# Exploring the Frontiers of Softmax: Provable Optimization, Applications in Diffusion Model, and Beyond

arXiv: https://arxiv.org/abs/2405.03251 · HTML: https://mlsanity.com/paper/2405.03251/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 42 of its 159 references that are in the map.
- First posted on arXiv 6 May 2024.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 42)

- [GPT-4 Technical Report](https://mlsanity.com/paper/2303.08774/) (2023)
- [LLaMA: Open and Efficient Foundation Language Models](https://mlsanity.com/paper/2302.13971/) (2023)
- [Llama 2: Open Foundation and Fine-Tuned Chat Models](https://mlsanity.com/paper/2307.09288/) (2023)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [PaLM: Scaling Language Modeling with Pathways](https://mlsanity.com/paper/2204.02311/) (2022)
- [Emergent Abilities of Large Language Models](https://mlsanity.com/paper/2206.07682/) (2022)
- [Linformer: Self-Attention with Linear Complexity](https://mlsanity.com/paper/2006.04768/) (2020)
- [In-context Learning and Induction Heads](https://mlsanity.com/paper/2209.11895/) (2022)
- [How to Train Your Energy-Based Models](https://mlsanity.com/paper/2101.03288/) (2021)
- [Feature Learning in Infinite-Width Neural Networks](https://mlsanity.com/paper/2011.14522/) (2020)
- [Monitoring AI-Modified Content at Scale: A Case Study on the Impact of ChatGPT on AI Conference Peer Reviews](https://mlsanity.com/paper/2403.07183/) (2024)
- [Scaling Limits of Wide Neural Networks with Weight Sharing: Gaussian Process Behavior, Gradient Independence, and Neural Tangent Kernel Derivation](https://mlsanity.com/paper/1902.04760/) (2019)
- [Stochastic Gradient Descent Optimizes Over-parameterized Deep ReLU Networks](https://mlsanity.com/paper/1811.08888/) (2018)
- [Error Bounds for Flow Matching Methods](https://mlsanity.com/paper/2305.16860/) (2023)
- [Training Dynamics of Multi-Head Softmax Attention for In-Context Learning: Emergence, Convergence, and Optimality](https://mlsanity.com/paper/2402.19442/) (2024)
- [Generalization Guarantees for Neural Networks via Harnessing the Low-rank Structure of the Jacobian](https://mlsanity.com/paper/1906.05392/) (2019)
- [Quadratic Suffices for Over-parametrization via Matrix Chernoff Bound](https://mlsanity.com/paper/1906.03593/) (2019)
- [A Short Survey of Viewing Large Language Models in Legal Aspect](https://mlsanity.com/paper/2303.09136/) (2023)
- [Attention Scheme Inspired Softmax Regression](https://mlsanity.com/paper/2304.10411/) (2023)
- [Towards Understanding Hierarchical Learning: Benefits of Neural Representations](https://mlsanity.com/paper/2006.13436/) (2020)
- [Theoretical Insights for Diffusion Guidance: A Case Study for Gaussian Mixture Models](https://mlsanity.com/paper/2403.01639/) (2024)
- [An Over-parameterized Exponential Regression](https://mlsanity.com/paper/2303.16504/) (2023)
- [Learning general Gaussian mixtures with efficient score matching](https://mlsanity.com/paper/2404.18893/) (2024)
- [Learning Mixtures of Gaussians Using Diffusion Models](https://mlsanity.com/paper/2404.18869/) (2024)
- [Quantifying the Benefit of Using Differentiable Learning over Tangent Kernels](https://mlsanity.com/paper/2103.01210/) (2021)
- [The Closeness of In-Context Learning and Weight Shifting for Softmax Regression](https://mlsanity.com/paper/2304.13276/) (2023)
- [PolySketchFormer: Fast Transformers via Sketching Polynomial Kernels](https://mlsanity.com/paper/2310.01655/) (2023)
- [In-Context Learning for Attention Scheme: from Single Softmax Regression to Multiple Softmax Regression via a Tensor Trick](https://mlsanity.com/paper/2307.02419/) (2023)
- [A Fast Optimization View: Reformulating Single Layer Attention in LLM Based on Tensor and SVM Trick, and Solving It in Matrix Multiplication Time](https://mlsanity.com/paper/2309.07418/) (2023)
- [The Fine-Grained Complexity of Gradient Computation for Training Large Language Models](https://mlsanity.com/paper/2402.04497/) (2024)

## Built on by (most cited first, 15 of 15)

- [Why Larger Language Models Do In-context Learning Differently?](https://mlsanity.com/paper/2405.19592/) (2024)
- [Unraveling the Smoothness Properties of Diffusion Models: A Gaussian Mixture Perspective](https://mlsanity.com/paper/2405.16418/) (2024)
- [Bypassing the Exponential Dependency: Looped Transformers Efficiently Learn In-context by Multi-step Gradient Descent](https://mlsanity.com/paper/2410.11268/) (2024)
- [On Statistical Rates of Conditional Diffusion Transformers: Approximation, Estimation and Minimax Optimality](https://mlsanity.com/paper/2411.17522/) (2024)
- [Differential Privacy Mechanisms in Neural Tangent Kernel Regression](https://mlsanity.com/paper/2407.13621/) (2024)
- [Towards Infinite-Long Prefix in Transformer](https://mlsanity.com/paper/2406.14036/) (2024)
- [On Computational Limits of FlowAR Models: Expressivity and Efficiency](https://mlsanity.com/paper/2502.16490/) (2025)
- [Fast RoPE Attention: Combining the Polynomial Method and Fast Fourier Transform](https://mlsanity.com/paper/2505.11892/) (2025)
- [Only Large Weights (And Not Skip Connections) Can Prevent the Perils of Rank Collapse](https://mlsanity.com/paper/2505.16284/) (2025)
- [Unlocking the Theory Behind Scaling 1-Bit Neural Networks](https://mlsanity.com/paper/2411.01663/) (2024)
- [HOFAR: High-Order Augmentation of Flow Autoregressive Transformers](https://mlsanity.com/paper/2503.08032/) (2025)
- [When Can We Solve the Weighted Low Rank Approximation Problem in Truly Subquadratic Time?](https://mlsanity.com/paper/2502.16912/) (2025)
- [Dynamics of Spontaneous Topic Changes in Next Token Prediction with Self-Attention](https://mlsanity.com/paper/2501.06382/) (2025)
- [Unifying Learning Dynamics and Generalization in Transformers Scaling Law](https://mlsanity.com/paper/2512.22088/) (2025)
- [CoGraM: Context-sensitive granular optimization method with rollback for robust model fusion](https://mlsanity.com/paper/2512.03610/) (2025)

## Abstract (arXiv)

The softmax activation function plays a crucial role in the success of large language models (LLMs), particularly in the self-attention mechanism of the widely adopted Transformer architecture. However, the underlying learning dynamics that contribute to the effectiveness of softmax remain largely unexplored. As a step towards better understanding, this paper provides a theoretical study of the optimization and generalization properties of two-layer softmax neural networks, providing theoretical insights into their superior performance as other activation functions, such as ReLU and exponential. Leveraging the Neural Tangent Kernel (NTK) framework, our analysis reveals that the normalization effect of the softmax function leads to a good perturbation property of the induced NTK matrix, resulting in a good convex region of the loss landscape. Consequently, softmax neural networks can learn the target function in the over-parametrization regime. To demonstrate the broad applicability of our theoretical findings, we apply them to the task of learning score estimation functions in diffusion models, a promising approach for generative modeling. Our analysis shows that gradient-based algorithms can learn the score function with a provable accuracy. Our work provides a deeper understanding of the effectiveness of softmax neural networks and their potential in various domains, paving the way for further advancements in natural language processing and beyond.