The Elementary Shortest-Path Problem(ESPP) seeks a minimum cost path from s to t that visits each vertex at most once. The presence of negative-cost cycles renders the problem NP-hard. We present a probabilistic method for finding near-optimal ESPP, enabled by an unsupervised graph neural network that jointly learns node value estimates and edge-selection probabilities via a surrogate loss function. The loss provides a high probability certificate of finding near-optimal ESPP solutions by simultaneously reducing negative-cost cycles and embedding the desired algorithmic alignment. At inference time, a decoding algorithm transforms the learned edge probabilities into an elementary path. Experiments on graphs of up to 100 nodes show that the proposed method surpasses both unsupervised baselines and classical heuristics, while exhibiting high performance in cross-size and cross-topology generalization on unseen synthetic graphs.

The drive to develop artificial neural networks that efficiently utilize resources has generated significant interest in bio-inspired Spiking Neural Networks (SNNs). These networks are particularly attractive due to their potential in applications requiring low power and memory. This potential is further enhanced by the ability to perform online local learning, enabling them to adapt to dynamic environments. This requires the model to be adaptive in a self-supervised manner. While self-supervised learning has seen great success in many deep learning domains, its application for online local learning in multi-layer SNNs remains underexplored. In this paper, we introduce the "EchoSpike Predictive Plasticity" (ESPP) learning rule, a pioneering online local learning rule designed to leverage hierarchical temporal dynamics in SNNs through predictive and contrastive coding. We validate the effectiveness of this approach using benchmark datasets, demonstrating that it performs on par with current state-of-the-art supervised learning rules. The temporal and spatial locality of ESPP makes it particularly well-suited for low-cost neuromorphic processors, representing a significant advancement in developing biologically plausible self-supervised learning models for neuromorphic computing at the edge.