Bern2Edge: A Neurosymbolic Compiler for Edge Deployment via Bernstein Polynomial Networks

Malak Gamal El-Din; Yifan Zhang; Yasser Shoukry; Sitao Huang; Salma Elmalaki

cs.LGAug 20, 2026

Bern2Edge: A Neurosymbolic Compiler for Edge Deployment via Bernstein Polynomial Networks

Authors: Malak Gamal El-Din, Yifan Zhang, Yasser Shoukry, Sitao Huang, Salma Elmalaki

Organizations: Department of Electrical Engineering and Computer Science, University of California, Irvine, CA 92697 USA

Abstract

Deploying high-accuracy neural networks on resource-constrained edge devices remains challenging, as existing approaches treat training, compression, and hardware synthesis as separate stages, leaving a gap between software-trained models and efficient end-to-end deployment with limited support for interpretability. We propose Bern2Edge, an end-to-end framework that uses knowledge distillation to convert a pretrained teacher feed-forward network into hardware-efficient representations via Bernstein polynomial activations. This representation enables two deployment paths: (i) a high-fidelity LUT-based realization that preserves model fidelity under compression, and (ii) a symbolic rule-based representation derived from Bernstein activation geometry, enabling interpretable inference with explicit input-space constraints. The resulting BNNs achieve up to 2.12 percentage-point (pp) accuracy improvement over ReLU under identical compression constraints. At the system level, Bern2Edge achieves up to 99.8% latency reduction and 95.2% BRAM reduction relative to a W8A8 quantized teacher on an AMD Xilinx KV260 FPGA, while maintaining accuracy within 0.5 pp, and further deploys on a low-power Spartan-7 XC7S15 FPGA. The rule-based path reduces DSP usage by up to 89.0% at a cost of 1.5 pp in total accuracy.

Figures & tables

Fig. 1 : Overview of Bern2Edge : A high-accuracy teacher model is distilled into a compressed BNN student via KD. The resulting representation is synthesized and deployed via either exact LUT-based realization or symbolic rule extraction.

Fig. 2 : Decision boundaries learned by a BNN (left) and a ReLU model (right) of equal capacity on the synthetic Two Moons dataset [ 24 ] . The BNN captures the curved structure of the data, while the ReLU model produces a simpler, predominantly linear boundary.

Panel of Figure 3 — Fig. 3 : Per-neuron LUT behavior vs. table size on Covertype: test accuracy (top) and error rate (bottom) vs. LUT entries per neuron. Error% is the fraction of samples where LUT output differs from full-precision inference.

Fig. 4 : Bernstein activation curves with analytically derived regime breakpoints. Six representative neuron activations from a BNN (

h=64

) trained on Adult [ 26 ] , with breakpoints from derivative roots and inflection points marked.

Fig. 5 : Activation-geometry-based rule formation. A selected regime in normalized activation space ( $t$ ) maps through $z$ -space to an affine constraint in input space, forming interpretable oblique regions.

Fig. 6 : Activation-geometry rule regions on the Two Moons dataset [ 24 ] . Rules align with the curved decision boundary via Bernstein-derived breakpoints, enabling compact partitioning.

Dataset	Arch.	Act.	Acc. (%)	CE Loss	$\Delta$ Acc / $\Delta$ CE	Latency	DSPs	BRAMs	LUTs
HIGGS-Small	$\{28,16,2\}$	ReLU	70.98 $\pm$ 0.20	0.553	$+0.94$ / $-0.013$	85	108	0	5397
	$\{28,16,2\}$	Bern	71.92 $\pm$ 0.26	0.540	$+0.94$ / $-0.013$	87	106	2	5682
	$\{28,16,8,2\}$	ReLU	71.84 $\pm$ 0.24	0.540	$+0.48$ / $-0.008$	121	156	0	6852
	$\{28,16,8,2\}$	Bern	72.32 $\pm$ 0.18	0.532	$+0.48$ / $-0.008$	125	135	3	7294
	$\{28,128,2\}$	ReLU	72.08 $\pm$ 0.27	0.524	$+0.45$ / $-0.012$	337	108	34	5114
	$\{28,128,2\}$	Bern	72.53 $\pm$ 0.18	0.512	$+0.45$ / $-0.012$	339	103	44	5410

TABLE I : Accuracy and synthesis metrics on AMD KV260 (5-fold CV). Latency in clock cycles.

\Delta

Acc /

\Delta

CE: BNN minus ReLU.

Panel of Figure 8 — Fig. 7 : Test accuracy vs. KD weight $\alpha$ ( $T{=}2$ ) on HIGGS-Small for students with single layer $h=16$ (top) and $h=128$ (bottom). Dashed line: teacher baseline ( $72.32\%$ ).

Latency	BRAMs	Bernstein		ReLU		$\Delta$ Acc
(cyc)		Acc (%)	(cyc, BRAM)	Acc (%)	(cyc, BRAM)	(pp)
${\leq}200$	${\leq}30$	76.50 $\pm$ 0.44	(189, 26)	74.93 $\pm$ 0.08	(188, 25)	$+1.57$
${\leq}400$	${\leq}55$	82.98 $\pm$ 0.16	(397, 52)	81.87 $\pm$ 0.37	(396, 49)	$+1.11$
${\leq}1000$	${\leq}120$	91.09 $\pm$ 0.27	(963, 105)	88.97 $\pm$ 0.82	(960, 113)	$+2.12$
${\leq}2500$	${\leq}140$	95.05 $\pm$ 0.05	(2481, 136)	93.53 $\pm$ 0.18	(2477, 137)	$+1.52$
${\leq}7500$	${\leq}180$	96.39 $\pm$ 0.02	(7433, 179)	95.62 $\pm$ 0.12	(7429, 169)	$+0.77$

TABLE II : Covertype accuracy under matched latency and BRAM budgets. Parentheses: actual synthesis values (cycles, BRAMs).

Panel of Figure 10 — Fig. 8 : FPGA resource utilization on Adult ( $h=128$ ) for naive Bernstein evaluation and LUT-based inference across polynomial degrees $n\in\{3,5,7,9,11\}$ .

Arch	BNN Acc. (%)	$\alpha_{\text{sc}}$	Rules	$\overline{\ell}$	Cov. (%)	Cov. Acc. (%)	Acc t (%)
$\{14,16,2\}$	84.33	0.5	37	1.76	78.2	86.8	82.6
$\{14,16,2\}$	84.33	0.1	45	1.67	91.5	84.5	81.7
$\{14,32,2\}$	84.42	0.5	45	2.04	84.9	87.2	83.2
$\{14,32,2\}$	84.42	0.1	53	2.02	94.0	85.3	82.1
$\{14,128,2\}$	84.63	0.5	61	1.64	89.5	85.9	82.8
$\{14,128,2\}$	84.63	0.1	85	1.75	95.8	85.3	82.0

TABLE III : Rule extraction on Adult across five BNN architectures.

\alpha_{\text{conf}}

: conflict penalty constant = 0.1. CART fallback.

\alpha_{\text{sc}}

: same-coverage penalty;

\overline{\ell}

: avg. conditions per rule; Cov.: coverage; Cov. Acc.: covered accuracy; Acc t : total accuracy.

Arch	Method	Acc (%)	Latency	DSP	BRAM	LUT	FF
$\{14,16,2\}$	LUT	84.33	77	77	2	4,329	5,582
$\{14,16,2\}$	Rules	82.59	183	30	5	2,997	2,290
$\{14,32,2\}$	LUT	84.42	119	109	4	6,019	8,333
$\{14,32,2\}$	Rules	83.16	215	30	8	3,009	2,218
$\{14,128,2\}$	LUT	84.63	376	301	28	15,609	22,865
$\{14,128,2\}$	Rules	82.76	279	30	8	3,002	2,217

TABLE IV : LUT-based BNNs vs. rule networks HW results on Adult (AMD KV260).

Type	Method	Rules	$\overline{\ell}$	Acc (%)	Fid (%)
Decompositional	DeepRED [ 19 ]	$5143\pm 9799$	5.43	78.7	89.3
	REM-D [ 35 ]	$3617\pm 6748$	5.41	78.6	89.4
	ECLAIRE [ 20 ]	$396\pm 75$	3.82	84.6	89.4
Bern2Edge	no-KD ( $\alpha_{\text{sc}}{=}0.1$ )	$46\pm 4$	1.84	84.2	91.8
	no-KD ( $\alpha_{\text{sc}}{=}0.5$ )	$\mathbf{24\pm 3}$	$\mathbf{1.81}$	82.9	94.3
	KD ( $\alpha_{\text{sc}}{=}0.1$ )	$44\pm 4$	$1.86$	$\mathbf{84.4}$	$91.8$

TABLE V : Rule extraction on MAGIC Gamma Telescope. Baselines from Zarlenga et al. [ 20 ] (mean

\pm

std, 5 folds); Decompositional: rule extraction from intermediate layers;

\overline{\ell}

: avg. conditions per rule; Fid: fidelity to baseline network.

Method	Acc (%)	Latency (cycles)	DSPs	BRAMs	URAMs
HIGGS-Small
Teacher (W8A8)	72.3	6,981	186	50	–
Bern2Edge (LUT)	72.3	125 ( $\downarrow$ 98.2%)	135 ( $\downarrow$ 27.4%)	3 ( $\downarrow$ 94.0%)	–
Covertype
Teacher (W8A8)	96.9	91,634	389	128	52
Bern2Edge (LUT)	96.4	7,433 ( $\downarrow$ 91.9%)	326 ( $\downarrow$ 16.2%)	149 ( $\uparrow$ 16.4%)	3 ( $\downarrow$ 94.2%)

TABLE VI : Post-synthesis accuracy and end-to-end hardware results for Bern2Edge across three datasets. Percentages indicate change relative to the W8A8 teacher baseline. Rule deployment path reported for Adult only.

Model	Config.	LUT	FF	DSP	BRAM	Lat.	Pwr	Acc	Acc fp
					18K	(cyc)	(mW)	(%)	(%)
Rules	R50	1,086	844	16	6	340	62	83.02	—
Rules	R29	710	765	16	4	156	59	83.01	—
Bern. $\{14,h,2\}$	$h{=}4$	553	883	18	1	61	51	84.18	84.25
	$h{=}8$	635	804	19	1	82	63	84.57	84.64
	$h{=}16$	662	777	19	1	115	63	84.88	84.86

TABLE VII : Deployment on XC7S15. R50 and R29 denote the 50-rule and 29-rule symbolic classifiers, respectively. Acc is test accuracy; Acc fp the fp32 reference for BNNs.

$\alpha_{\text{conf}}$	$\alpha_{\text{sc}}$	Conf.	Cov. (%)	Cov. Acc. (%)	Acc t (%)	Rules	$\overline{\ell}$
0.1	0.1	308	94.01	85.26	82.10	53	2.02
1.0	0.1	156	92.60	84.67	82.39	50	2.06
0.1	0.3	136	87.15	86.72	83.16	53	2.06
1.0	0.3	111	86.99	86.77	83.16	52	2.04
0.1	0.5	121	84.86	87.18	83.16	45	2.04
1.0	0.5	106	84.42	87.18	83.39	46	2.09

TABLE VIII : Joint penalty sweep on Adult (

h=32

, BNN acc.

84.42

%). Conf.: conflicting samples; Cov.: coverage; Cov. Acc.: covered accuracy; Acc t : total accuracy;

\overline{\ell}

: avg. conditions per rule.

Panel of Figure 17 — Fig. 9 : Coverage and covered accuracy as functions of the two rule-extraction penalty parameters on Adult, averaged across five network architectures ( $h\in\{16,32,128,16{\times}8,32{\times}16\}$ ).

Panel of Figure 18 — Fig. 10 : BRAM and total accuracy vs. sparsity threshold $k$ on Adult ( $h=32$ ).

Fallback	Acc.	Acc ${}_{\text{unc}}$	Fid ${}_{\text{unc}}$	LUT	FF	DSP	Lat.
Linear Regression	81.67	67.4	84.1	1749	520	10	13
Full BNN	82.91	71.9	100.0	1887	2318	40	68
Small BNN	82.23	71.3	90.5	1129	2178	38	46
CART	83.02	72.1	87.7	714	471	0	26

TABLE IX : Fallback strategy ablation on the Adult dataset (

h=32

\alpha_{\text{sc}}=0.5

\alpha_{\text{conf}}=0.1

). Resource columns report the fallback component only. Acc

{}_{\text{unc}}

and Fid

{}_{\text{unc}}

are on uncovered samples. All accuracy and fidelity values in %.

	Empirical (%)		Certified (%)
$c$	Rule Acc.	Fid.	Rules	BNN	ReLU
0.00	100.0	98.2	100.0	100.0	100.0
0.01	99.6	97.8	83.4	82.1	80.1
0.03	99.0	97.2	59.4	41.3	27.1
0.05	98.4	96.4	42.7	16.0	12.4
0.10	97.0	94.7	22.7	6.6	4.8

TABLE X : Robustness certification on MAGIC. Empirical columns show rule accuracy and rule–BNN fidelity under sampled noise; certified columns show provably stable fractions for rules and networks.

			Geographic shift					Temporal shift (CA)
System	Metric	ID (CA-18)	GEO-AVG	$\Delta$	MS	WY	WV	TEMP-AVG	$\Delta$	2019	2021	2022
ReLU Teacher	Acc. (%)	81.28 $\pm$ 0.09	74.32 $\pm$ 0.64	$-$ 7.0	73.73	74.61	74.61	80.00 $\pm$ 0.09	$-$ 1.3	80.74	79.82	79.44
ReLU	Acc. (%)	80.55 $\pm$ 0.15	74.10 $\pm$ 0.76	$-$ 6.5	73.42	74.45	74.43	79.37 $\pm$ 0.19	$-$ 1.2	80.03	79.23	78.84
BNN	Acc. (%)	80.89 $\pm$ 0.29	74.45 $\pm$ 0.69	$-$ 6.4	73.79	74.77	74.79	79.81 $\pm$ 0.36	$-$ 1.1	80.49	79.67	79.26
Rules	Coverage (%)	90.09 $\pm$ 0.86	85.82 $\pm$ 1.38	$-$ 4.3	86.10	85.07	86.29	90.24 $\pm$ 0.78	$+$ 0.2	90.01	90.35	90.35
	Covered acc. (%)	80.29 $\pm$ 0.20	76.41 $\pm$ 1.07	$-$ 3.9	77.31	74.82	77.09	78.67 $\pm$ 0.42	$-$ 1.6	79.38	78.62	78.00

TABLE XI : Distribution-shift robustness on ACS Income, training on CA-2018. ID = held-out CA-2018 test set; GEO-AVG and TEMP-AVG are means over shifted conditions;

\Delta

= AVG

-

ID (pp). Means

\pm

std over 5 seeds.

		TinyBERT4	$h{=}600$		$h{=}312$
Scope	Metric	TinyBERT4	GeLU	Bern	GeLU	Bern
	SST-2 Acc. (%)	90.37	90.02	90.48	89.11	90.02
FFN	Lat. (cycles)	6,413,520	3,282,640 ( $\downarrow$ 48.8%)	3,278,288 ( $\downarrow$ 48.9%)	1,789,648 ( $\downarrow$ 72.1%)	1,785,296 ( $\downarrow$ 72.2%)
	DSP	121	121 (0.0%)	80 ( $\downarrow$ 33.9%)	121 (0.0%)	80 ( $\downarrow$ 33.9%)
	BRAM	155	82 ( $\downarrow$ 47.1%)	128 ( $\downarrow$ 17.4%)	80 ( $\downarrow$ 48.4%)	108 ( $\downarrow$ 30.3%)
	FF	17,746	19,580 ( $\uparrow$ 10.3%)	15,654 ( $\downarrow$ 11.8%)	19,528 ( $\uparrow$ 10.0%)	15,568 ( $\downarrow$ 12.3%)

TABLE XII : SST-2 accuracy, latency, and FPGA resource utilization for full 4-layer FFN substitution. FFN : FFN sublayers only; Full : complete encoder. Resource reductions are relative to TinyBERT4.

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Federated Learning of Binary Neural Networks: Enabling Low-Cost Inference

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Federated Learning (FL) preserves privacy by distributing training across devices. However, using DNNs is computationally intensive at the low-powered edge during inference. Edge deployment demands models that simultaneously optimize memory footprint and computational efficiency, a dilemma where conventional DNNs fail by exceeding resource limits. Traditional post-training binarization reduces model size but suffers from severe accuracy loss due to quantization errors. To address these challenges, we propose FedBNN, a rotation-aware binary neural network framework that learns binary representations directly during local training. By encoding each weight as a single bit

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