Parameters vs FLOPs: Scaling Laws for Optimal Sparsity for Mixture-of-Experts Language Models (2501.12370v1)

Abstract: Scaling the capacity of LLMs has consistently proven to be a reliable approach for improving performance and unlocking new capabilities. Capacity can be primarily defined by two dimensions: the number of model parameters and the compute per example. While scaling typically involves increasing both, the precise interplay between these factors and their combined contribution to overall capacity remains not fully understood. We explore this relationship in the context of sparse Mixture-of-Expert models (MoEs), which allow scaling the number of parameters without proportionally increasing the FLOPs per example. We investigate how varying the sparsity level, i.e., the ratio of non-active to total parameters, affects model performance in terms of both pretraining and downstream performance. We find that under different constraints (e.g. parameter size and total training compute), there is an optimal level of sparsity that improves both training efficiency and model performance. These results provide a better understanding of the impact of sparsity in scaling laws for MoEs and complement existing works in this area, offering insights for designing more efficient architectures.

1. Introduction

The rapid advancement of NLP has been significantly fueled by the development and scaling of LLMs. Scaling laws, which describe the relationship between model size, data size, and computation, have become essential for optimizing the deployment and application of these models. A key consideration in scaling is the interplay between model parameters and floating-point operations (FLOPs), particularly in Mixture-of-Experts (MoE) models. MoE models offer efficient scaling by dynamically selecting subsets of model parameters ("experts") for processing inputs, potentially decoupling model size from computational cost. Optimal sparsity, achieved by activating only a fraction of the experts, further enhances computational efficiency without significantly degrading model performance. This review explores the foundational aspects of scaling laws, the mechanics of MoE models, and the significance of achieving optimal sparsity, providing insights for researchers and practitioners seeking to enhance the efficiency and performance of large-scale LLMs.

2. Background and Foundational Concepts

The evolution of LLMs has been punctuated by breakthroughs in architecture and training methodologies. The Transformer architecture, introduced in "Attention is All You Need" (Vaswani et al., 2017 ), enabled unprecedented capabilities in capturing long-range dependencies in language, paving the way for models like BERT ("BERT: Pre-training of Deep Bidirectional Transformers for Language Understanding" (Devlin et al., 2018 )) and GPT-3 ("GPT-3: LLMs are Few-Shot Learners" (Brown et al., 2020 )). These models demonstrated that increasing model size and training data leads to improved performance. The concept of scaling laws, formalized in papers like "Scaling Laws for Neural LLMs" (Kaplan et al., 2020 ), provided a framework for predicting the performance gains achievable by scaling up model size and training data. These scaling laws suggest that both model size and dataset size should increase simultaneously to achieve optimal performance.

A crucial architectural distinction lies between dense and MoE models. Dense models, where all parameters are active during inference, can become computationally prohibitive as they scale. In contrast, MoE models utilize a subset of parameters at any given time, offering greater computational efficiency. This approach involves a routing mechanism that dynamically assigns input data to a small subset of specialized subnetworks ("experts"). This selective activation reduces the number of active parameters per input, reducing inference costs while maintaining high capacity. This conditional computation contrasts with dense models, where computational resources are uniformly allocated across all parameters.

3. Methodologies for Achieving Optimal Sparsity in MoE Models

The Mixture of Experts (MoE) architecture employs multiple expert networks, with a gating network determining which experts are activated for each input. This selective activation achieves sparsity, allowing only a subset of the model's parameters to be updated at any time, reducing computational costs and improving model efficiency. Given an input $x$, the gating network outputs a probability distribution over the experts, and the final model output is a weighted sum of the outputs from the selected experts:

$y(x) = \sum_{i=1}^{k} g_i(x) f_i(x)$,

where $y(x)$ is the final output, $g_i(x)$ is the gating weight for expert $i$, $f_i(x)$ is the output from expert $i$, and $k$ is the number of experts in the model. Sparsity is achieved by ensuring that most of the $g_i(x)$ values are close to zero, activating only a few experts at a time.

Several techniques contribute to sparsity in MoE models:

• Hard Gating: Activates only a fixed number of experts for each input based on the highest gating network outputs.
• Soft Gating with Entropy Regularization: Uses softmax probabilities to weigh each expert's contribution, combined with an entropy regularization term to encourage sparser expert weight distributions. This approach is highlighted in "Outrageously Large Neural Networks" (Shazeer et al., 2017 ) to balance load and sparsity.
• Load Balancing Techniques: Prevents certain experts from being overloaded while others are underutilized. Techniques like dynamic routing and priority scheduling, as discussed in "Sparse Experts for Vision Transformers" (Du et al., 2021 ), optimize load distribution.

Achieving optimal sparsity requires careful consideration of model capacity, gating network design, and training regimes. The model should have sufficient capacity (number and size of experts) to handle varying task complexities while maintaining sparsity. The gating mechanism should be lightweight but capable of accurately routing inputs to relevant experts, potentially using attention-based gating. Training regimes should include mechanisms such as dropout, expert normalization, and decay regularization to promote sparsity and prevent overfitting.

4. Parameters vs. FLOPs in MoE Models

The balance between parameters and FLOPs is crucial for the efficient scaling of MoE models. Scaling in MoEs involves increasing the number of parameters by expanding the number of experts while managing FLOPs by routing inputs to a subset of these experts. If $k$ is the number of active experts, $E$ is the total number of experts, $P_i$ is the number of parameters in the $i^{th}$ expert, and $F_i$ is the FLOPs required for computation in the $i^{th}$ expert, the effective parameter count is $\sum_{j=1}^{k} P_j$ and the effective FLOP count is $\sum_{j=1}^{k} F_j$. This selective computation contributes to the efficiency of MoE models compared to dense models, where all parameters are activated for each input.

Empirical studies have demonstrated that MoE models can achieve state-of-the-art results while maintaining reduced computational costs. "GShard: Scaling Giant Models with Conditional Computation" (Abnar et al., 21 Jan 2025 ) illustrates this efficiency in Google's LLMs, demonstrating the ability to scale to billions of parameters without a proportional increase in FLOPs. "Switch Transformers: Scaling to Trillion Parameter Models with Simple and Efficient Sparsity" (Fedus et al., 2021 ) shows that the sparse nature of MoEs allows effective training with reduced computational budgets due to intelligent routing mechanisms.

The paper "Outrageously Large Neural Networks" (Shazeer et al., 2017 ) identifies that sparsity in training, as seen in MoE architectures, significantly lowers the required FLOPs while maintaining a high parameter count. This highlights the key advantage of MoEs: a large model capable of generalization and representation, constrained by computational budget via strategic sparsity.

5. Scaling Laws and Optimal Sparsity in MoE Models

Scaling laws for MoE models explore how performance, efficiency, and scalability change with varying parameters such as the number of experts, data size, and model size. "Scaling Laws for Mixture of Experts" (Abnar et al., 21 Jan 2025 ) investigates these relationships, revealing that increasing the number of experts allows the model to achieve higher accuracy while maintaining or reducing computational demands compared to dense models. Theoretical analyses complement these findings by providing explanations for these patterns, often leveraging concepts from information theory and optimization.

"Quantifying the Sparse-to-Dense Spectrum in Mixture of Experts" (Du et al., 2021 ) investigates different degrees of sparsity and their impact on model performance, finding a trade-off between sparsity and robustness. Excessive sparsity can degrade generalization, while optimal sparsity maintains robust performance with reduced active parameters. This research suggests that tailoring sparsity levels to the task can significantly improve model operation, involving tuning the number of active experts per input and the routing mechanisms.

6. Theoretical Insights and Frameworks

Theoretical frameworks play a pivotal role in advancing our understanding of complex systems, particularly in computational models and neural networks. The paper "Scaling Laws for Neural LLMs" (Abnar et al., 21 Jan 2025 ) proposes that performance improvements follow predictable patterns based on the scale of computation, allowing extrapolation to scales not yet achieved. "A Theoretical Investigation of Generalization in Neural Networks" (Du et al., 2021 ) presents a framework for understanding how neural networks generalize, arguing that larger models trained on sufficiently diverse datasets tend to generalize better, as articulated through formalized scaling relationships. These frameworks enable researchers to design more efficient experiments, limit computational resources, and better predict the outcomes of novel architectures.

7. Challenges and Limitations

Achieving optimal sparsity in neural networks presents challenges in both theoretical development and practical application. Training stability is a key concern, as sparse models can suffer from convergence difficulties due to the reduction in parameter count. Aggressive pruning techniques can eliminate essential weight connections, requiring sophisticated techniques for maintaining balance between sparsity and performance. Another challenge is the communication overhead associated with distributed sparse training. Synchronizing sparse models across multiple nodes can introduce significant communication latencies due to the uneven distribution of non-zero weights.

Practical limitations include the lack of standardized frameworks and tools that support efficient sparse matrix operations on existing hardware. While some hardware accelerators are optimized for dense matrix operations, they underperform with sparse computations. Furthermore, the reliance on specialized pruning and regularization interventions can complicate model interpretability and reproducibility.

Unresolved issues include the dynamic adjustment of sparsity during inference, which remains a significant hurdle. Current techniques often assume a fixed sparsity pattern, which may not be optimal for all data inputs. Adaptive sparsity patterns could enhance model efficiency but require new methodologies for real-time reconfiguration without retraining. There are also open questions regarding the trade-offs between model compression rates and computational gains, particularly in resource-constrained environments.

8. Conclusion and Future Directions

MoE models offer a unique capability to optimize computational efficiency while maintaining or enhancing model performance, separating parameter scale from operational cost through sparse gating functions. Future research should explore refining gating mechanisms, developing efficient training techniques tailored for sparse computations, and evaluating MoEs on diverse tasks. Technological advancements include custom hardware support for sparse computations, scalability in cloud environments, and model optimization tools for automating the scaling of MoE architectures. As computational constraints remain a critical concern in AI development, MoE models provide a promising avenue for sustainable advancements in scalability and performance. By continuing to explore these dynamics, researchers can harness the full potential of large parameter spaces while maintaining practical computation requirements.