φ-Balancing for Mixture-of-Experts Training

Sparse Mixture-of-Experts (MoE) models scale capacity by routing each token to a small subset of experts. However, their routers exhibit a fundamental trade-off: strong load balancing can suppress expert specialization, while aggressive diversity often causes routing collapse. We propose Hi-MoE, a grouped MoE framework that decomposes routing control into two coupled levels: (i) inter-group balancing that enforces fair traffic across expert groups, and (ii) intra-group specialization that promotes complementary expert behaviors while preventing within-group collapse. Our analysis provides a principled explanation of how our hierarchical objectives reshape the router, thereby promoting stable specialization and mitigating collapse. We observe consistent improvements over recent sparse-routing and grouped-MoE baselines across NLP and vision benchmarks, and confirm robustness via scaling studies (model size, expert count) and targeted ablations. In large-scale pre-training on 58B tokens, Hi-MoE-7B achieves a 5.6% perplexity reduction and a 40% improvement in expert balance over OLMoE-7B across diverse evaluation domains.

Load Balancing has emerged as a critical problem in expert-parallel distributed inference of Mixture-of-Experts (MoE) models. As routing distributions are typically skewed across experts, devices hosting lighter-loaded experts must idle to wait for the heaviest during expert computing, leading to inefficiency. Existing load-balancing approaches primarily rely on expert replication or migration within each layer, which introduce additional overhead and limit their flexibility and scalability. To address this problem, we propose EasyBalance, a cross-layer load balancing strategy that requires no modifications to the expert-device mapping, enabling instant adaptability and incurring essentially no additional overhead. Our key insights are that (1) experts of other layers can be viewed as naturally redundant for the current layer, and (2) cross-layer MoE workloads can be jointly executed to mitigate their individual imbalance. Based on these observations, EasyBalance greedily schedules a subset of cross-layer workloads to run at each MoE step and defers the remaining workloads for future balancing opportunities, effectively leveraging cross-layer imbalance mitigation. Extensive experiments across models, tasks, and configurations demonstrate that EasyBalance consistently accelerates distributed MoE inference, reducing GPU idling by mostly over 40%. Code is available at https://github.com/yize-wu/EasyInfra.

Mixture-of-Experts (MoE) models scale by activating only a small subset of experts per token. However, training such models remains challenging because top-$k$ routing is discrete and non-differentiable, requiring gradient estimators for expert selection whose design remains a central open problem. We introduce ProbMoE, a probabilistic routing framework that models expert selection as a distribution over cardinality-constrained expert subsets and formulates routing as probabilistic inference in this discrete subset space. We first propose ProbMoE Exact-$k$ routing, which samples $k$-expert subsets in the forward pass, and the backward pass uses gradients through each expert's exact marginal probability as a tractable surrogate for the true gradient. ProbMoE naturally generalizes to a dynamic-$k$ routing setting, where both training and inference constrain the routing cardinality to the same predefined range, allowing adaptive expert allocation per token. Across benchmarks and model backbones, ProbMoE Exact-$k$ achieves strong performance compared to competitive baselines, with improved expert utilization and routing diversity; ProbMoE Dynamic-$k$ achieves comparable performance with fewer activated experts.