Publications

2026

1. arXiv

   Minimax Estimation of Kernel Stein Discrepancy: Trace versus Hilbert-Schmidt Scales

   Davit Gogolashvili

   arXiv preprint arXiv:2607.03367, 2026

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   Kernel Stein Discrepancy (KSD) compares a sample to a fixed target distribution known only through its score, and is widely used for goodness-of-fit testing, sample quality assessment, and approximate inference. We study the estimation of $\operatorname{KSD}(P_0,P)$ from $n$ independent observations and identify the sharp spectral constant governing the minimax risk: it is the Hilbert-Schmidt norm of the Stein covariance operator $C_\star$, giving the minimax scale $\sqrt{\|C_\star\|_{\mathrm{HS}}/n}$. This scale is attained by the positive-part square-root U-statistic, whereas the standard plug-in V-statistic remains at the trace scale $\sqrt{\operatorname{tr}(C_\star)/n}$ and is therefore suboptimal by the fourth root of the effective rank of $C_\star$; for a Gaussian target with a fixed-bandwidth Gaussian kernel this factor is exponential in the dimension.

2. AISTATS

   Local Regression on Path Spaces with Signature Metrics

   Christian Bayer, Davit Gogolashvili, and Luca Pelizzari

   In Proceedings of The 29th International Conference on Artificial Intelligence and Statistics, 2026

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   We study nonparametric regression and classification for path-valued data. We introduce a functional Nadaraya-Watson estimator that combines the signature transform from rough path theory with local kernel regression. The signature transform provides a principled way to encode sequential data through iterated integrals, enabling direct comparison of paths in a natural metric space. Our approach leverages signature-induced distances within the classical kernel regression framework, achieving computational efficiency while avoiding the scalability bottlenecks of large-scale kernel matrix operations. We establish finite-sample convergence bounds demonstrating favorable statistical properties of signature-based distances compared to traditional metrics in infinite-dimensional settings. We propose robust signature variants that provide stability against outliers, enhancing practical performance. Applications to both synthetic and real-world data — including stochastic differential equation learning and time series classification — demonstrate competitive accuracy while offering significant computational advantages over existing methods.

3. arXiv

   Importance Weighting Correction of Regularized Least-Squares for Target Shift

   Davit Gogolashvili

   arXiv preprint arXiv:2210.09709, 2026

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   Importance weighting is a standard tool for correcting distribution shift, but its statistical behavior under target shift — where the label distribution changes between training and testing while the conditional distribution of inputs given the label remains stable — remains under-explored. We analyze importance-weighted kernel ridge regression under target shift and show that, because the weights depend only on the output variable, reweighting corrects the train-test mismatch without altering the input-space complexity that governs kernel generalization. Under standard RKHS regularity and capacity conditions and a mild Bernstein-type moment condition on the label weights, we obtain finite-sample guarantees showing that the estimator achieves the same convergence behavior as in the no-shift case, with shift severity affecting only the constants through weight moments. We complement these results with matching minimax lower bounds, establishing rate optimality and quantifying the unavoidable dependence on shift severity. We further study more general weighting schemes and prove that weight misspecification induces an irreducible bias: the estimator concentrates around an induced population regression function that generally differs from the desired test regression function unless the weights are accurate. Finally, we derive consequences for plug-in classification under target shift via standard calibration arguments.

2024

1. JMLR

   Estimating the Minimizer and the Minimum Value of a Regression Function under Passive Design

   Arya Akhavan, Davit Gogolashvili, and Alexandre B. Tsybakov

   Journal of Machine Learning Research, 2024

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   We propose a new method for estimating the minimizer $\boldsymbol{x}^*$ and the minimum value $f^*$ of a smooth and strongly convex regression function $f$ from the observations contaminated by random noise. Our estimator $\boldsymbol{z}_n$ of the minimizer $\boldsymbol{x}^*$ is based on a version of the projected gradient descent with the gradient estimated by a regularized local polynomial algorithm. Next, we propose a two-stage procedure for estimation of the minimum value $f^*$ of regression function $f$. At the first stage, we construct an accurate enough estimator of $\boldsymbol{x}^*$, which can be, for example, $\boldsymbol{z}_n$. At the second stage, we estimate the function value at the point obtained in the first stage using a rate optimal nonparametric procedure. We derive non-asymptotic upper bounds for the quadratic risk and optimization risk of $\boldsymbol{z}_n$, and for the risk of estimating $f^*$. We establish minimax lower bounds showing that, under certain choice of parameters, the proposed algorithms achieve the minimax optimal rates of convergence on the class of smooth and strongly convex functions.

2023

1. arXiv

   When is Importance Weighting Correction Needed for Covariate Shift Adaptation?

   Davit Gogolashvili, Matteo Zecchin, Motonobu Kanagawa, and 2 more authors

   arXiv preprint arXiv:2303.04020, 2023

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   This paper investigates when the importance weighting (IW) correction is needed to address covariate shift, a common situation in supervised learning where the input distributions of training and test data differ. Classic results show that the IW correction is needed when the model is parametric and misspecified. In contrast, recent results indicate that the IW correction may not be necessary when the model is nonparametric and well-specified. We examine the missing case in the literature where the model is nonparametric and misspecified, and show that the IW correction is needed for obtaining the best approximation of the true unknown function for the test distribution. We do this by analyzing IW-corrected kernel ridge regression, covering a variety of settings, including parametric and nonparametric models, well-specified and misspecified settings, and arbitrary weighting functions.

2022

1. Locally Smoothed Gaussian Process Regression

   Davit Gogolashvili, Bogdan Kozyrskiy, and Maurizio Filippone

   Procedia Computer Science, 2022

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   We develop a novel framework to accelerate Gaussian process regression (GPR). In particular, we consider localization kernels at each data point to down-weigh the contributions from other data points that are far away, and we derive the GPR model stemming from the application of such localization operation. Through a set of experiments, we demonstrate the competitive performance of the proposed approach compared to full GPR, other localized models, and deep Gaussian processes. Crucially, these performances are obtained with considerable speedups compared to standard global GPR due to the sparsification effect of the Gram matrix induced by the localization operation.