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!

We consider large matrices of low rank. We address the problem of recovering such matrices when most of the entries are unknown. Matrix completion finds applications in recommender systems. In this setting, the rows of the matrix may correspond to items and the columns may correspond to users. The known entries are the ratings given by users to some items.

We describe a family of global optimization procedures that automatically decompose optimization problems into smaller loosely coupled problems, then combine the solutions of these with message passing algorithms. We show empirically that these methods excel in avoiding local minima and produce better solutions with fewer function evaluations than existing global optimization methods. To develop these methods, we introduce a notion of coupling between variables of optimization that generalizes the notion of coupling that arises from factoring functions into terms that involve small subsets of the variables. Despite being more general, this notion of coupling is easier to verify empirically -- making structure estimation easy -- yet it allows us to migrate well-established inference methods on graphical models to the setting of global optimization.

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.

In many graph-based machine learning and data mining approaches, the quality of the graph is critical. However, in real-world applications, especially in semi-supervised learning and unsupervised learning, the evaluation of the quality of a graph is often expensive and sometimes even impossible, due the cost or the unavailability of ground truth. In this paper, we proposed a robust approach with convex optimization to forge'' a graph: with an input of a graph, to learn a graph with higher quality. Our major concern is that an ideal graph shall satisfy all the following constraints: non-negative, symmetric, low rank, and positive semidefinite. We develop a graph learning algorithm by solving a convex optimization problem and further develop an efficient optimization to obtain global optimal solutions with theoretical guarantees.

This paper presents a novel non-parametric approximate dynamic programming (ADP) algorithm that enjoys graceful, dimension-independent approximation and sample complexity guarantees. In particular, we establish both theoretically and computationally that our proposal can serve as a viable alternative to state-of-the-art parametric ADP algorithms, freeing the designer from carefully specifying an approximation architecture. We accomplish this by developing a kernel-based mathematical program for ADP. Via a computational study on a controlled queueing network, we show that our non-parametric procedure is competitive with parametric ADP approaches.

Diversified ranking is a fundamental task in machine learning. It is broadly applicable in many real world problems, e.g., information retrieval, team assembling, product search, etc. In this paper, we consider a generic setting where we aim to diversify the top-k ranking list based on an arbitrary relevance function and an arbitrary similarity function among all the examples. We formulate it as an optimization problem and show that in general it is NP-hard. Then, we show that for a large volume of the parameter space, the proposed objective function enjoys the diminishing returns property, which enables us to design a scalable, greedy algorithm to find the near-optimal solution. Experimental results on real data sets demonstrate the effectiveness of the proposed algorithm.

In stochastic optimal control the distribution of the exogenous noise is typically unknown and must be inferred from limited data before dynamic programming (DP)-based solution schemes can be applied. If the conditional expectations in the DP recursions are estimated via kernel regression, however, the historical sample paths enter the solution procedure directly as they determine the evaluation points of the cost-to-go functions. The resulting data-driven DP scheme is asymptotically consistent and admits efficient computational solution when combined with parametric value function approximations. If training data is sparse, however, the estimated cost-to-go functions display a high variability and an optimistic bias, while the corresponding control policies perform poorly in out-of-sample tests. To mitigate these small sample effects, we propose a robust data-driven DP scheme, which replaces the expectations in the DP recursions with worst-case expectations over a set of distributions close to the best estimate.

Variational inference algorithms provide the most effective framework for large-scale training of Bayesian nonparametric models. Stochastic online approaches are promising, but are sensitive to the chosen learning rate and often converge to poor local optima. We present a new algorithm, memoized online variational inference, which scales to very large (yet finite) datasets while avoiding the complexities of stochastic gradient. Our algorithm maintains finite-dimensional sufficient statistics from batches of the full dataset, requiring some additional memory but still scaling to millions of examples. Exploiting nested families of variational bounds for infinite nonparametric models, we develop principled birth and merge moves allowing non-local optimization.

We investigate two new optimization problems -- minimizing a submodular function subject to a submodular lower bound constraint (submodular cover) and maximizing a submodular function subject to a submodular upper bound constraint (submodular knapsack). We are motivated by a number of real-world applications in machine learning including sensor placement and data subset selection, which require maximizing a certain submodular function (like coverage or diversity) while simultaneously minimizing another (like cooperative cost). These problems are often posed as minimizing the difference between submodular functions [9, 23] which is in the worst case inapproximable. We show, however, that by phrasing these problems as constrained optimization, which is more natural for many applications, we achieve a number of bounded approximation guarantees. We also show that both these problems are closely related and, an approximation algorithm solving one can be used to obtain an approximation guarantee for the other.