In the field of multi-objective optimization algorithms, multi-objective Bayesian Global Optimization (MOBGO) is an important branch, in addition to evolutionary multi-objective optimization algorithms (EMOAs). MOBGO utilizes Gaussian Process Models learned from previous objective function evaluations to decide the next evaluation site by maximizing or minimizing an infill criterion. A common criterion in MOBGO is the Expected Hypervolume Improvement (EHVI), which shows a good performance on a wide range of problems, with respect to exploration and exploitation. However, so far it has been a challenge to calculate exact EHVI values efficiently. In this paper, an efficient algorithm for the computation of the exact EHVI for a generic case is proposed. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities are improved from previously $O (n^2)$ and $O(n^3)$, for two- and three-objective problems respectively, to $\Theta(n\log n)$, which is asymptotically optimal. This article generalizes the scheme in higher dimensional case by utilizing a new hyperbox decomposition technique, which was proposed by D{\"a}chert et al, EJOR, 2017. It also utilizes a generalization of the multilayered integration scheme that scales linearly in the number of hyperboxes of the decomposition. The speed comparison shows that the proposed algorithm in this paper significantly reduces computation time. Finally, this decomposition technique is applied in the calculation of the Probability of Improvement (PoI).

It is common to encounter situations where one must solve a sequence of similar computational problems. Running a standard algorithm with worst-case runtime guarantees on each instance will fail to take advantage of valuable structure shared across the problem instances. For example, when a commuter drives from work to home, there are typically only a handful of routes that will ever be the shortest path. A naive algorithm that does not exploit this common structure may spend most of its time checking roads that will never be in the shortest path. More generally, we can often ignore large swaths of the search space that will likely never contain an optimal solution. We present an algorithm that learns to maximally prune the search space on repeated computations, thereby reducing runtime while provably outputting the correct solution each period with high probability. Our algorithm employs a simple explore-exploit technique resembling those used in online algorithms, though our setting is quite different. We prove that, with respect to our model of pruning search spaces, our approach is optimal up to constant factors. Finally, we illustrate the applicability of our model and algorithm to three classic problems: shortest-path routing, string search, and linear programming. We present experiments confirming that our simple algorithm is effective at significantly reducing the runtime of solving repeated computations.

A wide array of machine learning problems are formulated as the minimization of the expectation of a convex loss function on some parameter space. Since the probability distribution of the data of interest is usually unknown, it is is often estimated from training sets, which may lead to poor out-of-sample performance. In this work, we bring new insights in this problem by using the framework which has been developed in quantitative finance for risk measures. We show that the original min-max problem can be recast as a convex minimization problem under suitable assumptions. We discuss several important examples of robust formulations, in particular by defining ambiguity sets based on $\varphi$-divergences and the Wasserstein metric.We also propose an efficient algorithm for solving the corresponding convex optimization problems involving complex convex constraints. Through simulation examples, we demonstrate that this algorithm scales well on real data sets.

Scarce data is a major challenge to scaling robot learning to truly complex tasks, as we need to generalize locally learned policies over different task contexts. Contextual policy search offers data-efficient learning and generalization by explicitly conditioning the policy on a parametric context space. In this paper, we further structure the contextual policy representation. We propose to factor contexts into two components: target contexts that describe the task objectives, e.g. target position for throwing a ball; and environment contexts that characterize the environment, e.g. initial position or mass of the ball. Our key observation is that experience can be directly generalized over target contexts. We show that this can be easily exploited in contextual policy search algorithms. In particular, we apply factorization to a Bayesian optimization approach to contextual policy search both in sampling-based and active learning settings. Our simulation results show faster learning and better generalization in various robotic domains. See our supplementary video: https://youtu.be/MNTbBAOufDY.

Image partitioning, or segmentation without semantics, is the task of decomposing an image into distinct segments, or equivalently to detect closed contours. Most prior work either requires seeds, one per segment; or a threshold; or formulates the task as multicut / correlation clustering, an NP-hard problem. Here, we propose a greedy algorithm for signed graph partitioning, the "Mutex Watershed". Unlike seeded watershed, the algorithm can accommodate not only attractive but also repulsive cues, allowing it to find a previously unspecified number of segments without the need for explicit seeds or a tunable threshold. We also prove that this simple algorithm solves to global optimality an objective function that is intimately related to the multicut / correlation clustering integer linear programming formulation. The algorithm is deterministic, very simple to implement, and has empirically linearithmic complexity. When presented with short-range attractive and long-range repulsive cues from a deep neural network, the Mutex Watershed gives the best results currently known for the competitive ISBI 2012 EM segmentation benchmark.

In this paper, we propose the use of a black-box optimization method called deterministic Mesh Adaptive Direct Search (MADS) algorithm with orthogonal directions (Ortho-MADS) for the selection of hyperparameters of Support Vector Machines with a Gaussian kernel. Different from most of the methods in the literature that exploit the properties of the data or attempt to minimize the accuracy of a validation dataset over the first quadrant of (C, gamma), the Ortho-MADS provides convergence proof. We present the MADS, followed by the Ortho-MADS, the dynamic stopping criterion defined by the MADS mesh size and two different search strategies (Nelder-Mead and Variable Neighborhood Search) that contribute to a competitive convergence rate as well as a mechanism to escape from undesired local minima. We have investigated the practical selection of hyperparameters for the Support Vector Machine with a Gaussian kernel, i.e., properly choose the hyperparameters gamma (bandwidth) and C (trade-off) on several benchmark datasets. The experimental results have shown that the proposed approach for hyperparameter tuning consistently finds comparable or better solutions, when using a common configuration, than other methods. We have also evaluated the accuracy and the number of function evaluations of the Ortho-MADS with the Nelder-Mead search strategy and the Variable Neighborhood Search strategy using the mesh size as a stopping criterion, and we have achieved accuracy that no other method for hyperparameters optimization could reach.

In this paper, we introduce a framework for solving finite-horizon multistage optimization problems under uncertainty in the presence of auxiliary data. We assume the joint distribution of the uncertain quantities is unknown, but noisy observations, along with observations of auxiliary covariates, are available. We utilize effective predictive methods from machine learning (ML), including $k$-nearest neighbors regression ($k$NN), classification and regression trees (CART), and random forests (RF), to develop specific methods that are applicable to a wide variety of problems. We demonstrate that our solution methods are asymptotically optimal under mild conditions. Additionally, we establish finite sample guarantees for the optimality of our method with $k$NN weight functions. Finally, we demonstrate the practicality of our approach with computational examples. We see a significant decrease in cost by taking into account the auxiliary data in the multistage setting.

In multi-task learning (MTL), related tasks learn jointly to improve generalization performance. To exploit the high learning speed of extreme learning machines (ELMs), we apply the ELM framework to the MTL problem, where the output weights of ELMs for all the tasks are learned collaboratively. We first present the ELM based MTL problem in the centralized setting, which is solved by the proposed MTL-ELM algorithm. Due to the fact that many data sets of different tasks are geo-distributed, decentralized machine learning is studied. We formulate the decentralized MTL problem based on ELM as majorized multi-block optimization with coupled bi-convex objective functions. To solve the problem, we propose the DMTL-ELM algorithm, which is a hybrid Jacobian and Gauss-Seidel Proximal multi-block alternating direction method of multipliers (ADMM). Further, to reduce the computation load of DMTL-ELM, DMTL-ELM with first-order approximation (FO-DMTL-ELM) is presented. Theoretical analysis shows that the convergence to the stationary point of DMTL-ELM and FO-DMTL-ELM can be guaranteed conditionally. Through simulations, we demonstrate the convergence of proposed MTL-ELM, DMTL-ELM, and FO-DMTL-ELM algorithms, and also show that they can outperform existing MTL methods. Moreover, by adjusting the dimension of hidden feature space, there exists a trade-off between communication load and learning accuracy for DMTL-ELM.

We propose a new algorithm for hyperparameter selection in machine learning algorithms. The algorithm is a novel modification of Harmonica, a spectral hyperparameter selection approach using sparse recovery methods. In particular, we show that a special encoding of hyperparameter space enables a natural group-sparse recovery formulation, which when coupled with HyperBand (a multi-armed bandit strategy) leads to improvement over existing hyperparameter optimization methods such as Successive Halving and Random Search. Experimental results on image datasets such as CIFAR-10 confirm the benefits of our approach.

Recursive stochastic algorithms have gained significant attention in the recent past due to data driven applications. Examples include stochastic gradient descent for solving large-scale optimization problems and empirical dynamic programming algorithms for solving Markov decision problems. These recursive stochastic algorithms approximates certain contraction operators and can be viewed within the framework of iterated random maps. Accordingly, we consider iterated random maps over a Polish space that simulates a contraction operator over that Polish space. Assume that the iterated maps are indexed by $n$ such that as $n\rightarrow\infty$, each realization of the random map converges (in some sense) to the contraction map it is simulating. We show that starting from the same initial condition, the distribution of the random sequence generated by the iterated random maps converge weakly to the trajectory generated by the contraction operator. We further show that under certain conditions, the time average of the random sequence converge to the spatial mean of the invariant distribution. We then apply these results to logistic regression, empirical value iteration, empirical Q value iteration, and empirical relative value iteration for finite state finite action MDPs.