Polar Code Based Federated Learning: Convergence Analysis and Resource Allocation

Han Xiao; Wei Kang; Nan Liu

cs.LGAug 14, 2026

Polar Code Based Federated Learning: Convergence Analysis and Resource Allocation

Authors: Han Xiao, Wei Kang, Nan Liu

Organizations: School of Information Science and Engineering, Southeast University, Nanjing 211189, China · National Mobile Communications Research Laboratory, Southeast University, Nanjing 211189, China

Abstract

Federated learning (FL) enables collaborative model training across distributed devices without sharing raw data; however, it faces significant communication bottlenecks and channel impairments in practice. Conventional network layer treatments either idealize the channel as error free or apply equal error protection (EEP) to transmitted model updates, failing to account for the inherently unequal importance of quantization bits within a single local model. To address this limitation, we propose a cross layer polar code based FL scheme that leverages the unequal error protection (UEP) property of polar codes under finite block lengths. Specifically, the proposed design selectively protects more significant quantization bits, thereby mitigating the detrimental effects of channel noise. We further provide a rigorous convergence analysis of the proposed scheme, deriving an upper bound on the convergence gap, which we then jointly optimize over the number of quantization bits and the polar code block length across all training iterations. Experimental results demonstrate that both constant and variable block length configurations of our polar code based scheme consistently achieve substantial performance gains over uncoded and LDPC-based EEP benchmarks, with the advantage becoming increasingly pronounced as the channel quality deteriorating. These findings confirm the efficacy of our cross-layer design in enhancing FL robustness and efficiency under realistic channel conditions.

Figures & tables

Fig. 2 : Flowchart of the learning procedure in a single round.

Panel of Figure 3 — Fig. 3 : Fitting results of 64 BECs.

Panel of Figure 4 — Fig. 4 : The fitted function of $N$ vs $\mu$ and $N$ vs $\sigma$ with $\epsilon=0.6$ .

Erasure Probability $\mathbf{\epsilon}$	$\mathbf{a_{\epsilon}}$	$\mathbf{b_{\epsilon}}$	$\mathbf{c_{\epsilon}}$	$\mathbf{d_{\epsilon}}$
0.1	0.9022	0.7252	0.0557	1.3703
0.2	0.8029	0.6396	0.0874	1.8729
0.3	0.7019	0.6178	0.1062	2.2381
0.4	0.6008	0.5765	0.1163	2.5415
0.5	0.500	0.500	0.1176	2.7372
0.6	0.3992	0.4265	0.1163	2.5415

TABLE I : Fitting results

a_{\epsilon},b_{\epsilon},c_{\epsilon},d_{\epsilon}

under different

\epsilon

N	32				64				128
Erasure Probability $\mathbf{\epsilon}$	0.2	0.4	0.6	0.8	0.2	0.4	0.6	0.8	0.2	0.4	0.6	0.8
$\mathbf{\mu=a_{\epsilon}N+b_{\epsilon}}$	26.332	19.802	13.201	6.666	52.02	39.03	25.98	12.98	103.411	77.479	51.524	25.607
$\mathbf{\sigma=c_{\epsilon}N+d_{\epsilon}}$	4.670	6.263	6.263	4.670	7.467	9.985	9.985	7.467	13.060	17.428	17.428	13.060
SSE	0.021	0.007	0.007	0.021	0.024	0.010	0.010	0.024	0.028	0.013	0.013	0.028
$\mathbf{R^{2}}$	0.994	0.999	0.999	0.994	0.997	0.999	0.999	0.997	0.998	0.999	0.999	0.998
$\mathbf{AR^{2}}$	0.993	0.999	0.999	0.993	0.997	0.999	0.999	0.997	0.998	0.999	0.999	0.998

TABLE II : Fitting performance for different number of channel uses.

Fig. 6 : The curve of objective function.

$\mathbf{\epsilon}$ $\backslash$ N	16	32	64	128
0.1	{3,4,…,12}	{3,4,…,23}	{3,4,…,51}	{3,4,…,102}
0.2	{3,4,…,8}	{3,4,…,19}	{3,4,…,39}	{3,4,…,77}
0.3	{3,4,5,6}	{3,4,…,13}	{3,4,…,26}	{3,4,…,60}
0.4	{3,4,5}	{3,4,…,8}	{3,4,…,18}	{3,4,…,45}
0.5	{2,3,4}	{3,4,5}	{3,4,…,11}	{3,4,…,29}
0.6	{2,3,4}	{3,4,5}	{3,4,5,6,7}	{3,4,5,…,18}

TABLE III : “good” choices of

n

under different

N

and

\epsilon

Fig. 7 : The convergence upper bound of the proposed scheme and benchmark scheme.

Panel of Figure 11 — Fig. 8 : The test accuracy of different schemes.

Panel of Figure 12 — Fig. 9 : The test accuracy of different schemes

Fig. 10 : The test accuracy after convergence of different schemes under different erase probabilities.

Unequal Error Protection

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Federated Learning Low-Bit Quantization Randomized Sketching

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Personalized federated learning (PFL) is a promising paradigm for collaborative learning over distributed devices, where edge nodes collaboratively train personalized models without sharing raw data. Although PFL addresses data heterogeneity by learning client-specific models, it still suffers from substantial uplink and downlink communication costs when exchanging high-dimensional parameters in bandwidth-constrained systems. Recent one-bit methods achieve extreme compression, but they usually rely on a single thresholding rule applied to the whole model. This design has two limitations. First, it overlooks layer-wise differences in parameter distributions and quantization sensitivities. Second, a single threshold provides only coarse binary information and cannot capture fine-grained variations in parameter distributions. To address these issues, we propose a communication-efficient PFL framework via layer-wise multi-threshold random sketching. In the proposed method, each layer is assigned its own set of quantization thresholds, so that the compressed representation can adapt to layer-specific statistics while using multiple intervals to provide a finer low-bit description of sketched parameters. The proposed method supports bidirectional communication using compact low-bit sketches and improves the communication-accuracy tradeoff compared with existing one-bit compression approaches.

Abstract

Unequal Error Protection

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