Optimization problems with rank constraints appear in many diverse fields such as control, machine learning and image analysis. Since the rank constraint is non-convex, these problems are often approximately solved via convex relaxations. Nuclear norm regularization is the prevailing convexifying technique for dealing with these types of problem. This paper introduces a family of low-rank inducing norms and regularizers which includes the nuclear norm as a special case. A posteriori guarantees on solving an underlying rank constrained optimization problem with these convex relaxations are provided. We evaluate the performance of the low-rank inducing norms on three matrix completion problems. In all examples, the nuclear norm heuristic is outperformed by convex relaxations based on other low-rank inducing norms. For two of the problems there exist low-rank inducing norms that succeed in recovering the partially unknown matrix, while the nuclear norm fails. These low-rank inducing norms are shown to be representable as semi-definite programs and to have cheaply computable proximal mappings. The latter makes it possible to also solve problems of large size with the help of scalable first-order methods. Finally, it is proven that our findings extend to the more general class of atomic norms. In particular, this allows us to solve corresponding vector-valued problems, as well as problems with other non-convex constraints.

Bayesian Optimization (BO) has become a core method for solving expensive black-box optimization problems. While much research focussed on the choice of the acquisition function, we focus on online length-scale adaption and the choice of kernel function. Instead of choosing hyperparameters in view of maximum likelihood on past data, we propose to use the acquisition function to decide on hyperparameter adaptation more robustly and in view of the future optimization progress. Further, we propose a particular kernel function that includes non-stationarity and local anisotropy and thereby implicitly integrates the efficiency of local convex optimization with global Bayesian optimization. Comparisons to state-of-the art BO methods underline the efficiency of these mechanisms on global optimization benchmarks.

This paper presents the design of a supervisory algorithm that monitors safety at road intersections and overrides drivers with a safe input when necessary. The design of the supervisor consists of two parts: safety verification and control design. Safety verification is the problem to determine if vehicles will be able to cross the intersection without colliding with current drivers' inputs. We translate this safety verification problem into a jobshop scheduling problem, which minimizes the maximum lateness and evaluates if the optimal cost is zero. The zero optimal cost corresponds to the case in which all vehicles can cross each conflict area without collisions. Computing the optimal cost requires solving a Mixed Integer Nonlinear Programming (MINLP) problem due to the nonlinear second-order dynamics of the vehicles. We therefore estimate this optimal cost by formulating two related Mixed Integer Linear Programming (MILP) problems that assume simpler vehicle dynamics. We prove that these two MILP problems yield lower and upper bounds of the optimal cost. We also quantify the worst case approximation errors of these MILP problems. We design the supervisor to override the vehicles with a safe control input if the MILP problem that computes the upper bound yields a positive optimal cost. We theoretically demonstrate that the supervisor keeps the intersection safe and is non-blocking. Computer simulations further validate that the algorithms can run in real time for problems of realistic size.

Training of one-vs.-rest SVMs can be parallelized over the number of classes in a straight forward way. Given enough computational resources, one-vs.-rest SVMs can thus be trained on data involving a large number of classes. The same cannot be stated, however, for the so-called all-in-one SVMs, which require solving a quadratic program of size quadratically in the number of classes. We develop distributed algorithms for two all-in-one SVM formulations (Lee et al. and Weston and Watkins) that parallelize the computation evenly over the number of classes. This allows us to compare these models to one-vs.-rest SVMs on unprecedented scale. The results indicate superior accuracy on text classification data.

Traditionally, optimization of task allocation problems considered only the costs involved in the allocation. However, there has been in recent years more attention to cases where cost should not always be the sole consideration (Campbell et al. 2008). There are circumstances when other criteria need to be taken into account as well during the decision making process. Fairness has been considered as one of the important additional criteria in many application domains (Ogryczak et al. 2005, Gopinathan and Li 2011, Bertsimas et al. 2012). Although there is no common definition for the term, there are two fairness criteria that are often used in the literature: the Nash bargaining criterion and the Rawlsian maximin criterion. The former is based on Nash's four axioms of pareto optimality, independency of irrelevant alternatives, symmetry, and invariance to affine transformations or equivalent utility representations (Nash 1950). The latter is based on Rawls' two principles of justice (Rawls 1971). Rawls' maximin criterion maximizes the welfare level of the worst-off group member and has therefore been used in allocation problems (Jaffe 1981, Kumar and Kleinberg 2000).

Scarce data is a major challenge to scaling robot learning to truly complex tasks, as we need to generalize locally learned policies over different "contexts". Bayesian optimization approaches to contextual policy search (CPS) offer data-efficient policy learning that generalize over a context space. We propose to improve data- efficiency by factoring typically considered contexts into two components: target- type contexts that correspond to a desired outcome of the learned behavior, e.g. target position for throwing a ball; and environment type contexts that correspond to some state of the environment, e.g. initial ball position or wind speed. Our key observation is that experience can be directly generalized over target-type contexts. Based on that we introduce Factored Contextual Policy Search with Bayesian Optimization for both passive and active learning settings. Preliminary results show faster policy generalization on a simulated toy problem.

Supervised topic models can help clinical researchers find interpretable cooccurence patterns in count data that are relevant for diagnostics. However, standard formulations of supervised Latent Dirichlet Allocation have two problems. First, when documents have many more words than labels, the influence of the labels will be negligible. Second, due to conditional independence assumptions in the graphical model the impact of supervised labels on the learned topic-word probabilities is often minimal, leading to poor predictions on heldout data. We investigate penalized optimization methods for training sLDA that produce interpretable topic-word parameters and useful heldout predictions, using recognition networks to speed-up inference. We report preliminary results on synthetic data and on predicting successful anti-depressant medication given a patient's diagnostic history.

There has been a lot of activity recently around revenue attribution - marketers want to develop a better understanding of their customer acquisition funnel and be able to measure progress against it. Most of this attention has been focused on the B2C space. However, less work has been done measuring the performance of B2B marketing activities. While Salesforce is an excellent platform for managing leads and campaigns, their business model is founded on developing a sales and marketing ecosystem comprising partnerships with specialist vendors that can provide more focused solutions to specific sales and marketing issues. As a result, companies such as Full Circle Insights, Bright Funnel and Bizable have emerged to fill the void in B2B marketing attribution by leveraging the Salesforce platform.

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.

Student-$t$ processes have recently been proposed as an appealing alternative non-parameteric function prior. They feature enhanced flexibility and predictive variance. In this work the use of Student-$t$ processes are explored for multi-objective Bayesian optimization. In particular, an analytical expression for the hypervolume-based probability of improvement is developed for independent Student-$t$ process priors of the objectives. Its effectiveness is shown on a multi-objective optimization problem which is known to be difficult with traditional Gaussian processes.