We study the problem of efficiently estimating the mean of a $k$-class random variable, $Y$, using a limited number of labels, $N$, in settings where the analyst has access to auxiliary information (i.e.: covariates) $X$ that may be informative about $Y$. We propose an active learning algorithm (PartiBandits) to estimate $\mathbb{E}[Y]$.

This paper introduces a novel kernel-based formulation of the Equalized Odds (EO) criterion, denoted as $\operatorname{EO}_k$, for fair representation learning (FRL) in supervised settings. The central goal of FRL is to mitigate discrimination regarding a sensitive attribute $S$ while preserving prediction accuracy for the target variable $Y$. Our proposed criterion enables a rigorous and interpretable quantification of three core fairness objectives: independence ($\widehat{Y} \perp S$), separation--also known as equalized odds ($\widehat{Y} \perp S \mid Y$), and calibration ($Y \perp S \mid \widehat{Y}$). Under both unbiased ($Y \perp S$) and biased ($Y \not \perp S$) conditions, we show that $\operatorname{EO}_k$ satisfies both independence and separation in the former, and uniquely preserves predictive accuracy while lower bounding independence and calibration in the latter, thereby offering a unified analytical characterization of the tradeoffs among these fairness criteria. We further define the empirical counterpart, $\widehat{\operatorname{EO}}_k$, a kernel-based statistic that can be computed in quadratic time, with linear-time approximations also available. A concentration inequality for $\widehat{\operatorname{EO}}_k$ is derived, providing performance guarantees and error bounds, which serve as practical certificates of fairness compliance. While our focus is on theoretical development, the results lay essential groundwork for principled and provably fair algorithmic design in future empirical studies.

Reinforcement learning (RL) algorithms are typically based on optimizing a Markov Decision Process (MDP) using the optimal Bellman equation. Recent studies have revealed that focusing the optimization of Bellman equations solely on in-sample actions tends to result in more stable optimization, especially in the presence of function approximation. Upon on these findings, in this paper, we propose an Empirical MDP Iteration (EMIT) framework.

We present a family $\{\widehat{\pi}_p\}_{p\ge 1}$ of pessimistic learning rules for offline learning of linear contextual bandits, relying on confidence sets with respect to different $\ell_p$ norms, where $\widehat{\pi}_2$ corresponds to Bellman-consistent pessimism (BCP), while $\widehat{\pi}_\infty$ is a novel generalization of lower confidence bound (LCB) to the linear setting. We show that the novel $\widehat{\pi}_\infty$ learning rule is, in a sense, adaptively optimal, as it achieves the minimax performance (up to log factors) against all $\ell_q$-constrained problems, and as such it strictly dominates all other predictors in the family, including $\widehat{\pi}_2$.

Labeled data often comes at a high cost as it may require recruiting human labelers or running costly experiments. At the same time, in many practical scenarios, one already has access to a partially labeled, potentially biased dataset that can help with the learning task at hand. Motivated by such settings, we formally initiate a study of ``semi-supervised active learning'' through the frame of linear regression. Here, the learner has access to a dataset $X \in \mathbb{R}^{(n_{\text{un}}+n_{\text{lab}}) \times d}$ composed of $n_{\text{un}}$ unlabeled examples that a learner can actively query, and $n_{\text{lab}}$ examples labeled a priori.

Reinforcement learning (RL) algorithms are typically based on optimizing a Markov Decision Process (MDP) using the optimal Bellman equation. Recent studies have revealed that focusing the optimization of Bellman equations solely on in-sample actions tends to result in more stable optimization, especially in the presence of function approximation. Upon on these findings, in this paper, we propose an Empirical MDP Iteration (EMIT) framework. For each of these empirical MDPs, it learns an estimated Q-function denoted as \widehat{Q} . The key strength is that by restricting the Bellman update to in-sample bootstrapping, each empirical MDP converges to a unique optimal \widehat{Q} function.

We present a family \{\widehat{\pi}_p\}_{p\ge 1} of pessimistic learning rules for offline learning of linear contextual bandits, relying on confidence sets with respect to different \ell_p norms, where \widehat{\pi}_2 corresponds to Bellman-consistent pessimism (BCP), while \widehat{\pi}_\infty is a novel generalization of lower confidence bound (LCB) to the linear setting. We show that the novel \widehat{\pi}_\infty learning rule is, in a sense, adaptively optimal, as it achieves the minimax performance (up to log factors) against all \ell_q -constrained problems, and as such it strictly dominates all other predictors in the family, including \widehat{\pi}_2 .

Labeled data often comes at a high cost as it may require recruiting human labelers or running costly experiments. At the same time, in many practical scenarios, one already has access to a partially labeled, potentially biased dataset that can help with the learning task at hand. Motivated by such settings, we formally initiate a study of semi-supervised active learning'' through the frame of linear regression. In this paper, we introduce an instance dependent parameter called the reduced rank, denoted \text{R}_X, and propose an efficient algorithm with query complexity O(\text{R}_X/\epsilon) . This result directly implies improved upper bounds for two important special cases: (i) active ridge regression, and (ii) active kernel ridge regression, where the reduced-rank equates to the statistical dimension'', \textsf{sd}_\lambda and effective dimension'', d_\lambda of the problem respectively, where \lambda \ge 0 denotes the regularization parameter. Finally, we introduce a distributional version of the problem as a special case of the agnostic formulation we consider earlier; here, for every X, we prove a matching instance-wise lower bound of \Omega (\text{R}_X / \epsilon) on the query complexity of any algorithm.

Reading comprehension is one of the fundamental skills for human, which one must learn systematically since the elementary school. Do you still remember how the worksheet of your reading class looks like? It usually consists of an article and few questions about its content. To answer these questions, you need to first gather information by collecting answer-related sentences from the article. Sometimes you can directly copy those original sentences from the article as the final answer.

A linear dynamical system (A,B,C,D) is equivalent to the system (TAT {-1}, TB, CT {-1}, D) for any invertible matrix T in terms of the behavior of the outputs. A little thought shows therefore that in its unrestricted parameterization the objective function cannot have a unique optimum. A common way of removing this redundancy is to impose a canonical form.