Gaussian processes~(Kriging) are interpolating data-driven models that are frequently applied in various disciplines. Often, Gaussian processes are trained on datasets and are subsequently embedded as surrogate models in optimization problems. These optimization problems are nonconvex and global optimization is desired. However, previous literature observed computational burdens limiting deterministic global optimization to Gaussian processes trained on few data points. We propose a reduced-space formulation for deterministic global optimization with trained Gaussian processes embedded. For optimization, the branch-and-bound solver branches only on the degrees of freedom and McCormick relaxations are propagated through explicit Gaussian process models. The approach also leads to significantly smaller and computationally cheaper subproblems for lower and upper bounding. To further accelerate convergence, we derive envelopes of common covariance functions for GPs and tight relaxations of acquisition functions used in Bayesian optimization including expected improvement, probability of improvement, and lower confidence bound. In total, we reduce computational time by orders of magnitude compared to state-of-the-art methods, thus overcoming previous computational burdens. We demonstrate the performance and scaling of the proposed method and apply it to Bayesian optimization with global optimization of the acquisition function and chance-constrained programming. The Gaussian process models, acquisition functions, and training scripts are available open-source within the "MeLOn - Machine Learning Models for Optimization" toolbox~(https://git.rwth-aachen.de/avt.svt/public/MeLOn).

Achieving the Bayes optimal binary classification rule subject to group fairness constraints is known to be reducible, in some cases, to learning a group-wise thresholding rule over the Bayes regressor. In this paper, we extend this result by proving that, in a broader setting, the Bayes optimal fair learning rule remains a group-wise thresholding rule over the Bayes regressor but with a (possible) randomization at the thresholds. This provides a stronger justification to the post-processing approach in fair classification, in which (1) a predictor is learned first, after which (2) its output is adjusted to remove bias. We show how the post-processing rule in this two-stage approach can be learned quite efficiently by solving an unconstrained optimization problem. The proposed algorithm can be applied to any black-box machine learning model, such as deep neural networks, random forests and support vector machines. In addition, it can accommodate many fairness criteria that have been previously proposed in the literature, such as equalized odds and statistical parity. We prove that the algorithm is Bayes consistent and motivate it, furthermore, via an impossibility result that quantifies the tradeoff between accuracy and fairness across multiple demographic groups. Finally, we conclude by validating the algorithm on the Adult benchmark dataset.

In this work, we address the problem of learning to seek novel policies in reinforcement learning tasks. Instead of following the multi-objective framework used in previous methods, we propose to rethink the problem under a novel perspective of constrained optimization. We first introduce a new metric to evaluate the difference between policies, and then design two practical novel policy seeking methods following the new perspective, namely the Constrained Task Novel Bisector (CTNB), and the Interior Policy Differentiation (IPD), corresponding to the feasible direction method and the interior point method commonly known in constrained optimization problems. Experimental comparisons on the MuJuCo control suite show our methods achieve substantial improvements over previous novelty-seeking methods in terms of both novelty and primal task performance.

As AI is deployed increasingly broadly, AI researchers are confronted with the moral implications of their work. The pursuit of simple objectives, such as minimizing error rates, maximizing resource efficiency, or decreasing response times, often results in systems that have unintended consequences when they confront the real world, such as discriminating against certain groups of people [34]. It would be helpful for AI researchers and practitioners to have a general set of principles with which to approach these problems [45, 41, 24, 16, 33]. One may ask why any moral decisions should be left to computers at all. There are multiple possible reasons. One is that the decision needs to be made so quickly that calling in a human for the decision is not feasible, as would be the case for a self-driving car having to make a split-second decision about whom to hit [13]. Another reason could be that each individual decision by itself is too insignificant to bother a human, even though all the decisions combined may be highly significant morally--for example, if we were to consider the moral impact of each advertisement shown online. A third reason is that the moral decision is hard to decouple from a computational problem that apparently exceeds human capabilities. This is the case in many machine learning applications (e.g., should this person be released on bail?

Path and motion planning for autonomous systems has long been an area of research within the robotics and artificial intelligence communities. This has led to the development of a number of frameworks which formulate planning tasks in terms of mathematical optimization problems, which can then be solved by utilizing approaches from optimization theory and optimal control [1, 2]. However, planning in complex domains can be a challenging problem, and requires the agents to spend time and computational resources in order to find solutions, leading to an intrinsic need to balance computational complexity and optimality [3, 4, 5, 6, 7]. Within the path-planning community, this observation has resulted in the development of a number of approaches, which aim to explicitly capture the interplay between complexity and optimality. For example, in [8, 5, 9], the authors utilize wavelets to obtain multi-resolution representations of a two-dimensional environment for path-planning.

This paper presents a novel neural network design that learns the heuristic for Large Neighborhood Search (LNS). LNS consists of a destroy operator and a repair operator that specify a way to carry out the neighborhood search to solve the Combinatorial Optimization problems. The proposed approach in this paper applies a Hierarchical Recurrent Graph Convolutional Network (HRGCN) as a LNS heuristic, namely Dynamic Partial Removal, with the advantage of adaptive destruction and the potential to search across a large scale, as well as the context-awareness in both spatial and temporal perspective. This model is generalized as an efficient heuristic approach to different combinatorial optimization problems, especially to the problems with relatively tight constraints. We apply this model to vehicle routing problem (VRP) in this paper as an example. The experimental results show that this approach outperforms the traditional LNS heuristics on the same problem as well. The source code is available at \href{https://github.com/water-mirror/DPR}{https://github.com/water-mirror/DPR}.

This paper tackles the age-old question of derivative free optimization in neural networks. This paper introduces AdaSwarm, a novel derivative-free optimizer to have similar or better performance to Adam but without "gradients". To support the AdaSwarm, a novel Particle Swarm Optimization Exponentially weighted Momentum PSO (EM-PSO), a derivative-free optimizer, is also proposed which tackles constrained and unconstrained single objective optimization problems and looks at applying the proposed momentum particle swarm optimization on benchmark test functions, engineering optimization problems and habitability scores for exoplanets which show speed and convergence of the technique. The EM-PSO is extended by approximating the gradient of a function at any point using the parameters of the particle swarm optimization. This is a novel technique to simulate gradient descent, an extremely popular method in the back-propagation algorithm, using the approximated gradients from the particle swarm optimization parameters. Mathematical proofs of gradient approximation by EM-PSO, thereby bypassing the gradient computation, are presented. The AdaSwarm is compared with various optimizers and the theory and algorithmic performance are supported by promising results.

We present a new approach to solve the sparse approximation or best subset selection problem, namely find a $k$-sparse vector ${\bf x}\in\mathbb{R}^d$ that minimizes the $\ell_2$ residual $\lVert A{\bf x}-{\bf y} \rVert_2$. We consider a regularized approach, whereby this residual is penalized by the non-convex $\textit{trimmed lasso}$, defined as the $\ell_1$-norm of ${\bf x}$ excluding its $k$ largest-magnitude entries. We prove that the trimmed lasso has several appealing theoretical properties, and in particular derive sparse recovery guarantees assuming successful optimization of the penalized objective. Next, we show empirically that directly optimizing this objective can be quite challenging. Instead, we propose a surrogate for the trimmed lasso, called the $\textit{generalized soft-min}$. This penalty smoothly interpolates between the classical lasso and the trimmed lasso, while taking into account all possible $k$-sparse patterns. The generalized soft-min penalty involves summation over $\binom{d}{k}$ terms, yet we derive a polynomial-time algorithm to compute it. This, in turn, yields a practical method for the original sparse approximation problem. Via simulations, we demonstrate its competitive performance compared to current state of the art.

In this paper, we consider optimizing a smooth, convex, lower semicontinuous function in Riemannian space with constraints. To solve the problem, we first convert it to a dual problem and then propose a general primal-dual algorithm to optimize the primal and dual variables iteratively. In each optimization iteration, we employ a proximal operator to search optimal solution in the primal space. We prove convergence of the proposed algorithm and show its non-asymptotic convergence rate. By utilizing the proposed primal-dual optimization technique, we propose a novel metric learning algorithm which learns an optimal feature transformation matrix in the Riemannian space of positive definite matrices. Preliminary experimental results on an optimal fund selection problem in fund of funds (FOF) management for quantitative investment showed its efficacy.

We consider a setting where we have a set of data samples that were not uniformly sampled from a population, or where they were sampled from a different population than the one from which we wish to draw some conclusions. A common approach is to assign weights to the samples, so the resulting weighted distribution is representative of the population we wish to study. Here representative means that with the weights, certain expected values or probabilities match or are close to known values for the population we wish to study. A a very simple example, consider a data set where each sample is associated with a person. Our data set is 70% female, whereas we'd like to draw conclusions about a population that is 50% female. A simple solution is to down-weight the female samples, and up-weight the male samples in our data set, so the weighted fraction of females is 50%. As a more sophisticated example, suppose we have multiple groups, for example various combinations of sex, age group, income level, and education, and our goal is to find weights for our samples so the fractions of all these groups matches or approximates known fractions in the population we wish to study. In this case, there will be many possible assignments of weights that match the given fractions, and we need to choose a reasonable one. One approach is to maximize the entropy of the weights, subject to matching the given fractions.