Trainable Bitwise Soft Quantization for Input Feature Compression

Karsten Schrödter; Jan Stenkamp; Nina Herrmann; Fabian Gieseke

cs.LGMar 5, 2026

Trainable Bitwise Soft Quantization for Input Feature Compression

Authors: Karsten Schrödter, Jan Stenkamp, Nina Herrmann, Fabian Gieseke

Organizations: University of Münster (Germany) · University of Copenhagen (Denmark)

Abstract

The growing demand for machine learning applications in the context of the Internet of Things calls for new approaches to optimize the use of limited compute and memory resources. Despite significant progress that has been made w.r.t. reducing model sizes and improving efficiency, many applications still require remote servers to provide the required resources. However, such approaches rely on transmitting data from edge devices to remote servers, which may not always be feasible due to bandwidth, latency, or energy constraints. We propose a task-specific, trainable feature quantization layer that compresses the input features of a neural network. This can significantly reduce the amount of data that needs to be transferred from the device to a remote server. In particular, the layer allows each input feature to be quantized to a user-defined number of bits, enabling a simple on-device compression at the time of data collection. The layer is designed to approximate step functions with sigmoids, enabling trainable quantization thresholds. By concatenating outputs from multiple sigmoids, introduced as bitwise soft quantization, it achieves trainable quantized values when integrated with a neural network. We compare our method to full-precision inference as well as to several quantization baselines. Experiments show that our approach outperforms standard quantization methods, while maintaining accuracy levels close to those of full-precision models. In particular, depending on the dataset, compression factors of $5\times$ to $16\times$ can be achieved compared to $32$ -bit input without significant performance loss.

Figures & tables

Panel of Figure 1 — Figure 1: The trainable feature quantization layer is integrated into a neural network to learn task-specific compressions for each input feature: (a) During training, the quantization layer (green rectangle) and the neural network are trained jointly on a remote server. The layer gives rise to an encoder-decoder composition $\quant_{i}=\dec_{i}\circ\enc_{i}$ for the $i$ -th input feature. (b) During inference, the encoder $\enc_{i}$ is used to encode the $i$ -th feature on the resource-constrained device using lightweight coding logic. The compressed features are then sent to the server, where the decoder $\dec_{i}$ is used to decode the $i$ -th feature. All decoded features are then used as input for the remaining neural network, which is executed on the remote server.

Figure 2: Schematic overview: Soft Quantization (

\quant^{\softmath}

) and Bitwise Soft Quantization (

\quant^{\bwmath,\softmath}

) are formed by summing or concatenating multiple soft step functions. Both are differentiable with respect to thresholds, enabling optimization during training. During inference, they are converted into Hard Quantization (

\quant

) and Bitwise Quantization (

\quant^{\bwmath}

) via rounding.

Figure 3: Example of a quantization with bit width

n=2

(i.e.,

M=3

thresholds) using minmax and quantile quantization on artificial skewed data. Left: Histogram of input data together with thresholds for minmax and quantile quantization. Center: Minmax and quantile quantization functions. Right: Histogram of quantized values for minmax and quantile quantization.

Name	Abbreviation	Source
Full Precision	FP	Baseline
Pre Minmax Quantization	Pr-MQ	Baseline
Pre Quantile Quantization	Pr-QQ	Baseline
Learnable Step Size Quantization	LSQ	[ 33 ]
Learnable Lookup Table with granule 4	LLT4	[ 34 ]
Learnable Lookup Table with granule 9	LLT9	[ 34 ]

Table 1: Models

Figure 4: MSE values per dataset per quantization level of the best-performing hyperparameter setting averaged over 10 data splits for all methods. The red line is the average over the full precision measurements.

Dataset	california	fried	superconduct	cpu_act	sulfur	wine	Mean
Bw-SQ	$0.195$	$0.061$	$0.086$	$0.020$	$0.172$	$0.543$	$0.180$
SQ	$+12.15\%$	$+3.74\%$	$+5.86\%$	$+28.34\%$	$+13.36\%$	$+7.36\%$	$+11.80\%$
Bw-MQ	$+57.43\%$	$+8.03\%$	$+16.42\%$	$+5.74\%$	$+21.21\%$	$+4.86\%$	$+18.95\%$
Bw-QQ	$+9.91\%$	$+6.64\%$	$-0.75\%$	$+112.85\%$	$+33.49\%$	$-1.36\%$	$+26.80\%$

Table 2: Average MSE of Bitwise Soft Quantization per dataset together with average MSE ratio of ablation methods against Bitwise Soft Quantization per dataset. Values below

0

indicate better performance than Bitwise Soft Quantization.

Appendix figures & tables17 assets

Supplementary material from the paper’s appendix.

Appendix

Bit Width	Number of	Encoding Latency	Energy Consumption	Energy Consumption
	Features	( $\mu s$ )	( $\mu Ah$ )	( $\mu J$ )
2	6	10	0.00016	2.11
2	81	50	0.00073	9.79
3	6	10	0.00016	2.11
3	81	70	0.00102	13.54
4	6	10	0.00016	2.14

Appendix

Table 3: Latency and energy requirements for encoding of input features to bit widths 2, 3, and 4 on a micrcontroller with ESP32-S3 CPU.

Byte size	Latency ( $ms$ )	Energy Consumption ( $\mu Ah$ )	Energy Consumption ( $\mu J$ )
8 bytes	21	0.6733	8968.34
32 bytes	140	4.0779	54318.13
128 byte	250	8.4043	111945.38

Appendix

Table 4: Energy and latency requirements for sending different amounts of data via CoAP protocol from a microcontroller equipped with ESP32-S3 CPU.

Dataset	Short Name	Source	$N$	$K$	$\bar{C}$	OpenML ID
California Housing	california	[ 38 ]	20640	8	8880	43979
Fried	fried	[ 39 , 40 ]	40768	10	1001	564
Superconduct	superconduct	[ 42 ]	21263	81	7449	43174
CPU Activity Database	cpu_act		8192	21	2192	197
Sulfur	sulfur	[ 41 ]	10081	6	8636	23515
Wine Quality	wine	[ 43 ]	6497	11	241	287

Appendix

Table 5: Datasets used where

N

denotes the size of the dataset,

K

the number of features and

\bar{C}

the average number of characteristics (i.e. distinct values) per feature.

Figure 5: Visualization of different temperature schedules. All schedules start with a temperature value $\tau_{init}=1$ and end with a temperature value $\tau_{end}$ after $n_{epochs}$ epochs.

Schedule	$\tau_{end}$
Schedule	0.0001	0.001	0.01
Linear	0.6315	0.6515	0.6421
Warmup	0.6425	0.6456	0.6604
Cosine	0.6401	0.6412	0.6320
Cyclical	0.6526	0.6473	0.6503
Exponential	0.6300	0.6201	0.6412

Appendix

Table 6: Average validation MSE of Bitwise Soft-Quantization on wine dataset for

2

bit quantization for different temperature schedules.

Schedule	$\tau_{end}$
Schedule	0.0001	0.001	0.01
Linear	0.1615	0.1790	0.1672
Warmup	0.1848	0.1741	0.1737
Cosine	0.1335	0.1336	0.1449
Cyclical	0.1595	0.1636	0.1896
Exponential	0.1197	0.1112	0.1230

Appendix

Table 7: Average validation MSE of Bitwise Soft-Quantization on superconduct dataset for

7

bit quantization for different temperature schedules.

Name	Values	Description
dropout rate	[0.0, 0.2, 0.4, 0.5]	dropout rate for neurons during training
learning rate	[0.001, 0.0001]	learning rate for optimizer
hidden layers	[5, 6, 8, 10]	number of hidden layers
hidden neurons	[32, 64, 128, 256, 512,	number of neurons per hidden layer
hidden neurons	1024, 2048, 4096, 8192]	number of neurons per hidden layer
num epochs	[30, 50, 70]	number of epochs to train

Appendix

Table 8: Search Space for Hyperparameter Tuning

Figure 6: Distribution of the k-fold values for the best average hyperparameter setup per quantization method for a 2-bit quantization level.

Figure 7: Distribution of the k-fold values for the best average hyperparameter setup per quantization method for a 3-bit quantization level.

Figure 8: Distribution of the k-fold values for the best average hyperparameter setup per quantization method for a 4-bit quantization level.

Figure 9: Distribution of the k-fold values for the best average hyperparameter setup per quantization method for a 5-bit quantization level.

Figure 10: Distribution of the k-fold values for the best average hyperparameter setup per quantization method for a 6-bit quantization level.

Figure 11: Distribution of the k-fold values for the best average hyperparameter setup per quantization method for a 7-bit quantization level.

Figure 12: Distribution of the k-fold values for the best average hyperparameter setup per quantization method for a 8-bit quantization level.

dataset	bits	FP	Comparison models					Ours
dataset	bits	FP	Pr-MQ	Pr-QQ	LLT4	LLT9	LSQ	Bw-SQ
california	2	0.186	0.593	0.338	0.424	0.396	0.356	0.290
	3	0.186	0.376	0.279	0.278	0.273	0.259	0.210
	4	0.186	0.326	0.213	0.219	0.222	0.216	0.185
	5	0.186	0.279	0.199	0.204	0.208	0.207	0.171
	6	0.186	0.254	0.189	0.197	0.197	0.195	0.173

Appendix

Table 9: Average MSE for selected hyperparameter setting per dataset and bit width. Minimal values of baseline and our models per row are bold, values within a range of 2.5% from the minimal value are underlined.

dataset	bits	FP	Comparison models					Ours
dataset	bits	FP	Pr-MQ	Pr-QQ	LLT4	LLT9	LSQ	Bw-SQ
california	2	[0.163, 0.210]	[0.565, 0.621]	[0.302, 0.374]	[0.387, 0.462]	[0.357, 0.435]	[0.322, 0.389]	[0.257, 0.323]
	3	[0.163, 0.210]	[0.349, 0.402]	[0.248, 0.310]	[0.241, 0.315]	[0.243, 0.303]	[0.226, 0.291]	[0.178, 0.241]
	4	[0.163, 0.210]	[0.298, 0.354]	[0.188, 0.238]	[0.196, 0.241]	[0.190, 0.254]	[0.189, 0.243]	[0.154, 0.216]
	5	[0.163, 0.210]	[0.245, 0.313]	[0.171, 0.226]	[0.178, 0.230]	[0.178, 0.239]	[0.177, 0.236]	[0.147, 0.194]
	6	[0.163, 0.210]	[0.219, 0.289]	[0.164, 0.213]	[0.163, 0.232]	[0.172, 0.221]	[0.163, 0.226]	[0.142, 0.204]

Appendix

Table 10: 95%-confidence intervals of

10

-fold results for selected hyperparameter settings per dataset and bit width.

dataset	bits	Comparison models			Ours
dataset	bits	SQ	Bw-MQ	Bw-QQ	Bw-SQ
california	2	0.288	0.595	0.329	0.290
	3	0.217	0.377	0.265	0.210
	4	0.214	0.312	0.207	0.185
	5	0.208	0.262	0.182	0.171
	6	0.200	0.230	0.177	0.173

Appendix

Table 11: Ablation Study: Average MSE for selected hyperparameter setting per dataset and bit width. Minimal values are bold, values within a range of 2.5% from the minimal value are underlined.

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