TY  - JOUR
A1  - Blanchard, Gilles
A1  - Zadorozhnyi, Oleksandr
T1  - Concentration of weakly dependent Banach-valued sums and applications to statistical learning methods
JF  - Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
N2  - We obtain a Bernstein-type inequality for sums of Banach-valued random variables satisfying a weak dependence assumption of general type and under certain smoothness assumptions of the underlying Banach norm. We use this inequality in order to investigate in the asymptotical regime the error upper bounds for the broad family of spectral regularization methods for reproducing kernel decision rules, when trained on a sample coming from a tau-mixing process.
KW  - Banach-valued process
KW  - Bernstein inequality
KW  - concentration
KW  - spectral regularization
KW  - weak dependence
Y1  - 2019
U6  - https://doi.org/10.3150/18-BEJ1095
SN  - 1350-7265
SN  - 1573-9759
VL  - 25
IS  - 4B
SP  - 3421
EP  - 3458
PB  - International Statistical Institute
CY  - Voorburg
ER  - 
TY  - THES
A1  - Zadorozhnyi, Oleksandr
T1  - Contributions to the theoretical analysis of the algorithms with adversarial and dependent data
N2  - In this work I present the concentration inequalities of Bernstein's type for the norms of Banach-valued random sums under a general 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 the 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. Afterwards, in the framework of sequential learning, the multi-armed bandit problem under $\cC-$mixing assumption on the arm's outputs is considered and the complete regret analysis of a version of Improved UCB algorithm is given. Lastly, 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).
KW  - Machine learning
KW  - nonparametric regression
KW  - kernel methods
KW  - regularisation
KW  - concentration inequalities
KW  - learning rates
KW  - sequential learning
KW  - multi-armed bandits
KW  - Sobolev spaces
Y1  - 2021
ER  -