Sabyasachi Chatterjee

Sabyasachi Chatterjee, Amit Acharya, Zvi Artstein

A computational tool for coarse-graining nonlinear systems of ordinary differential equations in time is discussed. Three illustrative model examples are worked out that demonstrate the range of capability of the method. This includes the averaging of Hamiltonian as well as dissipative microscopic dynamics whose `slow' variables, defined in a precise sense, can often display mixed slow-fast response as in relaxation oscillations, and dependence on initial conditions of the fast variables. Also covered is the case where the quasi-static assumption in solid mechanics is violated. The computational tool is demonstrated to capture all of these behaviors in an accurate and robust manner, with significant savings in time. A practically useful strategy for initializing short bursts of microscopic runs for the accurate computation of the evolution of slow variables is also developed.

Carlos Misael Madrid Padilla, Oscar Hernan Madrid Padilla, Sabyasachi Chatterjee

This work examines risk bounds for nonparametric distributional regression estimators. For convex-constrained distributional regression, general upper bounds are established for the continuous ranked probability score (CRPS) and the worst-case mean squared error (MSE) across the domain. These theoretical results are applied to isotonic and trend filtering distributional regression, yielding convergence rates consistent with those for mean estimation. Furthermore, a general upper bound is derived for distributional regression under non-convex constraints, with a specific application to neural network-based estimators. Comprehensive experiments on both simulated and real data validate the theoretical contributions, demonstrating their practical effectiveness.

Sabyasachi Chatterjee, Subhajit Goswami, Soumendu Sundar Mukherjee

Adaptive bandwidth selection is a fundamental challenge in nonparametric regression. This paper introduces a new bandwidth selection procedure inspired by the optimality criteria for $\ell_0$-penalized regression. Although similar in spirit to Lepski's method and its variants in selecting the largest interval satisfying an admissibility criterion, our approach stems from a distinct philosophy, utilizing criteria based on $\ell_2$-norms of interval projections rather than explicit point and variance estimates. We obtain non-asymptotic risk bounds for the local polynomial regression methods based on our bandwidth selection procedure which adapt (near-)optimally to the local Hölder exponent of the underlying regression function simultaneously at all points in its domain. Furthermore, we show that there is a single ideal choice of a global tuning parameter in each case under which the above-mentioned local adaptivity holds. The optimal risks of our methods derive from the properties of solutions to a new ``bandwidth selection equation'' which is of independent interest. We believe that the principles underlying our approach provide a new perspective to the classical yet ever relevant problem of locally adaptive nonparametric regression.

Sabyasachi Chatterjee, Subhajit Goswami

We consider the problem of estimating piecewise regular functions in an online setting, i.e., the data arrive sequentially and at any round our task is to predict the value of the true function at the next revealed point using the available data from past predictions. We propose a suitably modified version of a recently developed online learning algorithm called the sleeping experts aggregation algorithm. We show that this estimator satisfies oracle risk bounds simultaneously for all local regions of the domain. As concrete instantiations of the expert aggregation algorithm proposed here, we study an online mean aggregation and an online linear regression aggregation algorithm where experts correspond to the set of dyadic subrectangles of the domain. The resulting algorithms are near linear time computable in the sample size. We specifically focus on the performance of these online algorithms in the context of estimating piecewise polynomial and bounded variation function classes in the fixed design setup. The simultaneous oracle risk bounds we obtain for these estimators in this context provide new and improved (in certain aspects) guarantees even in the batch setting and are not available for the state of the art batch learning estimators.

Anamitra Chaudhuri, Sabyasachi Chatterjee

This paper formulates a general cross validation framework for signal denoising. The general framework is then applied to nonparametric regression methods such as Trend Filtering and Dyadic CART. The resulting cross validated versions are then shown to attain nearly the same rates of convergence as are known for the optimally tuned analogues. There did not exist any previous theoretical analyses of cross validated versions of Trend Filtering or Dyadic CART. To illustrate the generality of the framework we also propose and study cross validated versions of two fundamental estimators; lasso for high dimensional linear regression and singular value thresholding for matrix estimation. Our general framework is inspired by the ideas in Chatterjee and Jafarov (2015) and is potentially applicable to a wide range of estimation methods which use tuning parameters.

Sabyasachi Chatterjee, Subhajit Goswami

Proposed by Donoho (1997), Dyadic CART is a nonparametric regression method which computes a globally optimal dyadic decision tree and fits piecewise constant functions in two dimensions. In this article we define and study Dyadic CART and a closely related estimator, namely Optimal Regression Tree (ORT), in the context of estimating piecewise smooth functions in general dimensions in the fixed design setup. More precisely, these optimal decision tree estimators fit piecewise polynomials of any given degree. Like Dyadic CART in two dimensions, we reason that these estimators can also be computed in polynomial time in the sample size $N$ via dynamic programming. We prove oracle inequalities for the finite sample risk of Dyadic CART and ORT which imply tight risk bounds for several function classes of interest. Firstly, they imply that the finite sample risk of ORT of order $r \geq 0$ is always bounded by $C k \frac{\log N}{N}$ whenever the regression function is piecewise polynomial of degree $r$ on some reasonably regular axis aligned rectangular partition of the domain with at most $k$ rectangles. Beyond the univariate case, such guarantees are scarcely available in the literature for computationally efficient estimators. Secondly, our oracle inequalities uncover minimax rate optimality and adaptivity of the Dyadic CART estimator for function spaces with bounded variation. We consider two function spaces of recent interest where multivariate total variation denoising and univariate trend filtering are the state of the art methods. We show that Dyadic CART enjoys certain advantages over these estimators while still maintaining all their known guarantees.

Sabyasachi Chatterjee, Subhajit Goswami

2D Total Variation Denoising (TVD) is a widely used technique for image denoising. It is also an important nonparametric regression method for estimating functions with heterogenous smoothness. Recent results have shown the TVD estimator to be nearly minimax rate optimal for the class of functions with bounded variation. In this paper, we complement these worst case guarantees by investigating the adaptivity of the TVD estimator to functions which are piecewise constant on axis aligned rectangles. We rigorously show that, when the truth is piecewise constant, the ideally tuned TVD estimator performs better than in the worst case. We also study the issue of choosing the tuning parameter. In particular, we propose a fully data driven version of the TVD estimator which enjoys similar worst case risk guarantees as the ideally tuned TVD estimator.

Matt Bonakdarpour, Sabyasachi Chatterjee, Rina Foygel Barber et al.

Two methods are proposed for high-dimensional shape-constrained regression and classification. These methods reshape pre-trained prediction rules to satisfy shape constraints like monotonicity and convexity. The first method can be applied to any pre-trained prediction rule, while the second method deals specifically with random forests. In both cases, efficient algorithms are developed for computing the estimators, and experiments are performed to demonstrate their performance on four datasets. We find that reshaping methods enforce shape constraints without compromising predictive accuracy.

Yuancheng Zhu, Sabyasachi Chatterjee, John Duchi et al.

We extend the traditional worst-case, minimax analysis of stochastic convex optimization by introducing a localized form of minimax complexity for individual functions. Our main result gives function-specific lower and upper bounds on the number of stochastic subgradient evaluations needed to optimize either the function or its "hardest local alternative" to a given numerical precision. The bounds are expressed in terms of a localized and computational analogue of the modulus of continuity that is central to statistical minimax analysis. We show how the computational modulus of continuity can be explicitly calculated in concrete cases, and relates to the curvature of the function at the optimum. We also prove a superefficiency result that demonstrates it is a meaningful benchmark, acting as a computational analogue of the Fisher information in statistical estimation. The nature and practical implications of the results are demonstrated in simulations.