Valeriy Avanesov

Valeriy Avanesov

The ever-growing size of the datasets renders well-studied learning techniques, such as Kernel Ridge Regression, inapplicable, posing a serious computational challenge. Divide-and-conquer is a common remedy, suggesting to split the dataset into disjoint partitions, obtain the local estimates and average them, it allows to scale-up an otherwise ineffective base approach. In the current study we suggest a fully data-driven approach to quantify uncertainty of the averaged estimator. Namely, we construct simultaneous element-wise confidence bands for the predictions yielded by the averaged estimator on a given deterministic prediction set. The novel approach features rigorous theoretical guaranties for a wide class of base learners with Kernel Ridge regression being a special case. As a by-product of our analysis we also obtain a sup-norm consistency result for the divide-and-conquer Kernel Ridge Regression. The simulation study supports the theoretical findings.

Valeriy Avanesov

Gaussian Process Regression and Kernel Ridge Regression are popular nonparametric regression approaches. Unfortunately, they suffer from high computational complexity rendering them inapplicable to the modern massive datasets. To that end a number of approximations have been suggested, some of them allowing for a distributed implementation. One of them is the divide and conquer approach, splitting the data into a number of partitions, obtaining the local estimates and finally averaging them. In this paper we suggest a novel computationally efficient fully data-driven algorithm, quantifying uncertainty of this method, yielding frequentist $L_2$-confidence bands. We rigorously demonstrate validity of the algorithm. Another contribution of the paper is a minimax-optimal high-probability bound for the averaged estimator, complementing and generalizing the known risk bounds.

Valeriy Avanesov

In $\mathcal{X}$-armed bandit problem an agent sequentially interacts with environment which yields a reward based on the vector input the agent provides. The agent's goal is to maximise the sum of these rewards across some number of time steps. The problem and its variations have been a subject of numerous studies, suggesting sub-linear and some times optimal strategies. The given paper introduces a novel variation of the problem. We consider an environment, which can abruptly change its behaviour an unknown number of times. To that end we propose a novel strategy and prove it attains sub-linear cumulative regret. Moreover, in case of highly smooth relation between an action and the corresponding reward, the method is nearly optimal. The theoretical result are supported by experimental study.