In this paradigm of centralised training for decentralised execution, QMIX [25] is a popular Qlearning algorithm with state-of-the-art performance ontheStarCraft Multi-Agent Challenge [26]. QMIX represents the optimal joint action value function using a monotonicmixing function of per-agent utilities.

Methods that train machine learning models on decentralized data offer many advantages over traditional centralized approaches in core aspects such as data ownership, privacy, fault tolerance and scalability [12,33]. Manycurrent efforts inthis direction come under the banner offederated learning [17, 29, 28, 12], where a central entity orchestrates the training and collects aggregate updates from the participating devices.

Failure case 2: Augerino depends on the used parameterisation of invariance. The full GGN approximation in Eq. 5 is inO(NP2C) for computingN matrix-products. The diagonalGGNapproximation would be inO(NPC)and computation of the log-determinant onlyO(P). Computing the log-determinant can be done efficiently inO(D3 +G3)by decomposing the Kronecker factors (Immer et al., 2021a). The last two terms dependent onS come up due to the aggregation ofaugmentation samples inour approximation, that is,the expectations overaandg in the second line of Eq. 15.

Motivated by these limitations,weintroduceahybridapproach,whereLLMcompletionsforagiven taskareusedtolearn atask-specific, context-free surrogatemodel, which isthen used to guide program synthesis.