arXiv subjects

Vladimir Spokoiny

Publications and source records attributed to Vladimir Spokoiny.

At least 37 records · Page 2Linked to original sources

arXiv2022

Adaptive Weights Community Detection

Due to the technological progress of the last decades, Community Detection has become a major topic in machine learning. However, there is still a huge gap between practical and theoretical results, as theoretically optimal procedures often lack a feasible implementation and vice versa. This paper aims to close this gap and presents a novel algorithm that is both numerically and statistically efficient. Our procedure uses a test of homogeneity to compute adaptive weights describing local communities. The approach was inspired by the Adaptive Weights Community Detection (AWCD) algorithm by Adamyan et al. (2019). This algorithm delivered some promising results on artificial and real-life data, but our theoretical analysis reveals its performance to be suboptimal on a stochastic block model. In particular, the involved estimators are biased and the procedure does not work for sparse graphs. We propose significant modifications, addressing both shortcomings and achieving a nearly optimal rate of strong consistency on the stochastic block model. Our theoretical results are illustrated and validated by numerical experiments.

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arXiv2022

Adaptive Manifold Clustering

Clustering methods seek to partition data such that elements are more similar to elements in the same cluster than to elements in different clusters. The main challenge in this task is the lack of a unified definition of a cluster, especially for high dimensional data. Different methods and approaches have been proposed to address this problem. This paper continues the study originated by Efimov, Adamyan and Spokoiny (2019) where a novel approach to adaptive nonparametric clustering called Adaptive Weights Clustering (AWC) was offered. The method allows analyzing high-dimensional data with an unknown number of unbalanced clusters of arbitrary shape under very weak modeling assumptions. The procedure demonstrates a state-of-the-art performance and is very efficient even for large data dimension D. However, the theoretical study in Efimov, Adamyan and Spokoiny (2019) is very limited and did not really address the question of efficiency. This paper makes a significant step in understanding the remarkable performance of the AWC procedure, particularly in high dimension. The approach is based on combining the ideas of adaptive clustering and manifold learning. The manifold hypothesis means that high dimensional data can be well approximated by a d-dimensional manifold for small d helping to overcome the curse of dimensionality problem and to get sharp bounds on the cluster separation which only depend on the intrinsic dimension d. We also address the problem of parameter tuning. Our general theoretical results are illustrated by some numerical experiments.

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arXiv2022

Dimension free non-asymptotic bounds on the accuracy of high dimensional Laplace approximation

This note attempts to revisit the classical results on Laplace approximation in a modern non-asymptotic and dimension free form. Such an extension is motivated by applications to high dimensional statistical and optimization problems. The established results provide explicit non-asymptotic bounds on the quality of a Gaussian approximation of the posterior distribution in total variation distance in terms of the so called \emph{effective dimension} $ p_G $. This value is defined as interplay between information contained in the data and in the prior distribution. In the contrary to prominent Bernstein - von Mises results, the impact of the prior is not negligible and it allows to keep the effective dimension small or moderate even if the true parameter dimension is huge or infinite. We also address the issue of using a Gaussian approximation with inexact parameters with the focus on replacing the Maximum a Posteriori (MAP) value by the posterior mean and design the algorithm of Bayesian optimization based on Laplace iterations. The results are specified to the case of nonlinear inverse problem.

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arXiv2022

Reinforced optimal control

Least squares Monte Carlo methods are a popular numerical approximation method for solving stochastic control problems. Based on dynamic programming, their key feature is the approximation of the conditional expectation of future rewards by linear least squares regression. Hence, the choice of basis functions is crucial for the accuracy of the method. Earlier work by some of us [Belomestny, Schoenmakers, Spokoiny, Zharkynbay. Commun.~Math.~Sci., 18(1):109-121, 2020](arXiv:1808.02341) proposes to reinforce the basis functions in the case of optimal stopping problems by already computed value functions for later times, thereby considerably improving the accuracy with limited additional computational cost. We extend the reinforced regression method to a general class of stochastic control problems, while considerably improving the method's efficiency, as demonstrated by substantial numerical examples as well as theoretical analysis.

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arXiv2022

High dimensional change-point detection: a complete graph approach

The aim of online change-point detection is for a accurate, timely discovery of structural breaks. As data dimension outgrows the number of data in observation, online detection becomes challenging. Existing methods typically test only the change of mean, which omit the practical aspect of change of variance. We propose a complete graph-based, change-point detection algorithm to detect change of mean and variance from low to high-dimensional online data with a variable scanning window. Inspired by complete graph structure, we introduce graph-spanning ratios to map high-dimensional data into metrics, and then test statistically if a change of mean or change of variance occurs. Theoretical study shows that our approach has the desirable pivotal property and is powerful with prescribed error probabilities. We demonstrate that this framework outperforms other methods in terms of detection power. Our approach has high detection power with small and multiple scanning window, which allows timely detection of change-point in the online setting. Finally, we applied the method to financial data to detect change-points in S&P 500 stocks.

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arXiv2022

Structure-adaptive manifold estimation

We consider a problem of manifold estimation from noisy observations. Many manifold learning procedures locally approximate a manifold by a weighted average over a small neighborhood. However, in the presence of large noise, the assigned weights become so corrupted that the averaged estimate shows very poor performance. We suggest a structure-adaptive procedure, which simultaneously reconstructs a smooth manifold and estimates projections of the point cloud onto this manifold. The proposed approach iteratively refines the weights on each step, using the structural information obtained at previous steps. After several iterations, we obtain nearly "oracle" weights, so that the final estimates are nearly efficient even in the presence of relatively large noise. In our theoretical study, we establish tight lower and upper bounds proving asymptotic optimality of the method for manifold estimation under the Hausdorff loss, provided that the noise degrades to zero fast enough.

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arXiv2020

Manifold-based time series forecasting

Prediction for high dimensional time series is a challenging task due to the curse of dimensionality problem. Classical parametric models like ARIMA or VAR require strong modeling assumptions and time stationarity and are often overparametrized. This paper offers a new flexible approach using recent ideas of manifold learning. The considered model includes linear models such as the central subspace model and ARIMA as particular cases. The proposed procedure combines manifold denoising techniques with a simple nonparametric prediction by local averaging. The resulting procedure demonstrates a very reasonable performance for real-life econometric time series. We also provide a theoretical justification of the manifold estimation procedure.

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arXiv2020

Adaptive Gradient Descent for Convex and Non-Convex Stochastic Optimization

In this paper we propose several adaptive gradient methods for stochastic optimization. Unlike AdaGrad-type of methods, our algorithms are based on Armijo-type line search and they simultaneously adapt to the unknown Lipschitz constant of the gradient and variance of the stochastic approximation for the gradient. We consider an accelerated and non-accelerated gradient descent for convex problems and gradient descent for non-convex problems. In the experiments we demonstrate superiority of our methods to existing adaptive methods, e.g. AdaGrad and Adam.

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arXiv2020

Accuracy of Gaussian approximation in nonparametric Bernstein -- von Mises Theorem

The prominent Bernstein -- von Mises (BvM) result claims that the posterior distribution after centering by the efficient estimator and standardizing by the square root of the total Fisher information is nearly standard normal. In particular, the prior completely washes out from the asymptotic posterior distribution. This fact is fundamental and justifies the Bayes approach from the frequentist viewpoint. In the nonparametric setup the situation changes dramatically and the impact of prior becomes essential even for the contraction of the posterior; see [vdV2008], [Bo2011], [CaNi2013,CaNi2014] for different models like Gaussian regression or i.i.d. model in different weak topologies. This paper offers another non-asymptotic approach to studying the behavior of the posterior for a special but rather popular and useful class of statistical models and for Gaussian priors. First we derive tight finite sample bounds on posterior contraction in terms of the so called effective dimension of the parameter space. Our main results describe the accuracy of Gaussian approximation of the posterior. In particular, we show that restricting to the class of all centrally symmetric credible sets around pMLE allows to get Gaussian approximation up to order (n^{-1}). We also show that the posterior distribution mimics well the distribution of the penalized maximum likelihood estimator (pMLE) and reduce the question of reliability of credible sets to consistency of the pMLE-based confidence sets. The obtained results are specified for nonparametric log-density estimation and generalized regression.

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arXiv2020

Bayesian inference for nonlinear inverse problems

Bayesian methods are actively used for parameter identification and uncertainty quantification when solving nonlinear inverse problems with random noise. However, there are only few theoretical results justifying the Bayesian approach. Recent papers, see e.g. \cite{Nickl2017,lu2017bernsteinvon} and references therein, illustrate the main difficulties and challenges in studying the properties of the posterior distribution in the nonparametric setup. This paper offers a new approach for study the frequentist properties of the nonparametric Bayes procedures. The idea of the approach is to relax the nonlinear structural equation by introducing an auxiliary functional parameter and replacing the structural equation with a penalty and by imposing a prior on the auxiliary parameter. For the such extended model, we state sharp bounds on posterior concentration and on the accuracy of the penalized MLE and on Gaussian approximation of the posterior, and a number of further results. All the bounds are given in terms of effective dimension, and we show that the proposed calming device does not significantly affect this value.

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arXiv2019

An adaptive multiclass nearest neighbor classifier

We consider a problem of multiclass classification, where the training sample $S_n = \{(X_i, Y_i)\}_{i=1}^n$ is generated from the model $\mathbb P(Y = m | X = x) = η_m(x)$, $1 \leq m \leq M$, and $η_1(x), \dots, η_M(x)$ are unknown $α$-Holder continuous functions.Given a test point $X$, our goal is to predict its label. A widely used $\mathsf k$-nearest-neighbors classifier constructs estimates of $η_1(X), \dots, η_M(X)$ and uses a plug-in rule for the prediction. However, it requires a proper choice of the smoothing parameter $\mathsf k$, which may become tricky in some situations. In our solution, we fix several integers $n_1, \dots, n_K$, compute corresponding $n_k$-nearest-neighbor estimates for each $m$ and each $n_k$ and apply an aggregation procedure. We study an algorithm, which constructs a convex combination of these estimates such that the aggregated estimate behaves approximately as well as an oracle choice. We also provide a non-asymptotic analysis of the procedure, prove its adaptation to the unknown smoothness parameter $α$ and to the margin and establish rates of convergence under mild assumptions.

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arXiv2019

Optimal stopping via reinforced regression

In this note we propose a new approach towards solving numerically optimal stopping problems via reinforced regression based Monte Carlo algorithms. The main idea of the method is to reinforce standard linear regression algorithms in each backward induction step by adding new basis functions based on previously estimated continuation values. The proposed methodology is illustrated by a numerical example from mathematical finance.

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arXiv2019

Bayesian inference for spectral projectors of the covariance matrix

Let $X_1, \ldots, X_n$ be i.i.d. sample in $\mathbb{R}^p$ with zero mean and the covariance matrix $\mathbf{Σ^*}$. The classical PCA approach recovers the projector $\mathbf{P^*_{\mathcal{J}}}$ onto the principal eigenspace of $\mathbf{Σ^*}$ by its empirical counterpart $\mathbf{\widehat{P}_{\mathcal{J}}}$. Recent paper [Koltchinskii, Lounici (2017)] investigated the asymptotic distribution of the Frobenius distance between the projectors $\| \mathbf{\widehat{P}_{\mathcal{J}}} - \mathbf{P^*_{\mathcal{J}}} \|_2$, while [Naumov et al. (2017)] offered a bootstrap procedure to measure uncertainty in recovering this subspace $\mathbf{P^*_{\mathcal{J}}}$ even in a finite sample setup. The present paper considers this problem from a Bayesian perspective and suggests to use the credible sets of the pseudo-posterior distribution on the space of covariance matrices induced by the conjugated Inverse Wishart prior as sharp confidence sets. This yields a numerically efficient procedure. Moreover, we theoretically justify this method and derive finite sample bounds on the corresponding coverage probability. Contrary to [Koltchinskii, Lounici (2017), Naumov et al. (2017)], the obtained results are valid for non-Gaussian data: the main assumption that we impose is the concentration of the sample covariance $\mathbf{\widehatΣ}$ in a vicinity of $\mathbf{Σ^*}$. Numerical simulations illustrate good performance of the proposed procedure even on non-Gaussian data in a rather challenging regime.

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arXiv2019

Gaussian processes with multidimensional distribution inputs via optimal transport and Hilbertian embedding

In this work, we investigate Gaussian Processes indexed by multidimensional distributions. While directly constructing radial positive definite kernels based on the Wasserstein distance has been proven to be possible in the unidimensional case, such constructions do not extend well to the multidimensional case as we illustrate here. To tackle the problem of defining positive definite kernels between multivariate distributions based on optimal transport, we appeal instead to Hilbert space embeddings relying on optimal transport maps to a reference distribution, that we suggest to take as a Wasserstein barycenter. We characterize in turn radial positive definite kernels on Hilbert spaces, and show that the covariance parameters of virtually all parametric families of covariance functions are microergodic in the case of (infinite-dimensional) Hilbert spaces. We also investigate statistical properties of our suggested positive definite kernels on multidimensional distributions, with a focus on consistency when a population Wasserstein barycenter is replaced by an empirical barycenter and additional explicit results in the special case of Gaussian distributions. Finally, we study the Gaussian process methodology based on our suggested positive definite kernels in regression problems with multidimensional distribution inputs, on simulation data stemming both from synthetic examples and from a mechanical engineering test case.

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arXiv2019

Statistical inference for Bures-Wasserstein barycenters

In this work we introduce the concept of Bures-Wasserstein barycenter $Q_*$, that is essentially a Fréchet mean of some distribution $\mathbb{P}$ supported on a subspace of positive semi-definite Hermitian operators $\mathbb{H}_{+}(d)$. We allow a barycenter to be restricted to some affine subspace of $\mathbb{H}_{+}(d)$ and provide conditions ensuring its existence and uniqueness. We also investigate convergence and concentration properties of an empirical counterpart of $Q_*$ in both Frobenius norm and Bures-Wasserstein distance, and explain, how obtained results are connected to optimal transportation theory and can be applied to statistical inference in quantum mechanics.

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arXiv2018

Multivariate Brenier cumulative distribution functions and their application to non-parametric testing

In this work we introduce a novel approach of construction of multivariate cumulative distribution functions, based on cyclical-monotone mapping of an original measure $μ\in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ to some target measure $ν\in \mathcal{P}^{ac}_2(\mathbb{R}^d)$ , supported on a convex compact subset of $\mathbb{R}^d$. This map is referred to as $ν$-Brenier distribution function ($ν$-BDF), whose counterpart under the one-dimensional setting $d = 1$ is an ordinary CDF, with $ν$ selected as $\mathcal{U}[0, 1]$, a uniform distribution on $[0, 1]$. Following one-dimensional frame-work, a multivariate analogue of Glivenko-Cantelli theorem is provided. A practical applicability of the theory is then illustrated by the development of a non-parametric pivotal two-sample test, that is rested on $2$-Wasserstein distance.

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arXiv2018

Instrumental variables regression

IV regression in the context of a re-sampling is considered in the work. Comparatively, the contribution in the development is a structural identification in the IV model. The work also contains a multiplier-bootstrap justification.

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arXiv2018

Toolbox: Gaussian comparison on Eucledian balls

In the work a characterization of difference of multivariate Gaussian measures is found on the family of centered Eucledian balls. In particular, it helps to bound corresponding Kolmogorov distance.

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