We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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Convex Analysisの二,三の進展について

Suppose that one observes an incomplete subset of entries selected from a low-rank matrix. When is it possible to complete the matrix and recover the entries that have not been seen? We demonstrate that in very general settings, one can perfectly recover all of the missing entries from most sufficiently large subsets by solving a convex programming problem that finds the matrix with the minimum nuclear norm agreeing with the observed entries. The techniques used in this analysis draw upon parallels in the field of compressed sensing, demonstrating that objects other than signals and images can be perfectly reconstructed from very limited information.

/pdf/exact-matrix-completion-via-convex-optimization-15hkb1t9u7.pdf

Exact matrix completion via convex optimization

Discover New Methods for Dealing with High-Dimensional Data A sparse statistical model has only a small number of nonzero parameters or weights; therefore, it is much easier to estimate and interpret than a dense model. Statistical Learning with Sparsity: The Lasso and Generalizations presents methods that exploit sparsity to help recover the underlying signal in a set of data. Top experts in this rapidly evolving field, the authors describe the lasso for linear regression and a simple coordinate descent algorithm for its computation. They discuss the application of 1 penalties to generalized linear models and support vector machines, cover generalized penalties such as the elastic net and group lasso, and review numerical methods for optimization. They also present statistical inference methods for fitted (lasso) models, including the bootstrap, Bayesian methods, and recently developed approaches. In addition, the book examines matrix decomposition, sparse multivariate analysis, graphical models, and compressed sensing. It concludes with a survey of theoretical results for the lasso. In this age of big data, the number of features measured on a person or object can be large and might be larger than the number of observations. This book shows how the sparsity assumption allows us to tackle these problems and extract useful and reproducible patterns from big datasets. Data analysts, computer scientists, and theorists will appreciate this thorough and up-to-date treatment of sparse statistical modeling.

Statistical Learning with Sparsity: The Lasso and Generalizations

Matrix Factorization Techniques for Recommender Systems

This work provides a simplified proof of the statistical minimax optimality of (iterate averaged) stochastic gradient descent (SGD), for the special case of least squares. This result is obtained by analyzing SGD as a stochastic process and by sharply characterizing the stationary covariance matrix of this process. The finite rate optimality characterization captures the constant factors and addresses model mis-specification.

A Markov Chain Theory Approach to Characterizing the Minimax Optimality of Stochastic Gradient Descent (for Least Squares)

We study the first-order convex optimization problem, where we have black-box access to a (not necessarily smooth) function $f:\mathbb{R}^n \to \mathbb{R}$ and its (sub)gradient. Our goal is to find an $\epsilon$-approximate minimum of $f$ starting from a point that is distance at most $R$ from the true minimum. If $f$ is $G$-Lipschitz, then the classic gradient descent algorithm solves this problem with $O((GR/\epsilon)^{2})$ queries. Importantly, the number of queries is independent of the dimension $n$ and gradient descent is optimal in this regard: No deterministic or randomized algorithm can achieve better complexity that is still independent of the dimension $n$. 
In this paper we reprove the randomized lower bound of $\Omega((GR/\epsilon)^{2})$ using a simpler argument than previous lower bounds. We then show that although the function family used in the lower bound is hard for randomized algorithms, it can be solved using $O(GR/\epsilon)$ quantum queries. We then show an improved lower bound against quantum algorithms using a different set of instances and establish our main result that in general even quantum algorithms need $\Omega((GR/\epsilon)^2)$ queries to solve the problem. Hence there is no quantum speedup over gradient descent for black-box first-order convex optimization without further assumptions on the function family.

No quantum speedup over gradient descent for non-smooth convex optimization

We consider the problem of computing the squareroot of a positive semidefinite (PSD) matrix. Several fast algorithms (some based on eigenvalue decomposition and some based on Taylor expansion) are known to solve this problem. 
In this paper, we propose another way to solve this problem: a natural algorithm performing gradient descent on a non-convex formulation of the matrix squareroot problem. We show that on an $n\times n$ input PSD matrix ${M}$, if the initial point is well conditioned, then the algorithm finds an $\epsilon$-accurate solution in $O\left(\kappa^{3/2} \log \frac{\left\|{M}\right\|_F}{\epsilon}\right)$ iterations, where $\kappa$ is the condition number of $M$. Each iteration involves three matrix multiplications (and does not use either matrix inversions or solutions of linear system), giving a total run time of $O\left(n^{\omega}\kappa^{3/2}\log\frac{\left\|{M}\right\|_F}{\epsilon}\right)$, where $\omega$ is the matrix multiplication exponent. Furthermore we show that our algorithm is robust to errors in each iteration. We also show a lower bound of $\Omega(\kappa)$ iterations for our algorithm demonstrating that the dependence of our result on $\kappa$ is necessary. 
Existing analyses of similar algorithms (e.g., Newton's method) require commutativity of the input matrix with each iterate of the algorithm which is ensured by choosing the starting iterate carefully. Our analysis, on the other hand, is much more general and does not require each iterate to commute with the input matrix. Consequently, our result guarantees convergence from a wide range of starting points. 
More generally, our result demonstrates that non-convex optimization can be a viable approach to obtaining fast and robust algorithms. Our argument is quite general and we believe it will find application in designing such algorithms for other problems in numerical linear algebra.

Computing Matrix Squareroot via Non Convex Local Search

We consider the problem of outlier robust PCA (OR-PCA) where the goal is to recover principal directions despite the presence of outlier data points. That is, given a data matrix $M^*$, where $(1-\alpha)$ fraction of the points are noisy samples from a low-dimensional subspace while $\alpha$ fraction of the points can be arbitrary outliers, the goal is to recover the subspace accurately. Existing results for \OR-PCA have serious drawbacks: while some results are quite weak in the presence of noise, other results have runtime quadratic in dimension, rendering them impractical for large scale applications. 
In this work, we provide a novel thresholding based iterative algorithm with per-iteration complexity at most linear in the data size. Moreover, the fraction of outliers, $\alpha$, that our method can handle is tight up to constants while providing nearly optimal computational complexity for a general noise setting. For the special case where the inliers are obtained from a low-dimensional subspace with additive Gaussian noise, we show that a modification of our thresholding based method leads to significant improvement in recovery error (of the subspace) even in the presence of a large fraction of outliers.

Thresholding Based Efficient Outlier Robust PCA

We consider the problem of online learning and its application to solving minimax games. For the online learning problem, Follow the Perturbed Leader (FTPL) is a widely studied algorithm which enjoys the optimal $O(T^{1/2})$ worst-case regret guarantee for both convex and nonconvex losses. In this work, we show that when the sequence of loss functions is predictable, a simple modification of FTPL which incorporates optimism can achieve better regret guarantees, while retaining the optimal worst-case regret guarantee for unpredictable sequences. A key challenge in obtaining these tighter regret bounds is the stochasticity and optimism in the algorithm, which requires different analysis techniques than those commonly used in the analysis of FTPL. The key ingredient we utilize in our analysis is the dual view of perturbation as regularization. While our algorithm has several applications, we consider the specific application of minimax games. For solving smooth convex-concave games, our algorithm only requires access to a linear optimization oracle. For Lipschitz and smooth nonconvex-nonconcave games, our algorithm requires access to an optimization oracle which computes the perturbed best response. In both these settings, our algorithm solves the game up to an accuracy of $O(T^{-1/2})$ using $T$ calls to the optimization oracle. An important feature of our algorithm is that it is highly parallelizable and requires only $O(T^{1/2})$ iterations, with each iteration making $O(T^{1/2})$ parallel calls to the optimization oracle.

/pdf/follow-the-perturbed-leader-optimism-and-fast-parallel-dupgk1euop.pdf

Praneeth Netrapalli

Papers

A Markov Chain Theory Approach to Characterizing the Minimax Optimality of Stochastic Gradient Descent (for Least Squares)

No quantum speedup over gradient descent for non-smooth convex optimization

Computing Matrix Squareroot via Non Convex Local Search

Thresholding Based Efficient Outlier Robust PCA

Follow the Perturbed Leader: Optimism and Fast Parallel Algorithms for Smooth Minimax Games