# X-MoD: Practical Scaling Laws for Sparse-Depth Routing Beyond Mixture-of-Depths

arXiv: https://arxiv.org/abs/2609.34212 · HTML: https://mlsanity.com/paper/2609.34212/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Expert routing strategies, one of 12 directions in Mixture-of-experts.
- Builds on 40 of its 46 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Placed in

- Mixture-of-experts / [Expert routing strategies](https://mlsanity.com/moe/expert-routing-strategies/)

## Builds on (most cited first, 30 of 40)

- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Think you have Solved Question Answering? Try ARC, the AI2 Reasoning Challenge](https://mlsanity.com/paper/1803.05457/) (2018)
- [HellaSwag: Can a Machine Really Finish Your Sentence?](https://mlsanity.com/paper/1905.07830/) (2019)
- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Gemini 1.5: Unlocking multimodal understanding across millions of tokens of context](https://mlsanity.com/paper/2403.05530/) (2024)
- [Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity](https://mlsanity.com/paper/2101.03961/) (2021)
- [Training Compute-Optimal Large Language Models](https://mlsanity.com/paper/2203.15556/) (2022)
- [PIQA: Reasoning about Physical Commonsense in Natural Language](https://mlsanity.com/paper/1911.11641/) (2019)
- [Efficient Streaming Language Models with Attention Sinks](https://mlsanity.com/paper/2309.17453/) (2023)
- [BoolQ: Exploring the Surprising Difficulty of Natural Yes/No Questions](https://mlsanity.com/paper/1905.10044/) (2019)
- [GShard: Scaling Giant Models with Conditional Computation and Automatic Sharding](https://mlsanity.com/paper/2006.16668/) (2020)
- [GQA: Training Generalized Multi-Query Transformer Models from Multi-Head Checkpoints](https://mlsanity.com/paper/2305.13245/) (2023)
- [The LAMBADA dataset: Word prediction requiring a broad discourse context](https://mlsanity.com/paper/1606.06031/) (2016)
- [GLaM: Efficient Scaling of Language Models with Mixture-of-Experts](https://mlsanity.com/paper/2112.06905/) (2021)
- [DeepSeekMoE: Towards Ultimate Expert Specialization in Mixture-of-Experts Language Models](https://mlsanity.com/paper/2401.06066/) (2024)
- [H2O: Heavy-Hitter Oracle for Efficient Generative Inference of Large Language Models](https://mlsanity.com/paper/2306.14048/) (2023)
- [Model Tells You What to Discard: Adaptive KV Cache Compression for LLMs](https://mlsanity.com/paper/2310.01801/) (2023)
- [Native Sparse Attention: Hardware-Aligned and Natively Trainable Sparse Attention](https://mlsanity.com/paper/2502.11089/) (2025)
- [LayerSkip: Enabling Early Exit Inference and Self-Speculative Decoding](https://mlsanity.com/paper/2404.16710/) (2024)
- [Effective Long-Context Scaling of Foundation Models](https://mlsanity.com/paper/2309.16039/) (2023)
- [Mixture-of-Depths: Dynamically allocating compute in transformer-based language models](https://mlsanity.com/paper/2404.02258/) (2024)
- [Hierarchical Reasoning Model](https://mlsanity.com/paper/2506.21734/) (2025)
- [Scaling Laws for Fine-Grained Mixture of Experts](https://mlsanity.com/paper/2402.07871/) (2024)
- [Mixture-of-Recursions: Learning Dynamic Recursive Depths for Adaptive Token-Level Computation](https://mlsanity.com/paper/2507.10524/) (2025)
- [Not All Layers of LLMs Are Necessary During Inference](https://mlsanity.com/paper/2403.02181/) (2024)
- [MoLe-VLA: Dynamic Layer-skipping Vision Language Action Model via Mixture-of-Layers for Efficient Robot Manipulation](https://mlsanity.com/paper/2503.20384/) (2025)
- [MoMa: Efficient Early-Fusion Pre-training with Mixture of Modality-Aware Experts](https://mlsanity.com/paper/2407.21770/) (2024)
- [LoopFormer: Elastic-Depth Looped Transformers for Latent Reasoning via Shortcut Modulation](https://mlsanity.com/paper/2602.11451/) (2026)
- [Inner Thinking Transformer: Leveraging Dynamic Depth Scaling to Foster Adaptive Internal Thinking](https://mlsanity.com/paper/2502.13842/) (2025)
- [MoE++: Accelerating Mixture-of-Experts Methods with Zero-Computation Experts](https://mlsanity.com/paper/2410.07348/) (2024)

## Abstract (arXiv)

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.