Statistical learning theory for dependent data such as time series and possibly other dependency structures.

Non-stationary, non-asymptotic bounds please. Keywords: Ergodic, α-, β-mixing.

Mohri and Kuznetsov have done lots of work here; See, e.g. their NIPS2016 tutorial, or KuMo16.

There seem to be a lot of types of ergodic/mixing results, about which I know as yet nothing. Notably KuMo15 and KuMo16 try to go beyond this setup.

Overview in McSS11a:

Yu (Yu94) sets forth many of the uniform ergodic theorems that are needed to derive generalization error bounds for stochastic processes. Meir (Meir00) is one of the first papers to construct risk bounds for time series. […]

More recently, others have provided PAC results for non-IID data. Steinwart and Christmann (StCh09) prove an oracle inequality for generic regularized empirical risk minimization algorithms learning from α-mixing processes, a fairly general sort of weak serial dependence, getting learning rates for least-squares support vector machines (SVMs) close to the optimal IID rates. Mohri and Rostamizadeh (MoRo09b) prove stability-based generalization bounds when the data are stationary and φ-mixing or β-mixing, strictly generalizing IID results and applying to all stable learning algorithms. […] Karandikar and Vidyasagar (KaVi00) show that if an algorithm is “subadditive” and yields a predictor whose risk can be upper bounded when the data are IID, then the same algorithm yields predictors whose risk can be bounded if data are β-mixing. They use this result to derive generalization error bounds in terms of the learning rates for IID data and the β-mixing coefficients.

See also Geer02.

## Refs

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*Dependence Modeling*, 1, 65–93. DOI. - AlWi12
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*Bernoulli*. - CKMY16
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