In recent years, sparse principal component analysis has emerged as an extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed; some of these are fast to compute, while others are known to achieve the minimax optimal rate over certain Gaussian or sub-Gaussian classes. In this paper, we show that, under a widely-believed assumption from computational complexity theory, there is a fundamental trade-off between statistical and computational performance in this problem. More precisely, working with new, larger classes satisfying a restricted covariance concentration condition, we show that there is an effective sample size regime in which no randomised polynomial time algorithm can achieve the minimax optimal rate. We also study the theoretical performance of a (polynomial time) variant of the well-known semidefinite relaxation estimator, revealing a subtle interplay between statistical and computational efficiency.

Ben Recht spoke about optimization a few days ago at the Simons Institute. His talk was a highly entertaining tour de force through about a semester of convex optimization. You should go watch it. It's easy to spend a semester of convex optimization on various guises of gradient descent alone. Simply pick one of the following variants and work through the specifics of the analysis: conjugate, accelerated, projected, conditional, mirrored, stochastic, coordinate, online.

Model ensembling is a very powerful technique to increase accuracy on a variety of ML tasks. In this article I will share my ensembling approaches for Kaggle Competitions. For the first part we look at creating ensembles from submission files. The second part will look at creating ensembles through stacked generalization/blending. I answer why ensembling reduces the generalization error. Finally I show different methods of ensembling, together with their results and code to try it out for yourself. This is how you win ML competitions: you take other peoples' work and ensemble them together." The most basic and convenient way to ensemble is to ensemble Kaggle submission CSV files. You only need the predictions on the test set for these methods -- no need to retrain a model. This makes it a quick way to ensemble already existing model predictions, ideal when teaming up. Let's see why model ensembling reduces error rate and why it works better to ensemble low-correlated model ...

I think that gradient descent is one of those things which suffers a lot from Dunning Kruger. People will often learning about things like gradient descent, and feel ok, I know this. When in reality, there is such a wealth beneath the surface.

Solving real world problems with data science concepts is so exciting and it yields so fun. A famous dialogue you could listen from the data science people. It could be true if we add it's so challenging at the end of the dialogue. The foremost challenge starts from categorising the problem itself. The first level of categorising could be whether supervised or unsupervised learning.

This topic doesn't have much to do with nicer code, but there is probably some overlap in interest. However, some of the topics that we cover arise naturally here, so read on! MCMC is simply an algorithm for sampling from a distribution. The term stands for "Markov Chain Monte Carlo", because it is a type of "Monte Carlo" (i.e., a random) method that uses "Markov chains" (we'll discuss these later). MCMC is just one type of Monte Carlo method, although it is possible to view many other commonly used methods as simply special cases of MCMC.

Many machine learning tasks can be formulated as Regularized Empirical Risk Minimization (R-ERM), and solved by optimization algorithms such as gradient descent (GD), stochastic gradient descent (SGD), and stochastic variance reduction (SVRG). Conventional analysis on these optimization algorithms focuses on their convergence rates during the training process, however, people in the machine learning community may care more about the generalization performance of the learned model on unseen test data. In this paper, we investigate on this issue, by using stability as a tool. In particular, we decompose the generalization error for R-ERM, and derive its upper bound for both convex and non-convex cases. In convex cases, we prove that the generalization error can be bounded by the convergence rate of the optimization algorithm and the stability of the R-ERM process, both in expectation (in the order of $\mathcal{O}((1/n)+\mathbb{E}\rho(T))$, where $\rho(T)$ is the convergence error and $T$ is the number of iterations) and in high probability (in the order of $\mathcal{O}\left(\frac{\log{1/\delta}}{\sqrt{n}}+\rho(T)\right)$ with probability $1-\delta$). For non-convex cases, we can also obtain a similar expected generalization error bound. Our theorems indicate that 1) along with the training process, the generalization error will decrease for all the optimization algorithms under our investigation; 2) Comparatively speaking, SVRG has better generalization ability than GD and SGD. We have conducted experiments on both convex and non-convex problems, and the experimental results verify our theoretical findings.

The paper studies the solution of stochastic optimization problems in which approximations to the gradient and Hessian are obtained through subsampling. We first consider Newton-like methods that employ these approximations and discuss how to coordinate the accuracy in the gradient and Hessian to yield a superlinear rate of convergence in expectation. The second part of the paper analyzes an inexact Newton method that solves linear systems approximately using the conjugate gradient (CG) method, and that samples the Hessian and not the gradient (the gradient is assumed to be exact). We provide a complexity analysis for this method based on the properties of the CG iteration and the quality of the Hessian approximation, and compare it with a method that employs a stochastic gradient iteration instead of the CG method. We report preliminary numerical results that illustrate the performance of inexact subsampled Newton methods on machine learning applications based on logistic regression.

We propose a nonparametric statistical test for goodness-of-fit: given a set of samples, the test determines how likely it is that these were generated from a target density function. The measure of goodness-of-fit is a divergence constructed via Stein's method using functions from a Reproducing Kernel Hilbert Space. Our test statistic is based on an empirical estimate of this divergence, taking the form of a V-statistic in terms of the log gradients of the target density and the kernel. We derive a statistical test, both for i.i.d. and non-i.i.d. samples, where we estimate the null distribution quantiles using a wild bootstrap procedure. We apply our test to quantifying convergence of approximate Markov Chain Monte Carlo methods, statistical model criticism, and evaluating quality of fit vs model complexity in nonparametric density estimation.

There are various parametric models for analyzing pairwise comparison data, including the Bradley-Terry-Luce (BTL) and Thurstone models, but their reliance on strong parametric assumptions is limiting. In this work, we study a flexible model for pairwise comparisons, under which the probabilities of outcomes are required only to satisfy a natural form of stochastic transitivity. This class includes parametric models including the BTL and Thurstone models as special cases, but is considerably more general. We provide various examples of models in this broader stochastically transitive class for which classical parametric models provide poor fits. Despite this greater flexibility, we show that the matrix of probabilities can be estimated at the same rate as in standard parametric models. On the other hand, unlike in the BTL and Thurstone models, computing the minimax-optimal estimator in the stochastically transitive model is non-trivial, and we explore various computationally tractable alternatives. We show that a simple singular value thresholding algorithm is statistically consistent but does not achieve the minimax rate. We then propose and study algorithms that achieve the minimax rate over interesting sub-classes of the full stochastically transitive class. We complement our theoretical results with thorough numerical simulations.