The statistical inference problem within the risk minimization framework is then to compute an approximate minimizer to the above program when given access to samples D n = . Θ * = arg min θ ∈ Θ R ( θ ) ≡ arg min θ ∈ Θ E z ∼ P, (1)where L ¯ is an appropriate loss function, R is the population risk and Θ is the set of feasible parameters. Despite this progress, however, the statistical methods with the strongest robustness guarantees are computationally intractable, for instance those based on non-convex M-estimators (Huber, 1981), l 1-tournaments (Devroye and Györfi, 1985 Yatracos, 1985 Donoho and Liu, 1988) and notions of depth (Mizera, 2002 Gao, 2017 Chen et al., 2018). These have led to rich statistical concepts such as the influence function, the breakdown point and the Huber ε-contamination model, to assess the robustness of estimators. Strong model assumptions are rarely met in practice, and this has motivated the development of robust inferential procedures, and which has a rich history in statistics with seminal contributions due to Box ( 1953), Tukey ( 1975), Huber ( 1981), Hampel ( 2011) and several others. ![]() In classical analyses of statistical estimators, statistical guarantees are derived under strong model assumptions, and in most cases these guarantees hold only in the absence of arbitrary outliers, and other deviations from the model assumptions. ![]() ![]() Heavy tails, Huber contamination, Outliers, Robust gradients, Robustness 1 Introduction
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