Papers › Error Feedback Shines when Features are Rare
Error Feedback Shines when Features are Rare
Peter Richtárik, Elnur Gasanov, Konstantin Burlachenko
We provide the first proof that gradient descent (GD) with greedy sparsification (TopK) and error feedback (EF) can obtain better communication complexity than vanilla GD when solving the distributed optimization problem min_(x∈ℝᵈ) f(x)=1/n∑ᵢ₌₁ⁿ fᵢ(x), where n = # of clients, d = # of features, and f₁,…,fₙ are smooth nonconvex functions. Despite intensive research since 2014 when EF was first proposed by Seide et al., this problem remained open until now. We show that EF shines in the regime when features are rare, i.e., when each feature is present in the data owned by a small number of clients only. To illustrate our main result, we show that in order to find a random vector x̂ such that ‖∇f(x̂) ‖² ≤ε in expectation, GD with the Top1 sparsifier and EF requires O (( L+r √( c/n min( c/n maxᵢ Lᵢ², 1/n∑ᵢ₌₁ⁿ Lᵢ² ) )) 1/ε ) bits to be communicated by each worker to the server only, where L is the smoothness constant of f, Lᵢ is the smoothness constant of fᵢ, c is the maximal number of clients owning any feature (1≤c ≤n), and r is the maximal number of features owned by any client (1≤r ≤d). Clearly, the communication complexity improves as c decreases (i.e., as features become more rare), and can be much better than the O(r L 1/ε) communication complexity of GD in the same regime.
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