We design algorithms for fitting a high-dimensional statistical model to a large, sparse network without revealing sensitive information of individual members. Given a sparse input graph $G$, our algorithms output a node-differentially private nonparametric block model approximation. By node-differentially private, we mean that our output hides the insertion or removal of a vertex and all its adjacent edges. If $G$ is an instance of the network obtained from a generative nonparametric model defined in terms of a graphon $W$, our model guarantees consistency: as the number of vertices tends to infinity, the output of our algorithm converges to $W$ in an appropriate version of the $L_2$ norm. In particular, this means we can estimate the sizes of all multi-way cuts in $G$. Our results hold as long as $W$ is bounded, the average degree of $G$ grows at least like the log of the number of vertices, and the number of blocks goes to infinity at an appropriate rate. We give explicit error bounds in terms of the parameters of the model; in several settings, our bounds improve on or match known nonprivate results.

We consider a sequential learning problem with Gaussian payoffs and side information: after selecting an action $i$, the learner receives information about the payoff of every action $j$ in the form of Gaussian observations whose mean is the same as the mean payoff, but the variance depends on the pair $(i,j)$ (and may be infinite). The setup allows a more refined information transfer from one action to another than previous partial monitoring setups, including the recently introduced graph-structured feedback case. For the first time in the literature, we provide non-asymptotic problem-dependent lower bounds on the regret of any algorithm, which recover existing asymptotic problem-dependent lower bounds and finite-time minimax lower bounds available in the literature. We also provide algorithms that achieve the problem-dependent lower bound (up to some universal constant factor) or the minimax lower bounds (up to logarithmic factors).

The iteration complexity of the block-coordinate descent (BCD) type algorithm has been under extensive investigation. It was recently shown that for convex problems the classical cyclic BCGD (block coordinate gradient descent) achieves an O(1/r) complexity (r is the number of passes of all blocks). However, such bounds are at least linearly depend on $K$ (the number of variable blocks), and are at least $K$ times worse than those of the gradient descent (GD) and proximal gradient (PG) methods.In this paper, we close such theoretical performance gap between cyclic BCD and GD/PG. First we show that for a family of quadratic nonsmooth problems, the complexity bounds for cyclic Block Coordinate Proximal Gradient (BCPG), a popular variant of BCD, can match those of the GD/PG in terms of dependency on $K$ (up to a \log^2(K) factor). Second, we establish an improved complexity bound for Coordinate Gradient Descent (CGD) for general convex problems which can match that of GD in certain scenarios. Our bounds are sharper than the known bounds as they are always at least $K$ times worse than GD. {Our analyses do not depend on the update order of block variables inside each cycle, thus our results also apply to BCD methods with random permutation (random sampling without replacement, another popular variant).

Matrix factorization (MF) collaborative filtering is an effective and widely used method in recommendation systems. However, the problem of finding an optimal trade-off between exploration and exploitation (otherwise known as the bandit problem), a crucial problem in collaborative filtering from cold-start, has not been previously addressed.In this paper, we present a novel algorithm for online MF recommendation that automatically combines finding the most relevantitems with exploring new or less-recommended items.Our approach, called Particle Thompson Sampling for Matrix-Factorization, is based on the general Thompson sampling framework, but augmented with a novel efficient online Bayesian probabilistic matrix factorization method based on the Rao-Blackwellized particle filter.Extensive experiments in collaborative filtering using several real-world datasets demonstrate that our proposed algorithm significantly outperforms the current state-of-the-arts.

Tree structured group Lasso (TGL) is a powerful technique in uncovering the tree structured sparsity over the features, where each node encodes a group of features. It has been applied successfully in many real-world applications. However, with extremely large feature dimensions, solving TGL remains a significant challenge due to its highly complicated regularizer. In this paper, we propose a novel Multi-Layer Feature reduction method (MLFre) to quickly identify the inactive nodes (the groups of features with zero coefficients in the solution) hierarchically in a top-down fashion, which are guaranteed to be irrelevant to the response. Thus, we can remove the detected nodes from the optimization without sacrificing accuracy. The major challenge in developing such testing rules is due to the overlaps between the parents and their children nodes. By a novel hierarchical projection algorithm, MLFre is able to test the nodes independently from any of their ancestor nodes. Moreover, we can integrate MLFre---that has a low computational cost---with any existing solvers. Experiments on both synthetic and real data sets demonstrate that the speedup gained by MLFre can be orders of magnitude.

We study the performance of standard online learning algorithms when the feedback isdelayed by an adversary. We show that online-gradient-descent [1] and follow-the-perturbed-leader [2] achieve regret O( D)in the delayed setting, where D is the sum of delays of each round's feedback. This bound collapses to an optimal O( T) bound in the usual setting of no delays (where D T). Our main contribution is to show that standard algorithms for online learning already have simple regret bounds in the most general setting of delayed feedback, making adjustments to the analysis and not to the algorithms themselves. Our results help affirm and clarify the success of recent algorithms in optimization and machine learning that operate in a delayed feedback model.

In addition to identifying the content within a single image, relating images and generating related images are critical tasks for image understanding. Recently, deep convolutional networks have yielded breakthroughs in producing image labels, annotations and captions, but have only just begun to be used for producing high-quality image outputs. In this paper we develop a novel deep network trained end-to-end to perform visual analogy making, which is the task of transforming a query image according to an example pair of related images. Solving this problem requires both accurately recognizing a visual relationship and generating a transformed query image accordingly. Inspired by recent advances in language modeling, we propose to solve visual analogies by learning to map images to a neural embedding in which analogical reasoning is simple, such as by vector subtraction and addition. In experiments, our model effectively models visual analogies on several datasets: 2D shapes, animated video game sprites, and 3D car models.

Training deep feature hierarchies to solve supervised learning tasks has achieving state of the art performance on many problems in computer vision. However, a principled way in which to train such hierarchies in the unsupervised setting has remained elusive. In this work we suggest a new architecture and loss for training deep feature hierarchies that linearize the transformations observed in unlabelednatural video sequences. This is done by training a generative model to predict video frames. We also address the problem of inherent uncertainty in prediction by introducing a latent variables that are non-deterministic functions of the input into the network architecture.

Practitioners of Bayesian statistics have long depended on Markov chain Monte Carlo (MCMC) to obtain samples from intractable posterior distributions. Unfortunately, MCMC algorithms are typically serial, and do not scale to the large datasets typical of modern machine learning. The recently proposed consensus Monte Carlo algorithm removes this limitation by partitioning the data and drawing samples conditional on each partition in parallel (Scott et al, 2013). A fixed aggregation function then combines these samples, yielding approximate posterior samples. We introduce variational consensus Monte Carlo (VCMC), a variational Bayes algorithm that optimizes over aggregation functions to obtain samples from a distribution that better approximates the target. The resulting objective contains an intractable entropy term; we therefore derive a relaxation of the objective and show that the relaxed problem is blockwise concave under mild conditions. We illustrate the advantages of our algorithm on three inference tasks from the literature, demonstrating both the superior quality of the posterior approximation and the moderate overhead of the optimization step. Our algorithm achieves a relative error reduction (measured against serial MCMC) of up to 39% compared to consensus Monte Carlo on the task of estimating 300-dimensional probit regression parameter expectations; similarly, it achieves an error reduction of 92% on the task of estimating cluster comembership probabilities in a Gaussian mixture model with 8 components in 8 dimensions. Furthermore, these gains come at moderate cost compared to the runtime of serial MCMC, achieving near-ideal speedup in some instances.

Matrix completion problems have been widely studied under special low dimensional structures such as low rank or structure induced by decomposable norms. 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 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 \textit{\modified}~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 including the recently proposed spectral $k$-support norm.