Decentralized sparsity learning has attracted a significant amount of attention recently due to its rapidly growing applications. To obtain the robust and sparse estimators, a natural idea is to adopt the non-smooth median loss combined with a $\ell_1$ sparsity regularizer. However, most of the existing methods suffer from slow convergence performance caused by the {\em double} non-smooth objective. To accelerate the computation, in this paper, we proposed a decentralized surrogate median regression (deSMR) method for efficiently solving the decentralized sparsity learning problem. We show that our proposed algorithm enjoys a linear convergence rate with a simple implementation. We also investigate the statistical guarantee, and it shows that our proposed estimator achieves a near-oracle convergence rate without any restriction on the number of network nodes. Moreover, we establish the theoretical results for sparse support recovery. Thorough numerical experiments and real data study are provided to demonstrate the effectiveness of our method.
翻译:最近,由于应用的迅速增长,分散化的广度学习引起了大量关注。为了获得强健和稀少的测算器,自然的想法是采用非悬浮中位损失加上1美元聚度调节器。然而,大多数现有方法都因非分散化目标而出现缓慢的趋同性效果。为了加速计算,我们在本文件中提出了一种分散化替代中位回归(deSMR)方法,以有效解决分散化的聚度学习问题。我们用简单的实施来表明我们提议的算法具有线性趋同率。我们还调查了统计保证,它表明我们提议的估算器在网络节点数量上没有限制地实现了近乎乎乎乎的趋同率。此外,我们为微弱的支持恢复建立了理论结果。提供了细数实验和实际数据研究,以证明我们的方法的有效性。