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A convex optimization model predicts an output from an input by solving a convex optimization problem. The class of convex optimization models is large, and includes as special cases many well-known models like linear and logistic regression. We propose a heuristic for learning the parameters in a convex optimization model given a dataset of input-output pairs, using recently developed methods for differentiating the solution of a convex optimization problem with respect to its parameters. We describe three general classes of convex optimization models, maximum a posteriori (MAP) models, utility maximization models, and agent models, and present a numerical experiment for each.
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We consider the decision-making framework of online convex optimization with a very large number of experts. This setting is ubiquitous in contextual and reinforcement learning problems, where the size of the policy class renders enumeration and search within the policy class infeasible. Instead, we consider generalizing the methodology of online boosting. We define a weak learning algorithm as a mechanism that guarantees multiplicatively approximate regret against a base class of experts. In this access model, we give an efficient boosting algorithm that guarantees near-optimal regret against the convex hull of the base class. We consider both full and partial (a.k.a. bandit) information feedback models. We also give an analogous efficient boosting algorithm for the i.i.d. statistical setting. Our results simultaneously generalize online boosting and gradient boosting guarantees to contextual learning model, online convex optimization and bandit linear optimization settings.
Bandit Convex Optimization (BCO) is a fundamental framework for modeling sequential decision-making with partial information, where the only feedback available to the player is the one-point or two-point function values. In this paper, we investigate BCO in non-stationary environments and choose the emph{dynamic regret} as the performance measure, which is defined as the difference between the cumulative loss incurred by the algorithm and that of any feasible comparator sequence. Let $T$ be the time horizon and $P_T$ be the path-length of the comparator sequence that reflects the non-stationarity of environments. We propose a novel algorithm that achieves $O(T^{3/4}(1+P_T)^{1/2})$ and $O(T^{1/2}(1+P_T)^{1/2})$ dynamic regret respectively for the one-point and two-point feedback models. The latter result is optimal, matching the $Omega(T^{1/2}(1+P_T)^{1/2})$ lower bound established in this paper. Notably, our algorithm is more adaptive to non-stationary environments since it does not require prior knowledge of the path-length $P_T$ ahead of time, which is generally unknown.
We consider a novel setting of zeroth order non-convex optimization, where in addition to querying the function value at a given point, we can also duel two points and get the point with the larger function value. We refer to this setting as optimization with dueling-choice bandits since both direct queries and duels are available for optimization. We give the COMP-GP-UCB algorithm based on GP-UCB (Srinivas et al., 2009), where instead of directly querying the point with the maximum Upper Confidence Bound (UCB), we perform a constrained optimization and use comparisons to filter out suboptimal points. COMP-GP-UCB comes with theoretical guarantee of $O(frac{Phi}{sqrt{T}})$ on simple regret where $T$ is the number of direct queries and $Phi$ is an improved information gain corresponding to a comparison based constraint set that restricts the search space for the optimum. In contrast, in the direct query only setting, $Phi$ depends on the entire domain. Finally, we present experimental results to show the efficacy of our algorithm.
The non-negative matrix factorization (NMF) model with an additional orthogonality constraint on one of the factor matrices, called the orthogonal NMF (ONMF), has been found a promising clustering model and can outperform the classical K-means. However, solving the ONMF model is a challenging optimization problem because the coupling of the orthogonality and non-negativity constraints introduces a mixed combinatorial aspect into the problem due to the determination of the correct status of the variables (positive or zero). Most of the existing methods directly deal with the orthogonality constraint in its original form via various optimization techniques, but are not scalable for large-scale problems. In this paper, we propose a new ONMF based clustering formulation that equivalently transforms the orthogonality constraint into a set of norm-based non-convex equality constraints. We then apply a non-convex penalty (NCP) approach to add them to the objective as penalty terms, leading to a problem that is efficiently solvable. One smooth penalty formulation and one non-smooth penalty formulation are respectively studied. We build theoretical conditions for the penalized problems to provide feasible stationary solutions to the ONMF based clustering problem, as well as proposing efficient algorithms for solving the penalized problems of the two NCP methods. Experimental results based on both synthetic and real datasets are presented to show that the proposed NCP methods are computationally time efficient, and either match or outperform the existing K-means and ONMF based methods in terms of the clustering performance.
Recently, there has been an increasing interest in designing distributed convex optimization algorithms under the setting where the data matrix is partitioned on features. Algorithms under this setting sometimes have many advantages over those under the setting where data is partitioned on samples, especially when the number of features is huge. Therefore, it is important to understand the inherent limitations of these optimization problems. In this paper, with certain restrictions on the communication allowed in the procedures, we develop tight lower bounds on communication rounds for a broad class of non-incremental algorithms under this setting. We also provide a lower bound on communication rounds for a class of (randomized) incremental algorithms.
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