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Alexander Rakhlin

Publications and source records attributed to Alexander Rakhlin.

At least 73 records · Page 4Linked to original sources
arXiv2020

On Suboptimality of Least Squares with Application to Estimation of Convex Bodies

We develop a technique for establishing lower bounds on the sample complexity of Least Squares (or, Empirical Risk Minimization) for large classes of functions. As an application, we settle an open problem regarding optimality of Least Squares in estimating a convex set from noisy support function measurements in dimension $d\geq 6$. Specifically, we establish that Least Squares is mimimax sub-optimal, and achieves a rate of $\tildeΘ_d(n^{-2/(d-1)})$ whereas the minimax rate is $Θ_d(n^{-4/(d+3)})$.

math.ST↗
arXiv2020

Data Driven Estimation of Stochastic Switched Linear Systems of Unknown Order

We address the problem of learning the parameters of a mean square stable switched linear systems (SLS) with unknown latent space dimension, or \textit{order}, from its noisy input--output data. In particular, we focus on learning a good lower order approximation of the underlying model allowed by finite data. Motivated by subspace-based algorithms in system theory, we construct a Hankel-like matrix from finite noisy data using ordinary least squares. Such a formulation circumvents the non-convexities that arise in system identification, and allows for accurate estimation of the underlying SLS as data size increases. Since the model order is unknown, the key idea of our approach is model order selection based on purely data dependent quantities. We construct Hankel-like matrices from data of dimension obtained from the order selection procedure. By exploiting tools from theory of model reduction for SLS, we obtain suitable approximations via singular value decomposition (SVD) and show that the system parameter estimates are close to a balanced truncated realization of the underlying system with high probability.

eess.SY↗
arXiv2020

Learning nonlinear dynamical systems from a single trajectory

We introduce algorithms for learning nonlinear dynamical systems of the form $x_{t+1}=σ(Θ^{\star}x_t)+\varepsilon_t$, where $Θ^{\star}$ is a weight matrix, $σ$ is a nonlinear link function, and $\varepsilon_t$ is a mean-zero noise process. We give an algorithm that recovers the weight matrix $Θ^{\star}$ from a single trajectory with optimal sample complexity and linear running time. The algorithm succeeds under weaker statistical assumptions than in previous work, and in particular i) does not require a bound on the spectral norm of the weight matrix $Θ^{\star}$ (rather, it depends on a generalization of the spectral radius) and ii) enjoys guarantees for non-strictly-increasing link functions such as the ReLU. Our analysis has two key components: i) we give a general recipe whereby global stability for nonlinear dynamical systems can be used to certify that the state-vector covariance is well-conditioned, and ii) using these tools, we extend well-known algorithms for efficiently learning generalized linear models to the dependent setting.

cs.LG↗
arXiv2020

Nonparametric Finite Time LTI System Identification

We address the problem of learning the parameters of a stable linear time invariant (LTI) system or linear dynamical system (LDS) with unknown latent space dimension, or order, from a single time--series of noisy input-output data. We focus on learning the best lower order approximation allowed by finite data. Motivated by subspace algorithms in systems theory, where the doubly infinite system Hankel matrix captures both order and good lower order approximations, we construct a Hankel-like matrix from noisy finite data using ordinary least squares. This circumvents the non-convexities that arise in system identification, and allows accurate estimation of the underlying LTI system. Our results rely on careful analysis of self-normalized martingale difference terms that helps bound identification error up to logarithmic factors of the lower bound. We provide a data-dependent scheme for order selection and find an accurate realization of system parameters, corresponding to that order, by an approach that is closely related to the Ho-Kalman subspace algorithm. We demonstrate that the proposed model order selection procedure is not overly conservative, i.e., for the given data length it is not possible to estimate higher order models or find higher order approximations with reasonable accuracy.

eess.SY↗
arXiv2020

Optimality of Maximum Likelihood for Log-Concave Density Estimation and Bounded Convex Regression

In this paper, we study two problems: (1) estimation of a $d$-dimensional log-concave distribution and (2) bounded multivariate convex regression with random design with an underlying log-concave density or a compactly supported distribution with a continuous density. First, we show that for all $d \ge 4$ the maximum likelihood estimators of both problems achieve an optimal risk of $Θ_d(n^{-2/(d+1)})$ (up to a logarithmic factor) in terms of squared Hellinger distance and $L_2$ squared distance, respectively. Previously, the optimality of both these estimators was known only for $d\le 3$. We also prove that the $ε$-entropy numbers of the two aforementioned families are equal up to logarithmic factors. We complement these results by proving a sharp bound $Θ_d(n^{-2/(d+4)})$ on the minimax rate (up to logarithmic factors) with respect to the total variation distance. Finally, we prove that estimating a log-concave density - even a uniform distribution on a convex set - up to a fixed accuracy requires the number of samples \emph{at least} exponential in the dimension. We do that by improving the dimensional constant in the best known lower bound for the minimax rate from $2^{-d}\cdot n^{-2/(d+1)}$ to $c\cdot n^{-2/(d+1)}$ (when $d\geq 2$).

math.ST↗
arXiv2020

Adaptive Online Learning

We propose a general framework for studying adaptive regret bounds in the online learning framework, including model selection bounds and data-dependent bounds. Given a data- or model-dependent bound we ask, "Does there exist some algorithm achieving this bound?" We show that modifications to recently introduced sequential complexity measures can be used to answer this question by providing sufficient conditions under which adaptive rates can be achieved. In particular each adaptive rate induces a set of so-called offset complexity measures, and obtaining small upper bounds on these quantities is sufficient to demonstrate achievability. A cornerstone of our analysis technique is the use of one-sided tail inequalities to bound suprema of offset random processes. Our framework recovers and improves a wide variety of adaptive bounds including quantile bounds, second-order data-dependent bounds, and small loss bounds. In addition we derive a new type of adaptive bound for online linear optimization based on the spectral norm, as well as a new online PAC-Bayes theorem that holds for countably infinite sets.

cs.LG↗
arXiv2020

On the Multiple Descent of Minimum-Norm Interpolants and Restricted Lower Isometry of Kernels

We study the risk of minimum-norm interpolants of data in Reproducing Kernel Hilbert Spaces. Our upper bounds on the risk are of a multiple-descent shape for the various scalings of $d = n^α$, $α\in(0,1)$, for the input dimension $d$ and sample size $n$. Empirical evidence supports our finding that minimum-norm interpolants in RKHS can exhibit this unusual non-monotonicity in sample size; furthermore, locations of the peaks in our experiments match our theoretical predictions. Since gradient flow on appropriately initialized wide neural networks converges to a minimum-norm interpolant with respect to a certain kernel, our analysis also yields novel estimation and generalization guarantees for these over-parametrized models. At the heart of our analysis is a study of spectral properties of the random kernel matrix restricted to a filtration of eigen-spaces of the population covariance operator, and may be of independent interest.

math.ST↗
arXiv2019

Size-Independent Sample Complexity of Neural Networks

We study the sample complexity of learning neural networks, by providing new bounds on their Rademacher complexity assuming norm constraints on the parameter matrix of each layer. Compared to previous work, these complexity bounds have improved dependence on the network depth, and under some additional assumptions, are fully independent of the network size (both depth and width). These results are derived using some novel techniques, which may be of independent interest.

cs.LG↗
arXiv2019

$\ell_{\infty}$ Vector Contraction for Rademacher Complexity

We show that the Rademacher complexity of any $\mathbb{R}^{K}$-valued function class composed with an $\ell_{\infty}$-Lipschitz function is bounded by the maximum Rademacher complexity of the restriction of the function class along each coordinate, times a factor of $\tilde{O}(\sqrt{K})$.

cs.LG↗
arXiv2019

Breast Tumor Cellularity Assessment using Deep Neural Networks

Breast cancer is one of the main causes of death worldwide. Histopathological cellularity assessment of residual tumors in post-surgical tissues is used to analyze a tumor's response to a therapy. Correct cellularity assessment increases the chances of getting an appropriate treatment and facilitates the patient's survival. In current clinical practice, tumor cellularity is manually estimated by pathologists; this process is tedious and prone to errors or low agreement rates between assessors. In this work, we evaluated three strong novel Deep Learning-based approaches for automatic assessment of tumor cellularity from post-treated breast surgical specimens stained with hematoxylin and eosin. We validated the proposed methods on the BreastPathQ SPIE challenge dataset that consisted of 2395 image patches selected from whole slide images acquired from 64 patients. Compared to expert pathologist scoring, our best performing method yielded the Cohen's kappa coefficient of 0.70 (vs. 0.42 previously known in literature) and the intra-class correlation coefficient of 0.89 (vs. 0.83). Our results suggest that Deep Learning-based methods have a significant potential to alleviate the burden on pathologists, enhance the diagnostic workflow, and, thereby, facilitate better clinical outcomes in breast cancer treatment.

eess.IV↗
arXiv2019

Fisher-Rao Metric, Geometry, and Complexity of Neural Networks

We study the relationship between geometry and capacity measures for deep neural networks from an invariance viewpoint. We introduce a new notion of capacity --- the Fisher-Rao norm --- that possesses desirable invariance properties and is motivated by Information Geometry. We discover an analytical characterization of the new capacity measure, through which we establish norm-comparison inequalities and further show that the new measure serves as an umbrella for several existing norm-based complexity measures. We discuss upper bounds on the generalization error induced by the proposed measure. Extensive numerical experiments on CIFAR-10 support our theoretical findings. Our theoretical analysis rests on a key structural lemma about partial derivatives of multi-layer rectifier networks.

cs.LG↗
arXiv2019

Near optimal finite time identification of arbitrary linear dynamical systems

We derive finite time error bounds for estimating general linear time-invariant (LTI) systems from a single observed trajectory using the method of least squares. We provide the first analysis of the general case when eigenvalues of the LTI system are arbitrarily distributed in three regimes: stable, marginally stable, and explosive. Our analysis yields sharp upper bounds for each of these cases separately. We observe that although the underlying process behaves quite differently in each of these three regimes, the systematic analysis of a self--normalized martingale difference term helps bound identification error up to logarithmic factors of the lower bound. On the other hand, we demonstrate that the least squares solution may be statistically inconsistent under certain conditions even when the signal-to-noise ratio is high.

eess.SY↗
arXiv2019

Just Interpolate: Kernel "Ridgeless" Regression Can Generalize

In the absence of explicit regularization, Kernel "Ridgeless" Regression with nonlinear kernels has the potential to fit the training data perfectly. It has been observed empirically, however, that such interpolated solutions can still generalize well on test data. We isolate a phenomenon of implicit regularization for minimum-norm interpolated solutions which is due to a combination of high dimensionality of the input data, curvature of the kernel function, and favorable geometric properties of the data such as an eigenvalue decay of the empirical covariance and kernel matrices. In addition to deriving a data-dependent upper bound on the out-of-sample error, we present experimental evidence suggesting that the phenomenon occurs in the MNIST dataset.

math.ST↗
arXiv2018

Consistency of Interpolation with Laplace Kernels is a High-Dimensional Phenomenon

We show that minimum-norm interpolation in the Reproducing Kernel Hilbert Space corresponding to the Laplace kernel is not consistent if input dimension is constant. The lower bound holds for any choice of kernel bandwidth, even if selected based on data. The result supports the empirical observation that minimum-norm interpolation (that is, exact fit to training data) in RKHS generalizes well for some high-dimensional datasets, but not for low-dimensional ones.

stat.ML↗
arXiv2018

Pediatric Bone Age Assessment Using Deep Convolutional Neural Networks

Skeletal bone age assessment is a common clinical practice to diagnose endocrine and metabolic disorders in child development. In this paper, we describe a fully automated deep learning approach to the problem of bone age assessment using data from Pediatric Bone Age Challenge organized by RSNA 2017. The dataset for this competition is consisted of 12.6k radiological images of left hand labeled by the bone age and sex of patients. Our approach utilizes several deep learning architectures: U-Net, ResNet-50, and custom VGG-style neural networks trained end-to-end. We use images of whole hands as well as specific parts of a hand for both training and inference. This approach allows us to measure importance of specific hand bones for the automated bone age analysis. We further evaluate performance of the method in the context of skeletal development stages. Our approach outperforms other common methods for bone age assessment.

cs.CV↗
arXiv2018

Automatic Instrument Segmentation in Robot-Assisted Surgery Using Deep Learning

Semantic segmentation of robotic instruments is an important problem for the robot-assisted surgery. One of the main challenges is to correctly detect an instrument's position for the tracking and pose estimation in the vicinity of surgical scenes. Accurate pixel-wise instrument segmentation is needed to address this challenge. In this paper we describe our winning solution for MICCAI 2017 Endoscopic Vision SubChallenge: Robotic Instrument Segmentation. Our approach demonstrates an improvement over the state-of-the-art results using several novel deep neural network architectures. It addressed the binary segmentation problem, where every pixel in an image is labeled as an instrument or background from the surgery video feed. In addition, we solve a multi-class segmentation problem, where we distinguish different instruments or different parts of an instrument from the background. In this setting, our approach outperforms other methods in every task subcategory for automatic instrument segmentation thereby providing state-of-the-art solution for this problem. The source code for our solution is made publicly available at https://github.com/ternaus/robot-surgery-segmentation

cs.CV↗
arXiv2018

Angiodysplasia Detection and Localization Using Deep Convolutional Neural Networks

Accurate detection and localization for angiodysplasia lesions is an important problem in early stage diagnostics of gastrointestinal bleeding and anemia. Gold-standard for angiodysplasia detection and localization is performed using wireless capsule endoscopy. This pill-like device is able to produce thousand of high enough resolution images during one passage through gastrointestinal tract. In this paper we present our winning solution for MICCAI 2017 Endoscopic Vision SubChallenge: Angiodysplasia Detection and Localization its further improvements over the state-of-the-art results using several novel deep neural network architectures. It address the binary segmentation problem, where every pixel in an image is labeled as an angiodysplasia lesions or background. Then, we analyze connected component of each predicted mask. Based on the analysis we developed a classifier that predict angiodysplasia lesions (binary variable) and a detector for their localization (center of a component). In this setting, our approach outperforms other methods in every task subcategory for angiodysplasia detection and localization thereby providing state-of-the-art results for these problems. The source code for our solution is made publicly available at https://github.com/ternaus/angiodysplasia-segmentatio

cs.CV↗