On the number of modes of Gaussian kernel density estimators

We consider the Gaussian kernel density estimator with bandwidth $\beta^{-\frac12}$ of $n$ iid Gaussian samples. Using the Kac-Rice formula and an Edgeworth expansion, we prove that the expected number of modes on the real line scales as $\Theta(\sqrt{\beta\log\beta})$ as $\beta,n\to\infty$ provided $n^c\lesssim \beta\lesssim n^{2-c}$ for some constant $c>0$. An impetus behind this investigation is to determine the number of clusters to which Transformers are drawn in a metastable state.

ProAgent: Building Proactive Cooperative AI with Large Language Models

Building AIs with adaptive behaviors in human-AI cooperation stands as a pivotal focus in AGI research. Current methods for developing cooperative agents predominantly rely on learning-based methods, where policy generalization heavily hinges on past interactions with specific teammates. These approaches constrain the agent's capacity to recalibrate its strategy when confronted with novel teammates. We propose \textbf{ProAgent}, a novel framework that harnesses large language models (LLMs) to fashion a \textit{pro}active \textit{agent} empowered with the ability to anticipate teammates' forthcoming decisions and formulate enhanced plans for itself. ProAgent excels at cooperative reasoning with the capacity to dynamically adapt its behavior to enhance collaborative efforts with teammates. Moreover, the ProAgent framework exhibits a high degree of modularity and interpretability, facilitating seamless integration to address a wide array of coordination scenarios. Experimental evaluations conducted within the framework of \textit{Overcook-AI} unveil the remarkable performance superiority of ProAgent, outperforming five methods based on self-play and population-based training in cooperation with AI agents. Further, when cooperating with human proxy models, its performance exhibits an average improvement exceeding 10\% compared to the current state-of-the-art, COLE. The advancement was consistently observed across diverse scenarios involving interactions with both AI agents of varying characteristics and human counterparts. These findings inspire future research for human-robot collaborations. For a hands-on demonstration, please visit \url{https://pku-proagent.github.io}.

Randomly Initialized One-Layer Neural Networks Make Data Linearly Separable

Recently, neural networks have been shown to perform exceptionally well in transforming two arbitrary sets into two linearly separable sets. Doing this with a randomly initialized neural network is of immense interest because the associated computation is cheaper than using fully trained networks. In this paper, we show that, with sufficient width, a randomly initialized one-layer neural network transforms two sets into two linearly separable sets with high probability. Furthermore, we provide explicit bounds on the required width of the neural network for this to occur. Our first bound is exponential in the input dimension and polynomial in all other parameters, while our second bound is independent of the input dimension, thereby overcoming the curse of dimensionality. We also perform an experimental study comparing the separation capacity of randomly initialized one-layer and two-layer neural networks. With correctly chosen biases, our study shows for low-dimensional data, the two-layer neural network outperforms the one-layer network. However, the opposite is observed for higher-dimensional data.