We consider a class of sparsity-inducing regularization terms based on submodular functions. While previous work has focused on non-decreasing functions, we explore symmetric submodular functions and their \lova extensions. We show that the Lovasz extension may be seen as the convex envelope of a function that depends on level sets (i.e., the set of indices whose corresponding components of the underlying predictor are greater than a given constant): this leads to a class of convex structured regularization terms that impose prior knowledge on the level sets, and not only on the supports of the underlying predictors. We provide a unified set of optimization algorithms, such as proximal operators, and theoretical guarantees (allowed level sets and recovery conditions). By selecting specific submodular functions, we give a new interpretation to known norms, such as the total variation; we also define new norms, in particular ones that are based on order statistics with application to clustering and outlier detection, and on noisy cuts in graphs with application to change point detection in the presence of outliers.

A majority of approximate dynamic programming approaches to the reinforcement learning problem can be categorized into greedy value function methods and value-based policy gradient methods. The former approach, although fast, is well known to be susceptible to the policy oscillation phenomenon. We take a fresh view to this phenomenon by casting a considerable subset of the former approach as a limiting special case of the latter. We explain the phenomenon in terms of this view and illustrate the underlying mechanism with artificial examples. We also use it to derive the constrained natural actor-critic algorithm that can interpolate between the aforementioned approaches. In addition, it has been suggested in the literature that the oscillation phenomenon might be subtly connected to the grossly suboptimal performance in the Tetris benchmark problem of all attempted approximate dynamic programming methods. We report empirical evidence against such a connection and in favor of an alternative explanation. Finally, we report scores in the Tetris problem that improve on existing dynamic programming based results.

The L_1 regularized Gaussian maximum likelihood estimator has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In contrast to other state-of-the-art methods that largely use first order gradient information, our algorithm is based on Newton's method and employs a quadratic approximation, but with some modifications that leverage the structure of the sparse Gaussian MLE problem. We show that our method is superlinearly convergent, and also present experimental results using synthetic and real application data that demonstrate the considerable improvements in performance of our method when compared to other state-of-the-art methods.

Using the $\ell_1$-norm to regularize the estimation of the parameter vector of a linear model leads to an unstable estimator when covariates are highly correlated. In this paper, we introduce a new penalty function which takes into account the correlation of the design matrix to stabilize the estimation. This norm, called the trace Lasso, uses the trace norm of the selected covariates, which is a convex surrogate of their rank, as the criterion of model complexity. We analyze the properties of our norm, describe an optimization algorithm based on reweighted least-squares, and illustrate the behavior of this norm on synthetic data, showing that it is more adapted to strong correlations than competing methods such as the elastic net.

Increasingly, optimization problems in machine learning, especially those arising from high-dimensional statistical estimation, have a large number of variables. Modern statistical estimators developed over the past decade have statistical or sample complexity that depends only weakly on the number of parameters when there is some structure to the problem, such as sparsity. A central question is whether similar advances can be made in their computational complexity as well. In this paper, we propose strategies that indicate that such advances can indeed be made. In particular, we investigate the greedy coordinate descent algorithm, and note that performing the greedy step efficiently weakens the costly dependence on the problem size provided the solution is sparse. We then propose a suite of methods that perform these greedy steps efficiently by a reduction to nearest neighbor search. We also devise a more amenable form of greedy descent for composite non-smooth objectives; as well as several approximate variants of such greedy descent. We develop a practical implementation of our algorithm that combines greedy coordinate descent with locality sensitive hashing. Without tuning the latter data structure, we are not only able to significantly speed up the vanilla greedy method, but also outperform cyclic descent when the problem size becomes large. Our results indicate the effectiveness of our nearest neighbor strategies, and also point to many open questions regarding the development of computational geometric techniques tailored towards first-order optimization methods.

In this paper, we propose the first exact algorithm for minimizing the difference of two submodular functions (D.S.), i.e., the discrete version of the D.C. programming problem. The developed algorithm is a branch-and-bound-based algorithm which responds to the structure of this problem through the relationship between submodularity and convexity. The D.S. programming problem covers a broad range of applications in machine learning because this generalizes the optimization of a wide class of set functions. We empirically investigate the performance of our algorithm, and illustrate the difference between exact and approximate solutions respectively obtained by the proposed and existing algorithms in feature selection and discriminative structure learning.

Knowledge-based support vector machines (KBSVMs) incorporate advice from domain experts, which can improve generalization significantly. A major limitation that has not been fully addressed occurs when the expert advice is imperfect, which can lead to poorer models. We propose a model that extends KBSVMs and is able to not only learn from data and advice, but also simultaneously improve the advice. The proposed approach is particularly effective for knowledge discovery in domains with few labeled examples. The proposed model contains bilinear constraints, and is solved using two iterative approaches: successive linear programming and a constrained concave-convex approach. Experimental results demonstrate that these algorithms yield useful refinements to expert advice, as well as improve the performance of the learning algorithm overall.

Mini-batch algorithms have recently received significant attention as a way to speed-up stochastic convex optimization problems. In this paper, we study how such algorithms can be improved using accelerated gradient methods. We provide a novel analysis, which shows how standard gradient methods may sometimes be insufficient to obtain a significant speed-up. We propose a novel accelerated gradient algorithm, which deals with this deficiency, and enjoys a uniformly superior guarantee. We conclude our paper with experiments on real-world datasets, which validates our algorithm and substantiates our theoretical insights.

Most previous research on image categorization has focused on medium-scale data sets, while large-scale image categorization with millions of images from thousands of categories remains a challenge. With the emergence of structured large-scale dataset such as the ImageNet, rich information about the conceptual relationships between images, such as a tree hierarchy among various image categories, become available. As human cognition of complex visual world benefits from underlying semantic relationships between object classes, we believe a machine learning system can and should leverage such information as well for better performance. In this paper, we employ such semantic relatedness among image categories for large-scale image categorization. Specifically, a category hierarchy is utilized to properly define loss function and select common set of features for related categories. An efficient optimization method based on proximal approximation and accelerated parallel gradient method is introduced. Experimental results on a subset of ImageNet containing 1.2 million images from 1000 categories demonstrate the effectiveness and promise of our proposed approach.

We present an optimization approach for linear SVMs based on a stochastic primal-dual approach, where the primal step is akin to an importance-weighted SGD, and the dual step is a stochastic update on the importance weights. This yields an optimization method with a sublinear dependence on the training set size, and the first method for learning linear SVMs with runtime less then the size of the training set required for learning!