Artur Goldman

Denis Belomestny, Artur Goldman, Alexey Naumov et al.

In this paper, we propose a variance reduction approach for Markov chains based on additive control variates and the minimization of an appropriate estimate for the asymptotic variance. We focus on the particular case when control variates are represented as deep neural networks. We derive the optimal convergence rate of the asymptotic variance under various ergodicity assumptions on the underlying Markov chain. The proposed approach relies upon recent results on the stochastic errors of variance reduction algorithms and function approximation theory.

Louis Claeys, Artur Goldman, Zebang Shen et al.

High-dimensional stochastic optimal control (SOC) becomes harder with longer planning horizons: existing methods scale linearly in the horizon $T$, with performance often deteriorating exponentially. We overcome these limitations for a subclass of linearly-solvable SOC problems-those whose uncontrolled drift is the gradient of a potential. In this setting, the Hamilton-Jacobi-Bellman equation reduces to a linear PDE governed by an operator $\mathcal{L}$. We prove that, under the gradient drift assumption, $\mathcal{L}$ is unitarily equivalent to a Schrödinger operator $\mathcal{S} = -Δ+ \mathcal{V}$ with purely discrete spectrum, allowing the long-horizon control to be efficiently described via the eigensystem of $\mathcal{L}$. This connection provides two key results: first, for a symmetric linear-quadratic regulator (LQR), $\mathcal{S}$ matches the Hamiltonian of a quantum harmonic oscillator, whose closed-form eigensystem yields an analytic solution to the symmetric LQR with \emph{arbitrary} terminal cost. Second, in a more general setting, we learn the eigensystem of $\mathcal{L}$ using neural networks. We identify implicit reweighting issues with existing eigenfunction learning losses that degrade performance in control tasks, and propose a novel loss function to mitigate this. We evaluate our method on several long-horizon benchmarks, achieving an order-of-magnitude improvement in control accuracy compared to state-of-the-art methods, while reducing memory usage and runtime complexity from $\mathcal{O}(Td)$ to $\mathcal{O}(d)$.

Artur Goldman, Nikita Puchkin, Valeriia Shcherbakova et al.

We suggest a novel procedure for online change point detection. Our approach expands an idea of maximizing a discrepancy measure between points from pre-change and post-change distributions. This leads to a flexible procedure suitable for both parametric and nonparametric scenarios. We prove non-asymptotic bounds on the average running length of the procedure and its expected detection delay. The efficiency of the algorithm is illustrated with numerical experiments on synthetic and real-world data sets.