Including covariant information, such as position, force, velocity or spin is important in many tasks in computational physics and chemistry. We introduce Steerable E(3) Equivariant Graph Neural Networks (SEGNNs) that generalise equivariant graph networks, such that node and edge attributes are not restricted to invariant scalars, but can contain covariant information, such as vectors or tensors. This model, composed of steerable MLPs, is able to incorporate geometric and physical information in both the message and update functions. Through the definition of steerable node attributes, the MLPs provide a new class of activation functions for general use with steerable feature fields. We discuss ours and related work through the lens of equivariant non-linear convolutions, which further allows us to pin-point the successful components of SEGNNs: non-linear message aggregation improves upon classic linear (steerable) point convolutions; steerable messages improve upon recent equivariant graph networks that send invariant messages. We demonstrate the effectiveness of our method on several tasks in computational physics and chemistry and provide extensive ablation studies.

Generative Adversarial Networks (GAN) have become one of the most successful frameworks for unsupervised generative modeling. As GANs are difficult to train much research has focused on this. However, very little of this research has directly exploited gametheoretic techniques. We introduce Generative Adversarial Network Games (GANGs), which explicitly model a finite zero-sum game between a generator (G) and classifier (C) that use mixed strategies. The size of these games precludes exact solution methods, therefore we define resource-bounded best responses (RBBRs), and a resourcebounded Nash Equilibrium (RB-NE) as a pair of mixed strategies such that neither G or C can find a better RBBR. The RB-NE solution concept is richer than the notion of 'local Nash equilibria' in that it captures not only failures of escaping local optima of gradient descent, but applies to any approximate best response computations, including methods with random restarts. To validate our approach, we solve GANGs with the Parallel Nash Memory algorithm, which provably monotonically converges to an RB-NE. We compare our results to standard GAN setups, and demonstrate that our method deals well with typical GAN problems such as mode collapse, partial mode coverage and forgetting.

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Save for some special cases, current training methods for Generative Adversarial Networks (GANs) are at best guaranteed to converge to a 'local Nash equilibrium' (LNE). Such LNEs, however, can be arbitrarily far from an actual Nash equilibrium (NE), which implies that there are no guarantees on the quality of the found generator or classifier. This paper proposes to model GANs explicitly as finite games in mixed strategies, thereby ensuring that every LNE is an NE. With this formulation, we propose a solution method that is proven to monotonically converge to a resourcebounded Nash equilibrium (RB-NE): by increasing computational resources we can find better solutions. We empirically demonstrate that our method is less prone to typical GAN problems such as mode collapse, and produces solutions that are less exploitable than those produced by GANs and MGANs, and closely resemble theoretical predictions about NEs. * This version supersedes our earlier arXiv paper (Oliehoek et al., 2017).

Due to the capability of deep learning to perform well in high dimensional problems, deep reinforcement learning agents perform well in challenging tasks such as Atari 2600 games. However, clearly explaining why a certain action is taken by the agent can be as important as the decision itself. Deep reinforcement learning models, as other deep learning models, tend to be opaque in their decision-making process. In this work, we propose to make deep reinforcement learning more transparent by visualizing the evidence on which the agent bases its decision. In this work, we emphasize the importance of producing a justification for an observed action, which could be applied to a black-box decision agent.