches such as BSQ and CSQ learn bit masks per bit, but they increase trainable parameters and GPU memory because they rely on explicit bit-level parameter splitting. MS-Q2P arises from MSQ’s reformulation of this problem: instead of splitting parameters by bit, MSQ treats bit sparsification as differentiable LSB extraction and regularization directly over the original weights. This avoids per-bit trainable variables while still creating a pathway from quantization to pruning at the bit level (Han et al., 30 Jul 2025).

In that sense, MS-Q2P is not a separate training framework from MSQ, but the mechanism by which MSQ converts continuous shrinkage of LSB contributions into discrete bit removal. A plausible implication is that the main novelty lies less in mixed-precision quantization per se than in how mixed precision is reached dynamically during training.

2. Round-Clamp quantization and differentiable LSB extraction

MSQ operates on normalized weights $W \in [0,1]$. Its quantizer is the round-clamp quantizer

$$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$

where $\lceil \cdot \rceil$ denotes standard rounding to nearest integer and the $\min$ term clamps to the valid code range $[0, 2^n - 1]$ (Han et al., 30 Jul 2025).

The defining choice is the use of the scale $2^n$ rather than $2^n - 1$. According to the method description, this centers the $(n-1)$-bit bin boundaries at the midpoints of the $n$-bit bins. The consequence is improved “rounding to nearest LSB-zero bin” behavior and the correction of misalignment issues observed with the standard DoReFa quantizer (Han et al., 30 Jul 2025).

Differentiability is provided by the straight-through estimator. In the forward pass,

$W_n = q_r(W; n),$

while in the backward pass the surrogate gradient is

$$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$0

This allows training with discrete forward values while treating rounding and clamping as identity in backpropagation, which is the standard QAT mechanism used here (Han et al., 30 Jul 2025).

The second ingredient is differentiable LSB extraction. For a block of $$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$1 LSBs, MSQ uses the continuous proxy

$$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$2

Under the round-clamp alignment, this equals the $$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$3-LSB contribution and remains differentiable with respect to $$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$4 under STE. The method describes this as a bipartite bit slicing view: an $$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$5-bit quantized value is decomposed into its top $$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$6 MSBs and its $$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$7 LSBs, but without introducing additional bit variables (Han et al., 30 Jul 2025).

3. LSB sparsification objective

MS-Q2P proceeds by driving these extracted LSB contributions toward zero. Across layers, the regularizer is

$$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$8

The total objective is

$$q_r(W; n) = \frac{1}{2^n - 1} \cdot \min(\lceil 2^n \cdot W \rceil, 2^n - 1),$$9

where $\lceil \cdot \rceil$0 is the standard task loss under quantized forward computation and $\lceil \cdot \rceil$1 controls the compression-accuracy trade-off (Han et al., 30 Jul 2025).

Under the STE treatment of $\lceil \cdot \rceil$2 in the backward pass, the regularization gradient is approximated as

$\lceil \cdot \rceil$3

This pushes weights toward bins whose $\lceil \cdot \rceil$4-LSB block is zero. The round-clamp bin alignment is central here: the stated purpose is to ensure that the gradients move weights toward LSB-zero bins rather than simply biasing them downward, which is described as a failure mode of standard DoReFa scaling (Han et al., 30 Jul 2025).

The method summary further states that the objective $\lceil \cdot \rceil$5 forms a convex surrogate on $\lceil \cdot \rceil$6 that continuously shrinks the $\lceil \cdot \rceil$7-LSB block until it becomes negligible for most weights of a layer. This suggests that MS-Q2P converts bit pruning from an explicit combinatorial operation into a regularized geometric migration of weights between aligned quantization bins.

4. Hessian-guided pruning and the training pipeline

The pruning stage is driven by a Hessian-based sensitivity estimate following HAWQ-V2. For layer $\lceil \cdot \rceil$8,

$\lceil \cdot \rceil$9

where $\min$0 is the Hessian trace with respect to that layer’s weights and the norm term measures quantization perturbation. Layers with lower $\min$1 are treated as less sensitive and can tolerate more aggressive precision reduction (Han et al., 30 Jul 2025).

Pruning is triggered by an LSB non-zero rate $\min$2, defined as the fraction of weights whose current $\min$3-LSB block is non-zero. When $\min$4, the layer is pruned by lowering its bit-width according to

$\min$5

with

$\min$6

This is the core MS-Q2P rule: continuous LSB shrinkage creates bit-level sparsity, and once sparsity is sufficient, one or two bits are removed at once depending on Hessian sensitivity (Han et al., 30 Jul 2025).

The algorithmic procedure is specified as follows. Inputs include the dataset $\min$7, an initial model $\min$8 with per-layer bit-widths $\min$9, regularization strength $[0, 2^n - 1]$0, pruning interval $[0, 2^n - 1]$1, pruning threshold $[0, 2^n - 1]$2, and target compression $[0, 2^n - 1]$3. During each epoch, each layer is quantized with round-clamp, task loss is computed, $[0, 2^n - 1]$4-LSB contributions are formed as

$[0, 2^n - 1]$5

and optimization proceeds on $[0, 2^n - 1]$6 using SGD and a cosine schedule. At pruning intervals, if the current compression $[0, 2^n - 1]$7, the method computes $[0, 2^n - 1]$8, evaluates $[0, 2^n - 1]$9, measures $2^n$0, sets $2^n$1, and prunes any layer satisfying $2^n$2. Once $2^n$3 is reached, both LSB regularization and pruning are disabled, and standard QAT fine-tuning continues under the finalized mixed-precision configuration (Han et al., 30 Jul 2025).

This training loop is the operational form of MS-Q2P. Quantization creates a differentiable representation of removable bits; regularization suppresses those bits; pruning converts suppression into an actual reduction in layer precision.

5. Mixed-precision assignment and efficiency characteristics

A defining property of MS-Q2P is that mixed precision emerges without bit splitting. MSQ keeps one floating-point weight per parameter, computes LSB blocks on the fly, and regularizes them, rather than allocating per-bit trainable variables, masks, or gates. The summary characterizes the memory profile as $2^n$4 for MSQ, versus $2^n$5 for bit-splitting baselines with average bit-width $2^n$6 (Han et al., 30 Jul 2025).

The reported reductions in trainable parameters are substantial. For ResNet-50, BSQ and CSQ require 204.8M trainable parameters, whereas MSQ uses 25.6M. For ResNet-18, the comparison is 93.52M versus 11.69M; for ResNet-20, 2.16M versus 0.27M. These are the basis for the reported reduction of up to $2^n$7 in trainable parameters (Han et al., 30 Jul 2025).

Training-time reductions are similarly emphasized. The stated examples are 128.76 h for BSQ, 346.12 h for CSQ, and 24.30 h for MSQ on ResNet-50; and 63.92 h for BSQ, 220.32 h for CSQ, and 21.11 h for MSQ on ResNet-18. The paper summary reports up to 86% reduction in training time, as well as total training time reductions of $2^n$8 versus BSQ and $2^n$9 versus CSQ on ResNet-50 (Han et al., 30 Jul 2025).

The same section attributes throughput gains to larger feasible batch sizes before out-of-memory failure. One example given is ResNet-50 with batch size up to 256 for MSQ versus 32 for DoReFa baselines. The claimed explanation is straightforward: on-the-fly computation of $2^n - 1$0 and periodic Hessian traces are lighter than per-bit forward and backward passes, and simultaneous multi-bit pruning shortens the number of pruning epochs needed to reach the target compression (Han et al., 30 Jul 2025).

6. Empirical behavior, scope, and limitations

The reported results cover CNNs and vision transformers. On ResNet-20 for CIFAR-10, MSQ achieves 92.17% at $2^n - 1$1 compression for $2^n - 1$2, 92.00% at $2^n - 1$3 for $2^n - 1$4, and 90.22% at $2^n - 1$5 for $2^n - 1$6. On ImageNet, the mixed-precision ResNet-18 result is $2^n - 1$7 compression with 69.74% Top-1, compared with CSQ mixed precision at $2^n - 1$8 and 69.73%; for ResNet-50, MSQ reports $2^n - 1$9 and 75.32%, versus CSQ at $(n-1)$0 and 75.47% (Han et al., 30 Jul 2025).

For MobileNetV3-Large on ImageNet, the reported MSQ points are $(n-1)$1, 74.29% and $(n-1)$2, 73.58%. For ViTs fine-tuned from OFQ 4-bit checkpoints with activation quantized to 8-bit, the reported values are DeiT-T at $(n-1)$3, 74.74%; DeiT-S at $(n-1)$4, 80.64%; and Swin-T at $(n-1)$5, 81.38%. These results are presented as evidence that the same pathway extends beyond conventional CNN backbones (Han et al., 30 Jul 2025).

The training setup is also specified. CIFAR-10 experiments with ResNet-20 are trained from scratch using SGD with initial learning rate 0.1, warm-start cosine annealing, and 400 epochs. ImageNet experiments on ResNet-18/50, MobileNetV3, and ViTs fine-tune from FP or OFQ checkpoints with SGD initial learning rate 0.01, warm-start cosine annealing, and 100 epochs. Activations are kept floating-point for CNNs on ImageNet, while ViT activations are quantized to 8-bit (Han et al., 30 Jul 2025).

Ablation findings attribute faster and smoother pruning to Hessian guidance. Without Hessian information, layers prune unevenly, require more epochs, and show larger loss spikes at pruning steps. With Hessian guidance, $(n-1)$6 adapts between 1 and 2 according to layer sensitivity, reducing per-step accuracy drops and reaching $(n-1)$7 sooner. The resulting bit schemes are described as more balanced than BSQ’s, which can overly prune particular layers and in some cases drive them to 0-bit under strong per-bit penalties (Han et al., 30 Jul 2025).

The stated limitations are practical rather than conceptual. Pruning steps cause transient accuracy drops; if validation loss spikes, the recommendation is to reduce $(n-1)$8 or prune less aggressively by choosing smaller $(n-1)$9 or larger $n$0. Early and mid-training sensitivity can differ from high-precision pre-training, so recalculating $n$1 at each pruning interval is said to be important. First and last layers, as well as attention projections in ViTs, tend to be more sensitive and often end up with smaller $n$2 and higher final $n$3. Peak memory can still rise with large batch sizes because activations and optimizer state remain substantial, even though per-bit variables are absent (Han et al., 30 Jul 2025).

In summary, MS-Q2P names the mechanism by which MSQ transforms quantization into pruning at the bit level: round-clamp quantization produces stable, STE-compatible LSB extraction; $n$4 regularization drives those LSB contributions toward zero; Hessian-guided rules remove one or two bits when sparsity and sensitivity criteria are satisfied. The resulting mixed-precision assignment is dynamic and data-driven, and the reported outcome is competitive accuracy and compression with up to $n$5 fewer trainable parameters and up to 86% shorter training time than earlier explicit bit-level quantization methods (Han et al., 30 Jul 2025).