arXiv subjects

Dmitrii M. Ostrovskii

Publications and source records attributed to Dmitrii M. Ostrovskii.

9 recordsLinked to original sources

arXiv2026

Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain

We study the problem of finding approximate first-order stationary points in optimization problems of the form $\min_{x \in X} \max_{y \in Y} f(x,y)$, where the sets $X,Y$ are convex and $Y$ is compact. The objective function $f$ is smooth, but assumed neither convex in $x$ nor concave in $y$. Our approach relies upon replacing the function $f(x,\cdot)$ with its $k$th order Taylor approximation (in $y$) and finding a near-stationary point in the resulting surrogate problem. To guarantee its success, we establish the following result: let the Euclidean diameter of $Y$ be small in terms of the target accuracy $\varepsilon$, namely $O(\varepsilon^{\frac{2}{k+1}})$ for $k \in \mathbb{N}$ and $O(\varepsilon)$ for $k = 0$, with the constant factors controlled by certain regularity parameters of $f$; then any $\varepsilon$-stationary point in the surrogate problem remains $O(\varepsilon)$-stationary for the initial problem. Moreover, we show that these upper bounds are nearly optimal: the aforementioned reduction provably fails when the diameter of $Y$ is larger. For $0 \le k \le 2$ the surrogate function can be efficiently maximized in $y$; our general approximation result then leads to efficient algorithms for finding a near-stationary point in nonconvex-nonconcave min-max problems, for which we also provide convergence guarantees.

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arXiv2026

Entropy-Smooth Convex Optimization Cannot Be Accelerated

We prove an $Ω(L/T)$ lower bound for the convergence rate of minimization in the class of functions that are convex and $L$-smooth relative to negative entropy on the standard $d$-simplex, valid for every first-order method when $d = Ω(T^2)$. In particular, this shows that mirror descent is optimal up to a logarithmic factor in this class. This may be surprising due to the fact that accelerated methods are readily available under the assumption of smoothness in $\ell_1$-norm. While Dragomir et al. (Mathematical Programming, 2022) have already showed that acceleration might be impossible under relative smoothness, their prox-function is pathological and constructed together with the hard instance. In contrast, we show non-acceleration for a specific prox-function with particularly favorable structure. We also extend the result to the quantum setting, proving the same lower bound in the class of functions $L$-smooth relative to negative von Neumann entropy on the spectrahedron of $d \times d$ Hermitian positive-semidefinite matrices with unit trace.

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arXiv2026

Near-Optimal and Tractable Estimation under Shift-Invariance

How hard is it to estimate a discrete-time signal $(x_{1}, ..., x_{n}) \in \mathbb{C}^n$ satisfying an unknown linear recurrence relation of order $s$ and observed in i.i.d. complex Gaussian noise? The class of all such signals is parametric but extremely rich: it contains all exponential polynomials over $\mathbb{C}$ with total degree $s$, including harmonic oscillations with $s$ arbitrary frequencies. Geometrically, this class corresponds to the projection onto $\mathbb{C}^{n}$ of the union of all shift-invariant subspaces of $\smash{\mathbb{C}^\mathbb{Z}}$ of dimension $s$. We show that the statistical complexity of this class, as measured by the squared minimax radius of the $(1-δ)$-confidence $\ell_2$-ball, is nearly the same as for the class of $s$-sparse signals, namely $\smash{O\left(s\log(en) + \log(δ^{-1})\right) \cdot \log^2(es) \cdot \log(en/s).}$ Moreover, the corresponding near-minimax estimator is tractable, and it can be used to build a test statistic with a near-minimax detection threshold in the associated detection problem. These statistical results rely upon a simple analytic observation: the interpretation of the Fourier coefficients of the Christoffel function of any shift-invariant subspace of $\smash{\mathbb{C}^\mathbb{Z}}$ as a reproducing filter with the smallest possible spectrum in all $\ell_p$-norms, $p \in [1,\infty]$, at once.

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arXiv2026

Admittance Matrix Concentration Inequalities for Understanding Uncertain Power Networks

This paper presents conservative probabilistic bounds for the spectrum of the admittance matrix and classical linear power flow models under uncertain network parameters; for example, probabilistic line contingencies. Our proposed approach imports tools from probability theory, such as concentration inequalities for random matrices. This provides a theoretical framework for understanding error bounds of common approximations of the AC power flow equations under parameter uncertainty, including the DC and LinDistFlow approximations. Additionally, we show that the upper bounds scale as functions of nodal criticality. This network-theoretic quantity captures how uncertainty concentrates at critical nodes for use in contingency analysis. We validate these bounds on IEEE test networks, demonstrating that they correctly capture the scaling behavior of spectral perturbations up to conservative constants.

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arXiv2025

Efficient Network Reconfiguration by Randomized Switching

We present an algorithm that efficiently computes nearly-optimal solutions to a class of combinatorial reconfiguration problems on weighted, undirected graphs. Inspired by societally relevant applications in networked infrastructure systems, these problems consist of simultaneously finding an unreweighted sparsified graph and nodal potentials that satisfy fixed demands, where the objective is to minimize some congestion criterion, e.g., a Laplacian quadratic form. These are mixed-integer nonlinear programming problems that are NP-hard in general. To circumvent these challenges, instead of solving for a single best configuration, the proposed randomized switching algorithm seeks to design a distribution of configurations that, when sampled, ensures that congestion concentrates around its optimum. We show that the proposed congestion metric is a generalized self-concordant function in the space of switching probabilities, which enables the use of efficient and simple conditional gradient methods. We implement our algorithm and show that it outperforms a state-of-the-art commercial mixed-integer second-order cone programming (MISOCP) solver by orders of magnitude over a large range of problem sizes.

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arXiv2025

Amplitude maximization in stable systems, Schur positivity, and some conjectures on polynomial interpolation

For $r > 0$ and integers $t \ge n > 0$, we consider the following problem: maximize the amplitude $|x_t|$ at time $t$, over all complex solutions $x = (x_0, x_1, \dots)$ of arbitrary homogeneous linear difference equations of order $n$ with the characteristic roots in the disc $\{z \in \mathbb{C}: |z| \le r\}$, and with initial values $x_0, \dots, x_{n-1}$ in the unit disc. We find that for any triple $t,n,r$, the maximum is attained with coinciding roots on the boundary circle; in particular, this implies that the peak amplitude $\sup_{t \ge n} |x_t|$ can be maximized explicitly, by studying a unique equation with the characteristic polynomial $(z-r)^n$. Moreover, the optimality of the cophase root configuration holds for origin-centered polydiscs. To prove this result, we first reduce the problem to a certain interpolation problem over monomials, then solve the latter by leveraging the theory of symmetric functions and identifying the associated Schur positivity structure. We also discuss the implications for more general Reinhardt domains. Finally, we study the problem of estimating the derivatives of a real entire function from its values at $n/2$ pairs of complex conjugate points in the unit disc. We propose conjectures on the extremality of the monomial $z^n$, and restate them in terms of Schur polynomials.

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arXiv2025

Efficient and Near-Optimal Online Portfolio Selection

In the problem of online portfolio selection as formulated by Cover (1991), the trader repeatedly distributes her capital over $ d $ assets in each of $ T > 1 $ rounds, with the goal of maximizing the total return. Cover proposed an algorithm, termed Universal Portfolios, that performs nearly as well as the best (in hindsight) static assignment of a portfolio, with an $ O(d\log(T)) $ regret in terms of the logarithmic return. Without imposing any restrictions on the market this guarantee is known to be worst-case optimal, and no other algorithm attaining it has been discovered so far. Unfortunately, Cover's algorithm crucially relies on computing certain $ d $-dimensional integral which must be approximated in any implementation; this results in a prohibitive $ \tilde O(d^4(T+d)^{14}) $ per-round runtime for the fastest known implementation due to Kalai and Vempala (2002). We propose an algorithm for online portfolio selection that admits essentially the same regret guarantee as Universal Portfolios -- up to a constant factor and replacement of $ \log(T) $ with $ \log(T+d) $ -- yet has a drastically reduced runtime of $ \tilde O(d^2(T+d)) $ per round. The selected portfolio minimizes the current logarithmic loss regularized by the log-determinant of its Hessian -- equivalently, the hybrid logarithmic-volumetric barrier of the polytope specified by the asset return vectors. As such, our work reveals surprising connections of online portfolio selection with two classical topics in optimization theory: cutting-plane and interior-point algorithms.

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arXiv2021

Near-Optimal Procedures for Model Discrimination with Non-Disclosure Properties

Let $θ_0,θ_1 \in \mathbb{R}^d$ be the population risk minimizers associated to some loss $\ell:\mathbb{R}^d\times \mathcal{Z}\to\mathbb{R}$ and two distributions $\mathbb{P}_0,\mathbb{P}_1$ on $\mathcal{Z}$. The models $θ_0,θ_1$ are unknown, and $\mathbb{P}_0,\mathbb{P}_1$ can be accessed by drawing i.i.d samples from them. Our work is motivated by the following model discrimination question: "What sizes of the samples from $\mathbb{P}_0$ and $\mathbb{P}_1$ allow to distinguish between the two hypotheses $θ^*=θ_0$ and $θ^*=θ_1$ for given $θ^*\in\{θ_0,θ_1\}$?" Making the first steps towards answering it in full generality, we first consider the case of a well-specified linear model with squared loss. Here we provide matching upper and lower bounds on the sample complexity as given by $\min\{1/Δ^2,\sqrt{r}/Δ\}$ up to a constant factor; here $Δ$ is a measure of separation between $\mathbb{P}_0$ and $\mathbb{P}_1$ and $r$ is the rank of the design covariance matrix. We then extend this result in two directions: (i) for general parametric models in asymptotic regime; (ii) for generalized linear models in small samples ($n\le r$) under weak moment assumptions. In both cases we derive sample complexity bounds of a similar form while allowing for model misspecification. In fact, our testing procedures only access $θ^*$ via a certain functional of empirical risk. In addition, the number of observations that allows us to reach statistical confidence does not allow to "resolve" the two models $-$ that is, recover $θ_0,θ_1$ up to $O(Δ)$ prediction accuracy. These two properties allow to use our framework in applied tasks where one would like to $\textit{identify}$ a prediction model, which can be proprietary, while guaranteeing that the model cannot be actually $\textit{inferred}$ by the identifying agent.

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arXiv2021

Efficient Search of First-Order Nash Equilibria in Nonconvex-Concave Smooth Min-Max Problems

We propose an efficient algorithm for finding first-order Nash equilibria in min-max problems of the form $\min_{x \in X}\max_{y\in Y} F(x,y)$, where the objective function is smooth in both variables and concave with respect to $y$; the sets $X$ and $Y$ are convex and "projection-friendly," and $Y$ is compact. Our goal is to find an $(\varepsilon_x,\varepsilon_y)$-first-order Nash equilibrium with respect to a stationarity criterion that is stronger than the commonly used proximal gradient norm. The proposed approach is fairly simple: we perform approximate proximal-point iterations on the primal function, with inexact oracle provided by Nesterov's algorithm run on the regularized function $F(x_t,\cdot)$, $x_t$ being the current primal iterate. The resulting iteration complexity is $O(\varepsilon_x{}^{-2} \varepsilon_y{}^{-1/2})$ up to a logarithmic factor. As a byproduct, the choice $\varepsilon_y = O(\varepsilon_x{}^2)$ allows for the $O(\varepsilon_x{}^{-3})$ complexity of finding an $\varepsilon_x$-stationary point for the standard Moreau envelope of the primal function. Moreover, when the objective is strongly concave with respect to $y$, the complexity estimate for our algorithm improves to $O(\varepsilon_x{}^{-2}{κ_y}^{1/2})$ up to a logarithmic factor, where $κ_y$ is the condition number appropriately adjusted for coupling. In both scenarios, the complexity estimates are the best known so far, and are only known for the (weaker) proximal gradient norm criterion. Meanwhile, our approach is "user-friendly:" (i) the algorithm is built upon running a variant of Nesterov's accelerated algorithm as subroutine and avoids extragradient steps; (ii) the convergence analysis recycles the well-known results on accelerated methods with inexact oracle. Finally, we extend the approach to non-Euclidean proximal geometries.

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