In this article we show that the asymptotic outcomes of both shallow and deep neural networks such as those used in BloombergGPT to generate economic time series are exactly the Nash equilibria of a non-potential game. We then design and analyze deep neural network algorithms that converge to these equilibria. The methodology is extended to federated deep neural networks between clusters of regional servers and on-device clients. Finally, the variational inequalities behind large language models including encoder-decoder related transformers are established.

翻译：本文表明，浅层和深层神经网络（如BloombergGPT中用于生成经济时间序列的神经网络）的渐近结果恰好是非势博弈的纳什均衡。我们随后设计并分析了收敛于这些均衡的深度神经网络算法。该方法可扩展至区域服务器集群与终端设备客户端之间的联邦深度神经网络。最后，本文建立了包含编码器-解码器相关Transformer在内的大语言模型背后的变分不等式。