Contributions to the theoretical analysis of algorithms with adversarial and dependent data.

08.07.2021, 10h30
Verteidigung Dissertation / PhD Defence

Oleksandr Zadorozhnyi (Potsdam)

I overview the results contained in my thesis that studies theoretical guarantees of both sequential and batch learning algorithms that work with adversarial and dependent data. Firstly, I present concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under functional weak-dependency assumption (the so-called $\cC-$mixing). The latter is then used to prove, in the asymptotic framework, excess risk upper bounds of the regularised Hilbert valued statistical learning rules under τ-mixing assumption on the underlying training sample. These results (of the batch statistical setting) are then supplemented with the regret analysis over the classes of Sobolev balls of the type of kernel ridge regression algorithm in the setting of online nonparametric regression with arbitrary data sequences. Here, in particular, a question of robustness of the kernel-based forecaster is investigated. From the other side, probabilistic inequalities of the first part are extended to the case of deviations (both of Azuma-Hoeffding's and of Burkholder's type) to the partial sums of real-valued weakly dependent random fields (under the type of projective dependence condition). Lastly, in the framework of sequential learning, the multi-armed bandit problem under $\cC-$mixing assumption on the arm's outputs is considered and complete regret analysis of a version of Improved UCB algorithm is given.

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