Our proposed method significantly reduces communication overhead in federated learning. This method poses a trade-off between time and memory complexity. We also provide detailed information about the optimization hyperparameters e.g. In this section, we explore the effect of fitness sparsification i.e. selecting top-k fitness values from the To enable a fair and insightful comparison between the two population sizes, our focus was on assessing performance based on the number of members remaining post-sparsification rather than directly contrasting sparsification rates. Our results underline the crucial role that population size plays in exploring optimal solutions, overshadowing even the significance of compression rate.

Recently, flow-based generative models have shown superior efficiency compared to diffusion models. In this paper, we study rectified flow models, which constrain transport trajectories to be linear from the base distribution to the data distribution. This structural restriction greatly accelerates sampling, often enabling high-quality generation with a single Euler step. Under standard assumptions on the neural network classes used to parameterize the velocity field and data distribution, we prove that rectified flows achieve sample complexity $\tilde{O}(\varepsilon^{-2})$. This improves on the best known $O(\varepsilon^{-4})$ bounds for flow matching model and matches the optimal rate for mean estimation. Our analysis exploits the particular structure of rectified flows: because the model is trained with a squared loss along linear paths, the associated hypothesis class admits a sharply controlled localized Rademacher complexity. This yields the improved, order-optimal sample complexity and provides a theoretical explanation for the strong empirical performance of rectified flow models.

In spite of several claims stating that some models are more interpretable than others --e.g., linear models are more interpretable than deep neural networks-- we still lack a principled notion of interpretability that allows us to formally compare among different classes of models. We make a step towards such a theory by studying whether folklore interpretability claims have a correlate in terms of computational complexity theory. We focus on post-hoc explainability queries that, intuitively, attempt to answer why individual inputs are classified in a certain way by a given model. In a nutshell, we say that a class C1 of models is more interpretable than another class C2, if the computational complexity of answering post-hoc queries for models in C2 is higher than for C1. We prove that this notion provides a good theoretical counterpart to current beliefs on the interpretability of models; in particular, we show that under our definition and assuming standard complexity-theoretical assumptions (such as P!=NP), both linear and tree-based models are strictly more interpretable than neural networks. Our complexity analysis, however, does not provide a clear-cut difference between linear and tree-based models, as we obtain different results depending on the particular {post-hoc explanations} considered. Finally, by applying a finer complexity analysis based on parameterized complexity, we are able to prove a theoretical result suggesting that shallow neural networks are more interpretable than deeper ones.

A relevant and well-written technical paper, which presents a variation of the classic active learning setting coined'auditing'. This is generally defined by non-uniform cost of labels and not knowing the cost of a label a priori to the label query. The aim of the paper is to compare the complexity of auditing versus (standard) active learning, i.e., how many labels are required. The authors accomplishes this by deriving bounds on the complexity for the new variant and compares to standard active learning where they show interesting results. The authors focuses on two (tractable) cases: 1) They consider the active learning complexity as the total number of label queries.

We thank all reviewers for their helpful comments and suggestions! We will address all minor issues. We will add references in the revision. UMG ( π) has cycles. We havn't thought about checking it from data, which is an interesting statistical Theorem 1 is effectively known.

We sincerely thank the reviewers for their time and constructive comments. According to our complexity study in Theorem 1 and Section 3.2, maintaining a small active set is The correlation between features may affect the efficiency of Thunder, but it does not impact the algorithm's safety. According to the derivation in Section 2.1, the stop condition regarding feature recruiting given in Lemma 1 The current algorithm complexity analysis in the supplemental file ignores the sampling steps. The sampling strategy does not reduce or break the algorithm's We will take the reviewers' suggestions and show more results on the effectiveness of sampling. We thank the reviewers again for their insightful comments on writing.

We would like to thank all the reviewers for your helpful comments and suggestions. As shown in Appendix A.3, the layer-wise GCN network has the highest computational complexity in the computational propagation flow. Please see the response in Common Response 2 . For fair comparison we only report the result on semi-supervised task. Please see the response in Common Response 2 .

EPDiff) in Eq.(2) to compute