Since the attention mechanism is rarely used within the GP literature, we provide a brief review of the topic in this section. Below we follow the description of attention as given by Vaswani et al. [8], including extensions to self-attention and multi-head self-attention. The dot-product attention mechanism [8] takes as input a set of queries, keys and values. The queries and keys have dimension Dz and the values have dimension Dv which may differ from Dz. The operation of dot-product attention then generates weights from the queries and keys which are used to produce a linear mapping of the input values.

In many applications, we desire neural networks to exhibit invariance or equivari-ance to certain groups due to symmetries inherent in the data. Recently, frame-averaging methods emerged to be a unified framework for attaining symmetries efficiently by averaging over input-dependent subsets of the group, i.e., frames. What we currently lack is a principled understanding of the design of frames.

Since the attention mechanism is rarely used within the GP literature, we provide a brief review of thetopicinthissection.

The Centralized Training with Decentralized Execution (CTDE) paradigm has gained significant attention in multi-agent reinforcement learning (MARL) and is the foundation of many recent algorithms. However, decentralized policies operate under partial observability and often yield suboptimal performance compared to centralized policies, while fully centralized approaches typically face scalability challenges as the number of agents increases. We propose Centralized Permutation Equivariant (CPE) learning, a centralized training and execution framework that employs a fully centralized policy to overcome these limitations. Our approach leverages a novel permutation equivariant architecture, Global-Local Permutation Equivariant (GLPE) networks, that is lightweight, scalable, and easy to implement. Experiments show that CPE integrates seamlessly with both value decomposition and actor-critic methods, substantially improving the performance of standard CTDE algorithms across cooperative benchmarks including MPE, SMAC, and RWARE, and matching the performance of state-of-the-art RWARE implementations.