The following lemma given in [2] is useful for the proof of Theorem 1. Lemma 1. [2] Given a stochastic matrix H = 0 0 0 h The following propositions are used to prove this theorem. In this case, there is not enough observations to achieve an upper confidence bound using Proposition 2. The randomized UCB for this case has also an exact confidence as illustrated below: Pr{UCB In the second equality, the boundedness of the means of the arms and Proposition 1 were utilized. The steps in this proof closely follows the proof of Theorem 7.1 in [3]. Let us define a'good' event as G We are going to show 1. The next step is to bound the probability of the second set in (3).

The following lemma given in [2] is useful for the proof of Theorem 1. Lemma 1. [2] Given a stochastic matrix H = 0 0 0 h The following propositions are used to prove this theorem. In this case, there is not enough observations to achieve an upper confidence bound using Proposition 2. The randomized UCB for this case has also an exact confidence as illustrated below: Pr{UCB In the second equality, the boundedness of the means of the arms and Proposition 1 were utilized. The steps in this proof closely follows the proof of Theorem 7.1 in [3]. Let us define a'good' event as G We are going to show 1. The next step is to bound the probability of the second set in (3).

An elementary data analysis task is to count the number of unique elements occurring in a dataset.

The widespread use of generative models has created a feedback loop, in which each generation of models is trained on data partially produced by its predecessors. This process has raised concerns about \emph{model collapse}: A critical degradation in performance caused by repeated training on synthetic data. However, different analyses in the literature have reached different conclusions as to the severity of model collapse. As such, it remains unclear how concerning this phenomenon is, and under which assumptions it can be avoided. To address this, we theoretically study model collapse for maximum likelihood estimation (MLE), in a natural setting where synthetic data is gradually added to the original data set. Under standard assumptions (similar to those long used for proving asymptotic consistency and normality of MLE), we establish non-asymptotic bounds showing that collapse can be avoided even as the fraction of real data vanishes. On the other hand, we prove that some assumptions (beyond MLE consistency) are indeed necessary: Without them, model collapse can occur arbitrarily quickly, even when the original data is still present in the training set. To the best of our knowledge, these are the first rigorous examples of iterative generative modeling with accumulating data that rapidly leads to model collapse.

We introduce the Best Group Identification problem in a multi-objective multi-armed bandit setting, where an agent interacts with groups of arms with vector-valued rewards. The performance of a group is determined by an efficiency vector which represents the group's best attainable rewards across different dimensions. The objective is to identify the set of optimal groups in the fixed-confidence setting. We investigate two key formulations: group Pareto set identification, where efficiency vectors of optimal groups are Pareto optimal and linear best group identification, where each reward dimension has a known weight and the optimal group maximizes the weighted sum of its efficiency vector's entries. For both settings, we propose elimination-based algorithms, establish upper bounds on their sample complexity, and derive lower bounds that apply to any correct algorithm. Through numerical experiments, we demonstrate the strong empirical performance of the proposed algorithms.

In theoretical analysis of deep learning, discovering which features of deep learning lead to good performance is an important task. In this paper, using the framework for analyzing the generalization error developed in Suzuki (2018), we derive a fast learning rate for deep neural networks with more general activation functions. In Suzuki (2018), assuming the scale invariance of activation functions, the tight generalization error bound of deep learning was derived. They mention that the scale invariance of the activation function is essential to derive tight error bounds. Whereas the rectified linear unit (ReLU; Nair and Hinton, 2010) satisfies the scale invariance, the other famous activation functions including the sigmoid and the hyperbolic tangent functions, and the exponential linear unit (ELU; Clevert et al., 2016) does not satisfy this condition. The existing analysis indicates a possibility that a deep learning with the non scale invariant activations may have a slower convergence rate of $O(1/\sqrt{n})$ when one with the scale invariant activations can reach a rate faster than $O(1/\sqrt{n})$. In this paper, without the scale invariance of activation functions, we derive the tight generalization error bound which is essentially the same as that of Suzuki (2018). From this result, at least in the framework of Suzuki (2018), it is shown that the scale invariance of the activation functions is not essential to get the fast rate of convergence. Simultaneously, it is also shown that the theoretical framework proposed by Suzuki (2018) can be widely applied for analysis of deep learning with general activation functions.

A standard approach in pattern classification is to estimate the distributions of the label classes, and then to apply the Bayes classifier to the estimates of the distributions in order to classify unlabeled examples. As one might expect, the better our estimates of the label class distributions, the better the resulting classifier will be. In this paper we make this observation precise by identifying risk bounds of a classifier in terms of the quality of the estimates of the label class distributions. We show how PAC learnability relates to estimates of the distributions that have a PAC guarantee on their $L_1$ distance from the true distribution, and we bound the increase in negative log likelihood risk in terms of PAC bounds on the KL-divergence. We give an inefficient but general-purpose smoothing method for converting an estimated distribution that is good under the $L_1$ metric into a distribution that is good under the KL-divergence.