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Empirical process theory for i.i.d. observations has emerged as a ubiquitoustool for understanding the generalization properties of various statisticalproblems. However, in many applications where the data exhibit temporaldependencies (e.g., in finance, medical imaging, weather forecasting etc.), thecorresponding empirical processes are much less understood. Motivated by thisobservation, we present a general bound on the expected supremum of empiricalprocesses under standard $\beta/\rho$-mixing assumptions. Unlike most priorwork, our results cover both the long and the short-range regimes ofdependence. Our main result shows that a non-trivial trade-off between thecomplexity of the underlying function class and the dependence among theobservations characterizes the learning rate in a large class of nonparametricproblems. This trade-off reveals a new phenomenon, namely that even underlong-range dependence, it is possible to attain the same rates as in the i.i.d.setting, provided the underlying function class is complex enough. Wedemonstrate the practical implications of our findings by analyzing variousstatistical estimators in both fixed and growing dimensions. Our main examplesinclude a comprehensive case study of generalization error bounds innonparametric regression over smoothness classes in fixed as well as growingdimension using neural nets, shape-restricted multivariate convex regression,estimating the optimal transport (Wasserstein) distance between two probabilitydistributions, and classification under the Mammen-Tsybakov margin condition --all under appropriate mixing assumptions. In the process, we also developbounds on $L_r$ ($1\le r\le 2$)-localized empirical processes with dependentobservations, which we then leverage to get faster rates for (a) tuning-freeadaptation, and (b) set-structured learning problems.

Trade-off Between Dependence and Complexity for Nonparametric Learning -- an Empirical Process Approach