Neural scaling laws--power-law relationships between generalization errors and characteristics of deep learning models--are vital tools for developing reliable models while managing limited resources. Although the success of large language models highlights the importance of these laws, their application to deep regression models remains largely unexplored. Here, we empirically investigate neural scaling laws in deep regression using a parameter estimation model for twisted van der Waals magnets. We observe power-law relationships between the loss and both training dataset size and model capacity across a wide range of values, employing various architectures--including fully connected networks, residual networks, and vision transformers. Furthermore, the scaling exponents governing these relationships range from 1 to 2, with specific values depending on the regressed parameters and model details. The consistent scaling behaviors and their large scaling exponents suggest that the performance of deep regression models can improve substantially with increasing data size.

Self-supervised learning~(SSL) is essential to obtain foundation models in NLP and CV domains via effectively leveraging knowledge in large-scale unlabeled data. The reason for its success is that a suitable SSL design can help the model to follow the neural scaling law, i.e., the performance consistently improves with increasing model and dataset sizes. However, it remains a mystery whether existing SSL in the graph domain can follow the scaling behavior toward building Graph Foundation Models~(GFMs) with large-scale pre-training. In this study, we examine whether existing graph SSL techniques can follow the neural scaling behavior with the potential to serve as the essential component for GFMs. Our benchmark includes comprehensive SSL technique implementations with analysis conducted on both the conventional SSL setting and many new settings adopted in other domains. Surprisingly, despite the SSL loss continuously decreasing, no existing graph SSL techniques follow the neural scaling behavior on the downstream performance. The model performance only merely fluctuates on different data scales and model scales. Instead of the scales, the key factors influencing the performance are the choices of model architecture and pretext task design. This paper examines existing SSL techniques for the feasibility of Graph SSL techniques in developing GFMs and opens a new direction for graph SSL design with the new evaluation prototype. Our code implementation is available online to ease reproducibility on https://github.com/GraphSSLScaling/GraphSSLScaling.

HDA* is a simple, parallelization of A* where work is asynchronously distributed among the nodes by a global hash function. Using up to 1024 cores on a large distributed memory cluster, we evaluate HDA* for a domain-independent planner as well an application-specific 24-puzzle solver. We show that HDA* scales fairly well on a large cluster using up to 1024 cores. Our analysis of the scaling behavior shows that on a cluster of multicore nodes, using only a subset of the available cores and leaving some cores idle can, surprisingly, lead to better results.