Pre-training Large Language Models (LLMs) from scratch at larger scales yields remarkable performance but incurs substantially high training costs. Depth up-scaling provides an efficient alternative by inserting new layers into a pre-trained LLM, avoiding training from scratch. However, most existing methods copying or averaging base layers for new layer, which misalign functionally corresponding neurons, leading to neuron permutation mismatch that harms performance. To address this issue, we propose Optimal Transport Depth Up-Scaling (OT-DUS), which leverages Optimal Transport (OT) theory to align and fuse functionally corresponding neurons module by module in adjacent base layers for new layer construction. OT-DUS achieves better overall performance in both general and specialized domains than existing methods for continual pre-training and supervised fine-tuning across different model sizes and model families. Our analysis of insertion strategies further finds that inserting new layers at higher positions yields not only stronger performance but also improved training efficiency.

Deduction, induction, and abduction are fundamental reasoning paradigms, core for human logical thinking. Although improving Large Language Model (LLM) reasoning has attracted significant research efforts, the extent to which the fundamental paradigms induce generalization has yet to be systematically explored. In this study, we shed light on how the interplay between these core paradigms influences LLMs’ reasoning behavior. To this end, we first collect a new dataset of reasoning trajectories from symbolic tasks, each targeting one of the three fundamental paradigms, to abstract from concrete world knowledge. Then, we investigate effective ways for inducing these skills into LLMs.We experiment with a battery of methods including simple fine-tuning, and more complex approaches to increase model depth, or transform a dense model to a mixture-of-experts. We comprehensively evaluate induced models on realistic out-of-domain tasks, that are entirely formulated in natural language and contain real-world knowledge. Our results reveal that our approach yields strong generalizability with substantial performance gains (up to 14.60) across realistic tasks.