Theworddistributions of topics, their optimal transports to the word distributions of documents, and the embeddings of words are learned in a unified framework. When learning thetopic model, weleverage adistilled underlying distance matrix toupdate the topic distributions and smoothly calculate the corresponding optimal transports.

First, we rewrite Eq. (3) as max

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Fortheformer setting, an extensive literature has explored the statistical and computational aspects of learning Booleanfunctions[Angluin,1992,HellersteinandServedio,2007].

Learning in real-world multiagent tasks is challenging due to the usual partial observability ofeach agent. Previous efforts alleviate thepartial observability by historical hidden states with Recurrent Neural Networks, however, they do not consider themultiagent characters thateither themultiagent observationconsists ofanumber ofobject entities orthe action space shows clear entity interactions.

We study the embedding dimension of distance comparison data in two settings: contrastive learning and k -nearest neighbors ( k -NN).

Intensive algorithmic efforts have been made to enable the rapid improvements of certificated robustness for complex ML models recently. However, current robustness certification methods are only able to certify under a limited perturbation radius. Given that existingpure data-driven statistical approaches have reached a bottleneck, in this paper, we propose to integrate statistical ML modelswithknowledge (expressed aslogical rules) asareasoningcomponent using Markovlogic networks (MLN), so as to further improvethe overall certified robustness.