We consider distributed stochastic optimization problems that are solved with master/workers computation architecture. Statistical arguments allow to exploit statistical similarity and approximate this problem by a finite-sum problem, for which we propose an inexact accelerated cubicregularized Newton's method that achieves lower communication complexity bound for this setting and improves upon existing upper bound. We further exploit this algorithm to obtain convergence rate bounds for the original stochastic optimization problem and compare our bounds with the existing bounds in several regimes when the goal is to minimize the number of communication rounds and increase the parallelization by increasing the number of workers.

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Statistical preconditioning can be used to design fast methods for distributed large-scale empirical risk minimization problems, for strongly convex and smooth loss functions, allowing fewer communication rounds. Multiple worker nodes compute gradients in parallel, which are then used by the central node to update the parameter by solving an auxiliary (preconditioned) smallerscale optimization problem. However, previous works require an exact solution of an auxiliary optimization problem by the central node at every iteration, which may be impractical. This paper proposes a method that allows the inexact solution of the auxiliary problem, reducing the total computation time. Moreover, for loss functions with high-order smoothness, we exploit the structure of the auxiliary problem and propose a hyperfast second-order method with complexity Õ(κ 1/5 ), where κ is the local condition number.

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Electricity production currently generates approximately 25% of greenhouse gas emissions in the USA. Thus, increasing the amount of renewable energy is a key step to carbon neutrality. However, integrating a large amount of fluctuating renewable generation is a significant challenge for power grid operating and planning. Grid reliability, i.e. an ability to meet operational constraints under power fluctuations, is probably the most important of them. In this paper, we propose computationally efficient and accurate methods to estimate the probability of failure, i.e. reliability constraints violation, under a known distribution of renewable energy generation. To this end, we investigate an importance sampling approach, a flexible extension of Monte-Carlo methods, which adaptively changes the sampling distribution to generate more samples near the reliability boundary. The approach allows to estimate failure probability in real-time based only on a few dozens of random samples, compared to thousands required by the plain Monte-Carlo. Our study focuses on high voltage direct current power transmission grids with linear reliability constraints on power injections and line currents. We propose a novel theoretically justified physicsinformed adaptive importance sampling algorithm and compare its performance to state-of-the-art methods on multiple IEEE power grid test cases.