Federated learning is a useful framework for centralized learning from distributed data under practical considerations of heterogeneity, asynchrony, and privacy. Federated architectures are frequently deployed in deep learning settings, which generally give rise to non-convex optimization problems. Nevertheless, most existing analysis are either limited to convex loss functions, or only establish first-order stationarity, despite the fact that saddle-points, which are first-order stationary, are known to pose bottlenecks in deep learning. We draw on recent results on the second-order optimality of stochastic gradient algorithms in centralized and decentralized settings, and establish second-order guarantees for a class of federated learning algorithms.
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