We introduce a natural and mild restriction on the meta-level combination functions of the local trials.

The sample-averaged gradient of the CSO objective is biased due to its nested structure, and therefore requires a high sample complexity for convergence. We introduce a general stochastic extrapolation technique that effectively reduces the bias.

Training data attribution (TDA) techniques find influential training data for the model's prediction on the test data of interest.

Seq2Seq AP - 44.8 (e) Large vocabulary object detection.

In this paper, there are some equivalent forms of the generalization error we will study, e.g., Eq. (2) This lemma is a consequence of Lemma 2.1, with further utilizing some symmetric properties. Recall Eq. (1) in Lemma 2.1, E Note that Eq. (2) in the main text is from the second equation above, which is used to derive individual Notice that we do not change the definitions of any the random variable, e.g., This, as we have already seen in Eq. (5) in the main text, is used to derive hypotheses-conditioned CMI bounds in Section 4. It's easy to see that when To obtain Eq. (14), we let W This is used to derive supersample-conditioned CMI bounds in Section 4. It's easy to see that both Like all the previous information-theoretic bounds, the following lemma is widely used in our paper. We also invoke some other lemmas as given below. It's easy to verify that We note that the reason we introduce four types of SCH stability in Definition 2.1 is that solely using The basic set up is as follows. By Lemma A.3, we have E Recall Eq. (12) in Lemma A.1 and applying Jensen's inequality to the absolute function, the first The proof is nearly the same to the proof of Theorem 3.1, except that now the randomness of the algorithm is given for each DV auxiliary function, so the randomness of Similar to the proof of Theorem 3.1, we let We now prove the first bound. Lemma A.2, we have E By Lemma A.3, we have E Recall Eq. (14) in Lemma A.1 and by Jensen's inequality for the absolute function, the first bound is To prove the second bound, we return to Eq. (20), and take expectation over For the second part of Theorem 4.1, notice that it's valid to let The proof is similar to [18, Theorem 2.1].