ICML 2026 Papers

Sparse Mixture-of-Experts (MoE) language models enable conditional computation but face deployment challenges due to the memory wall: while few experts are activated per token, the entire model must reside in memory. Existing expert pruning methods primarily rely on independent ranking, failing to account for the complex inter-dependencies and redundancies between experts. In this paper, we formulate post-training MoE pruning as a reconstruction-driven subset selection problem, aiming to minimize layer-output distortion under a cardinality constraint. We introduce SCHUR-A*, an algorithm that leverages A* search to achieve globally optimal expert selection within each layer. To maintain computational tractability, we derive a novel, admissible heuristic upper bound using a Schur-complement-based relaxation of the reconstruction objective. This tight bound allows for aggressive pruning of the search space while mathematically guaranteeing optimality. Furthermore, we propose an automated strategy to balance fidelity and memory reduction across heterogeneous layers via knee-point detection. Extensive experiments on Qwen3-30B-A3B demonstrate that SCHUR-A* significantly outperforms greedy and ranking-based baselines, maintaining comparable performance even under aggressive pruning ratios.

Scaling up visual autoregressive models improves generation quality but incurs substantial inference costs. Mixture-of-Experts (MoE) architectures mitigate this issue through sparse activation and have proven effective in large language models. However, training MoE models from scratch remains prohibitively expensive, and dense-to-MoE conversion for visual autoregressive models is still underexplored. To enable *low-cost and high-quality dense-to-MoE conversion*, we propose **Prism-MoE**, an efficient framework for transforming pretrained dense visual autoregressive models into sparse MoE models. Prism-MoE consists of two key components. First, we introduce trajectory-consistent Initialization, which formulates expert initialization as a principled decomposition problem and preserves the generation trajectory of pretrained models. Second, we propose a confidence-adaptive sparse fine-tuning framework that aligns expert specialization with the information density of visual tokens via confidence-aware routing supervision. Experiments show that Prism-MoE achieves dense-to-MoE conversion with less than **10\%** of the standard training budget, while maintaining generation quality comparable to dense baselines with only **37.5\%** active parameters.

Upcycling pre-trained dense models into sparse Mixture-of-Experts (MoEs) efficiently increases model capacity but often suffers from poor expert specialization due to naive weight replication. We introduce Dirichlet-Prior Shaping Loss (DPSL), a novel router regularization technique that directly shapes routing probability distributions by matching expert assignments to a target Dirichlet prior resulting in enhanced expert specialization. DPSL enables encoding of inductive biases such as encouraging experts to focus on specific modalities or tasks, without requiring manual intervention. DPSL is a general tool applicable to any module that outputs categorical probability distributions, extending its utility beyond MoE training. Experiments on upcycled MoE vision-language models show that DPSL consistently outperforms upcycling strategies and regularization techniques across standard vision-language benchmarks, addressing the critical issue of poor specialization and fostering higher-performing models.

Sparse Mixture-of-Experts (SMoE) architectures are now widely deployed in state-of-the-art language and vision models, where conditional routing allows scaling to very large networks. However, this very Top-$k$ expert selection that enables conditional routing also renders the SMoE map inherently discontinuous. In the vicinity of these discontinuity surfaces, even inputs that are arbitrarily close may activate substantially different sets of experts resulting in significantly different outputs. In this work we give a rigorous geometric and stochastic analysis of these discontinuities. We first classify them by order, determined by the number of tied experts at a switching event. Using measure-theoretic slicing arguments, we establish asymptotic volume estimates for the thickened discontinuity surfaces, showing that lower-order discontinuity sets dominate, whereas higher-order ones occupy a vanishingly small relative volume. Next, modeling random perturbations in the input space via a diffusion process, we prove that the path eventually encounter a discontinuity, and moreover that the first hit almost surely occurs on an order-1 discontinuity with explicit finite-time probability bounds. We further derive occupation-time bounds that quantify the duration the random path spend in the neighborhoods of each discontinuity order. These theoretical results imply that inputs are more likely to lie near lower order discontinuities. Motivated by this insight, we propose a simple smoothing mechanism that can be directly applied to existing SMoEs, softly incorporating experts near discontinuities; our analysis guarantees that the added computational overhead remains small while providing localized smoothing near discontinuities, and experiments across language and vision tasks show that smoothing not only enforces continuity of the SMoE map but also enhances empirical performance.

Continual learning for visual question answering (VQA) is typically implemented by training one expert per task and routing each query using task-ID supervision. Yet continual VQA tasks overlap substantially: on the VQA-v2 task stream, a non-native expert outperforms the task’s own expert on $49.9\%$ of queries, so hard routing both wastes transferable knowledge and can be confidently wrong when mismatched. We propose a calibrated Bayesian mixture-of-experts that trains parameter-efficient per-task adapters, learns routing by directly maximizing expected VQA utility, and marginalizes expert identity at inference via Bayesian aggregation in a unified answer space; an entropy penalty prevents the utility objective from collapsing to one-hot routing, enabling evidence pooling across plausible experts. We reach $64.16$ accuracy with $0.63$ forgetting on VQA-v2 CL-LS ($+5.74\%$ accuracy, $-2.99$ forgetting vs. the strongest prior method), $78.81$ with $0.40$ forgetting on TDIUC CL-LS ($+3.10$, $-1.74$), and $83.41$ with $3.21$ forgetting on TDIUC CL-VS ($+1.58$, $-0.82$). Calibration also improves on VQA-v2, reducing ECE from $0.15$ to $0.07$.

Modern sparse language models typically achieve sparsity through Mixture-of-Experts (MoE) layers, which dynamically route tokens to dense MLP "experts." However, dynamic hard routing has a number of drawbacks, such as potentially poor hardware efficiency and needing auxiliary losses for stable training. In contrast, the tokenizer embedding table, which is natively sparse, largely avoids these issues by selecting a single embedding per token at the cost of not having contextual information. In this work, we introduce the Large Lookup Layer (L3), which generalizes embedding tables to model decoder layers as a means of further scaling sparsity. L3 layers use static token-based routing to aggregate a *set* of learned embeddings per token in a *context-dependent way*, allowing the model to efficiently balance memory and compute by caching information in embeddings. L3 has two main components: (1) a systems-friendly architecture that allows for fast training and CPU-offloaded inference with no overhead, and (2) an information-theoretic embedding allocation algorithm that effectively balances speed and quality. We empirically test L3 by training transformers with up to 2.6B active parameters and find that L3 strongly outperforms both dense models and iso-sparse MoEs in both language modeling and downstream tasks.

Modern deep learning is increasingly characterized by the use of open-weight foundation models that can be fine-tuned on specialized datasets. This has led to a proliferation of expert models and adapters, often shared via platforms like HuggingFace and AdapterHub. Model merging has recently emerged as an effective way to leverage these existing resources, enabling the composition of capabilities from different model checkpoints. A natural pipeline has thus formed to harness the benefits of transfer learning and amortize sunk training costs: models are pre-trained on general data, fine-tuned on specific tasks, and then multiple checkpoints are merged to obtain a more capable model. A prevailing assumption is that improvements at one stage of this pipeline propagate downstream, leading to gains at subsequent steps. In this work, we challenge that assumption by examining how expert fine-tuning affects model merging. We show that long fine-tuning of experts that optimizes for their individual performance leads to degraded merging performance across vision and language modalities, multiple model scales, and both fully fine-tuned and LoRA-adapted models. We trace this degradation to the memorization of a small set of difficult examples that dominate late fine-tuning steps. This causes negative parameter interference and encodes knowledge that is forgotten during merging. Finally, we demonstrate that task-dependent aggressive early stopping strategies can significantly improve model merging performance.

Pretraining produces a learned parameter vector that is typically treated as a starting point for further iterative adaptation. In this work, we instead view the outcome of pretraining as a distribution over parameter vectors, whose support already contains task-specific experts. We show that in smaller or insufficiently trained models such expert solutions occupy a negligible fraction of the volume of this distribution, making their discovery reliant on structured optimization methods such as gradient descent. In contrast, in large, well-pretrained models the density of task-experts increases dramatically, so that diverse specialists populate a substantial fraction of the neighborhood around the pretrained weights. Motivated by this perspective, we explore a simple, fully parallel post-training method that samples $N$ parameter vectors at random, selects the top $K$, and ensembles them via majority vote to combine complementary expertise. Despite its simplicity, this approach is competitive with standard post-training methods such as PPO, GRPO, and ES for contemporary large-scale models.

Mixture-of-Experts architectures have become the standard for efficient LLM scaling, typically employing expert parallelism to distribute experts across devices. However, the absence of explicit load balancing constraints during inference allows adversarial inputs to trigger severe routing concentration. We demonstrate that out-of-distribution prompts can manipulate the routing mechanism such that all tokens are routed to the same set of top-k experts, which creates computational bottlenecks on certain devices while forcing others to idle. This converts an efficiency mechanism into a denial-of-service attack vector, leading to violations of service-level agreements for time-to-first-token (TTFT). We propose RepetitionCurse, a black-box strategy to exploit this vulnerability. By identifying a universal flaw in MoE router behavior, RepetitionCurse constructs attack prompts using simple repetitive token patterns in a model-agnostic manner. On widely deployed MoE models hosted on 8-GPU clusters, our method increases TTFT by 20% to 148%, significantly degrading service quality.

Large language models usually face continuous domain shifts during testing, which degrade performance on unseen shifting domains. So, researchers propose continual test-time adaptation (CTTA) to adapt to evolving testing domains while preserving knowledge of previous domains, making adaptability-stability (A-S) balance. Existing CTTA methods are constrained by dense base models that encode knowledge from all domains into a global model, hardly achieving the A-S balance. We observe that the model sparsity of mixture-of-experts (MoE) models is better for achieving A–S balance than dense models. In CTTA, however, MoE faces difficulty in (1) correctly routing samples from unseen shifting domains and (2) capturing domain-level shifts. In this paper, we propose test-time mixture-of-experts routing (TiME) via asymmetric co-optimal transport (As-COOT): we model MoE routing in CTTA as a test-time allocation problem via COOT. To ensure reliable routing, we propose a semantic space alignment to align sample-expert distributions via bidirectional contrastive learning. To address COOT’s limitations in CTTA, we propose As-COOT, relaxing sample-side constraints while enforcing expert-side constraints to ensure noise robustness and balance expert load. Experiments show TiME outperforms baselines.

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. Code is available at: https://github.com/HengHugoZhao/ProbMoE.git

Optimizing data mixtures is is essential for unlocking the full potential of of large language models (LLMs), yet identifying the optimal composition remains computationally prohibitive due to reliance on heuristic trials or expensive proxy training. To address this, we introduce MergeMix, a novel appraoch that efficiently determines optimal data mixing ratios by repurposing model merging weights as a high-fidelity, low-cost performance proxy. By training domain-specific experts on minimal tokens and optimizing their merging weights against downstream benchmarks, MergeMix effectively optimizes the performance of data mixtures without incurring the cost of full-scale training. Extensive experiments on models with 8B and 16B parameters validate that MergeMix achieves performance comparable to or surpassing exhaustive manual tuning while drastically reducing search costs. Furthermore, MergeMix exhibits high rank consistency (Spearman $\rho > 0.9$) and strong cross-scale transferability, offering a scalable, automated solution for data mixture optimization.

This paper introduces Mixtures of Geodesic Factor Analyzers (MGFA) on Riemannian homogeneous spaces. MGFA uses a geodesic factor model within each mixture component, providing greater expressiveness than mixtures of Riemannian radial distributions and enabling clustering of manifold-valued data with anisotropic subpopulations. We establish root-$n$ consistency for the MGFA maximum likelihood estimator (MLE), thereby filling a theoretical gap for mixtures of Riemannian radial distributions as a special case. We also propose an iterative estimation algorithm and implement it on spheres, shape spaces, and hyperbolic spaces. Numerical experiments show that MGFA substantially outperforms competing methods in well-specified regimes while remaining robust under model misspecification. Finally, case studies on corpus callosum and left hippocampus shape datasets demonstrate MGFA’s effectiveness for both 2D contour and 3D shape analysis.

We introduce Mahalanobis-Pruned Mixture-of-Experts (MP-MoE), a novel routing framework that approaches expert selection from the perspective of ensemble pruning. Existing Mixture-of-Experts (MoE) routing strategies often suffer from representation collapse due to greedy top-k selection mechanisms or rely on complex auxiliary regularization terms that may compromise model performance. To address these issues, we formulate routing as a diversity-aware subset selection problem and optimize a Mahalanobis-distance-based objective that explicitly enhances expert diversity. Specifically, we demonstrate that the expert co-occurrence matrix effectively captures inter-expert correlations, allowing us to efficiently model the covariance structure required for distance computation without accessing expert parameters. Furthermore, we devise a greedy strategy for the routing mechanism, backed by theoretical approximation guarantees, rendering it a plug-and-play module with negligible overhead. MP-MoE increases wall-clock training time by approximately 3\%, while incurring no additional latency at inference time. Extensive experiments demonstrate that during the pre-training of the large language model, our method consistently outperforms the baseline by 1-3 percentage points across a broad range of benchmarks.

Mixture-of-Experts (MoE) architectures are evolving towards finer granularity to improve parameter efficiency. However, existing MoE designs face an inherent trade-off between the granularity of expert specialization and hardware execution efficiency. In this paper, we propose OmniMoE, a system-algorithm co-designed MoE framework that pushes granularity to the extreme with vector-level Atomic Experts, orchestrating their routing and execution at scale within a single MoE layer, while retaining a shared dense MLP for general-purpose processing. While this atomic design maximizes capacity, it poses severe challenges for routing complexity and memory access. To address these, OmniMoE adopts a system-algorithm co-design: (i) a Cartesian Product Router that decomposes the massive index space to reduce routing complexity from $O(N)$ to $O(\sqrt{N})$; and (ii) Expert-Centric Scheduling that inverts the execution order to turn scattered, memory-bound lookups into efficient dense matrix operations. Validated on seven benchmarks, OmniMoE (with 1.7B active parameters) achieves 50.9\% zero-shot accuracy across seven benchmarks, outperforming coarse-grained (e.g., DeepSeekMoE) and fine-grained (e.g., PEER) baselines. Crucially, OmniMoE reduces inference latency from 73ms to 6.7ms (a 10.9$\times$ speedup) compared to PEER, demonstrating that massive-scale fine-grained MoE can be fast and accurate.

Mixture-of-Experts Large Language Models (MoE-LLMs) achieve strong performance but incur substantial memory overhead due to massive expert parameters. Mixed-precision quantization mitigates this cost by allocating expert-wise bit-widths based on their importance, approaching the accuracy-memory Pareto frontier and enabling extreme low-bit quantization. However, existing methods rely on layer-wise importance estimation and overlook router shifts induced by quantization, resulting in suboptimal allocation and routing. In this work, we propose Global Expert-level Mixed-precision Quantization (GEMQ) to overcome these limitations via (1) a global linear-programming formulation that captures model-wide expert importance based on quantization error analysis, and (2) efficient router fine-tuning to adapt routing to quantized experts. These components are integrated into a progressive quantization framework that iteratively refines importance estimation and allocation. Experiments demonstrate that GEMQ significantly reduces memory and accelerates inference with minimal accuracy degradation.

We study the Awakening Crowd of Experts (ACE) problem, an online learning problem where the set of experts available to the learner grows at each round. ACE is a special case of the well-known sleeping experts problem (Kleinberg et al., 2010), where the number of experts is huge $(K=T)$. Existing results on sleeping experts preclude any learner from achieving a sublinear regret when the number of available experts is linear in $T$. Inspired by real-world applications, such as Q\&A platforms and social proof marketing, we thus focus on the awakening version of the sleeping experts problem, where a new expert arrives at every round and never leaves. We show that in the stochastic version of ACE, it is possible to obtain regret $\tilde{\mathcal{O}}(T^{2/3})$ using an unusual pessimism in the face of the uncertainty principle. Moreover, we characterize the dependence of the regret on the stability of an optimal strategy. For both results, we present matching lower bounds. Surprisingly, the adversarial version of ACE is sensibly harder. In particular, we provide a lower bound precluding sublinear $\alpha$-regret when the competitive ratio is constant. We provide an algorithm to face this crucial trade-off between competitive ratio and regret, and bound its $\alpha$-regret, almost matching the aforementioned lower bound. As a corollary, we get a $\tilde{\mathcal{O}}(\log(\log(T))$ competitive ratio when an optimal strategy enjoys a reward linear in $T$.

Mixture-of-Experts (MoE) architectures provide strong scalability for large language models, but their large expert parameter footprint poses challenges for efficient deployment. Expert pruning is widely used to reduce model size and inference cost; however, existing approaches are token-agnostic, treating all tokens equally when estimating expert importance. This uniform treatment dilutes the contributions of informative tokens and leads to suboptimal pruning decisions. To address this fundamental limitation, we propose **Step** (**S**elective **T**oken-guided **E**xpert **P**runing), a token-aware framework that rethinks expert pruning from the perspective of selective token guidance. By incorporating loss-aware expert evaluation and a lightweight knowledge-preserving mechanism, **Step** reduces information loss while removing redundant experts. Extensive experiments across different MoE architectures and model scales demonstrate the effectiveness of **Step**. On the 30B Qwen3 MoE model with 50\% expert sparsity, **Step** achieves nearly a 50\% reduction in memory usage with minimal performance degradation, delivers a 1.5$\times$ throughput improvement, and completes the entire pruning process within 10 minutes.

Lloyd's k-means algorithm is one of the most widely used clustering methods. We prove that in high-dimensional, high-noise settings, the algorithm exhibits catastrophic failure: with high probability, essentially every partition of the data is a fixed point. Consequently, Lloyd's algorithm simply returns its initial partition — even when the underlying clusters are trivially recoverable by other methods. In contrast, we prove that Hartigan's k-means algorithm does not exhibit this pathology. Our results show the stark difference between these algorithms and offer a theoretical explanation for the empirical difficulties often observed with k-means in high dimensions.