Hedonic Games (HGs) are a classical framework modeling coalition formation of strategic agents guided by their individual preferences.

This paper examines learning the optimal filtering policy, known as the Kalman gain, for a linear system with unknown noise covariance matrices using noisy output data. The learning problem is formulated as a stochastic policy optimization problem, aiming to minimize the output prediction error. This formulation provides a direct bridge between data-driven optimal control and, its dual, optimal filtering.

We conclude that equality in Lemma 3.1 holds if and only if w

It imposes lower and upper bound constraints on the number of facilities opened for each label, ensuring fair representation of all demographic groups by the selected facilities.

We still need to demonstrate that the properties in P AC-Bayes analysis hold for both the margin operator and the robust margin operator. Then we complete the proof of Lemma 6.1. The proof of Lemma 7.1 and 7.2 is similar. We provide the proof of Lemma 7.2 below. Lemma 7.1 follows the proof of Lemma 7.2 by replacing the robust margin operator by the margin Since the above bound holds for any x in the domain X, we can get the following a.s.: R The second inequality is the tail bound above.

After observing a sample, the analyst may update their estimate of the mean and variance of that group and choose the next group accordingly. The analyst's objective is to dynamically collect samples to minimize the

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