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We study the spectral gap of subgraphs of the hypercube induced by monotone subsets of vertices. For a monotone subset A ⊆ {0, 1}n of density μ(A), the previous best lower bound on the spectral gap, due to Cohen [Coh16], was γ ≳ μ(A)/n2, improving upon the earlier bound γ ≳ μ(A)2/n2 established by Ding and Mossel [DM14]. In this paper, we prove the optimal lower bound γ ≳ μ(A)/n. As a corollary, we improve the mixing time upper bound of the random walk on constant-density monotone sets from O(n3), as shown by Ding and Mossel, to O(n2). Along the way, we develop two new inequalities that may be of independent interest: (1) a directed L2-Poincar´e inequality on the hypercube, and (2) an “approximate” FKG inequality for monotone sets 

This paper introduces the concept of leakage-robust Bayesian persuasion. Situated between public Bayesian persuasion and private Bayesian persuasion, leakage-robust persuasion considers a setting where one or more signals privately communicated by a sender to the receivers may be leaked. We study the design of leakage-robust Bayesian persuasion schemes and quantify the price of robustness using two formalisms: - The first notion, k-worst-case persuasiveness, requires a signaling scheme to remain persuasive as long as each receiver observes no more than k leaked signals from other receivers. We quantify the Price of Robust Persuasiveness (PoRPk)— i.e., the gap in sender's utility as compared to the optimal private persuasion scheme—as Θ(min{2k,n}) for supermodular sender utilities and Θ(k) for submodular or XOS sender utilities, where n is the number of receivers. This result also establishes that in some instances, Θ(log k) leakages are sufficient for the utility of the optimal leakage-robust persuasion to degenerate to that of public persuasion. - The second notion, expected downstream utility robustness, relaxes the persuasiveness requirement and instead considers the impact on sender's utility resulting from receivers best responding to their observations. By quantifying the Price of Robust Downstream Utility (PoRU) as the gap between the sender's expected utility over the randomness in the leakage pattern as compared to private persuasion, our results show that, over several natural and structured distributions of leakage patterns, PoRU improves PoRP to Θ(k) or even Θ(1), where k is the maximum number of leaked signals observable to each receiver across leakage patterns in the distribution. En route to these results, we show that subsampling and masking serve as general-purpose algorithmic paradigms for transforming any private persuasion signaling scheme to one that is leakage-robust, with minmax optimal loss in sender's utility. A full version of this paper can be found at https://arxiv.org/abs/2411.16624. 

We extend the framework of augmented distribution testing (Aliakbarpour, Indyk, Rubinfeld, and Silwal, NeurIPS 2024) to the differentially private setting. This captures scenarios where a data ana- lyst must perform hypothesis testing tasks on sensitive data, but is able to leverage prior knowledge (public, but possibly erroneous or untrusted) about the data distribution. We design private algorithms in this augmented setting for three flagship distribution testing tasks, uniformity, identity, and closeness testing, whose sample complexity smoothly scales with the claimed quality of the auxiliary information. We complement our algorithms with information- theoretic lower bounds, showing that their sample complexity is optimal (up to logarithmic factors). Keywords: distribution testing, identity testing, closeness testing, differential privacy, learning- augmented algorithms