X-MoD: Practical Scaling Laws for Sparse-Depth Routing Beyond Mixture-of-Depths
cs.LG, cs.CL
Submitted: 2026-09-28
Updated: 2026-09-28
License: http://creativecommons.org/licenses/by/4.0/
The gist: Mixture-of-Depths (MoD) enables conditional computation across Transformer depth by routing only a subset of tokens through selected layers, but its original one-sparse--one-dense alternation tightly
Terminology
Abstract
Mixture-of-Depths (MoD) enables conditional computation across Transformer depth by routing only a subset of tokens through selected layers, but its original one-sparse--one-dense alternation tightly couples total capacity to active capacity and limits sparse-depth scaling. We introduce X-MoD, a scalable sparse-depth architecture that decouples token sparsity from anchor stride, allowing total parameter count to grow while keeping active-equivalent capacity nearly fixed. To make deep sparse routing trainable, X-MoD combines dense anchors with variance-scaled layer-wise gating and depth-wise token balancing. To make this regime analyzable and usable, we formulate sparse-depth routing as a conditional architecture-design problem: given compute, context length, and active-equivalent backbone size, how should the routing configuration be chosen? We develop a practical scaling-law framework by fitting X-MoD relative to FLOP-matched dense baselines, yielding an interpretable law that decomposes performance into sparse-capacity gain, sparse-context correction, and anchor-stride interaction. The law predicts validation loss across routing configurations and reveals how context length, model scale, and anchor stride shape sparse-depth performance. We validate the architecture and law through pretraining sweeps, held-out scaling-law prediction, ablations, downstream evaluations, and comparisons with Dense, MoD, and representative MoE baselines.
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