Saber Salehkaleybar, Arsalan Sharifnassab, S. Jamaloddin Golestani.
Year: 2021, Volume: 22, Issue: 189, Pages: 1−47
We consider distributed statistical optimization in one-shot setting, where there are $m$ machines each observing $n$ i.i.d. samples. Based on its observed samples, each machine sends a $B$-bit-long message to a server. The server then collects messages from all machines, and estimates a parameter that minimizes an expected convex loss function. We investigate the impact of communication constraint, $B$, on the expected error and derive a tight lower bound on the error achievable by any algorithm. We then propose an estimator, which we call Multi-Resolution Estimator (MRE), whose expected error (when $B\ge d\log mn$ where $d$ is the dimension of parameter) meets the aforementioned lower bound up to a poly-logarithmic factor in $mn$. The expected error of MRE, unlike existing algorithms, tends to zero as the number of machines ($m$) goes to infinity, even when the number of samples per machine ($n$) remains upper bounded by a constant. We also address the problem of learning under tiny communication budget, and present lower and upper error bounds for the case that the budget $B$ is a constant.