Robust Reinforcement Learning aims to find the optimal policy with some extent of robustness to environmental dynamics. Existing learning algorithms usually enable the robustness through disturbing the current state or simulating environmental parameters in a heuristic way, which lack quantified robustness to the system dynamics (i.e. transition probability). To overcome this issue, we leverage Wasserstein distance to measure the disturbance to the reference transition kernel. With Wasserstein distance, we are able to connect transition kernel disturbance to the state disturbance, i.e. reduce an infinite-dimensional optimization problem to a finite-dimensional risk-aware problem. Through the derived risk-aware optimal Bellman equation, we show the existence of optimal robust policies, provide a sensitivity analysis for the perturbations, and then design a novel robust learning algorithm--Wasserstein Robust Advantage Actor-Critic algorithm (WRAAC). The effectiveness of the proposed algorithm is verified in the Cart-Pole environment.

Simulators often require calibration inputs estimated from real world data and the quality of the estimate can significantly affect simulation output. Particularly when performing simulation optimisation to find an optimal solution, the uncertainty in the inputs significantly affects the quality of the found solution. One remedy is to search for the solution that has the best performance on average over the uncertain range of inputs yielding an optimal compromise solution. We consider the more general setting where a user may choose between either running simulations or instead collecting real world data. A user may choose an input and a solution and observe the simulation output, or instead query an external data source improving the input estimate enabling the search for a more focused, less compromised solution. We explicitly examine the trade-off between simulation and real data collection in order to find the optimal solution of the simulator with the true inputs. Using a value of information procedure, we propose a novel unified simulation optimisation procedure called Bayesian Information Collection and Optimisation (BICO) that, in each iteration, automatically determines which of the two actions (running simulations or data collection) is more beneficial. Numerical experiments demonstrate that the proposed algorithm is able to automatically determine an appropriate balance between optimisation and data collection.

The paper studies the performance of the Model-Agnostic Meta-Learning (MAML) algorithm as an optimization method. The goal is to determine the global convergence of MAML on sequential decision-making tasks possessing a common structure. We prove that the benign landscape of a single task leads to the global convergence of MAML in the single-task scenario and in the scenario of multiple structurally connected tasks. We also show that there is a two-task scenario that does not possess this global convergence property even for identical tasks. We analyze the landscape of the MAML objective on LQR tasks to determine what type of similarities in their structures enables the algorithm to converge to the globally optimal solution.

Bayesian optimization is a broadly applied methodology to optimize the expensive blackbox function. Despite its success, it still faces the challenge from the high-dimensional search space. To alleviate this problem, we propose a novel Bayesian optimization framework, which finds a low-dimensional space to perform Bayesian optimization through a semi-supervised, iterative, and embedding learning-based method (SILBO). SILBO incorporates both labeled and unlabeled points acquired from the acquisition function of Bayesian optimization to guide the learning of embedding space. To accelerate the learning procedure, we present a randomized method for generating the projection matrix. Furthermore, to map from the low-dimensional space to the high-dimensional original space, we propose two mapping strategies: SILBO-BU and SILBO-TD according to the evaluation overhead of the objective function. Experimental results on both synthetic function and hyperparameter optimization tasks demonstrate that SILBO outperforms the existing state-of-the-art high-dimensional Bayesian optimization methods.

We study the problem of differentially private constrained maximization of decomposable submodular functions. A submodular function is decomposable if it takes the form of a sum of submodular functions. The special case of maximizing a monotone, decomposable submodular function under cardinality constraints is known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al., 2008]. Previous work by Gupta et al. [2010] gave a differentially private algorithm for the CPP problem. We extend this work by designing differentially private algorithms for both monotone and non-monotone decomposable submodular maximization under general matroid constraints, with competitive utility guarantees. We complement our theoretical bounds with experiments demonstrating empirical performance, which improves over the differentially private algorithms for the general case of submodular maximization and is close to the performance of non-private algorithms.

The k-center problem is a fundamental problem we often face when considering complex service systems. Typical challenges include the placement of warehouses in logistics or positioning of servers for content delivery networks. We previously have proposed Dragoon as an effective algorithm to approach the k-center problem. This paper evaluates Dragoon with a focus on potential worst case behavior in comparison to other techniques. We use an evolutionary algorithm to generate instances of the k-center problem that are especially challenging for Dragoon. Ultimately, our experiments confirm the previous good results of Dragoon, however, we also can reliably find scenarios where it is clearly outperformed by other approaches.

Averting the effects of anthropogenic climate change requires a transition from fossil fuels to low-carbon technology. A way to achieve this is to decarbonize the electricity grid. However, further efforts must be made in other fields such as transport and heating for full decarbonization. This would reduce carbon emissions due to electricity generation, and also help to decarbonize other sources such as automotive and heating by enabling a low-carbon alternative. Carbon taxes have been shown to be an efficient way to aid in this transition. In this paper, we demonstrate how to to find optimal carbon tax policies through a genetic algorithm approach, using the electricity market agent-based model ElecSim. To achieve this, we use the NSGA-II genetic algorithm to minimize average electricity price and relative carbon intensity of the electricity mix. We demonstrate that it is possible to find a range of carbon taxes to suit differing objectives. Our results show that we are able to minimize electricity cost to below \textsterling10/MWh as well as carbon intensity to zero in every case. In terms of the optimal carbon tax strategy, we found that an increasing strategy between 2020 and 2035 was preferable. Each of the Pareto-front optimal tax strategies are at least above \textsterling81/tCO2 for every year. The mean carbon tax strategy was \textsterling240/tCO2.

We study a model for adversarial classification based on distributionally robust chance constraints. We show that under Wasserstein ambiguity, the model aims to minimize the conditional value-at-risk of the distance to misclassification, and we explore links to previous adversarial classification models and maximum margin classifiers. We also provide a reformulation of the distributionally robust model for linear classifiers, and show it is equivalent to minimizing a regularized ramp loss. Numerical experiments show that, despite the nonconvexity, standard descent methods appear to converge to the global minimizer for this problem. Inspired by this observation, we show that, for a certain benign distribution, the regularized ramp loss minimization problem has a single stationary point, at the global minimizer.

We explore why many recently proposed robust estimation problems are efficiently solvable, even though the underlying optimization problems are non-convex. We study the loss landscape of these robust estimation problems, and identify the existence of "generalized quasi-gradients". Whenever these quasi-gradients exist, a large family of low-regret algorithms are guaranteed to approximate the global minimum; this includes the commonly-used filtering algorithm. For robust mean estimation of distributions under bounded covariance, we show that any first-order stationary point of the associated optimization problem is an {approximate global minimum} if and only if the corruption level $\epsilon < 1/3$. Consequently, any optimization algorithm that aproaches a stationary point yields an efficient robust estimator with breakdown point $1/3$. With careful initialization and step size, we improve this to $1/2$, which is optimal. For other tasks, including linear regression and joint mean and covariance estimation, the loss landscape is more rugged: there are stationary points arbitrarily far from the global minimum. Nevertheless, we show that generalized quasi-gradients exist and construct efficient algorithms. These algorithms are simpler than previous ones in the literature, and for linear regression we improve the estimation error from $O(\sqrt{\epsilon})$ to the optimal rate of $O(\epsilon)$ for small $\epsilon$ assuming certified hypercontractivity. For mean estimation with near-identity covariance, we show that a simple gradient descent algorithm achieves breakdown point $1/3$ and iteration complexity $\tilde{O}(d/\epsilon^2)$.

Routing of multiple vehicles is an important and difficult problem with applications in the logistic domain [1], especially in the area of customer servicing [2]. In postal services, after-sales services, and in business to business delivery or pick up services one or more vehicles have to be efficiently routed towards customers. If customers can request services over time, the problem becomes dynamic: besides a set of fixed customers, new requests can appear at any point in time. Of course, it is desirable that as many customers as possible are serviced while the tour of any vehicle is kept short. However, it is usually infeasible (due to human resources, labor regulations, or other constraints) to service all customer requests. And clearly, the less customers are left unserviced, the longer the tours become. Thus, the problem is inherently multi-objective. Any efficient solution (smallest maximum tour across all vehicles) is a compromise between the desire to service as many customers as possible (e.g.