A model of sharing revenues among groups when group members are ranked several times is presented. The methodology is based on pairwise comparison matrices, allows for the use of any weighting method, and makes possible to tune the level of inequality. Our proposal is demonstrated on the example of Formula One prize money allocation among the constructors. We introduce an axiom called scale invariance, which requires the ranking of teams to be independent of the parameter controlling inequality. The eigenvector method is revealed to violate this condition in our dataset, while the row geometric mean method always satisfies it. The revenue allocation is not influenced by the arbitrary valuation given to the race prizes in the official points scoring system of Formula One.

Incompleteness is a common problem for existing knowledge graphs (KGs), and the completion of KG which aims to predict links between entities is challenging. Most existing KG completion methods only consider the direct relation between nodes and ignore the relation paths which contain useful information for link prediction. Recently, a few methods take relation paths into consideration but pay less attention to the order of relations in paths which is important for reasoning. In addition, these path-based models always ignore nonlinear contributions of path features for link prediction. To solve these problems, we propose a novel KG completion method named OPTransE. Instead of embedding both entities of a relation into the same latent space as in previous methods, we project the head entity and the tail entity of each relation into different spaces to guarantee the order of relations in the path. Meanwhile, we adopt a pooling strategy to extract nonlinear and complex features of different paths to further improve the performance of link prediction. Experimental results on two benchmark datasets show that the proposed model OPTransE performs better than state-of-the-art methods.

We consider the problem of controlling a large fleet of drones to deliver packages simultaneously across broad urban areas. To conserve their limited flight range, drones can seamlessly hop between and ride on top of public transit vehicles (e.g., buses and trams). We design a novel comprehensive algorithmic framework that strives to minimize the maximum time to complete any delivery. We address the multifaceted complexity of the problem through a two-layer approach. First, the upper layer assigns drones to package delivery sequences with a provably near-optimal polynomial-time task allocation algorithm. Then, the lower layer executes the allocation by periodically routing the fleet over the transit network while employing efficient bounded-suboptimal multi-agent pathfinding techniques tailored to our setting. We present extensive experiments supporting the efficiency of our approach on settings with up to $200$ drones, $5000$ packages, and large transit networks of up to $8000$ stops in San Francisco and the Washington DC area. Our results show that the framework can compute solutions within a few seconds (up to $2$ minutes for the largest settings) on commodity hardware, and that drones travel up to $450 \%$ of their flight range by using public transit.

A BSTRACT We present GQSA T, a branching heuristic in a Boolean SA T solver trained with value-based reinforcement learning (RL) using Graph Neural Networks for function approximation. Solvers using GQSA T are complete SA T solvers that either provide a satisfying assignment or a proof of unsatisfiability, which is required for many SA T applications. The branching heuristic commonly used in SA T solvers today suffers from bad decisions during their warm-up period, whereas GQSA T has been trained to examine the structure of the particular problem instance to make better decisions at the beginning of the search. Training GQSA T is data efficient and does not require elaborate dataset preparation or feature engineering to train. We train GQSA T on small SA T problems using RL interfacing with an existing SA T solver. We show that GQSA T is able to reduce the number of iterations required to solve SA T problems by 2-3X, and it generalizes to unsatisfiable SA T instances, as well as to problems with 5X more variables than it was trained on. We also show that, to a lesser extent, it generalizes to SA T problems from different domains by evaluating it on graph coloring. Our experiments show that augmenting SA T solvers with agents trained with RL and graph neural networks can improve performance on the SA T search problem. 1 I NTRODUCTION Boolean satisfiability (SA T) is an important problem for both industry and academia impacting various fields, including circuit design, computer security, artificial intelligence, automatic theorem proving, and combinatorial optimization. As a result, modern SA T solvers are well-crafted, sophisticated, reliable pieces of software that can scale to problems with hundreds of thousands of variables (Ohrimenko et al., 2009). SA T is known to be NPcomplete (Karp, 1972), and most state-of-the-art open-source and commercial solvers rely on multiple heuristics to speed up the exhaustive search, which is otherwise intractable. These heuristics are usually meticulously crafted using expert domain knowledge and are often iterated on using trial and error. In this paper, we investigate how we can use machine learning to improve upon an existing branching heuristic without leveraging domain expertise.

Given a certain complexity level, humanized agents may select from a wide range of possible tasks, with each activity corresponding to a transient goal. In general there will be no overarching credit assignment scheme allowing to compare available options with respect to expected utilities. For this situation we propose a task selection framework that is based on time allocation via emotional stationarity (TAES). Emotions are argued to correspond to abstract criteria, such as satisfaction, challenge and boredom, along which activities that have been carried out can be evaluated. The resulting timeline of experienced emotions is then compared with the `character' of the agent, which is defined in terms of a preferred distribution of emotional states. The long-term goal of the agent, to align experience with character, is achieved by optimizing the frequency for selecting the individual tasks. Upon optimization, the statistics of emotion experience becomes stationary.

Locally adaptive activation functions with slope recovery term for deep and physics-informed neural networks Ameya D. Jagtap 1, Kenji Kawaguchi 2 and George Em Karniadakis 1,3, 1 Division of Applied Mathematics, Brown University, 182 George Street, Providence, RI 02912, USA. 2 Massachusetts Institute of T echnology, 77 Massachusetts Ave, Cambridge, MA 02139, USA. 3 Pacific Northwest National Laboratory, Richland, WA 99354, USA.Abstract We propose two approaches of locally adaptive activation functions namely, layer-wise and neuron-wise locally adaptive activation functions, which improve the performance of deep and physics-informed neural networks. The local adaptation of activation function is achieved by introducing scalable hyper-parameters in each layer (layer-wise) and for every neuron separately (neuron-wise), and then optimizing it using the stochastic gradient descent algorithm. Introduction of neuron-wise activation function acts like a vector activation function as opposed to the traditional scalar activation function given by fixed, global and layer-wise activations. In order to further increase the training speed, an activation slope based slope recovery term is added in the loss function, which further accelerate convergence, thereby reducing the training cost. For numerical experiments, a nonlinear discontinuous function is approximated using a deep neural network with layer-wise and neuron-wise locally adaptive activation functions with and without the slope recovery term and compared with its global counterpart. Moreover, solution of the nonlinear Burgers equation, which exhibits steep gradients, is also obtained using the proposed methods. On the theoretical side, we prove that in the proposed method the gradient descent algorithms are not attracted to sub-optimal critical points or local minima under practical conditions on the initialization and learning rate. Furthermore, the proposed adaptive activation functions with the slope recovery are shown to accelerate the training process in standard deep learning benchmarks using CIFAR-10, CIFAR-100, SVHN, MNIST, KMNIST, Fashion-MNIST, and Semeion data sets with and without data augmentation. Keywords: Machine learning, bad minima, stochastic gradients, accelerated training, PINN, deep learning benchmarks. 1. Introduction In recent years, research on neural networks (NNs) has intensified around the world due to their successful applications in many diverse fields such as speech recognition [13], computer vision [16], natural language translation [25], etc. Training of NN is performed on data sets before using it in the actual applications.

O PTIMAL T RANSPORT, C YCLEGAN, AND P ENALIZED LS FOR U NSUPERVISEDL EARNING IN I NVERSE P ROB-LEMS Byeongsu Sim 1 Gyutaek Oh 2 Sungjun Lim 2 Jong Chul Y e 1,2 1 Department of Mathematical Sciences, KAIST, Daejeon, Republic of Korea 2 Department of Bio and Brain Engineering, KAIST, Daejeon, Republic of Korea A BSTRACT The penalized least squares (PLS) is a classic approach to inverse problems, where a regularization term is added to stabilize the solution. Optimal transport (OT) is another mathematical framework for computer vision tasks by providing means to transport one measure to another at minimal cost. Cycle-consistent generative adversarial network (cycleGAN) is a recent extension of GAN to learn target distributions with less mode collapsing behavior. Although similar in that no supervised training is required, the algorithms look different, so the mathematical relationship between these approaches is not clear. In this article, we provide an important advance to unveil the missing link. Specifically, we reveal that a cycle-GAN architecture can be derived as a dual formulation of the optimal transport problem, if the PLS with a deep learning penalty is used as a transport cost between the two probability measures from measurements and unknown images. This suggests that cycleGAN can be considered as stochastic generalization of classical PLS approaches. Our derivation is so general that various types of cy-cleGAN architecture can be easily derived by merely changing the transport cost.

While neural networks have acted as a strong unifying force in the design of modern AI systems, the neural network architectures themselves remain highly heterogeneous due to the variety of tasks to be solved. In this chapter, we explore how to adapt the Layer-wise Relevance Propagation (LRP) technique used for explaining the predictions of feed-forward networks to the LSTM architecture used for sequential data modeling and forecasting. The special accumulators and gated interactions present in the LSTM require both a new propagation scheme and an extension of the underlying theoretical framework to deliver faithful explanations.

The Analog Ensemble (AnEn) method tries to estimate the probability distribution of the future state of the atmosphere with a set of past observations that correspond to the best analogs of a deterministic Numerical Weather Prediction (NWP). This model post-processing method has been successfully used to improve the forecast accuracy for several weather-related applications including air quality, and short-term wind and solar power forecasting, to name a few. In order to provide a meaningful probabilistic forecast, the AnEn method requires storing a historical set of past predictions and observations in memory for a period of at least several months and spanning the seasons relevant for the prediction of interest. Although the memory and computing costs of the AnEn method are less expensive than using a brute-force dynamical ensemble approach, for a large number of stations and large datasets, the amount of memory required for AnEn can easily become prohibitive. Furthermore, in order to find the best analogs associated with a certain prediction produced by a NWP model, the current approach requires searching over the entire dataset by applying a certain metric. This approach requires applying the metric over the entire historical dataset, which may take a substantial amount of time. In this work, we investigate an alternative way to implement the AnEn method using deep generative models. By doing so, a generative model can entirely or partially replace the dataset of pairs of predictions and observations, reducing the amount of memory required to produce the probabilistic forecast by several orders of magnitude. Furthermore, the generative model can generate a meaningful set of analogs associated with a certain forecast in constant time without performing any search, saving a considerable amount of time even in the presence of huge historical datasets.

Existing graph embedding models often have weaknesses in exploiting graph structure similarities, potential dependencies among nodes and global network properties. To this end, we present U2GAN, a novel unsupervised model leveraging on the strength of the recently introduced universal self-attention network (Dehghani et al., 2019), to learn low-dimensional embeddings of graphs which can be used for graph classification. In particular, given an input graph, U2GAN first applies a self-attention computation, which is then followed by a recurrent transition to iteratively memorize its attention on vector representations of each node and its neighbors across each iteration. Thus, U2GAN can address the weaknesses in the existing models in order to produce plausible node embeddings whose sum is the final embedding of the whole graph. Experimental results show that our unsupervised U2GAN produces new state-of-the-art performances on a range of well-known benchmark datasets for the graph classification task. It even outperforms supervised methods in most of benchmark cases.