In this paper, we consider a class of possibly nonconvex, nonsmooth and non-Lipschitz optimization problems arising in many contemporary applications such as machine learning, variable selection and image processing. To solve this class of problems, we propose a proximal gradient method with extrapolation and line search (PGels). This method is developed based on a special potential function and successfully incorporates both extrapolation and non-monotone line search, which are two simple and efficient accelerating techniques for the proximal gradient method. Thanks to the line search, this method allows more flexibilities in choosing the extrapolation parameters and updates them adaptively at each iteration if a certain line search criterion is not satisfied. Moreover, with proper choices of parameters, our PGels reduces to many existing algorithms. We also show that, under some mild conditions, our line search criterion is well defined and any cluster point of the sequence generated by PGels is a stationary point of our problem. In addition, by assuming the Kurdyka-${\L}$ojasiewicz exponent of the objective in our problem, we further analyze the local convergence rate of two special cases of PGels, including the widely used non-monotone proximal gradient method as one case. Finally, we conduct some numerical experiments for solving the $\ell_1$ regularized logistic regression problem and the $\ell_{1\text{-}2}$ regularized least squares problem. Our numerical results illustrate the efficiency of PGels and show the potential advantage of combining two accelerating techniques.

Natural gradient descent is an optimization method traditionally motivated from the perspective of information geometry, and works well for many applications as an alternative to stochastic gradient descent. In this paper we critically analyze this method and its properties, and show how it can be viewed as a type of approximate 2nd-order optimization method, where the Fisher information matrix can be viewed as an approximation of the Hessian. This perspective turns out to have significant implications for how to design a practical and robust version of the method. Additionally, we make the following contributions to the understanding of natural gradient and 2nd-order methods: a thorough analysis of the convergence speed of stochastic natural gradient descent (and more general stochastic 2nd-order methods) as applied to convex quadratics, a critical examination of the oft-used "empirical" approximation of the Fisher matrix, and an analysis of the (approximate) parameterization invariance property possessed by natural gradient methods, which we show still holds for certain choices of the curvature matrix other than the Fisher, but notably not the Hessian.

Epanechnikov Mean Shift is a simple yet empirically very effective algorithm for clustering. It localizes the centroids of data clusters via estimating modes of the probability distribution that generates the data points, using the `optimal' Epanechnikov kernel density estimator. However, since the procedure involves non-smooth kernel density functions, the convergence behavior of Epanechnikov mean shift lacks theoretical support as of this writing---most of the existing analyses are based on smooth functions and thus cannot be applied to Epanechnikov Mean Shift. In this work, we first show that the original Epanechnikov Mean Shift may indeed terminate at a non-critical point, due to the non-smoothness nature. Based on our analysis, we propose a simple remedy to fix it. The modified Epanechnikov Mean Shift is guaranteed to terminate at a local maximum of the estimated density, which corresponds to a cluster centroid, within a finite number of iterations. We also propose a way to avoid running the Mean Shift iterates from every data point, while maintaining good clustering accuracies under non-overlapping spherical Gaussian mixture models. This further pushes Epanechnikov Mean Shift to handle very large and high-dimensional data sets. Experiments show surprisingly good performance compared to the Lloyd's K-means algorithm and the EM algorithm.

Robust Optimization has traditionally taken a pessimistic, or worst-case viewpoint of uncertainty which is motivated by a desire to find sets of optimal policies that maintain feasibility under a variety of operating conditions. In this paper, we explore an optimistic, or best-case view of uncertainty and show that it can be a fruitful approach. We show that these techniques can be used to address a wide variety of problems. First, we apply our methods in the context of robust linear programming, providing a method for reducing conservatism in intuitive ways that encode economically realistic modeling assumptions. Second, we look at problems in machine learning and find that this approach is strongly connected to the existing literature. Specifically, we provide a new interpretation for popular sparsity inducing non-convex regularization schemes. Additionally, we show that successful approaches for dealing with outliers and noise can be interpreted as optimistic robust optimization problems. Although many of the problems resulting from our approach are non-convex, we find that DCA or DCA-like optimization approaches can be intuitive and efficient.

Block Coordinate Update (BCU) methods enjoy low per-update computational complexity because every time only one or a few block variables would need to be updated among possibly a large number of blocks. They are also easily parallelized and thus have been particularly popular for solving problems involving large-scale dataset and/or variables. In this paper, we propose a primal-dual BCU method for solving linearly constrained convex program in multi-block variables. The method is an accelerated version of a primal-dual algorithm proposed by the authors, which applies randomization in selecting block variables to update and establishes an $O(1/t)$ convergence rate under weak convexity assumption. We show that the rate can be accelerated to $O(1/t^2)$ if the objective is strongly convex. In addition, if one block variable is independent of the others in the objective, we then show that the algorithm can be modified to achieve a linear rate of convergence. The numerical experiments show that the accelerated method performs stably with a single set of parameters while the original method needs to tune the parameters for different datasets in order to achieve a comparable level of performance.

This paper investigates, from information theoretic grounds, a learning problem based on the principle that any regularity in a given dataset can be exploited to extract compact features from data, i.e., using fewer bits than needed to fully describe the data itself, in order to build meaningful representations of a relevant content (multiple labels). We begin by introducing the noisy lossy source coding paradigm with the log-loss fidelity criterion which provides the fundamental tradeoffs between the \emph{cross-entropy loss} (average risk) and the information rate of the features (model complexity). Our approach allows an information theoretic formulation of the \emph{multi-task learning} (MTL) problem which is a supervised learning framework in which the prediction models for several related tasks are learned jointly from common representations to achieve better generalization performance. Then, we present an iterative algorithm for computing the optimal tradeoffs and its global convergence is proven provided that some conditions hold. An important property of this algorithm is that it provides a natural safeguard against overfitting, because it minimizes the average risk taking into account a penalization induced by the model complexity. Remarkably, empirical results illustrate that there exists an optimal information rate minimizing the \emph{excess risk} which depends on the nature and the amount of available training data. An application to hierarchical text categorization is also investigated, extending previous works.

This paper presents a sequential randomized lowrank matrix factorization approach for incrementally predicting values of an unknown function at test points using the Gaussian Processes framework. It is well-known that in the Gaussian processes framework, the computational bottlenecks are the inversion of the (regularized) kernel matrix and the computation of the hyper-parameters defining the kernel. The main contributions of this paper are two-fold. First, we formalize an approach to compute the inverse of the kernel matrix using randomized matrix factorization algorithms in a streaming scenario, i.e., data is generated incrementally over time. The metrics of accuracy and computational efficiency of the proposed method are compared against a batch approach based on use of randomized matrix factorization and an existing streaming approach based on approximating the Gaussian process by a finite set of basis vectors. Second, we extend the sequential factorization approach to a class of kernel functions for which the hyperparameters can be efficiently optimized. All results are demonstrated on two publicly available datasets.

Greedy optimization methods such as Matching Pursuit (MP) and Frank-Wolfe (FW) algorithms regained popularity in recent years due to their simplicity, effectiveness and theoretical guarantees. MP and FW address optimization over the linear span and the convex hull of a set of atoms, respectively. In this paper, we consider the intermediate case of optimization over the convex cone, parametrized as the conic hull of a generic atom set, leading to the first principled definitions of non-negative MP algorithms for which we give explicit convergence rates and demonstrate excellent empirical performance. In particular, we derive sublinear ($\mathcal{O}(1/t)$) convergence on general smooth and convex objectives, and linear convergence ($\mathcal{O}(e^{-t})$) on strongly convex objectives, in both cases for general sets of atoms. Furthermore, we establish a clear correspondence of our algorithms to known algorithms from the MP and FW literature. Our novel algorithms and analyses target general atom sets and general objective functions, and hence are directly applicable to a large variety of learning settings.

Iteratively reweighted $\ell_1$ algorithm is a popular algorithm for solving a large class of optimization problems whose objective is the sum of a Lipschitz differentiable loss function and a possibly nonconvex sparsity inducing regularizer. In this paper, motivated by the success of extrapolation techniques in accelerating first-order methods, we study how widely used extrapolation techniques such as those in [4,5,22,28] can be incorporated to possibly accelerate the iteratively reweighted $\ell_1$ algorithm. We consider three versions of such algorithms. For each version, we exhibit an explicitly checkable condition on the extrapolation parameters so that the sequence generated provably clusters at a stationary point of the optimization problem. We also investigate global convergence under additional Kurdyka-$\L$ojasiewicz assumptions on certain potential functions. Our numerical experiments show that our algorithms usually outperform the general iterative shrinkage and thresholding algorithm in [21] and an adaptation of the iteratively reweighted $\ell_1$ algorithm in [23, Algorithm 7] with nonmonotone line-search for solving random instances of log penalty regularized least squares problems in terms of both CPU time and solution quality.

This was a busy week, I had no time to read anything new, so I'm sharing a note that I wrote for myself, for no other reason than to understand things better. It's a kind of cookbook of various "transformations" you can apply to a machine learning problem to eventually turn it into something we know how to solve: seeking stable attractors of a tractable vector field. The typical setup is: you have some model parameters $\theta$. You seek to optimize some objective criterion, but the optimization problem is intractable or hard in one of the ways listed below. You then apply the corresponding transformation to your problem if you can.