Latent variable models are a powerful tool for addressing several tasks in machine learning. However, the algorithms for learning the parameters of latent variable models are prone to getting stuck in a bad local optimum. To alleviate this problem, we build on the intuition that, rather than considering all samples simultaneously, the algorithm should be presented with the training data in a meaningful order that facilitates learning. The order of the samples is determined by how easy they are. The main challenge is that often we are not provided with a readily computable measure of the easiness of samples. We address this issue by proposing a novel, iterative self-paced learning algorithm where each iteration simultaneously selects easy samples and learns a new parameter vector. The number of samples selected is governed by a weight that is annealed until the entire training data has been considered. We empirically demonstrate that the self-paced learning algorithm outperforms the state of the art method for learning a latent structural SVM on four applications: object localization, noun phrase coreference, motif finding and handwritten digit recognition.

Recent studies have shown that multiple kernel learning is very effective for object recognition, leading to the popularity of kernel learning in computer vision problems. In this work, we develop an efficient algorithm for multi-label multiple kernel learning (ML-MKL). We assume that all the classes under consideration share the same combination of kernel functions, and the objective is to find the optimal kernel combination that benefits all the classes. Although several algorithms have been developed for ML-MKL, their computational cost is linear in the number of classes, making them unscalable when the number of classes is large, a challenge frequently encountered in visual object recognition. We address this computational challenge by developing a framework for ML-MKL that combines the worst-case analysis with stochastic approximation. Our analysis shows that the complexity of our algorithm is $O(m^{1/3}\sqrt{ln m})$, where $m$ is the number of classes. Empirical studies with object recognition show that while achieving similar classification accuracy, the proposed method is significantly more efficient than the state-of-the-art algorithms for ML-MKL.

Likelihood ratio policy gradient methods have been some of the most successful reinforcement learning algorithms, especially for learning on physical systems. We describe how the likelihood ratio policy gradient can be derived from an importance sampling perspective. This derivation highlights how likelihood ratio methods under-use past experience by (a) using the past experience to estimate {\em only} the gradient of the expected return $U(\theta)$ at the current policy parameterization $\theta$, rather than to obtain a more complete estimate of $U(\theta)$, and (b) using past experience under the current policy {\em only} rather than using all past experience to improve the estimates. We present a new policy search method, which leverages both of these observations as well as generalized baselines---a new technique which generalizes commonly used baseline techniques for policy gradient methods. Our algorithm outperforms standard likelihood ratio policy gradient algorithms on several testbeds.

We consider a class of learning problems that involve a structured sparsity-inducing norm defined as the sum of $\ell_\infty$-norms over groups of variables. Whereas a lot of effort has been put in developing fast optimization methods when the groups are disjoint or embedded in a specific hierarchical structure, we address here the case of general overlapping groups. To this end, we show that the corresponding optimization problem is related to network flow optimization. More precisely, the proximal problem associated with the norm we consider is dual to a quadratic min-cost flow problem. We propose an efficient procedure which computes its solution exactly in polynomial time. Our algorithm scales up to millions of groups and variables, and opens up a whole new range of applications for structured sparse models. We present several experiments on image and video data, demonstrating the applicability and scalability of our approach for various problems.

Recent work in reinforcement learning has emphasized the power of L1 regularization to perform feature selection and prevent overfitting. We propose formulating the L1 regularized linear fixed point problem as a linear complementarity problem (LCP). This formulation offers several advantages over the LARS-inspired formulation, LARS-TD. The LCP formulation allows the use of efficient off-the-shelf solvers, leads to a new uniqueness result, and can be initialized with starting points from similar problems (warm starts). We demonstrate that warm starts, as well as the efficiency of LCP solvers, can speed up policy iteration. Moreover, warm starts permit a form of modified policy iteration that can be used to approximate a greedy" homotopy path, a generalization of the LARS-TD homotopy path that combines policy evaluation and optimization."

Many statistical $M$-estimators are based on convex optimization problems formed by the weighted sum of a loss function with a norm-based regularizer. We analyze the convergence rates of first-order gradient methods for solving such problems within a high-dimensional framework that allows the data dimension $d$ to grow with (and possibly exceed) the sample size $n$. This high-dimensional structure precludes the usual global assumptions---namely, strong convexity and smoothness conditions---that underlie classical optimization analysis. We define appropriately restricted versions of these conditions, and show that they are satisfied with high probability for various statistical models. Under these conditions, our theory guarantees that Nesterov's first-order method~\cite{Nesterov07} has a globally geometric rate of convergence up to the statistical precision of the model, meaning the typical Euclidean distance between the true unknown parameter $\theta^*$ and the optimal solution $\widehat{\theta}$. This globally linear rate is substantially faster than previous analyses of global convergence for specific methods that yielded only sublinear rates. Our analysis applies to a wide range of $M$-estimators and statistical models, including sparse linear regression using Lasso ($\ell_1$-regularized regression), group Lasso, block sparsity, and low-rank matrix recovery using nuclear norm regularization. Overall, this result reveals an interesting connection between statistical precision and computational efficiency in high-dimensional estimation.

We present a fast online solver for large scale maximum-flow problems as they occur in portfolio optimization, inventory management, computer vision, and logistics. Our algorithm solves an integer linear program in an online fashion. It exploits total unimodularity of the constraint matrix and a Lagrangian relaxation to solve the problem as a convex online game. The algorithm generates approximate solutions of max-flow problems by performing stochastic gradient descent on a set of flows. We apply the algorithm to optimize tier arrangement of over 80 Million web pages on a layered set of caches to serve an incoming query stream optimally. We provide an empirical demonstration of the effectiveness of our method on real query-pages data.

This paper proposes a simple and efficient finite difference method for implicit differentiationof marginal inference results in discrete graphical models. Given an arbitrary loss function, defined on marginals, we show that the derivatives of this loss with respect to model parameters can be obtained by running the inference procedure twice, on slightly perturbed model parameters. Thismethod can be used with approximate inference, with a loss function over approximate marginals. Convenient choices of loss functions makeit practical to fit graphical models with hidden variables, high treewidth and/or model misspecification.

We consider the tree structured group Lasso where the structure over the features can be represented as a tree with leaf nodes as features and internal nodes as clusters of the features. The structured regularization with a pre-defined tree structure is based on a group-Lasso penalty, where one group is defined for each node in the tree. Such a regularization can help uncover the structured sparsity, which is desirable for applications with some meaningful tree structures on the features. However, the tree structured group Lasso is challenging to solve due to the complex regularization. In this paper, we develop an efficient algorithm for the tree structured group Lasso. One of the key steps in the proposed algorithm is to solve the Moreau-Yosida regularization associated with the grouped tree structure. The main technical contributions of this paper include (1) we show that the associated Moreau-Yosida regularization admits an analytical solution, and (2) we develop an efficient algorithm for determining the effective interval for the regularization parameter. Our experimental results on the AR and JAFFE face data sets demonstrate the efficiency and effectiveness of the proposed algorithm.

We present policy gradient results within the framework of linearly-solvable MDPs. For the first time, compatible function approximators and natural policy gradients are obtained by estimating the cost-to-go function, rather than the (much larger) state-action advantage function as is necessary in traditional MDPs. We also develop the first compatible function approximators and natural policy gradients for continuous-time stochastic systems.