Byleveraging thehigh-order topological information ofdata,weareable to collect most of the clean data and train a high-quality model. Theoretically we prove that this topological approach is guaranteed to collect the clean data with high probability.

However, the resulting methods often suffer from high computational complexity which has reduced their practical applicability. For example, in the case of multiclass logistic regression, the aggregating forecaster (Foster et al. (2018)) achievesaregret ofO(log(Bn))whereas Online Newton Step achieves O(eBlog(n))obtaining adouble exponential gaininB (aboundonthenormof comparativefunctions).

Based ontheprinciple ofoptimism inthefaceofuncertainty(OFU) [56,49,10],OFU-RL achievestheglobal optimality by ensuring that the optimistically biased value is close to the real value in the long run. Based on Thompson Sampling [62], Posterior Sampling RL (PSRL) [57, 42, 43] explores by greedily optimizing the policy in an MDP which is sampled from the posterior distribution over MDPs.

Reasoning system dynamics is one of the most important analytical approaches for many scientific studies.