This paper studies sampling error bounds for denoising diffusion probabilistic models (DDPMs) in the 2-Wasserstein distance. Our contributions are threefold. (i) Under general Lipschitz-type conditions on the score function and for a broad class of variance schedules, including the cosine schedule, we establish sharp upper bounds that are optimal in both the dimension and the number of steps, and recover several sharp error bounds previously obtained in the literature. (ii) We prove that the same Lipschitz-type conditions, which encompass those commonly imposed on the (learned) score, imply a logarithmic Sobolev inequality and hence a quadratic transportation cost inequality for the DDPM. As a consequence, in settings covered by existing work, an optimal Wasserstein bound, up to a logarithmic factor, follows from the recently obtained sharp error bound in the Kullback-Leibler divergence under geometric-type variance schedules. (iii) We show that for general log-concave target distributions, the optimal Wasserstein error bound remains attainable even without a quadratic transportation cost inequality for the target. Our analysis is based on viewing the DDPM sampler as a discretization of the Föllmer process rather than the conventional reverse Ornstein-Uhlenbeck process.

We study the problem of (ϵ,δ)-certified machine unlearning for minimax models. Most of the existing works focus on unlearning from standard statistical learning models that have a single variable and their unlearning steps hinge on the direct Hessian-based conventional Newton update. We develop a new (ϵ,δ)-certified machine unlearning algorithm for minimax models. It proposes a minimax unlearning step consisting of a total Hessian-based complete Newton update and the Gaussian mechanism borrowed from differential privacy. To obtain the unlearning certification, our method injects calibrated Gaussian noises by carefully analyzing the "sensitivity" of the minimax unlearning step (i.e., the closeness between the minimax unlearning variables and the retraining-from-scratch variables).

We study the problem of (ϵ,δ)-certified machine unlearning for minimax models. Most of the existing works focus on unlearning from standard statistical learning models that have a single variable and their unlearning steps hinge on the direct Hessian-based conventional Newton update. We develop a new (ϵ,δ)-certified machine unlearning algorithm for minimax models. It proposes a minimax unlearning step consisting of a total Hessian-based complete Newton update and the Gaussian mechanism borrowed from differential privacy. To obtain the unlearning certification, our method injects calibrated Gaussian noises by carefully analyzing the "sensitivity" of the minimax unlearning step (i.e., the closeness between the minimax unlearning variables and the retraining-from-scratch variables).

While there remains a small gap between our main lower bound of Theorem 3 and the deterministic quantised gradient descent of Section 6, we can show that the gap cannot be closed by improved deterministic algorithms where the coordinator learns value of objective function F(x) in addition to the minimiser x. That is, our quantised gradient descent is the communication-optimal deterministic algorithm for variant (1) for objectives with constant condition number. Recall that in the N-player equality over universe of size d, denoted by EQd,N, each player i is given an input bi 2{ 0,1}d, and the task is to decide if all players have the same input. It is known [33] that the deterministic communication complexity of EQd,N is CC(EQd,N)= ( Nd). Theorem 8. Given parameters N, d, ", 0 and = 0N satisfying d /" = (1), any deterministic protocol solving (1) for quadratic input functions x 7! 0kx x0k22 has communication complexity Nd log( d/"), if the coordinator is also required to output estimate r 2 R for the minimum function value such that Assume is a deterministic protocol solving (1) with communication complexity C .We show that can then solve N-party equality over a universe of size D = ( dlog( d/")), implying C = ( ND)= Nd log( d/") . More specifically, let S be the set given by Lemma 2 with =(2 "/)1/2, and let D = dlog|S|e = (dlog( d/")). Note that since we assume d /" = (1), the set S has at least two elements and D 1.

We study the prophet inequality, a fundamental problem in online decision-making and optimal stopping, in a practical setting where rewards are observed only through noisy realizations and reward distributions are unknown. At each stage, the decision-maker receives a noisy reward whose true value follows a linear model with an unknown latent parameter, and observes a feature vector drawn from a distribution. To address this challenge, we propose algorithms that integrate learning and decision-making via lower-confidence-bound (LCB) thresholding. In the i.i.d.\ setting, we establish that both an Explore-then-Decide strategy and an $\varepsilon$-Greedy variant achieve the sharp competitive ratio of $1 - 1/e$, under a mild condition on the optimal value. For non-identical distributions, we show that a competitive ratio of $1/2$ can be guaranteed against a relaxed benchmark. Moreover, with limited window access to past rewards, the tight ratio of $1/2$ against the optimal benchmark is achieved.

Bayesian methods lie at the heart of modern data science and provide a powerful scaffolding for estimation in data-constrained settings and principled quantification and propagation of uncertainty. Yet in many real-world use cases where these methods are deployed, there is a natural need to preserve the privacy of the individuals whose data is being scrutinized. While a number of works have attempted to approach the problem of differentially private Bayesian estimation through either reasoning about the inherent privacy of the posterior distribution or privatizing off-the-shelf Bayesian methods, these works generally do not come with rigorous utility guarantees beyond low-dimensional settings. In fact, even for the prototypical tasks of Gaussian mean estimation and linear regression, it was unknown how close one could get to the Bayes-optimal error with a private algorithm, even in the simplest case where the unknown parameter comes from a Gaussian prior. In this work, we give the first efficient algorithms for both of these problems that achieve mean-squared error $(1+o(1))\mathrm{OPT}$ and additionally show that both tasks exhibit an intriguing computational-statistical gap. For Bayesian mean estimation, we prove that the excess risk achieved by our method is optimal among all efficient algorithms within the low-degree framework, yet is provably worse than what is achievable by an exponential-time algorithm. For linear regression, we prove a qualitatively similar lower bound. Our algorithms draw upon the privacy-to-robustness framework of arXiv:2212.05015, but with the curious twist that to achieve private Bayes-optimal estimation, we need to design sum-of-squares-based robust estimators for inherently non-robust objects like the empirical mean and OLS estimator. Along the way we also add to the sum-of-squares toolkit a new kind of constraint based on short-flat decompositions.

Risdefined X Rd and is randomvectorP .

However, their data structure requires a significantly increased super-linear storage space, as well as super-linear preprocessing time. These limitations inhibit the practical applicability of their approach on large datasets.

The algorithm [62] consider Algorithm 2 for the stochastic generalized linear bandit problem. Assume thatθ is the true parameter of the reward model. Then we consider the lower bounds. For fj(A) = 12(ej1eTj2 +ej2eTj1),A with j1 j2, fj(Ai) is only 1 wheni = j and 0 otherwise. With Claim D.12 and Claim D.11 we get that g C q To get 1), we writeVl = [v1, vl] Rd l and V l = [vl+1, vk].

In the deterministic setting where the data is deterministically given without any probabilistic assumptions, significant advances inDP linear regression has been made [77,57,68, 16, 7, 83, 31, 67, 82, 71]. In the randomized settings where each example{xi,yi} is drawn i.i.d. We explain the closely related ones in Section 2.3, with analysis when the covariance matrixhasaspectralgap. The resulting utility guarantees are the same as those from [23], which are discussedinSection2.3. When privacy is not required, we know from Theorem 2.2 that under Assumptions A.1-A.3, we can achieve an error rate of O(κ p V/n).