Saptarshi Chakraborty

Federico Di Gennaro, Saptarshi Chakraborty, Nikita Zhivotovskiy

This paper studies the Exponential Weights (EW) algorithm with an isotropic Gaussian prior for online logistic regression. We show that the near-optimal worst-case regret bound $O(d\log(Bn))$ for EW, established by Kakade and Ng (2005) against the best linear predictor of norm at most $B$, can be achieved with total worst-case computational complexity $O(B^3 n^5)$. This substantially improves on the $O(B^{18}n^{37})$ complexity of prior work achieving the same guarantee (Foster et al., 2018). Beyond efficiency, we analyze the large-$B$ regime under linear separability: after rescaling by $B$, the EW posterior converges as $B\to\infty$ to a standard Gaussian truncated to the version cone. Accordingly, the predictor converges to a solid-angle vote over separating directions and, on every fixed-margin slice of this cone, the mode of the corresponding truncated Gaussian is aligned with the hard-margin SVM direction. Using this geometry, we derive non-asymptotic regret bounds showing that once $B$ exceeds a margin-dependent threshold, the regret becomes independent of $B$ and grows only logarithmically with the inverse margin. Overall, our results show that EW can be both computationally tractable and geometrically adaptive in online classification.

Saptarshi Chakraborty, Quentin Berthet, Peter L. Bartlett

Despite the remarkable empirical success of score-based diffusion models, their statistical guarantees remain underdeveloped. Existing analyses often provide pessimistic convergence rates that do not reflect the intrinsic low-dimensional structure common in real data, such as that arising in natural images. In this work, we study the statistical convergence of score-based diffusion models for learning an unknown distribution $μ$ from finitely many samples. Under mild regularity conditions on the forward diffusion process and the data distribution, we derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance. Unlike prior results, our guarantees hold for all $p \ge 1$ and require only a finite-moment assumption on $μ$, without compact-support, manifold, or smooth-density conditions. Specifically, given $n$ i.i.d.\ samples from $μ$ with finite $q$-th moment and appropriately chosen network architectures, hyperparameters, and discretization schemes, we show that the expected Wasserstein-$p$ error between the learned distribution $\hatμ$ and $μ$ scales as $\mathbb{E}\, \mathbb{W}_p(\hatμ,μ) = \widetilde{O}\!\left(n^{-1 / d^\ast_{p,q}(μ)}\right),$ where $d^\ast_{p,q}(μ)$ is the $(p,q)$-Wasserstein dimension of $μ$. Our results demonstrate that diffusion models naturally adapt to the intrinsic geometry of data and mitigate the curse of dimensionality, since the convergence rate depends on $d^\ast_{p,q}(μ)$ rather than the ambient dimension. Moreover, our theory conceptually bridges the analysis of diffusion models with that of GANs and the sharp minimax rates established in optimal transport. The proposed $(p,q)$-Wasserstein dimension also extends classical Wasserstein dimension notions to distributions with unbounded support, which may be of independent theoretical interest.

Saptarshi Chakraborty, Peter L. Bartlett

Despite the remarkable empirical successes of Generative Adversarial Networks (GANs), the theoretical guarantees for their statistical accuracy remain rather pessimistic. In particular, the data distributions on which GANs are applied, such as natural images, are often hypothesized to have an intrinsic low-dimensional structure in a typically high-dimensional feature space, but this is often not reflected in the derived rates in the state-of-the-art analyses. In this paper, we attempt to bridge the gap between the theory and practice of GANs and their bidirectional variant, Bi-directional GANs (BiGANs), by deriving statistical guarantees on the estimated densities in terms of the intrinsic dimension of the data and the latent space. We analytically show that if one has access to $n$ samples from the unknown target distribution and the network architectures are properly chosen, the expected Wasserstein-1 distance of the estimates from the target scales as $O\left( n^{-1/d_μ} \right)$ for GANs and $\tilde{O}\left( n^{-1/(d_μ+\ell)} \right)$ for BiGANs, where $d_μ$ and $\ell$ are the upper Wasserstein-1 dimension of the data-distribution and latent-space dimension, respectively. The theoretical analyses not only suggest that these methods successfully avoid the curse of dimensionality, in the sense that the exponent of $n$ in the error rates does not depend on the data dimension but also serve to bridge the gap between the theoretical analyses of GANs and the known sharp rates from optimal transport literature. Additionally, we demonstrate that GANs can effectively achieve the minimax optimal rate even for non-smooth underlying distributions, with the use of interpolating generator networks.

Saptarshi Chakraborty, Peter L. Bartlett

Variational Autoencoders (VAEs) have gained significant popularity among researchers as a powerful tool for understanding unknown distributions based on limited samples. This popularity stems partly from their impressive performance and partly from their ability to provide meaningful feature representations in the latent space. Wasserstein Autoencoders (WAEs), a variant of VAEs, aim to not only improve model efficiency but also interpretability. However, there has been limited focus on analyzing their statistical guarantees. The matter is further complicated by the fact that the data distributions to which WAEs are applied - such as natural images - are often presumed to possess an underlying low-dimensional structure within a high-dimensional feature space, which current theory does not adequately account for, rendering known bounds inefficient. To bridge the gap between the theory and practice of WAEs, in this paper, we show that WAEs can learn the data distributions when the network architectures are properly chosen. We show that the convergence rates of the expected excess risk in the number of samples for WAEs are independent of the high feature dimension, instead relying only on the intrinsic dimension of the data distribution.

Saptarshi Chakraborty, Peter L. Bartlett

Despite significant research on the optimization aspects of federated learning, the exploration of generalization error, especially in the realm of heterogeneous federated learning, remains an area that has been insufficiently investigated, primarily limited to developments in the parametric regime. This paper delves into the generalization properties of deep federated regression within a two-stage sampling model. Our findings reveal that the intrinsic dimension, characterized by the entropic dimension, plays a pivotal role in determining the convergence rates for deep learners when appropriately chosen network sizes are employed. Specifically, when the true relationship between the response and explanatory variables is described by a $β$-Hölder function and one has access to $n$ independent and identically distributed (i.i.d.) samples from $m$ participating clients, for participating clients, the error rate scales at most as $\Tilde{O}((mn)^{-2β/(2β+ \bar{d}_{2β}(λ))})$, whereas for non-participating clients, it scales as $\Tilde{O}(Δ\cdot m^{-2β/(2β+ \bar{d}_{2β}(λ))} + (mn)^{-2β/(2β+ \bar{d}_{2β}(λ))})$. Here $\bar{d}_{2β}(λ)$ denotes the corresponding $2β$-entropic dimension of $λ$, the marginal distribution of the explanatory variables. The dependence between the two stages of the sampling scheme is characterized by $Δ$. Consequently, our findings not only explicitly incorporate the ``heterogeneity" of the clients, but also highlight that the convergence rates of errors of deep federated learners are not contingent on the nominal high dimensionality of the data but rather on its intrinsic dimension.

Saptarshi Chakraborty, Peter L. Bartlett

Recent advances have revealed that the rate of convergence of the expected test error in deep supervised learning decays as a function of the intrinsic dimension and not the dimension $d$ of the input space. Existing literature defines this intrinsic dimension as the Minkowski dimension or the manifold dimension of the support of the underlying probability measures, which often results in sub-optimal rates and unrealistic assumptions. In this paper, we consider supervised deep learning when the response given the explanatory variable is distributed according to an exponential family with a $β$-Hölder smooth mean function. We consider an entropic notion of the intrinsic data-dimension and demonstrate that with $n$ independent and identically distributed samples, the test error scales as $\tilde{\mathcal{O}}\left(n^{-\frac{2β}{2β+ \bar{d}_{2β}(λ)}}\right)$, where $\bar{d}_{2β}(λ)$ is the $2β$-entropic dimension of $λ$, the distribution of the explanatory variables. This improves on the best-known rates. Furthermore, under the assumption of an upper-bounded density of the explanatory variables, we characterize the rate of convergence as $\tilde{\mathcal{O}}\left( d^{\frac{2\lfloorβ\rfloor(β+ d)}{2β+ d}}n^{-\frac{2β}{2β+ d}}\right)$, establishing that the dependence on $d$ is not exponential but at most polynomial. We also demonstrate that when the explanatory variable has a lower bounded density, this rate in terms of the number of data samples, is nearly optimal for learning the dependence structure for exponential families.

Saptarshi Chakraborty, Debolina Paul, Swagatam Das

Mean shift is a simple interactive procedure that gradually shifts data points towards the mode which denotes the highest density of data points in the region. Mean shift algorithms have been effectively used for data denoising, mode seeking, and finding the number of clusters in a dataset in an automated fashion. However, the merits of mean shift quickly fade away as the data dimensions increase and only a handful of features contain useful information about the cluster structure of the data. We propose a simple yet elegant feature-weighted variant of mean shift to efficiently learn the feature importance and thus, extending the merits of mean shift to high-dimensional data. The resulting algorithm not only outperforms the conventional mean shift clustering procedure but also preserves its computational simplicity. In addition, the proposed method comes with rigorous theoretical convergence guarantees and a convergence rate of at least a cubic order. The efficacy of our proposal is thoroughly assessed through experimental comparison against baseline and state-of-the-art clustering methods on synthetic as well as real-world datasets.

Debolina Paul, Saptarshi Chakraborty, Swagatam Das et al.

Kernel $k$-means clustering is a powerful tool for unsupervised learning of non-linearly separable data. Since the earliest attempts, researchers have noted that such algorithms often become trapped by local minima arising from non-convexity of the underlying objective function. In this paper, we generalize recent results leveraging a general family of means to combat sub-optimal local solutions to the kernel and multi-kernel settings. Called Kernel Power $k$-Means, our algorithm makes use of majorization-minimization (MM) to better solve this non-convex problem. We show the method implicitly performs annealing in kernel feature space while retaining efficient, closed-form updates, and we rigorously characterize its convergence properties both from computational and statistical points of view. In particular, we characterize the large sample behavior of the proposed method by establishing strong consistency guarantees. Its merits are thoroughly validated on a suite of simulated datasets and real data benchmarks that feature non-linear and multi-view separation.

Debolina Paul, Saptarshi Chakraborty, Didong Li et al.

Even with the rise in popularity of over-parameterized models, simple dimensionality reduction and clustering methods, such as PCA and k-means, are still routinely used in an amazing variety of settings. A primary reason is the combination of simplicity, interpretability and computational efficiency. The focus of this article is on improving upon PCA and k-means, by allowing non-linear relations in the data and more flexible cluster shapes, without sacrificing the key advantages. The key contribution is a new framework for Principal Elliptical Analysis (PEA), defining a simple and computationally efficient alternative to PCA that fits the best elliptical approximation through the data. We provide theoretical guarantees on the proposed PEA algorithm using Vapnik-Chervonenkis (VC) theory to show strong consistency and uniform concentration bounds. Toy experiments illustrate the performance of PEA, and the ability to adapt to non-linear structure and complex cluster shapes. In a rich variety of real data clustering applications, PEA is shown to do as well as k-means for simple datasets, while dramatically improving performance in more complex settings.

Saptarshi Chakraborty, Debolina Paul, Swagatam Das et al.

Despite its well-known shortcomings, $k$-means remains one of the most widely used approaches to data clustering. Current research continues to tackle its flaws while attempting to preserve its simplicity. Recently, the \textit{power $k$-means} algorithm was proposed to avoid trapping in local minima by annealing through a family of smoother surfaces. However, the approach lacks theoretical justification and fails in high dimensions when many features are irrelevant. This paper addresses these issues by introducing \textit{entropy regularization} to learn feature relevance while annealing. We prove consistency of the proposed approach and derive a scalable majorization-minimization algorithm that enjoys closed-form updates and convergence guarantees. In particular, our method retains the same computational complexity of $k$-means and power $k$-means, but yields significant improvements over both. Its merits are thoroughly assessed on a suite of real and synthetic data experiments.

Saptarshi Chakraborty, Swagatam Das

We propose the Lasso Weighted $k$-means ($LW$-$k$-means) algorithm as a simple yet efficient sparse clustering procedure for high-dimensional data where the number of features ($p$) can be much larger compared to the number of observations ($n$). In the $LW$-$k$-means algorithm, we introduce a lasso-based penalty term, directly on the feature weights to incorporate feature selection in the framework of sparse clustering. $LW$-$k$-means does not make any distributional assumption of the given dataset and thus, induces a non-parametric method for feature selection. We also analytically investigate the convergence of the underlying optimization procedure in $LW$-$k$-means and establish the strong consistency of our algorithm. $LW$-$k$-means is tested on several real-life and synthetic datasets and through detailed experimental analysis, we find that the performance of the method is highly competitive against some state-of-the-art procedures for clustering and feature selection, not only in terms of clustering accuracy but also with respect to computational time.