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.

We present a new method for searching optimal hyperparameters among several tasks and several criteria. Multi-Task Multi Criteria method (MTMC) provides several Pareto-optimal solutions, among which one solution is selected with given criteria significance coefficients. The article begins with a mathematical formulation of the problem of choosing optimal hyperparameters. Then, the steps of the MTMC method that solves this problem are described. The proposed method is evaluated on the image classification problem using a convolutional neural network. The article presents optimal hyperparameters for various criteria significance coefficients.

In supervised learning using kernel methods, we encounter a large-scale finite-sum minimization over a reproducing kernel Hilbert space(RKHS). Often times large-scale finite-sum problems can be solved using efficient variants of Newton's method where the Hessian is approximated via sub-samples. In RKHS, however, the dependence of the penalty function to kernel makes standard sub-sampling approaches inapplicable, since the gram matrix is not readily available in a low-rank form. In this paper, we observe that for this class of problems, one can naturally use kernel approximation to speed up the Newton's method. Focusing on randomized features for kernel approximation, we provide a novel second-order algorithm that enjoys local superlinear convergence and global convergence in the high probability sense. The key to our analysis is showing that the approximated Hessian via random features preserves the spectrum of the original Hessian. We provide numerical experiments verifying the efficiency of our approach, compared to variants of sub-sampling methods.

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 consider an abstract class of optimization problems that are parameterized concavely in a single parameter, and show that the solution path along the parameter can always be approximated with accuracy $\varepsilon 0$ by a set of size $O(1/\sqrt{\varepsilon})$. A lower bound of size $\Omega (1/\sqrt{\varepsilon})$ shows that the upper bound is tight up to a constant factor. We also devise an algorithm that calls a step-size oracle and computes an approximate path of size $O(1/\sqrt{\varepsilon})$. Finally, we provide an implementation of the oracle for soft-margin support vector machines, and a parameterized semi-definite program for matrix completion. Papers published at the Neural Information Processing Systems Conference.

We develop and analyze stochastic optimization algorithms for problems in which the expected loss is strongly convex, and the optimum is (approximately) sparse. Previous approaches are able to exploit only one of these two structures, yielding a $\order(\pdim/T)$ convergence rate for strongly convex objectives in $\pdim$ dimensions and $\order(\sqrt{\spindex( \log\pdim)/T})$ convergence rate when the optimum is $\spindex$-sparse. Our algorithm is based on successively solving a series of $\ell_1$-regularized optimization problems using Nesterov's dual averaging algorithm. We establish that the error of our solution after $T$ iterations is at most $\order(\spindex(\log\pdim)/T)$, with natural extensions to approximate sparsity. Our results apply to locally Lipschitz losses including the logistic, exponential, hinge and least-squares losses.

We consider derivative-free algorithms for stochastic optimization problems that use only noisy function values rather than gradients, analyzing their finite-sample convergence rates. We show that if pairs of function values are available, algorithms that use gradient estimates based on random perturbations suffer a factor of at most $\sqrt{\dim}$ in convergence rate over traditional stochastic gradient methods, where $\dim$ is the dimension of the problem. We complement our algorithmic development with information-theoretic lower bounds on the minimax convergence rate of such problems, which show that our bounds are sharp with respect to all problem-dependent quantities: they cannot be improved by more than constant factors. Papers published at the Neural Information Processing Systems Conference.

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! Papers published at the Neural Information Processing Systems Conference.

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.

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. Papers published at the Neural Information Processing Systems Conference.