The recent introduction of indefinite SVM by Luss and dAspremont [15] has effectively demonstrated SVM classification with a non-positive semi-definite kernel (indefinite kernel). This paper studies the properties of the objective function introduced there. In particular, we show that the objective function is continuously differentiable and its gradient can be explicitly computed. Indeed, we further show that its gradient is Lipschitz continuous. The main idea behind our analysis is that the objective function is smoothed by the penalty term, in its saddle (min-max) representation, measuring the distance between the indefinite kernel matrix and the proxy positive semi-definite one. Our elementary result greatly facilitates the application of gradient-based algorithms. Based on our analysis, we further develop Nesterovs smooth optimization approach [16,17] for indefinite SVM which has an optimal convergence rate for smooth problems. Experiments on various benchmark datasets validate our analysis and demonstrate the efficiency of our proposed algorithms.

Most algorithms for solving Markov decision processes rely on a discount factor, which ensures their convergence. It is generally assumed that using an artificially low discount factor will improve the convergence rate, while sacrificing the solution quality.We however demonstrate that using an artificially low discount factor may significantly improve the solution quality, when used in approximate dynamic programming. We propose two explanations of this phenomenon. The first justification followsdirectly from the standard approximation error bounds: using a lower discount factor may decrease the approximation error bounds. However, we also show that these bounds are loose, thus their decrease does not entirely justify the improved solution quality. We thus propose another justification: when the rewards are received only sporadically (as in the case of Tetris), we can derive tighter bounds, which support a significant improvement in the solution quality with a decreased discount factor.

We consider regularized stochastic learning and online optimization problems, where the objective function is the sum of two convex terms: one is the loss function of the learning task, and the other is a simple regularization term such as L1-norm for sparsity. We develop a new online algorithm, the regularized dual averaging method, that can explicitly exploit the regularization structure in an online setting. In particular, at each iteration, the learning variables are adjusted by solving a simple optimization problem that involves the running average of all past subgradients of the loss functions and the whole regularization term, not just its subgradient. This method achieves the optimal convergence rate and often enjoys a low complexity per iteration similar as the standard stochastic gradient method. Computational experiments are presented for the special case of sparse online learning using L1-regularization.

Despite the large amount of literature on upper bounds on complexity of convex analysis, surprisingly little is known about the fundamental hardness of these problems. The extensive use of convex optimization in machine learning and statistics makes such an understanding critical to understand fundamental computational limits of learning and estimation. In this paper, we study the complexity of stochastic convex optimization in an oracle model of computation. We improve upon known results and obtain tight minimax complexity estimates for some function classes. We also discuss implications of these results to the understanding the inherent complexity of large-scale learning and estimation problems.

Many popular optimization algorithms, like the Levenberg-Marquardt algorithm (LMA), use heuristic-based controllers'' that modulate the behavior of the optimizer during the optimization process. For example, in the LMA a damping parameter is dynamically modified based on a set rules that were developed using various heuristic arguments. Reinforcement learning (RL) is a machine learning approach to learn optimal controllers by examples and thus is an obvious candidate to improve the heuristic-based controllers implicit in the most popular and heavily used optimization algorithms. Improving the performance of off-the-shelf optimizers is particularly important for time-constrained optimization problems. For example the LMA algorithm has become popular for many real-time computer vision problems, including object tracking from video, where only a small amount of time can be allocated to the optimizer on each incoming video frame. Here we show that a popular modern reinforcement learning technique using a very simply state space can dramatically improve the performance of general purpose optimizers, like the LMA. Most surprisingly the controllers learned for a particular domain appear to work very well also on very different optimization domains. For example we used RL methods to train a new controller for the damping parameter of the LMA. This controller was trained on a collection of classic, relatively small, non-linear regression problems. The modified LMA performed better than the standard LMA on these problems. Most surprisingly, it also dramatically outperformed the standard LMA on a difficult large scale computer vision problem for which it had not been trained before. Thus the controller appeared to have extracted control rules that were not just domain specific but generalized across a wide range of optimization domains."

We present a multi-label multiple kernel learning (MKL) formulation, in which the data are embedded into a low-dimensional space directed by the instance-label correlations encoded into a hypergraph. We formulate the problem in the kernel-induced feature space and propose to learn the kernel matrix as a linear combination of a given collection of kernel matrices in the MKL framework. The proposed learning formulation leads to a non-smooth min-max problem, and it can be cast into a semi-infinite linear program (SILP). We further propose an approximate formulation with a guaranteed error bound which involves an unconstrained and convex optimization problem. In addition, we show that the objective function of the approximate formulation is continuously differentiable with Lipschitz gradient, and hence existing methods can be employed to compute the optimal solution efficiently. We apply the proposed formulation to the automated annotation of Drosophila gene expression pattern images, and promising results have been reported in comparison with representative algorithms.

Which ads should we display in sponsored search in order to maximize our revenue? How should we dynamically rank information sources to maximize value of information? These applications exhibit strong diminishing returns: Selection of redundant ads and information sources decreases their marginal utility. We show that these and other problems can be formalized as repeatedly selecting an assignment of items to positions to maximize a sequence of monotone submodular functions that arrive one by one. We present an efficient algorithm for this general problem and analyze it in the no-regret model. Our algorithm is equipped with strong theoretical guarantees, with a performance ratio that converges to the optimal constant of 1-1/e. We empirically evaluate our algorithms on two real-world online optimization problems on the web: ad allocation with submodular utilities, and dynamically ranking blogs to detect information cascades.

The stochastic approximation method is behind the solution to many important, actively-studied problems in machine learning. Despite its far-reaching application, there is almost no work on applying stochastic approximation to learning problems with constraints. The reason for this, we hypothesize, is that no robust, widely-applicable stochastic approximation method exists for handling such problems. We propose that interior-point methods are a natural solution. We establish the stability of a stochastic interior-point approximation method both analytically and empirically, and demonstrate its utility by deriving an on-line learning algorithm that also performs feature selection via L1 regularization.

Covariance estimation for high dimensional vectors is a classically difficult problem in statistical analysis and machine learning due to limited sample size. In this paper, we propose a new approach to covariance estimation, which is based on constrained maximum likelihood (ML) estimation of the covariance. Specifically, the covariance is constrained to have an eigen decomposition which can be represented as a sparse matrix transform (SMT). The SMT is formed by a product of pairwise coordinate rotations known as Givens rotations. Using this framework, the covariance can be efficiently estimated using greedy minimization of the log likelihood function, and the number of Givens rotations can be efficiently computed using a cross-validation procedure. The estimator obtained using this method is always positive definite and well-conditioned even with limited sample size. Experiments on hyperspectral data show that SMT covariance estimation results in consistently better estimates of the covariance for a variety of different classes and sample sizes compared to traditional shrinkage estimators.

We consider robust least-squares regression with feature-wise disturbance. We show that this formulation leads to tractable convex optimization problems, and we exhibit a particular uncertainty set for which the robust problem is equivalent to $\ell_1$ regularized regression (Lasso). This provides an interpretation of Lasso from a robust optimization perspective. We generalize this robust formulation to consider more general uncertainty sets, which all lead to tractable convex optimization problems. Therefore, we provide a new methodology for designing regression algorithms, which generalize known formulations. The advantage is that robustness to disturbance is a physical property that can be exploited: in addition to obtaining new formulations, we use it directly to show sparsity properties of Lasso, as well as to prove a general consistency result for robust regression problems, including Lasso, from a unified robustness perspective.