The hope is that these predictions allow the algorithm to circumvent worst case lower bounds when the predictions are good, and approximately match them otherwise. The precise definitions and guarantees vary with different settings, but there have been significant successes in applying this framework for many different algorithmic problems, ranging from general online problems to classical graph algorithms (see Section 1.2 for a more detailed discussion of related work, and [35] for a survey). In all of these settings it turns out to be possible to define a "prediction" where the "quality" of the algorithm (competitive ratio, running time, etc.) depends the "error" of the prediction.

Imagine a large firm with multiple departments that plans a large recruitment. Candidates arrive one-by-one, and for each candidate the firm decides, based on her data (CV, skills, experience, etc), whether to summon her for an interview. The firm wants to recruit the best candidates while minimizing the number of interviews. We model such scenarios as an assignment problem between items (candidates) and categories (departments): the items arrive one-by-one in an online manner, and upon processing each item the algorithm decides, based on its value and the categories it can be matched with, whether to retain or discard it (this decision is irrevocable). The goal is to retain as few items as possible while guaranteeing that the set of retained items contains an optimal matching. We consider two variants of this problem: (i) in the first variant it is assumed that the $n$ items are drawn independently from an unknown distribution $D$.

We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples $(\bx, y)$ from an unknown distribution on $\R^d \times \{ \pm 1\}$, whose marginal distribution on $\bx$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with 0-1 loss $\opt+\eps$, where $\opt$ is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples $(\bx, y)$ from an unknown distribution on $\R^d \times \R$, whose marginal distribution on $\bx$ is the standard Gaussian and the labels $y$ can be arbitrary, the goal is to output a hypothesis with square loss $\opt+\eps$, where $\opt$ is the square loss of the best-fitting ReLU. We prove Statistical Query (SQ) lower bounds of $d^{\poly(1/\eps)}$ for both of these problems. Our SQ lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.

Other comments will be taken into account in the revision. Reviewer_45: Thank you for the positive feedback and constructive comments.

Flow Matching, a promising approach in generative modeling, has recently gained popularity. Relying on ordinary differential equations, it offers a simple and flexible alternative to diffusion models, which are currently the state-of-the-art. Despite its empirical success, the mathematical understanding of its statistical power so far is very limited. This is largely due to the sensitivity of theoretical bounds to the Lipschitz constant of the vector field which drives the ODE. In this work, we study the assumptions that lead to controlling this dependency. Based on these results, we derive a convergence rate for the Wasserstein $1$ distance between the estimated distribution and the target distribution which improves previous results in high dimensional setting. This rate applies to certain classes of unbounded distributions and particularly does not require $\log$-concavity.

Accurately analyzing data while preserving individual privacy is a fundamental challenge in statistical inference. Since its formulation nearly two decades ago, Differential Privacy (DP) [DMNS06] has emerged as the leading framework for privacy-preserving data analysis, providing strong mathematical privacy guarantees and gaining adoption by major entities such as the U.S. Census Bureau, Amazon [Ama24], Google [EPK14], Microsoft [DKY17], and Apple [Dif17; TVVKFSD17]. Unfortunately, DP guarantees often come at the cost of increased data requirements or computational resources, which has limited the widespread adoption of differential privacy in spite of its theoretical appeal. To address this issue, a recent line of work has investigated whether access to even small amounts of additional public data could help mitigate this loss of performance. Promising results for various tasks have been shown, both experimentally [KST20; LLHR24; BZHZK24; DORKSF24] and theoretically [BKS22; BBCKS23]. The use of additional auxiliary information is very enticing, as such access is available in many real-world applications: for example, hospitals handling sensitive patient data might leverage public datasets, records from different periods or locations, or synthetic data generated by machine learning models to improve analysis. Similarly, medical or socio-econonomic studies focusing on a minority or protected group can leverage statistical data from the overall population. However, integrating public data introduces its own challenges, as it often lacks guarantees regarding its accuracy or relevance to private datasets.

We study the fundamental problems of agnostically learning halfspaces and ReLUs under Gaussian marginals. In the former problem, given labeled examples (\bx, y) from an unknown distribution on \R d \times \{ \pm 1\}, whose marginal distribution on \bx is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with 0-1 loss \opt \eps, where \opt is the 0-1 loss of the best-fitting halfspace. In the latter problem, given labeled examples (\bx, y) from an unknown distribution on \R d \times \R, whose marginal distribution on \bx is the standard Gaussian and the labels y can be arbitrary, the goal is to output a hypothesis with square loss \opt \eps, where \opt is the square loss of the best-fitting ReLU. We prove Statistical Query (SQ) lower bounds of d {\poly(1/\eps)} for both of these problems. Our SQ lower bounds provide strong evidence that current upper bounds for these tasks are essentially best possible.

Semantic communication (SemCom) has emerged as a new paradigm for 6G communication, with deep learning (DL) models being one of the key drives to shift from the accuracy of bit/symbol to the semantics and pragmatics of data. Nevertheless, DL-based SemCom systems often face performance bottlenecks due to overfitting, poor generalization, and sensitivity to outliers. Furthermore, the varying-fading gains and noises with uncertain signal-to-noise ratios (SNRs) commonly present in wireless channels usually restrict the accuracy of semantic information transmission. Consequently, this paper constructs a latent diffusion model-enabled SemCom system, and proposes three improvements compared to existing works: i) To handle potential outliers in the source data, semantic errors obtained by projected gradient descent based on the vulnerabilities of DL models, are utilized to update the parameters and obtain an outlier-robust encoder. ii) A lightweight single-layer latent space transformation adapter completes one-shot learning at the transmitter and is placed before the decoder at the receiver, enabling adaptation for out-of-distribution data and enhancing human-perceptual quality. iii) An end-to-end consistency distillation (EECD) strategy is used to distill the diffusion models trained in latent space, enabling deterministic single or few-step real-time denoising in various noisy channels while maintaining high semantic quality. Extensive numerical experiments across different datasets demonstrate the superiority of the proposed SemCom system, consistently proving its robustness to outliers, the capability to transmit data with unknown distributions, and the ability to perform real-time channel denoising tasks while preserving high human perceptual quality, outperforming the existing denoising approaches in semantic metrics.