arXiv subjects

Suriya Gunasekar

Publications and source records attributed to Suriya Gunasekar.

34 records · Page 2Linked to original sources

arXiv2020

Kernel and Rich Regimes in Overparametrized Models

A recent line of work studies overparametrized neural networks in the "kernel regime," i.e. when the network behaves during training as a kernelized linear predictor, and thus training with gradient descent has the effect of finding the minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. Building on an observation by Chizat and Bach, we show how the scale of the initialization controls the transition between the "kernel" (aka lazy) and "rich" (aka active) regimes and affects generalization properties in multilayer homogeneous models. We also highlight an interesting role for the width of a model in the case that the predictor is not identically zero at initialization. We provide a complete and detailed analysis for a family of simple depth-$D$ models that already exhibit an interesting and meaningful transition between the kernel and rich regimes, and we also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.

cs.LG↗

arXiv2020

Implicit Bias in Deep Linear Classification: Initialization Scale vs Training Accuracy

We provide a detailed asymptotic study of gradient flow trajectories and their implicit optimization bias when minimizing the exponential loss over "diagonal linear networks". This is the simplest model displaying a transition between "kernel" and non-kernel ("rich" or "active") regimes. We show how the transition is controlled by the relationship between the initialization scale and how accurately we minimize the training loss. Our results indicate that some limit behaviors of gradient descent only kick in at ridiculous training accuracies (well beyond $10^{-100}$). Moreover, the implicit bias at reasonable initialization scales and training accuracies is more complex and not captured by these limits.

cs.LG↗

arXiv2020

Characterizing Implicit Bias in Terms of Optimization Geometry

We study the implicit bias of generic optimization methods, such as mirror descent, natural gradient descent, and steepest descent with respect to different potentials and norms, when optimizing underdetermined linear regression or separable linear classification problems. We explore the question of whether the specific global minimum (among the many possible global minima) reached by an algorithm can be characterized in terms of the potential or norm of the optimization geometry, and independently of hyperparameter choices such as step-size and momentum.

stat.ML↗

arXiv2020

Kernel and Rich Regimes in Overparametrized Models

A recent line of work studies overparametrized neural networks in the "kernel regime," i.e. when the network behaves during training as a kernelized linear predictor, and thus training with gradient descent has the effect of finding the minimum RKHS norm solution. This stands in contrast to other studies which demonstrate how gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms. Building on an observation by Chizat and Bach, we show how the scale of the initialization controls the transition between the "kernel" (aka lazy) and "rich" (aka active) regimes and affects generalization properties in multilayer homogeneous models. We provide a complete and detailed analysis for a simple two-layer model that already exhibits an interesting and meaningful transition between the kernel and rich regimes, and we demonstrate the transition for more complex matrix factorization models and multilayer non-linear networks.

cs.LG↗

arXiv2019

Lexicographic and Depth-Sensitive Margins in Homogeneous and Non-Homogeneous Deep Models

With an eye toward understanding complexity control in deep learning, we study how infinitesimal regularization or gradient descent optimization lead to margin maximizing solutions in both homogeneous and non-homogeneous models, extending previous work that focused on infinitesimal regularization only in homogeneous models. To this end we study the limit of loss minimization with a diverging norm constraint (the "constrained path"), relate it to the limit of a "margin path" and characterize the resulting solution. For non-homogeneous ensemble models, which output is a sum of homogeneous sub-models, we show that this solution discards the shallowest sub-models if they are unnecessary. For homogeneous models, we show convergence to a "lexicographic max-margin solution", and provide conditions under which max-margin solutions are also attained as the limit of unconstrained gradient descent.

stat.ML↗

arXiv2019

On preserving non-discrimination when combining expert advice

We study the interplay between sequential decision making and avoiding discrimination against protected groups, when examples arrive online and do not follow distributional assumptions. We consider the most basic extension of classical online learning: "Given a class of predictors that are individually non-discriminatory with respect to a particular metric, how can we combine them to perform as well as the best predictor, while preserving non-discrimination?" Surprisingly we show that this task is unachievable for the prevalent notion of "equalized odds" that requires equal false negative rates and equal false positive rates across groups. On the positive side, for another notion of non-discrimination, "equalized error rates", we show that running separate instances of the classical multiplicative weights algorithm for each group achieves this guarantee. Interestingly, even for this notion, we show that algorithms with stronger performance guarantees than multiplicative weights cannot preserve non-discrimination.

cs.LG↗

arXiv2019

Convergence of Gradient Descent on Separable Data

We provide a detailed study on the implicit bias of gradient descent when optimizing loss functions with strictly monotone tails, such as the logistic loss, over separable datasets. We look at two basic questions: (a) what are the conditions on the tail of the loss function under which gradient descent converges in the direction of the $L_2$ maximum-margin separator? (b) how does the rate of margin convergence depend on the tail of the loss function and the choice of the step size? We show that for a large family of super-polynomial tailed losses, gradient descent iterates on linear networks of any depth converge in the direction of $L_2$ maximum-margin solution, while this does not hold for losses with heavier tails. Within this family, for simple linear models we show that the optimal rates with fixed step size is indeed obtained for the commonly used exponentially tailed losses such as logistic loss. However, with a fixed step size the optimal convergence rate is extremely slow as $1/\log(t)$, as also proved in Soudry et al. (2018). For linear models with exponential loss, we further prove that the convergence rate could be improved to $\log (t) /\sqrt{t}$ by using aggressive step sizes that compensates for the rapidly vanishing gradients. Numerical results suggest this method might be useful for deep networks.

stat.ML↗

arXiv2019

Implicit Bias of Gradient Descent on Linear Convolutional Networks

We show that gradient descent on full-width linear convolutional networks of depth $L$ converges to a linear predictor related to the $\ell_{2/L}$ bridge penalty in the frequency domain. This is in contrast to linearly fully connected networks, where gradient descent converges to the hard margin linear support vector machine solution, regardless of depth.

cs.LG↗

arXiv2018

Unified View of Matrix Completion under General Structural Constraints

In this paper, we present a unified analysis of matrix completion under general low-dimensional structural constraints induced by {\em any} norm regularization. We consider two estimators for the general problem of structured matrix completion, and provide unified upper bounds on the sample complexity and the estimation error. Our analysis relies on results from generic chaining, and we establish two intermediate results of independent interest: (a) in characterizing the size or complexity of low dimensional subsets in high dimensional ambient space, a certain partial complexity measure encountered in the analysis of matrix completion problems is characterized in terms of a well understood complexity measure of Gaussian widths, and (b) it is shown that a form of restricted strong convexity holds for matrix completion problems under general norm regularization. Further, we provide several non-trivial examples of structures included in our framework, notably the recently proposed spectral $k$-support norm.

stat.ML↗

arXiv2017

Learning Non-Discriminatory Predictors

We consider learning a predictor which is non-discriminatory with respect to a "protected attribute" according to the notion of "equalized odds" proposed by Hardt et al. [2016]. We study the problem of learning such a non-discriminatory predictor from a finite training set, both statistically and computationally. We show that a post-hoc correction approach, as suggested by Hardt et al, can be highly suboptimal, present a nearly-optimal statistical procedure, argue that the associated computational problem is intractable, and suggest a second moment relaxation of the non-discrimination definition for which learning is tractable.

cs.LG↗

arXiv2017

Implicit Regularization in Matrix Factorization

We study implicit regularization when optimizing an underdetermined quadratic objective over a matrix $X$ with gradient descent on a factorization of $X$. We conjecture and provide empirical and theoretical evidence that with small enough step sizes and initialization close enough to the origin, gradient descent on a full dimensional factorization converges to the minimum nuclear norm solution.

stat.ML↗

arXiv2016

Preference Completion from Partial Rankings

We propose a novel and efficient algorithm for the collaborative preference completion problem, which involves jointly estimating individualized rankings for a set of entities over a shared set of items, based on a limited number of observed affinity values. Our approach exploits the observation that while preferences are often recorded as numerical scores, the predictive quantity of interest is the underlying rankings. Thus, attempts to closely match the recorded scores may lead to overfitting and impair generalization performance. Instead, we propose an estimator that directly fits the underlying preference order, combined with nuclear norm constraints to encourage low--rank parameters. Besides (approximate) correctness of the ranking order, the proposed estimator makes no generative assumption on the numerical scores of the observations. One consequence is that the proposed estimator can fit any consistent partial ranking over a subset of the items represented as a directed acyclic graph (DAG), generalizing standard techniques that can only fit preference scores. Despite this generality, for supervision representing total or blockwise total orders, the computational complexity of our algorithm is within a $\log$ factor of the standard algorithms for nuclear norm regularization based estimates for matrix completion. We further show promising empirical results for a novel and challenging application of collaboratively ranking of the associations between brain--regions and cognitive neuroscience terms.

stat.ML↗

arXiv2016

Identifiable Phenotyping using Constrained Non-Negative Matrix Factorization

This work proposes a new algorithm for automated and simultaneous phenotyping of multiple co-occurring medical conditions, also referred as comorbidities, using clinical notes from the electronic health records (EHRs). A basic latent factor estimation technique of non-negative matrix factorization (NMF) is augmented with domain specific constraints to obtain sparse latent factors that are anchored to a fixed set of chronic conditions. The proposed anchoring mechanism ensures a one-to-one identifiable and interpretable mapping between the latent factors and the target comorbidities. Qualitative assessment of the empirical results by clinical experts suggests that the proposed model learns clinically interpretable phenotypes while being predictive of 30 day mortality. The proposed method can be readily adapted to any non-negative EHR data across various healthcare institutions.

stat.ML↗

arXiv2016

Phenotyping using Structured Collective Matrix Factorization of Multi--source EHR Data

The increased availability of electronic health records (EHRs) have spearheaded the initiative for precision medicine using data driven approaches. Essential to this effort is the ability to identify patients with certain medical conditions of interest from simple queries on EHRs, or EHR-based phenotypes. Existing rule--based phenotyping approaches are extremely labor intensive. Instead, dimensionality reduction and latent factor estimation techniques from machine learning can be adapted for phenotype extraction with no (or minimal) human supervision. We propose to identify an easily interpretable latent space shared across various sources of EHR data as potential candidates for phenotypes. By incorporating multiple EHR data sources (e.g., diagnosis, medications, and lab reports) available in heterogeneous datatypes in a generalized \textit{Collective Matrix Factorization (CMF)}, our methods can generate rich phenotypes. Further, easy interpretability in phenotyping application requires sparse representations of the candidate phenotypes, for example each phenotype derived from patients' medication and diagnosis data should preferably be represented by handful of diagnosis and medications, ($5$--$10$ active components). We propose a constrained formulation of CMF for estimating sparse phenotypes. We demonstrate the efficacy of our model through an extensive empirical study on EHR data from Vanderbilt University Medical Center.

stat.AP↗

arXiv2015

Exponential Family Matrix Completion under Structural Constraints

We consider the matrix completion problem of recovering a structured matrix from noisy and partial measurements. Recent works have proposed tractable estimators with strong statistical guarantees for the case where the underlying matrix is low--rank, and the measurements consist of a subset, either of the exact individual entries, or of the entries perturbed by additive Gaussian noise, which is thus implicitly suited for thin--tailed continuous data. Arguably, common applications of matrix completion require estimators for (a) heterogeneous data--types, such as skewed--continuous, count, binary, etc., (b) for heterogeneous noise models (beyond Gaussian), which capture varied uncertainty in the measurements, and (c) heterogeneous structural constraints beyond low--rank, such as block--sparsity, or a superposition structure of low--rank plus elementwise sparseness, among others. In this paper, we provide a vastly unified framework for generalized matrix completion by considering a matrix completion setting wherein the matrix entries are sampled from any member of the rich family of exponential family distributions; and impose general structural constraints on the underlying matrix, as captured by a general regularizer $\mathcal{R}(.)$. We propose a simple convex regularized $M$--estimator for the generalized framework, and provide a unified and novel statistical analysis for this general class of estimators. We finally corroborate our theoretical results on simulated datasets.

stat.ML↗

arXiv2015

Consistent Collective Matrix Completion under Joint Low Rank Structure

We address the collective matrix completion problem of jointly recovering a collection of matrices with shared structure from partial (and potentially noisy) observations. To ensure well--posedness of the problem, we impose a joint low rank structure, wherein each component matrix is low rank and the latent space of the low rank factors corresponding to each entity is shared across the entire collection. We first develop a rigorous algebra for representing and manipulating collective--matrix structure, and identify sufficient conditions for consistent estimation of collective matrices. We then propose a tractable convex estimator for solving the collective matrix completion problem, and provide the first non--trivial theoretical guarantees for consistency of collective matrix completion. We show that under reasonable assumptions stated in Section 3.1, with high probability, the proposed estimator exactly recovers the true matrices whenever sample complexity requirements dictated by Theorem 1 are met. The sample complexity requirement derived in the paper are optimum up to logarithmic factors, and significantly improve upon the requirements obtained by trivial extensions of standard matrix completion. Finally, we propose a scalable approximate algorithm to solve the proposed convex program, and corroborate our results through simulated experiments.

stat.ML↗