Minimax Excess Risk of First-Order Methods for Statistical Learning with Data-Dependent Oracles
arxiv(2023)
摘要
In this paper, our aim is to analyse the generalization capabilities of
first-order methods for statistical learning in multiple, different yet
related, scenarios including supervised learning, transfer learning, robust
learning and federated learning. To do so, we provide sharp upper and lower
bounds for the minimax excess risk of strongly convex and smooth statistical
learning when the gradient is accessed through partial observations given by a
data-dependent oracle. This novel class of oracles can query the gradient with
any given data distribution, and is thus well suited to scenarios in which the
training data distribution does not match the target (or test) distribution. In
particular, our upper and lower bounds are proportional to the smallest mean
square error achievable by gradient estimators, thus allowing us to easily
derive multiple sharp bounds in the aforementioned scenarios using the
extensive literature on parameter estimation.
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