a dual interpretation of our algorithm as sequentially maintaining a *dense* distribution over *all* (possibly *improper*) policies, where a policy is defined as a deterministic decision function mapping contexts to actions. We analyze how the realizability assumption enables us to establish uniform-convergence-type results for some *implicit* quantities in the universal policy space, regardless of the huge capacity of the universal policy space. Note that while the dual interpretation itself is not easy to compute in the universal policy space, it is only applied for the purpose of analysis and has nothing to do with our original algorithm’s implementation. Through this lens, we find that our algorithm’s dual interpretation satisfies a series of sufficient conditions for optimal contextual bandit learning. Our identified sufficient conditions for optimal contextual bandit learning in the universal policy space are built on the previous work of [@dudik2011efficient], [@agarwal2012contextual] and [@agarwal2014taming] — the first one is colloquially referred to as the “monster paper” by its authors due to its complexity ([link](https://hunch.net/~jl/projects/interactive/index.html)), and the third one is titled as “taming the monster” by its authors due to its improved computational efficiency. Since our algorithm achieve all the conditions required for regret optimality in the universal policy space in 