Prioritizing Gradient Sign Over Modulus: An Importance-Aware Framework for Wireless Federated Learning

Yiyang Yue; Jiacheng Yao; Wei Xu; Zhaohui Yang; George K. Karagiannidis; Dusit Niyato

cs.LGMar 11, 2026

Prioritizing Gradient Sign Over Modulus: An Importance-Aware Framework for Wireless Federated Learning

Authors: Yiyang Yue, Jiacheng Yao, Wei Xu, Zhaohui Yang, George K. Karagiannidis, Dusit Niyato

Organizations: National Mobile Communications Research Laboratory (NCRL), Southeast University, Nanjing 210096, China · Purple Mountain Laboratories, Nanjing 211111, China · Zhejiang Lab, Hangzhou 311121, China · College of Information Science and Electronic Engineering, Zhejiang University, Hangzhou, Zhejiang 310027, China ·

Abstract

Wireless federated learning (FL) facilitates collaborative training of artificial intelligence (AI) models to support ubiquitous intelligent applications at the wireless edge. However, the inherent constraints of limited wireless resources inevitably lead to unreliable communication, which poses a significant challenge to wireless FL. To overcome this challenge, we propose Sign-Prioritized FL (SP-FL), a novel framework that improves wireless FL by prioritizing the transmission of important gradient information through uneven resource allocation. Specifically, recognizing the importance of descent direction in model updating, we transmit gradient signs in individual packets and allow their reuse for gradient descent if the remaining gradient modulus cannot be correctly recovered. To further improve the reliability of transmission of important information, we formulate a hierarchical resource allocation problem based on the importance disparity at both the packet and device levels, optimizing bandwidth allocation across multiple devices and power allocation between sign and modulus packets. To make the problem tractable, the one-step convergence behavior of SP-FL, which characterizes data importance at both levels in an explicit form, is analyzed. We then propose an alternating optimization algorithm to solve this problem using the Newton-Raphson method and successive convex approximation (SCA). Simulation results confirm the superiority of SP-FL, especially in resource-constrained scenarios, demonstrating up to 9.96% higher testing accuracy on the CIFAR-10 dataset compared to existing methods.

Figures & tables

Fig. 1: Illustration of the proposed SP-FL.

$\displaystyle G(\alpha_{k,n},\beta_{k,n})\!\triangleq$	$\displaystyle\!\left(\!-4p_{k,n}\!+\!p_{k,n}^{2}\!+\!L\eta\frac{p_{k,n}}{q_{k,n}}\!\right)\!\left\\|\mathbf{g}_{k,n}\!\right\\|^{2}\!+\!\left(\!-2p_{k,n}\!+\!p_{k,n}^{2}\!+\!L\eta\frac{1-p_{k,n}}{q_{k,n}}\!\right)\!\left\\|\bar{\mathbf{g}}\right\\|^{2}\!+\!\left(6p_{k,n}-2p_{k,n}^{2}\right)\upsilon_{k,n}+L\eta\frac{p_{k,n}}{q_{k,n}}\delta^{2}_{k,n}$
$\displaystyle=$	$\displaystyle 2\!\left(\!-2\!\left\\|\mathbf{g}_{k,n}\!\right\\|^{2}\!-\!\left\\|\bar{\mathbf{g}}\right\\|^{2}\!+\!3\upsilon_{k,n}\!\right)\!\exp\left(\!\frac{H_{v}(\beta_{k,n})}{1-\alpha_{k,n}}\!\right)\!+\!\left(\!\left\\|\mathbf{g}_{k,n}\!\right\\|^{2}\!+\!\left\\|\bar{\mathbf{g}}\right\\|^{2}\!-\!2\upsilon_{k,n}\!\right)\!\exp\left(\!\frac{2H_{v}(\beta_{k,n})}{1-\alpha_{k,n}}\!\right)$
	$\displaystyle+\!L\eta(\left\\|\mathbf{g}_{k,n}\right\\|^{2}\!-\!\left\\|\bar{\mathbf{g}}\right\\|^{2}\!+\!\delta^{2}_{k,n})\!\exp\left(\!\frac{H_{v}(\beta_{k,n})}{1-\alpha_{k,n}}\!-\!\frac{H_{s}(\beta_{k,n})}{\alpha_{k,n}}\!\right)\!+\!L\eta\left\\|\bar{\mathbf{g}}\right\\|^{2}\exp\left(\!-\frac{H_{s}(\beta_{k,n})}{\alpha_{k,n}}\!\right)$
$\displaystyle\triangleq$	$\displaystyle A_{k,n}\!\exp\!\left(\!\frac{H_{v}\!(\!\beta_{k,n}\!)}{1\!-\!\alpha_{k,n}}\!\right)\!+\!B_{k,n}\!\exp\!\left(\!\frac{2H_{v}\!(\!\beta_{k,n}\!)}{1\!-\!\alpha_{k,n}}\!\right)\!+\!C_{k,n}\!\exp\!\left(\!\frac{H_{v}\!(\!\beta_{k,n}\!)}{1-\alpha_{k,n}}\!-\!\frac{H_{s}\!(\!\beta_{k,n}\!)}{\alpha_{k,n}}\!\right)\!+\!D_{k,n}\exp\!\left(\!-\frac{H_{s}\!(\!\beta_{k,n}\!)}{\alpha_{k,n}}\!\right)\!$	(27)

Table 2

		$\displaystyle A_{k,n}\exp\left(\frac{H_{v}(\beta_{k,n})}{1-x_{i}}\right)\frac{H_{v}(\beta_{k,n})}{(1-x_{i})^{2}}+B_{k,n}\exp\left(\frac{2H_{v}(\beta_{k,n})}{1-x_{i}}\right)\frac{2H_{v}(\beta_{k,n})}{(1-x_{i})^{2}}$
		$\displaystyle+C_{k,n}\exp\left(\frac{H_{v}(\beta_{k,n})}{1-x_{i}}-\frac{H_{s}(\beta_{k,n})}{x_{i}}\right)\left(\frac{H_{v}(\beta_{k,n})}{(1-x_{i})^{2}}+\frac{H_{s}(\beta_{k,n})}{x_{i}^{2}}\right)+D_{k,n}\exp\left(-\frac{H_{s}(\beta_{k,n})}{x_{i}}\right)\frac{H_{s}(\beta_{k,n})}{x_{i}^{2}}=0$		(31)

Table 3

Fig. 2: The upper bound and exact value of the loss function with IID and non-IID data.

Fig. 3: Performance comparison under varying non-IID levels.

Fig. 4: Performance comparison with low-complexity optimization method.

Fig. 5: Performance comparison under different compensation designs.

Fig. 6: Performance comparison with retransmission designs.

Fig. 7: Test accuracy versus transmit power.

Fig. 8: Test accuracy versus transmission latency thresholds.

Fig. 9: Test accuracy versus the number of participating devices.

Fig. 10: Test accuracy versus the number of quantization bits.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

$\displaystyle B_{2}\!=\!$	$\displaystyle{\sum_{k=1}^{K}\!\mathbb{E}\!\left[\!\left\\|\!\frac{C(\mathbf{g}_{k,n})\!\cdot s(\mathbf{g}_{k,n})\!\odot\!\hat{Q}_{v}(\!\mathbf{g}_{k,n}\!)}{q_{k,n}}\!-\!p_{k,n}\mathbf{g}_{k,n}\!-\!(\!1\!-\!p_{k,n}\!)s(\!\mathbf{g}_{k,n}\!)\!\odot\!\bar{\mathbf{g}}\right\\|^{2}\right]}\!+\!{\sum_{k=1}^{K}\!\left\\|p_{k,n}\mathbf{g}_{k,n}\!+\!(\!1\!-\!p_{k,n})s(\mathbf{g}_{k,n})\!\odot\!\bar{\mathbf{g}}\!-\!\mathbf{g}_{n}\right\\|^{2}}$
$\displaystyle=$	$\displaystyle\underbrace{\sum_{k=1}^{K}\left\\|(1-p_{k,n})(s(\mathbf{g}_{k,n})\odot\bar{\mathbf{g}}-\mathbf{g}_{k,n})+\mathbf{g}_{k,n}-\mathbf{g}_{n}\right\\|^{2}}_{C_{1}}+\underbrace{\sum_{k=1}^{K}(1-q_{k,n})\left\\|p_{k,n}\mathbf{g}_{k,n}+(1-p_{k,n})s(\mathbf{g}_{k,n})\odot\bar{\mathbf{g}}\right\\|^{2}}_{C_{2}}$
	$\displaystyle+\underbrace{\sum_{k=1}^{K}q_{k,n}\mathbb{E}\left[\left\\|\frac{s(\mathbf{g}_{k,n})\odot\hat{Q}_{v}(\mathbf{g}_{k,n})}{q_{k,n}}-p_{k,n}\mathbf{g}_{k,n}-(1-p_{k,n})s(\mathbf{g}_{k,n})\odot\bar{\mathbf{g}}\right\\|^{2}\right]}_{C_{3}}$	(60)

Appendix

Table 13

$\displaystyle C_{3}\!=$	$\displaystyle\!\sum_{k=1}^{K}p_{k\!,n}q_{k\!,n}\mathbb{E}\!\left[\!\left\\|\!\frac{Q(\!\mathbf{g}_{k\!,n}\!)}{q_{k,n}}\!-\!p_{k\!,n}\mathbf{g}_{k\!,n}\!-\!(\!1\!-\!p_{k,n}\!)s(\mathbf{g}_{k\!,n}\!)\!\odot\!\bar{\mathbf{g}}\!\right\\|^{2}\!\right]\!+\!\sum_{k=1}^{K}\!(1\!-\!p_{k\!,n}\!)q_{k,n}\!\left\\|\frac{s(\!\mathbf{g}_{k\!,n}\!)\!\odot\!\bar{\mathbf{g}}}{q_{k\!,n}}\!-\!p_{k\!,n}\mathbf{g}_{k\!,n}\!-\!(\!1\!-\!p_{k\!,n}\!)s(\!\mathbf{g}_{k\!,n}\!)\!\odot\!\bar{\mathbf{g}}\right\\|^{2}$
$\displaystyle=$	$\displaystyle\sum_{k=1}^{K}\frac{p_{k,n}}{q_{k,n}}\mathbb{E}\left[\left\\|Q(\mathbf{g}_{k,n})-\mathbf{g}_{k,n}\right\\|^{2}\right]+\sum_{k=1}^{K}p_{k,n}q_{k,n}\left\\|\frac{1-p_{k,n}q_{k,n}}{q_{k,n}}\mathbf{g}_{k,n}-(1-p_{k,n})s(\mathbf{g}_{k,n})\odot\bar{\mathbf{g}}\right\\|^{2}$
	$\displaystyle+\sum_{k=1}^{K}(1-p_{k,n})q_{k,n}\left\\|\frac{1-(1-p_{k,n})q_{k,n}}{q_{k,n}}s(\mathbf{g}_{k,n})\odot\bar{\mathbf{g}}-p_{k,n}\mathbf{g}_{k,n}\right\\|^{2}$
$\displaystyle\overset{(\mathrm{a})}{\operatorname*{\leq}}$	$\displaystyle\!\underbrace{\sum_{k=1}^{K}\!p_{k\!,\!n}\!q_{k\!,\!n}\!\left\\|\!\frac{1\!-\!p_{k\!,\!n}q_{k\!,\!n}}{q_{k\!,\!n}}\!\mathbf{g}_{k\!,\!n}\!-\!(\!1\!-\!p_{k\!,\!n}\!)\!s(\!\mathbf{g}_{k\!,\!n}\!)\!\odot\!\bar{\mathbf{g}}\!\right\\|^{2}}_{E_{1}}\!+\!\underbrace{\sum_{k=1}^{K}(\!1\!-\!p_{k\!,n}\!)q_{k\!,\!n}\!\left\\|\!\frac{1\!-\!(\!1\!-\!p_{k\!,\!n}\!)q_{k\!,\!n}}{q_{k\!,\!n}}\!s(\!\mathbf{g}_{k\!,\!n}\!)\!\odot\!\bar{\mathbf{g}}\!-\!p_{k\!,\!n}\mathbf{g}_{k\!,\!n}\!\right\\|^{2}}_{E_{2}}\!+\!\sum_{k=1}^{K}\frac{p_{k\!,\!n}}{q_{k\!,\!n}}\!\delta_{k\!,\!n}^{2}$	(63)

Appendix

Table 14

	$\displaystyle G^{\prime}(\alpha_{k,n})=$	$\displaystyle A_{k,n}\exp\left(\frac{H_{v}(\beta_{k,n})}{1-\alpha_{k,n}}\right)\frac{H_{v}(\beta_{k,n})}{(1-\alpha_{k,n})^{2}}+B_{k,n}\exp\left(\frac{2H_{v}(\beta_{k,n})}{1-\alpha_{k,n}}\right)\frac{2H_{v}(\beta_{k,n})}{(1-\alpha_{k,n})^{2}}$
		$\displaystyle+C_{k,n}\exp\left(\frac{H_{v}(\beta_{k,n})}{1-\alpha_{k,n}}-\frac{H_{s}(\beta_{k,n})}{\alpha_{k,n}}\right)\left(\frac{H_{v}(\beta_{k,n})}{(1-\alpha_{k,n})^{2}}+\frac{H_{s}(\beta_{k,n})}{\alpha_{k,n}^{2}}\right)+D_{k,n}\exp\left(-\frac{H_{s}(\beta_{k,n})}{\alpha_{k,n}}\right)\frac{H_{s}(\beta_{k,n})}{\alpha_{k,n}^{2}}$		(69)

Appendix

Table 15

Federated Learning

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In wireless federated learning (FL), data heterogeneity and multiple local updates induce client drift, degrading model convergence. It is further affected by unreliable wireless links, as transmission errors may invalidate model updates. To address these challenges, we propose a federated random walk averaging (FedRW) framework, which is a variant of federated averaging (FedAvg) that mitigates data heterogeneity by updating models along random walk (RW) paths and aggregating them at the server. Model parameters are transmitted in packets with retransmission support to improve training quality by mitigating wireless errors along RW paths. Meanwhile, wireless transmission delays hinder the exploration of FedRW. To this end, we formulate a joint optimization problem that integrates learning, RW path selection, and transmission parameter tuning, aiming to minimize the training loss under delay constraints. By deriving an upper bound on the expected convergence of FedRW over unreliable wireless networks, we reduce the problem to a general form agnostic to task type and model architecture. A distributed solution is then proposed, in which the server or clients optimize packet size and maximum number of retransmissions locally, and efficiently select reliable and expandable next-hop nodes via a resilience-aware beam search with dynamic pruning. Simulation results show that FedRW achieves 2.26%-9% higher accuracy than state-of-the-art baselines under high data heterogeneity. Furthermore, the jointly optimized FedRW yields at least 2.78% higher accuracy and faster convergence compared to baselines.

Abstract

Federated Learning

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