The decomposition idea is discussed more extensively by Seita et al. 2017 (for achieving subsampling without any

Theoretical analysis confirms that the proposed algorithm achieves sub-linear regret in relation to the number of rounds and arms.

We thank all reviewers for their valuable feedback and comments. Please find our responses below. Reviewer 1 - Explanation in the introduction: we strive for clarity and we appreciate this comment. We thank the reviewer for pointing this out. This can be done in many ways as discussed in Appendix C. The theoretical value used for the bounds is rather conservative however.

We provide a new understanding of the stochastic gradient bandit algorithm by showing that it converges to a globally optimal policy almost surely using any constant learning rate. This result demonstrates that the stochastic gradient algorithm continues to balance exploration and exploitation appropriately even in scenarios where standard smoothness and noise control assumptions break down. The proofs are based on novel findings about action sampling rates and the relationship between cumulative progress and noise, and extend the current understanding of how simple stochastic gradient methods behave in bandit settings.

Results are presented in Fig a. Full details will be provided Experiments Please allow us to first justify the use of the HIL experiment. All of the following points will be clarified in the revised manuscript (V2). 'gridding' continuous state/action spaces in order to apply DP-based methods, citing relevant literature. Re: the approximations in 4.1, we attempted to discuss each approximation Re: the outliers in Fig 2a, This is an interesting question. This is why the cost of greedy and RRL differ at the first epoch.

Membership inference arises in these contexts as a potential auditing method for detecting unauthorized data usage. While some efforts have been made on diffusion models, they are not applicable to text-to-image diffusion models due to the high computation overhead and enhanced generalization capabilities. In this paper, we first identify a conditional overfitting phenomenon in text-to-image diffusion models, indicating that these models tend to overfit the conditional distribution of images given the corresponding text rather than the marginal distribution of images only. Based on this observation, we derive an analytical indicator, namely Conditional Likelihood Discrepancy (CLiD), to perform membership inference, which reduces the stochasticity in estimating memorization of individual samples. Experimental results demonstrate that our method significantly outperforms previous methods across various data distributions and dataset scales. Additionally, our method shows superior resistance to overfitting mitigation strategies, such as early stopping and data augmentation.

Dimension reduction (DR) is an important and widely studied technique in exploratory data analysis. However, traditional DR methods are not applicable to datasets with a contrastive structure, where data are split into a foreground group of interest (case or treatment group), and a background group (control group). This type of data, common in biomedical studies, necessitates contrastive dimension reduction (CDR) methods to effectively capture information unique to or enriched in the foreground group relative to the background group. Despite the development of various CDR methods, two critical questions remain underexplored: when should these methods be applied, and how can the information unique to the foreground group be quantified? In this work, we address these gaps by proposing a hypothesis test to determine the existence of contrastive information, and introducing a contrastive dimension estimator (CDE) to quantify the unique components in the foreground group. We provide theoretical support for our methods and validate their effectiveness through extensive simulated, semi-simulated, and real experiments involving images, gene expressions, protein expressions, and medical sensors, demonstrating their ability to identify the unique information in the foreground group.

We thank all the reviewers for their valuable comments. We carefully address all the raised issues accordingly below. We appreciate your positive feedback. Thank you for the comments. We will proofread and improve the readability.

We thank the reviewers for their insightful feedback, and we appreciate the opportunity to improve our paper. We would like to emphasize that Theorem 1 is the most important contribution of our paper due to its generality. In the Gaussian case, our sample complexity result follows directly from the expression for the optimal loss. Response to Reviewer 2: We thank the reviewer for pointing us to Dohmatob's "Generalized No Free Lunch Theorem Finally, while Dohmatob's bounds become non-trivial only when the adversarial We will also add a clearer description of the "translate and pair in place" coupling. Comparisons with Sinha et al. are in Section 7 and we compare to Dohmatob above.

The aim of this paper is to address the challenge of gradual domain adaptation within a class of manifold-constrained data distributions.