In contrast to regularizationbased approaches, we formulate the adversarially robust learning problem as one of loss minimization with a Lipschitz constraint, and show that the saddle point of the associated Lagrangian is characterized by a Poisson equation with weighted Laplace operator. Further, the weighting for the Laplace operator is given by the Lagrange multiplier for the Lipschitz constraint, which modulates the sensitivity of the minimizer to perturbations.

Note that these games are two-player games, we select the runner in You-Shall-Not-Pass and kicker in Kick-And-Defend as our target agent. Section 4 mentioned that we download a well-trained policy for each game.

Deep reinforcement learning has shown great success in automatic policy learning for various sequential decision-making problems, such as training AI agents to defeat professional players in sophisticated games [74, 65, 24, 37] and controlling robots to accomplish complicated tasks [33, 38].

The last equation is derived as follows. Inaddition, we set the observation varianceσx to 0.25. Logistic(;µ,s) is the density function of a logistic distribution with the location parameterµand the scale parameters,andσ isthe logistic sigmoid function. Before each activation, we apply the layer normalization [Ba et al., 2016] to stabilize training. When the model has sufficiently high expressive power,b may diverge to infinity [Rezende and Viola, 2018], so we add a regularization term of(b+2ζ( b))/m to the loss function, wherem is the number of training examples.

However, FACMAClearnsacentralised butfactored critic,which combines per-agent utilities into the joint action-value function via a non-linear monotonic function, as inQMIX, apopular multi-agentQ-learning algorithm. However,unlikeQMIX, there are no inherent constraints on factoring the critic. We thus also employ a nonmonotonic factorisation and empirically demonstrate that its increased representational capacity allows it to solve some tasks that cannot be solved with monolithic, ormonotonically factored critics.