A Proofs of Theoretical Results

Lemma 1. F or any embedding f and finite N, we have L Theorem 3. F or any embedding f and finite N and M, we have e L By Jensen's inequality, we may push the absolute value inside the expectation to see that The outer expectation disappears since the tail probably bound of Theorem A.2 holds uniformly for all fixed x, x We still owe the reader a proof of Lemma A.2, which we give now. We then proceed to bound the right hand tail probability. Combining Lemma A.3 and Lemma A.4, with probability at least 1, for all f 2F, we have L Note the definition of g is slightly modified in this context. We again use the Adam optimizer with learning rate 0 . To implement the debiased objective, we only modify the "src/s2v-model.py"