Depth-CoT Tradeoffs in Transformers: A Look at Path Following

Modern transformers have been scaled up in size, data, training-time compute, and more recently test-time compute based on Chain-of-Thought (CoT), but the consequences of trading off these parameters are unknown. We construct explicit tradeoffs between two of these aspects: the number of layers in a transformer and the number of CoT tokens. Our first result is general: with mild assumptions and a single-token prompt, for any integer k, a transformer using soft attention and SwiGLU MLP layers with L layers and n CoT tokens can be converted to a transformer with at most ckL layers and ⌈n/k⌉ CoT tokens, and vice versa, although with an increase to the size of other parameters. We then a multi-token prompt path following task, finding that additional preprocessing from the multi-token prompt changes the depth-CoT relationship in this case. We construct an attention-only network with L layers and 2^(L−2) heads that can traverse 2^(L−1) − 1 edges along a path with a single CoT token, demonstrating at least an exponential relationship between increases in depth and reductions in the number of CoT tokens needed to traverse the entire path. This relationship for the path following task holds empirically as well, as demonstrated by training networks with up to 5 layers, with the tradeoff between depth and the maximum number of edge traversals in a single CoT token exactly matching our theoretical constructions.