ABSTRACT. We obtain an improved finite-sample guarantee on the linear convergence of stochastic gradient descent for smooth and strongly convex objectives, improving from a quadratic dependence on the conditioning (L/µ) 2 (where L is a bound on the smoothness and µ on the strong convexity) to a linear dependence on L/µ. Furthermore, we show how reweighting the sampling distribution (i.e. importance sampling) is necessary in order to further improve convergence, and obtain a linear dependence in the average smoothness, dominating previous results. We also discuss importance sampling for SGD more broadly and show how it can improve convergence also in other scenarios.Our results are based on a connection we make between SGD and the randomized Kaczmarz algorithm, which allows us to transfer ideas between the separate bodies of literature studying each of the two methods. In particular, we recast the randomized Kaczmarz algorithm as an instance of SGD, and apply our results to prove its exponential convergence, but to the solution of a weighted least squares problem rather than the original least squares problem. We then present a modified Kaczmarz algorithm with partially biased sampling which does converge to the original least squares solution with the same exponential convergence rate.

In the last decade, the use of simple rating and comparison surveys has proliferated on social and digital media platforms to fuel recommendations. These simple surveys and their extrapolation with machine learning algorithms like matrix factorization shed light on user preferences over large and growing pools of items, such as movies, songs and ads. Social scientists have a long history of measuring perceptions, preferences and opinions, often over smaller, discrete item sets with exhaustive rating or ranking surveys. This paper introduces simple surveys for social science application. We ran experiments to compare the predictive accuracy of both individual and aggregate comparative assessments using four types of simple surveys -pairwise comparisons and ratings on 2, 5 and continuous point scales in three distinct contexts -perceived Safety of Google Streetview Images, Likeability of Artwork, and Hilarity of Animal GIFs. Across contexts, we find that continuous scale ratings best predict individual assessments but consume the most time and cognitive effort. Binary choice surveys are quick and perform best to predict aggregate assessments, useful for collective decision tasks, but poorly predict personalized preferences, for which they are currently used by Netflix to recommend movies. Pairwise comparisons, by contrast, perform well to predict personal assessments, but poorly predict aggregate assessments despite being widely used to crowdsource ideas and collective preferences. We also demonstrate how findings from these surveys can be visualized in a low-dimensional space that reveals distinct respondent interpretations of questions asked in each context. We conclude by reflecting on differences between sparse, incomplete 'simple surveys' and their traditional survey counterparts in terms of efficiency, information elicited and settings in which knowing less about more may be critical for social science. is Professor at the Toyota Technological Institution of Chicago, and part time faculty in Computer Science and the Committee on Computational and Applied Mathematics at the University of Chicago. Srebro is interested in statistical and computational aspects of machine learning and their interaction. He has done theoretical work in statistical learning theory and in algorithms, devised novel learning models and optimization techniques, and has worked on applications in computational biology, text analysis, collaborative filtering and social science. Srebro obtained his PhD from the Massachusetts Institute of Technology in 2004.

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