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SubscribeLocMoE: A Low-overhead MoE for Large Language Model Training
The Mixtures-of-Experts (MoE) model is a widespread distributed and integrated learning method for large language models (LLM), which is favored due to its ability to sparsify and expand models efficiently. However, the performance of MoE is limited by load imbalance and high latency of All-To-All communication, along with relatively redundant computation owing to large expert capacity. Load imbalance may result from existing routing policies that consistently tend to select certain experts. The frequent inter-node communication in the All-To-All procedure also significantly prolongs the training time. To alleviate the above performance problems, we propose a novel routing strategy that combines load balance and locality by converting partial inter-node communication to that of intra-node. Notably, we elucidate that there is a minimum threshold for expert capacity, calculated through the maximal angular deviation between the gating weights of the experts and the assigned tokens. We port these modifications on the PanGu-Sigma model based on the MindSpore framework with multi-level routing and conduct experiments on Ascend clusters. The experiment results demonstrate that the proposed LocMoE reduces training time per epoch by 12.68% to 22.24% compared to classical routers, such as hash router and switch router, without impacting the model accuracy.
Mixture-of-Experts with Expert Choice Routing
Sparsely-activated Mixture-of-experts (MoE) models allow the number of parameters to greatly increase while keeping the amount of computation for a given token or a given sample unchanged. However, a poor expert routing strategy (e.g. one resulting in load imbalance) can cause certain experts to be under-trained, leading to an expert being under or over-specialized. Prior work allocates a fixed number of experts to each token using a top-k function regardless of the relative importance of different tokens. To address this, we propose a heterogeneous mixture-of-experts employing an expert choice method. Instead of letting tokens select the top-k experts, we have experts selecting the top-k tokens. As a result, each token can be routed to a variable number of experts and each expert can have a fixed bucket size. We systematically study pre-training speedups using the same computational resources of the Switch Transformer top-1 and GShard top-2 gating of prior work and find that our method improves training convergence time by more than 2x. For the same computational cost, our method demonstrates higher performance in fine-tuning 11 selected tasks in the GLUE and SuperGLUE benchmarks. For a smaller activation cost, our method outperforms the T5 dense model in 7 out of the 11 tasks.
M6-T: Exploring Sparse Expert Models and Beyond
Mixture-of-Experts (MoE) models can achieve promising results with outrageous large amount of parameters but constant computation cost, and thus it has become a trend in model scaling. Still it is a mystery how MoE layers bring quality gains by leveraging the parameters with sparse activation. In this work, we investigate several key factors in sparse expert models. We observe that load imbalance may not be a significant problem affecting model quality, contrary to the perspectives of recent studies, while the number of sparsely activated experts k and expert capacity C in top-k routing can significantly make a difference in this context. Furthermore, we take a step forward to propose a simple method called expert prototyping that splits experts into different prototypes and applies k top-1 routing. This strategy improves the model quality but maintains constant computational costs, and our further exploration on extremely large-scale models reflects that it is more effective in training larger models. We push the model scale to over 1 trillion parameters and implement it on solely 480 NVIDIA V100-32GB GPUs, in comparison with the recent SOTAs on 2048 TPU cores. The proposed giant model achieves substantial speedup in convergence over the same-size baseline.
MoE++: Accelerating Mixture-of-Experts Methods with Zero-Computation Experts
In this work, we aim to simultaneously enhance the effectiveness and efficiency of Mixture-of-Experts (MoE) methods. To achieve this, we propose MoE++, a general and heterogeneous MoE framework that integrates both Feed-Forward Network~(FFN) and zero-computation experts. Specifically, we introduce three types of zero-computation experts: the zero expert, copy expert, and constant expert, which correspond to discard, skip, and replace operations, respectively. This design offers three key advantages: (i) Low Computing Overhead: Unlike the uniform mixing mechanism for all tokens within vanilla MoE, MoE++ allows each token to engage with a dynamic number of FFNs, be adjusted by constant vectors, or even skip the MoE layer entirely. (ii) High Performance: By enabling simple tokens to utilize fewer FFN experts, MoE++ allows more experts to focus on challenging tokens, thereby unlocking greater performance potential than vanilla MoE. (iii) Deployment Friendly: Given that zero-computation experts have negligible parameters, we can deploy all zero-computation experts on each GPU, eliminating the significant communication overhead and expert load imbalance associated with FFN experts distributed across different GPUs. Moreover, we leverage gating residuals, enabling each token to consider the pathway taken in the previous layer when selecting the appropriate experts. Extensive experimental results demonstrate that MoE++ achieves better performance while delivering 1.1-2.1x expert forward throughput compared to a vanilla MoE model of the same size, which lays a solid foundation for developing advanced and efficient MoE-related models.
FuseMax: Leveraging Extended Einsums to Optimize Attention Accelerator Design
Attention for transformers is a critical workload that has recently received significant "attention" as a target for custom acceleration. Yet, while prior work succeeds in reducing attention's memory-bandwidth requirements, it creates load imbalance between attention operators (resulting in severe compute under-utilization) and requires on-chip memory that scales with sequence length (which is expected to grow over time). This paper ameliorates these issues, enabling attention with nearly 100% compute utilization, no off-chip memory traffic bottlenecks, and on-chip buffer size requirements that are independent of sequence length. The main conceptual contribution is to use a recently proposed abstraction -- the cascade of Einsums -- to describe, formalize and taxonomize the space of attention algorithms that appear in the literature. In particular, we show how Einsum cascades can be used to infer non-trivial lower bounds on the number of passes a kernel must take through its input data, which has implications for either required on-chip buffer capacity or memory traffic. We show how this notion can be used to meaningfully divide the space of attention algorithms into several categories and use these categories to inform our design process. Based on the above characterization, we propose FuseMax -- a novel mapping of attention onto a spatial array-style architecture. On attention, in an iso-area comparison, FuseMax achieves an average 6.7times speedup over the prior state-of-the-art FLAT while using 79% of the energy. Similarly, on the full end-to-end transformer inference, FuseMax achieves an average 5.3times speedup over FLAT using 83% of the energy.
MixLoRA: Enhancing Large Language Models Fine-Tuning with LoRA based Mixture of Experts
Large Language Models (LLMs) have showcased exceptional performance across a wide array of Natural Language Processing (NLP) tasks. Fine-tuning techniques are commonly utilized to tailor pre-trained models to specific applications. While methods like LoRA have effectively tackled GPU memory constraints during fine-tuning, their applicability is often restricted to limited performance, especially on multi-task. On the other hand, Mix-of-Expert (MoE) models, such as Mixtral 8x7B, demonstrate remarkable performance across multiple NLP tasks while maintaining a reduced parameter count. However, the resource requirements of these MoEs still challenging, particularly for consumer-grade GPUs only have limited VRAM. To address these challenge, we propose MixLoRA, an innovative approach aimed at constructing a resource-efficient sparse MoE model based on LoRA. MixLoRA inserts multiple LoRA-based experts within the feed-forward network block of a frozen pre-trained dense model through fine-tuning, employing a commonly used top-k router. Unlike other LoRA based MoE methods, MixLoRA enhances model performance by utilizing independently configurable attention-layer LoRA adapters, supporting the use of LoRA and its variants for the construction of experts, and applying auxiliary load balance loss to address the imbalance problem of the router. In experiments, MixLoRA achieves commendable performance across all evaluation metrics in both single-task and multi-task learning scenarios. Implemented within the m-LoRA framework, MixLoRA enables parallel fine-tuning of multiple mixture-of-experts models on a single 24GB consumer-grade GPU without quantization, thereby reducing GPU memory consumption by 41\% and latency during the training process by 17\%.
Auxiliary-Loss-Free Load Balancing Strategy for Mixture-of-Experts
For Mixture-of-Experts (MoE) models, an unbalanced expert load will lead to routing collapse or increased computational overhead. Existing methods commonly employ an auxiliary loss to encourage load balance, but a large auxiliary loss will introduce non-negligible interference gradients into training and thus impair the model performance. In order to control load balance while not producing undesired gradients during training, we propose Loss-Free Balancing, featured by an auxiliary-loss-free load balancing strategy. To be specific, before the top-K routing decision, Loss-Free Balancing will first apply an expert-wise bias to the routing scores of each expert. By dynamically updating the bias of each expert according to its recent load, Loss-Free Balancing can consistently maintain a balanced distribution of expert load. In addition, since Loss-Free Balancing does not produce any interference gradients, it also elevates the upper bound of model performance gained from MoE training. We validate the performance of Loss-Free Balancing on MoE models with up to 3B parameters trained on up to 200B tokens. Experimental results show that Loss-Free Balancing achieves both better performance and better load balance compared with traditional auxiliary-loss-controlled load balancing strategies.
Improvable Gap Balancing for Multi-Task Learning
In multi-task learning (MTL), gradient balancing has recently attracted more research interest than loss balancing since it often leads to better performance. However, loss balancing is much more efficient than gradient balancing, and thus it is still worth further exploration in MTL. Note that prior studies typically ignore that there exist varying improvable gaps across multiple tasks, where the improvable gap per task is defined as the distance between the current training progress and desired final training progress. Therefore, after loss balancing, the performance imbalance still arises in many cases. In this paper, following the loss balancing framework, we propose two novel improvable gap balancing (IGB) algorithms for MTL: one takes a simple heuristic, and the other (for the first time) deploys deep reinforcement learning for MTL. Particularly, instead of directly balancing the losses in MTL, both algorithms choose to dynamically assign task weights for improvable gap balancing. Moreover, we combine IGB and gradient balancing to show the complementarity between the two types of algorithms. Extensive experiments on two benchmark datasets demonstrate that our IGB algorithms lead to the best results in MTL via loss balancing and achieve further improvements when combined with gradient balancing. Code is available at https://github.com/YanqiDai/IGB4MTL.
A Theoretical Analysis of the Learning Dynamics under Class Imbalance
Data imbalance is a common problem in machine learning that can have a critical effect on the performance of a model. Various solutions exist but their impact on the convergence of the learning dynamics is not understood. Here, we elucidate the significant negative impact of data imbalance on learning, showing that the learning curves for minority and majority classes follow sub-optimal trajectories when training with a gradient-based optimizer. This slowdown is related to the imbalance ratio and can be traced back to a competition between the optimization of different classes. Our main contribution is the analysis of the convergence of full-batch (GD) and stochastic gradient descent (SGD), and of variants that renormalize the contribution of each per-class gradient. We find that GD is not guaranteed to decrease the loss for each class but that this problem can be addressed by performing a per-class normalization of the gradient. With SGD, class imbalance has an additional effect on the direction of the gradients: the minority class suffers from a higher directional noise, which reduces the effectiveness of the per-class gradient normalization. Our findings not only allow us to understand the potential and limitations of strategies involving the per-class gradients, but also the reason for the effectiveness of previously used solutions for class imbalance such as oversampling.
Exploring Weight Balancing on Long-Tailed Recognition Problem
Recognition problems in long-tailed data, in which the sample size per class is heavily skewed, have gained importance because the distribution of the sample size per class in a dataset is generally exponential unless the sample size is intentionally adjusted. Various methods have been devised to address these problems. Recently, weight balancing, which combines well-known classical regularization techniques with two-stage training, has been proposed. Despite its simplicity, it is known for its high performance compared with existing methods devised in various ways. However, there is a lack of understanding as to why this method is effective for long-tailed data. In this study, we analyze weight balancing by focusing on neural collapse and the cone effect at each training stage and found that it can be decomposed into an increase in Fisher's discriminant ratio of the feature extractor caused by weight decay and cross entropy loss and implicit logit adjustment caused by weight decay and class-balanced loss. Our analysis enables the training method to be further simplified by reducing the number of training stages to one while increasing accuracy.
FastSwitch: Optimizing Context Switching Efficiency in Fairness-aware Large Language Model Serving
Serving numerous users and requests concurrently requires good fairness in Large Language Models (LLMs) serving system. This ensures that, at the same cost, the system can meet the Service Level Objectives (SLOs) of more users , such as time to first token (TTFT) and time between tokens (TBT), rather than allowing a few users to experience performance far exceeding the SLOs. To achieve better fairness, the preemption-based scheduling policy dynamically adjusts the priority of each request to maintain balance during runtime. However, existing systems tend to overly prioritize throughput, overlooking the overhead caused by preemption-induced context switching, which is crucial for maintaining fairness through priority adjustments. In this work, we identify three main challenges that result in this overhead. 1) Inadequate I/O utilization. 2) GPU idleness. 3) Unnecessary I/O transmission during multi-turn conversations. Our key insight is that the block-based KV cache memory policy in existing systems, while achieving near-zero memory waste, leads to discontinuity and insufficient granularity in the KV cache memory. To respond, we introduce FastSwitch, a fairness-aware serving system that not only aligns with existing KV cache memory allocation policy but also mitigates context switching overhead. Our evaluation shows that FastSwitch outperforms the state-of-the-art LLM serving system vLLM with speedups of 1.4-11.2x across different tail TTFT and TBT.
Cephalo: Harnessing Heterogeneous GPU Clusters for Training Transformer Models
Training transformer models requires substantial GPU compute and memory resources. In homogeneous clusters, distributed strategies allocate resources evenly, but this approach is inefficient for heterogeneous clusters, where GPUs differ in power and memory. As high-end GPUs are costly and limited in availability, heterogeneous clusters with diverse GPU types are becoming more common. Existing methods attempt to balance compute across GPUs based on capacity but often underutilize compute due to memory constraints. We present Cephalo, a system that optimizes compute and memory usage by decoupling compute distribution from training state assignment. Cephalo outperforms state-of-the-art methods by achieving significantly higher training throughput while supporting larger models and batch sizes.
A systematic study of the class imbalance problem in convolutional neural networks
In this study, we systematically investigate the impact of class imbalance on classification performance of convolutional neural networks (CNNs) and compare frequently used methods to address the issue. Class imbalance is a common problem that has been comprehensively studied in classical machine learning, yet very limited systematic research is available in the context of deep learning. In our study, we use three benchmark datasets of increasing complexity, MNIST, CIFAR-10 and ImageNet, to investigate the effects of imbalance on classification and perform an extensive comparison of several methods to address the issue: oversampling, undersampling, two-phase training, and thresholding that compensates for prior class probabilities. Our main evaluation metric is area under the receiver operating characteristic curve (ROC AUC) adjusted to multi-class tasks since overall accuracy metric is associated with notable difficulties in the context of imbalanced data. Based on results from our experiments we conclude that (i) the effect of class imbalance on classification performance is detrimental; (ii) the method of addressing class imbalance that emerged as dominant in almost all analyzed scenarios was oversampling; (iii) oversampling should be applied to the level that completely eliminates the imbalance, whereas the optimal undersampling ratio depends on the extent of imbalance; (iv) as opposed to some classical machine learning models, oversampling does not cause overfitting of CNNs; (v) thresholding should be applied to compensate for prior class probabilities when overall number of properly classified cases is of interest.
Sampling Streaming Data with Parallel Vector Quantization -- PVQ
Accumulation of corporate data in the cloud has attracted more enterprise applications to the cloud creating data gravity. As a consequence, network traffic has become more cloud centric. This increase in cloud centric traffic poses new challenges in designing learning systems for streaming data due to class imbalance. The number of classes plays a vital role in the accuracy of the classifiers built from the data streams. In this paper, we present a vector quantization-based sampling method, which substantially reduces the class imbalance in data streams. We demonstrate its effectiveness by conducting experiments on network traffic and anomaly dataset with commonly used ML model building methods; Multilayered Perceptron on TensorFlow backend, Support Vector Machines, K-Nearest Neighbour, and Random Forests. We built models using parallel processing, batch processing, and randomly selecting samples. We show that the accuracy of classification models improves when the data streams are pre-processed with our method. We used out of the box hyper-parameters of these classifiers and auto sklearn for hyperparameter optimization.
Learned Best-Effort LLM Serving
Many applications must provide low-latency LLM service to users or risk unacceptable user experience. However, over-provisioning resources to serve fluctuating request patterns is often prohibitively expensive. In this work, we present a best-effort serving system that employs deep reinforcement learning to adjust service quality based on the task distribution and system load. Our best-effort system can maintain availability with over 10x higher client request rates, serves above 96% of peak performance 4.1x more often, and serves above 98% of peak performance 2.3x more often than static serving on unpredictable workloads. Our learned router is robust to shifts in both the arrival and task distribution. Compared to static serving, learned best-effort serving allows for cost-efficient serving through increased hardware utility. Additionally, we argue that learned best-effort LLM serving is applicable in wide variety of settings and provides application developers great flexibility to meet their specific needs.
Long-tailed Recognition by Routing Diverse Distribution-Aware Experts
Natural data are often long-tail distributed over semantic classes. Existing recognition methods tackle this imbalanced classification by placing more emphasis on the tail data, through class re-balancing/re-weighting or ensembling over different data groups, resulting in increased tail accuracies but reduced head accuracies. We take a dynamic view of the training data and provide a principled model bias and variance analysis as the training data fluctuates: Existing long-tail classifiers invariably increase the model variance and the head-tail model bias gap remains large, due to more and larger confusion with hard negatives for the tail. We propose a new long-tailed classifier called RoutIng Diverse Experts (RIDE). It reduces the model variance with multiple experts, reduces the model bias with a distribution-aware diversity loss, reduces the computational cost with a dynamic expert routing module. RIDE outperforms the state-of-the-art by 5% to 7% on CIFAR100-LT, ImageNet-LT and iNaturalist 2018 benchmarks. It is also a universal framework that is applicable to various backbone networks, long-tailed algorithms, and training mechanisms for consistent performance gains. Our code is available at: https://github.com/frank-xwang/RIDE-LongTailRecognition.
Large Scale Incremental Learning
Modern machine learning suffers from catastrophic forgetting when learning new classes incrementally. The performance dramatically degrades due to the missing data of old classes. Incremental learning methods have been proposed to retain the knowledge acquired from the old classes, by using knowledge distilling and keeping a few exemplars from the old classes. However, these methods struggle to scale up to a large number of classes. We believe this is because of the combination of two factors: (a) the data imbalance between the old and new classes, and (b) the increasing number of visually similar classes. Distinguishing between an increasing number of visually similar classes is particularly challenging, when the training data is unbalanced. We propose a simple and effective method to address this data imbalance issue. We found that the last fully connected layer has a strong bias towards the new classes, and this bias can be corrected by a linear model. With two bias parameters, our method performs remarkably well on two large datasets: ImageNet (1000 classes) and MS-Celeb-1M (10000 classes), outperforming the state-of-the-art algorithms by 11.1% and 13.2% respectively.
Deeper Insights into Weight Sharing in Neural Architecture Search
With the success of deep neural networks, Neural Architecture Search (NAS) as a way of automatic model design has attracted wide attention. As training every child model from scratch is very time-consuming, recent works leverage weight-sharing to speed up the model evaluation procedure. These approaches greatly reduce computation by maintaining a single copy of weights on the super-net and share the weights among every child model. However, weight-sharing has no theoretical guarantee and its impact has not been well studied before. In this paper, we conduct comprehensive experiments to reveal the impact of weight-sharing: (1) The best-performing models from different runs or even from consecutive epochs within the same run have significant variance; (2) Even with high variance, we can extract valuable information from training the super-net with shared weights; (3) The interference between child models is a main factor that induces high variance; (4) Properly reducing the degree of weight sharing could effectively reduce variance and improve performance.
Locality-aware Fair Scheduling in LLM Serving
Large language model (LLM) inference workload dominates a wide variety of modern AI applications, ranging from multi-turn conversation to document analysis. Balancing fairness and efficiency is critical for managing diverse client workloads with varying prefix patterns. Unfortunately, existing fair scheduling algorithms for LLM serving, such as Virtual Token Counter (VTC), fail to take prefix locality into consideration and thus suffer from poor performance. On the other hand, locality-aware scheduling algorithms in existing LLM serving frameworks tend to maximize the prefix cache hit rate without considering fair sharing among clients. This paper introduces the first locality-aware fair scheduling algorithm, Deficit Longest Prefix Match (DLPM), which can maintain a high degree of prefix locality with a fairness guarantee. We also introduce a novel algorithm, Double Deficit LPM (D^2LPM), extending DLPM for the distributed setup that can find a balance point among fairness, locality, and load-balancing. Our extensive evaluation demonstrates the superior performance of DLPM and D^2LPM in ensuring fairness while maintaining high throughput (up to 2.87times higher than VTC) and low per-client (up to 7.18times lower than state-of-the-art distributed LLM serving system) latency.
On The Fairness Impacts of Hardware Selection in Machine Learning
In the machine learning ecosystem, hardware selection is often regarded as a mere utility, overshadowed by the spotlight on algorithms and data. This oversight is particularly problematic in contexts like ML-as-a-service platforms, where users often lack control over the hardware used for model deployment. How does the choice of hardware impact generalization properties? This paper investigates the influence of hardware on the delicate balance between model performance and fairness. We demonstrate that hardware choices can exacerbate existing disparities, attributing these discrepancies to variations in gradient flows and loss surfaces across different demographic groups. Through both theoretical and empirical analysis, the paper not only identifies the underlying factors but also proposes an effective strategy for mitigating hardware-induced performance imbalances.
POLCA: Power Oversubscription in LLM Cloud Providers
Recent innovation in large language models (LLMs), and their myriad use-cases have rapidly driven up the compute capacity demand for datacenter GPUs. Several cloud providers and other enterprises have made substantial plans of growth in their datacenters to support these new workloads. One of the key bottleneck resources in datacenters is power, and given the increasing model sizes of LLMs, they are becoming increasingly power intensive. In this paper, we show that there is a significant opportunity to oversubscribe power in LLM clusters. Power oversubscription improves the power efficiency of these datacenters, allowing more deployable servers per datacenter, and reduces the deployment time, since building new datacenters is slow. We extensively characterize the power consumption patterns of a variety of LLMs and their configurations. We identify the differences between the inference and training power consumption patterns. Based on our analysis of these LLMs, we claim that the average and peak power utilization in LLM clusters for inference should not be very high. Our deductions align with the data from production LLM clusters, revealing that inference workloads offer substantial headroom for power oversubscription. However, the stringent set of telemetry and controls that GPUs offer in a virtualized environment, makes it challenging to have a reliable and robust power oversubscription mechanism. We propose POLCA, our framework for power oversubscription that is robust, reliable, and readily deployable for GPU clusters. Using open-source models to replicate the power patterns observed in production, we simulate POLCA and demonstrate that we can deploy 30% more servers in the same GPU cluster for inference, with minimal performance loss
LIFL: A Lightweight, Event-driven Serverless Platform for Federated Learning
Federated Learning (FL) typically involves a large-scale, distributed system with individual user devices/servers training models locally and then aggregating their model updates on a trusted central server. Existing systems for FL often use an always-on server for model aggregation, which can be inefficient in terms of resource utilization. They may also be inelastic in their resource management. This is particularly exacerbated when aggregating model updates at scale in a highly dynamic environment with varying numbers of heterogeneous user devices/servers. We present LIFL, a lightweight and elastic serverless cloud platform with fine-grained resource management for efficient FL aggregation at scale. LIFL is enhanced by a streamlined, event-driven serverless design that eliminates the individual heavy-weight message broker and replaces inefficient container-based sidecars with lightweight eBPF-based proxies. We leverage shared memory processing to achieve high-performance communication for hierarchical aggregation, which is commonly adopted to speed up FL aggregation at scale. We further introduce locality-aware placement in LIFL to maximize the benefits of shared memory processing. LIFL precisely scales and carefully reuses the resources for hierarchical aggregation to achieve the highest degree of parallelism while minimizing the aggregation time and resource consumption. Our experimental results show that LIFL achieves significant improvement in resource efficiency and aggregation speed for supporting FL at scale, compared to existing serverful and serverless FL systems.
Seagull: An Infrastructure for Load Prediction and Optimized Resource Allocation
Microsoft Azure is dedicated to guarantee high quality of service to its customers, in particular, during periods of high customer activity, while controlling cost. We employ a Data Science (DS) driven solution to predict user load and leverage these predictions to optimize resource allocation. To this end, we built the Seagull infrastructure that processes per-server telemetry, validates the data, trains and deploys ML models. The models are used to predict customer load per server (24h into the future), and optimize service operations. Seagull continually re-evaluates accuracy of predictions, fallback to previously known good models and triggers alerts as appropriate. We deployed this infrastructure in production for PostgreSQL and MySQL servers across all Azure regions, and applied it to the problem of scheduling server backups during low-load time. This minimizes interference with user-induced load and improves customer experience.
"Why did the Model Fail?": Attributing Model Performance Changes to Distribution Shifts
Machine learning models frequently experience performance drops under distribution shifts. The underlying cause of such shifts may be multiple simultaneous factors such as changes in data quality, differences in specific covariate distributions, or changes in the relationship between label and features. When a model does fail during deployment, attributing performance change to these factors is critical for the model developer to identify the root cause and take mitigating actions. In this work, we introduce the problem of attributing performance differences between environments to distribution shifts in the underlying data generating mechanisms. We formulate the problem as a cooperative game where the players are distributions. We define the value of a set of distributions to be the change in model performance when only this set of distributions has changed between environments, and derive an importance weighting method for computing the value of an arbitrary set of distributions. The contribution of each distribution to the total performance change is then quantified as its Shapley value. We demonstrate the correctness and utility of our method on synthetic, semi-synthetic, and real-world case studies, showing its effectiveness in attributing performance changes to a wide range of distribution shifts.