Specifically,inPAC-Bayesian approach, [45,26,4,22]provide the generalization bounds of order O( 1 n). In implicit embedding approach, [12, 13, 52, 11, 58, 7] provide the convergence rate of orderO( 1n1/4), and [53] of orderO( 1 n). In the factor graph decomposition approach, [18, 51] present the generalization upper bounds of orderO( 1 n).

Touse y, distances challenge sentences.

The existing theory bounds for multi-label learning, which preserve the coupling among different components, are invalid for LSRL.

Generative models for counterfactual outcomes face two key sources of bias. Confounding bias arises when approaches fail to account for systematic differences between those who receive the intervention and those who do not. Misspecification bias arises when methods attempt to address confounding through estimation of an auxiliary model, but specify it incorrectly. We introduce DoubleGen, a doubly robust framework that modifies generative modeling training objectives to mitigate these biases. The new objectives rely on two auxiliaries -- a propensity and outcome model -- and successfully address confounding bias even if only one of them is correct. We provide finite-sample guarantees for this robustness property. We further establish conditions under which DoubleGen achieves oracle optimality -- matching the convergence rates standard approaches would enjoy if interventional data were available -- and minimax rate optimality. We illustrate DoubleGen with three examples: diffusion models, flow matching, and autoregressive language models.

Expected improvement (EI) is one of the most widely used acquisition functions in Bayesian optimization (BO). Despite its proven empirical success in applications, the cumulative regret upper bound of EI remains an open question. In this paper, we analyze the classic noisy Gaussian process expected improvement (GP-EI) algorithm. We consider the Bayesian setting, where the objective is a sample from a GP. Three commonly used incumbents, namely the best posterior mean incumbent (BPMI), the best sampled posterior mean incumbent (BSPMI), and the best observation incumbent (BOI) are considered as the choices of the current best value in GP-EI. We present for the first time the cumulative regret upper bounds of GP-EI with BPMI and BSPMI. Importantly, we show that in both cases, GP-EI is a no-regret algorithm for both squared exponential (SE) and Matérn kernels. Further, we present for the first time that GP-EI with BOI either achieves a sublinear cumulative regret upper bound or has a fast converging noisy simple regret bound for SE and Matérn kernels. Our results provide theoretical guidance to the choice of incumbent when practitioners apply GP-EI in the noisy setting. Numerical experiments are conducted to validate our findings.

While this does not match the lower bound of Theorem 3 exactly, it scales in a similar manner.

This paper intends to answer the three interesting questions.

In this paper, we explore the knowledge transfer under the setting of matrix completion, which aims to enhance the estimation of a low-rank target matrix with auxiliary data available. We propose a transfer learning procedure given prior information on which source datasets are favorable. We study its convergence rates and prove its minimax optimality. Our analysis reveals that with the source matrices close enough to the target matrix, out method outperforms the traditional method using the single target data. In particular, we leverage the advanced sharp concentration inequalities introduced in \cite{brailovskaya2024universality} to eliminate a logarithmic factor in the convergence rate, which is crucial for proving the minimax optimality. When the relevance of source datasets is unknown, we develop an efficient detection procedure to identify informative sources and establish its selection consistency. Simulations and real data analysis are conducted to support the validity of our methodology.