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Capturing the underlying structural causal relations represented by Directed Acyclic Graphs (DAGs) has been a fundamental task in various AI disciplines. Causal DAG learning via the continuous optimization framework has recently achieved promising performance in terms of accuracy and efficiency. However, most methods make strong assumptions of homoscedastic noise, i.e., exogenous noises have equal variances across variables, observations, or even both. The noises in real data usually violate both assumptions due to the biases introduced by different data collection processes. To address the heteroscedastic noise issue, we introduce relaxed implementable sufficient conditions and prove the identifiability of a general class of SEM subject to those conditions. Based on the identifiable general SEM, we propose a novel formulation for DAG learning which accounts for the noise variance variation across variables and observations. We then propose an effective two-phase iterative DAG learning algorithm to address the increasing optimization difficulties and learn a causal DAG from data with heteroscedastic variables noise under varying variance. We show significant empirical gains of the proposed approaches over state-of-the-art methods on both synthetic data and real data.

 

Atmospheric photochemistry on Titan continuously transforms methane and nitrogen gases into various organic compounds. This study explores the fate of these molecules when they land on Titan's surface. Our analytical exploration reveals that most simple organics found in Titan's atmosphere, including all nitriles, triple‐bonded hydrocarbons, and benzene, land as solids. Only a few compounds are in the liquid phase, while only ethylene remains gaseous. For the simple organics that land as solids, we further examine their interactions with Titan's lake liquids. Utilizing principles of buoyancy, we found that flotation can be achieved via porosity‐induced (25%–60% porosity) or capillary force‐induced buoyancy for hydrogen cyanide ices on ethane‐rich lakes. Otherwise, these ices would sink and become lakebed sediments. By evaluating the timescale of flotation, our findings suggest that porosity‐induced flotation of millimeter‐sized and larger sediments is the only plausible mechanism for floating solids to explain the transient “magic islands” phenomena on Titan's lakes.

 

Markov games model interactions among multiple players in a stochastic, dynamic environment. Each player in a Markov game maximizes its expected total discounted reward, which depends upon the policies of the other players. We formulate a class of Markov games, termed affine Markov games, where an affine reward function couples the players’ actions. We introduce a novel solution concept, the soft-Bellman equilibrium, where each player is boundedly rational and chooses a soft-Bellman policy rather than a purely rational policy as in the well-known Nash equilibrium concept. We provide conditions for the existence and uniqueness of the soft-Bellman equilibrium and propose a nonlinear least-squares algorithm to compute such an equilibrium in the forward problem. We then solve the inverse game problem of inferring the players’ reward parameters from observed state-action trajectories via a projected-gradient algorithm. Experiments in a predator-prey OpenAI Gym environment show that the reward parameters inferred by the proposed algorithm outper- form those inferred by a baseline algorithm: they reduce the Kullback-Leibler divergence between the equilibrium policies and observed policies by at least two orders of magnitude.