Minimax Bounds for Distributed Logistic Regression
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We consider a distributed logistic regression problem where labeled data pairs (X_i,Y_i)∈R^d×{-1,1} for i=1,...,n are distributed across multiple machines in a network and must be communicated to a centralized estimator using at most k bits per labeled pair. We assume that the data X_i come independently from some distribution P_X, and that the distribution of Y_i conditioned on X_i follows a logistic model with some parameter θ∈R^d. By using a Fisher information argument, we give minimax lower bounds for estimating θ under different assumptions on the tail of the distribution P_X. We consider both ℓ^2 and logistic losses, and show that for the logistic loss our sub-Gaussian lower bound is order-optimal and cannot be improved.
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