#### Asynchronous Distributed Optimization with Redundancy in Cost Functions

##### Shuo Liu, Nirupam Gupta, Nitin H. Vaidya

This paper considers the problem of asynchronous distributed multi-agent optimization on server-based system architecture. In this problem, each agent has a local cost, and the goal for the agents is to collectively find a minimum of their aggregate cost. A standard algorithm to solve this problem is the iterative distributed gradient-descent (DGD) method being implemented collaboratively by the server and the agents. In the synchronous setting, the algorithm proceeds from one iteration to the next only after all the agents complete their expected communication with the server. However, such synchrony can be expensive and even infeasible in real-world applications. We show that waiting for all the agents is unnecessary in many applications of distributed optimization, including distributed machine learning, due to redundancy in the cost functions (or {\em data}). Specifically, we consider a generic notion of redundancy named $(r,\epsilon)$-redundancy implying solvability of the original multi-agent optimization problem with $\epsilon$ accuracy, despite the removal of up to $r$ (out of total $n$) agents from the system. We present an asynchronous DGD algorithm where in each iteration the server only waits for (any) $n-r$ agents, instead of all the $n$ agents. Assuming $(r,\epsilon)$-redundancy, we show that our asynchronous algorithm converges to an approximate solution with error that is linear in $\epsilon$ and $r$. Moreover, we also present a generalization of our algorithm to tolerate some Byzantine faulty agents in the system. Finally, we demonstrate the improved communication efficiency of our algorithm through experiments on MNIST and Fashion-MNIST using the benchmark neural network LeNet.

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