Emerging applications in multi-agent environments such as internet-of-things, networked sensing, autonomous systems and federated learning, call for decentralized algorithms for finite-sum optimizations that are resource-efficient in terms of both computation and communication. In this paper, we consider the prototypical setting where the agents work collaboratively to minimize the sum of local loss functions by only communicating with their neighbors over a predetermined network topology. We develop a new algorithm, called DEcentralized STochastic REcurSive gradient methodS (DESTRESS) for nonconvex finite-sum optimization, which matches the optimal incremental first-order oracle (IFO) complexity of centralized algorithms for finding first-order stationary points, while maintaining communication efficiency. Detailed theoretical and numerical comparisons corroborate that the resource efficiencies of DESTRESS improve upon prior decentralized algorithms over a wide range of parameter regimes. DESTRESS leverages several key algorithm design ideas including randomly activated stochastic recursive gradient updates with mini-batches for local computation, gradient tracking with extra mixing (i.e., multiple gossiping rounds) for per-iteration communication, together with careful choices of hyper-parameters and new analysis frameworks to provably achieve a desirable computation-communication trade-off.
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