Learning Rate Transfer for Hybrid Transformer-SSM Architectures

We study learning rate (LR) scaling for hybrid architectures combining Transformer and State-Space Model (SSM) blocks, a class adopted by several recent production language models. In particular, we focus on the gap between the theoretical scaling rules derived for SSMs under zero-order-hold (ZOH) discretization at infinite width with proportional state size, and the field-standard practical implementations using simplified-ZOH Mamba at fixed state size. Surprisingly, in this practical regime hybrid architectures achieve a near-zero LR transfer gap across widths 256-2048 and depths 4-32 at sub-billion-parameter scale using only the original $\mu$P prescription, even though SSM operations fall outside its Tensor Programs representability conditions and every parameterization we test fails the standard coordinate-check diagnostic of $\mu$P correctness. We attribute this to a two-condition decomposition of LR transfer in hybrid architectures: a global update-to-weight invariance, enforced by $\mu$P's initialization and LR scaling; and a local per-component balance, provided by AdamW's $g/\sqrt{v}$ per-parameter normalization. Our observations show that the optimal LR is invariant to width up to $8\times$, as well as to depth, sequence length, batch size, and Transformer-to-SSM ratio, and transfers to Nemotron-H, a production hybrid outside our custom architecture set. We hope these findings fill the gap between theoretical scaling rules and practical hybrid implementations, and stimulate further research toward bridging it.