arXiv subjects

Insu Han

Publications and source records attributed to Insu Han.

25 records · Page 2Linked to original sources

arXiv2021

Scalable Learning and MAP Inference for Nonsymmetric Determinantal Point Processes

Determinantal point processes (DPPs) have attracted significant attention in machine learning for their ability to model subsets drawn from a large item collection. Recent work shows that nonsymmetric DPP (NDPP) kernels have significant advantages over symmetric kernels in terms of modeling power and predictive performance. However, for an item collection of size $M$, existing NDPP learning and inference algorithms require memory quadratic in $M$ and runtime cubic (for learning) or quadratic (for inference) in $M$, making them impractical for many typical subset selection tasks. In this work, we develop a learning algorithm with space and time requirements linear in $M$ by introducing a new NDPP kernel decomposition. We also derive a linear-complexity NDPP maximum a posteriori (MAP) inference algorithm that applies not only to our new kernel but also to that of prior work. Through evaluation on real-world datasets, we show that our algorithms scale significantly better, and can match the predictive performance of prior work.

cs.LG↗

arXiv2021

Random Features for the Neural Tangent Kernel

The Neural Tangent Kernel (NTK) has discovered connections between deep neural networks and kernel methods with insights of optimization and generalization. Motivated by this, recent works report that NTK can achieve better performances compared to training neural networks on small-scale datasets. However, results under large-scale settings are hardly studied due to the computational limitation of kernel methods. In this work, we propose an efficient feature map construction of the NTK of fully-connected ReLU network which enables us to apply it to large-scale datasets. We combine random features of the arc-cosine kernels with a sketching-based algorithm which can run in linear with respect to both the number of data points and input dimension. We show that dimension of the resulting features is much smaller than other baseline feature map constructions to achieve comparable error bounds both in theory and practice. We additionally utilize the leverage score based sampling for improved bounds of arc-cosine random features and prove a spectral approximation guarantee of the proposed feature map to the NTK matrix of two-layer neural network. We benchmark a variety of machine learning tasks to demonstrate the superiority of the proposed scheme. In particular, our algorithm can run tens of magnitude faster than the exact kernel methods for large-scale settings without performance loss.

cs.LG↗

arXiv2020

Polynomial Tensor Sketch for Element-wise Function of Low-Rank Matrix

This paper studies how to sketch element-wise functions of low-rank matrices. Formally, given low-rank matrix A = [Aij] and scalar non-linear function f, we aim for finding an approximated low-rank representation of the (possibly high-rank) matrix [f(Aij)]. To this end, we propose an efficient sketching-based algorithm whose complexity is significantly lower than the number of entries of A, i.e., it runs without accessing all entries of [f(Aij)] explicitly. The main idea underlying our method is to combine a polynomial approximation of f with the existing tensor sketch scheme for approximating monomials of entries of A. To balance the errors of the two approximation components in an optimal manner, we propose a novel regression formula to find polynomial coefficients given A and f. In particular, we utilize a coreset-based regression with a rigorous approximation guarantee. Finally, we demonstrate the applicability and superiority of the proposed scheme under various machine learning tasks.

cs.LG↗

arXiv2018

Stochastic Chebyshev Gradient Descent for Spectral Optimization

A large class of machine learning techniques requires the solution of optimization problems involving spectral functions of parametric matrices, e.g. log-determinant and nuclear norm. Unfortunately, computing the gradient of a spectral function is generally of cubic complexity, as such gradient descent methods are rather expensive for optimizing objectives involving the spectral function. Thus, one naturally turns to stochastic gradient methods in hope that they will provide a way to reduce or altogether avoid the computation of full gradients. However, here a new challenge appears: there is no straightforward way to compute unbiased stochastic gradients for spectral functions. In this paper, we develop unbiased stochastic gradients for spectral-sums, an important subclass of spectral functions. Our unbiased stochastic gradients are based on combining randomized trace estimators with stochastic truncation of the Chebyshev expansions. A careful design of the truncation distribution allows us to offer distributions that are variance-optimal, which is crucial for fast and stable convergence of stochastic gradient methods. We further leverage our proposed stochastic gradients to devise stochastic methods for objective functions involving spectral-sums, and rigorously analyze their convergence rate. The utility of our methods is demonstrated in numerical experiments.

cs.LG↗

arXiv2017

Faster Greedy MAP Inference for Determinantal Point Processes

Determinantal point processes (DPPs) are popular probabilistic models that arise in many machine learning tasks, where distributions of diverse sets are characterized by matrix determinants. In this paper, we develop fast algorithms to find the most likely configuration (MAP) of large-scale DPPs, which is NP-hard in general. Due to the submodular nature of the MAP objective, greedy algorithms have been used with empirical success. Greedy implementations require computation of log-determinants, matrix inverses or solving linear systems at each iteration. We present faster implementations of the greedy algorithms by utilizing the complementary benefits of two log-determinant approximation schemes: (a) first-order expansions to the matrix log-determinant function and (b) high-order expansions to the scalar log function with stochastic trace estimators. In our experiments, our algorithms are orders of magnitude faster than their competitors, while sacrificing marginal accuracy.

cs.DM↗

arXiv2017

Approximating the Spectral Sums of Large-scale Matrices using Chebyshev Approximations

Computation of the trace of a matrix function plays an important role in many scientific computing applications, including applications in machine learning, computational physics (e.g., lattice quantum chromodynamics), network analysis and computational biology (e.g., protein folding), just to name a few application areas. We propose a linear-time randomized algorithm for approximating the trace of matrix functions of large symmetric matrices. Our algorithm is based on coupling function approximation using Chebyshev interpolation with stochastic trace estimators (Hutchinson's method), and as such requires only implicit access to the matrix, in the form of a function that maps a vector to the product of the matrix and the vector. We provide rigorous approximation error in terms of the extremal eigenvalue of the input matrix, and the Bernstein ellipse that corresponds to the function at hand. Based on our general scheme, we provide algorithms with provable guarantees for important matrix computations, including log-determinant, trace of matrix inverse, Estrada index, Schatten p-norm, and testing positive definiteness. We experimentally evaluate our algorithm and demonstrate its effectiveness on matrices with tens of millions dimensions.

cs.DS↗

arXiv2015

Large-scale Log-determinant Computation through Stochastic Chebyshev Expansions

Logarithms of determinants of large positive definite matrices appear ubiquitously in machine learning applications including Gaussian graphical and Gaussian process models, partition functions of discrete graphical models, minimum-volume ellipsoids, metric learning and kernel learning. Log-determinant computation involves the Cholesky decomposition at the cost cubic in the number of variables, i.e., the matrix dimension, which makes it prohibitive for large-scale applications. We propose a linear-time randomized algorithm to approximate log-determinants for very large-scale positive definite and general non-singular matrices using a stochastic trace approximation, called the Hutchinson method, coupled with Chebyshev polynomial expansions that both rely on efficient matrix-vector multiplications. We establish rigorous additive and multiplicative approximation error bounds depending on the condition number of the input matrix. In our experiments, the proposed algorithm can provide very high accuracy solutions at orders of magnitude faster time than the Cholesky decomposition and Schur completion, and enables us to compute log-determinants of matrices involving tens of millions of variables.

cs.DS↗