From ReLU to Softmax: A New Breakthrough in Transformer Approximation Theory

Introduction: Transformer's Theoretical Foundations Need Strengthening

Since its inception, the Transformer architecture has become the core engine powering large language models and numerous AI systems. However, compared to its enormous success in engineering practice, the academic community's theoretical understanding of Transformers' mathematical approximation capabilities — that is, what they "can actually compute precisely" — remains insufficient. Recently, a new paper published on arXiv (arXiv:2604.24878v1) proposed a systematic method for "translating" classical ReLU network approximation results into corresponding results under the Softmax attention mechanism, opening a new window for Transformer theoretical research.

Core Contribution: A "Translation Recipe" from ReLU to Softmax

The central idea of this research is elegant yet profound: the ReLU activation function, as the most fundamental nonlinear unit in neural networks, has accumulated decades of approximation theory results. The research team proposed a systematic conversion method they call a "recipe" that can directly transfer these mature ReLU approximation results to Softmax-based attention mechanisms.

The method has a remarkably broad scope of applicability, covering many common approximation targets. More importantly, unlike previous universal approximation theorems, this method can provide more economical and precise resource bounds for specific target functions — meaning it not only tells us whether a Transformer "can" approximate a given function, but also "how many resources" are needed to achieve a specified accuracy.

Technical Highlights: Approximation Analysis of Three Fundamental Operations

The paper demonstrates the power of this method through three classical computational primitives:

Multiplication

Multiplication is one of the basic operations that neural networks struggle to express directly. Using their conversion method, the research team provided specific resource estimates — including the number of layers, attention heads, and other parameters — required for a Softmax Transformer to approximate multiplication. This is of significant importance for understanding how Transformers perform arithmetic reasoning.

Reciprocal Computation

Reciprocal computation forms the foundation of division. The paper demonstrates a specific construction for Transformers to achieve reciprocal approximation through Softmax attention, a result that can be further extended to more complex rational function approximation.

Maximum and Minimum

Min/Max operations are indispensable in sorting, comparison, and other logical reasoning tasks. The research shows that the Softmax attention mechanism can approximate these operations with controllable resource overhead, providing theoretical support for explaining Transformer performance on reasoning tasks.

Academic Significance: Fine-Grained Analysis Beyond Universal Approximation

While traditional universal approximation theorems prove that neural networks can theoretically approximate any continuous function, such "existence" results often fail to provide practically useful resource estimates. The breakthrough of this research lies in making the leap from "can it be done" to "how much is enough."

Specifically, the advantages of this method are reflected in several aspects:

• Systematicity: It provides a general conversion framework rather than analyzing each problem individually
• Economy: The resource bounds are optimized for specific targets, avoiding the overestimation common in universal approximation
• Extensibility: As ReLU approximation theory continues to develop, new results can be automatically converted into corresponding conclusions in the Transformer domain through this "recipe"

This means that the rich body of results accumulated over the past several decades in feedforward neural network theory now has a "highway" leading to modern Transformer architectures.

Industry Impact and Future Outlook

Although this research leans heavily toward theory, its potential impact should not be underestimated. As large language models continue to grow in scale, the industry is paying increasing attention to the question of "how large does a model actually need to be." The fine-grained resource bounds provided by this research could offer more scientific theoretical guidance for model architecture design and scale selection.

Furthermore, understanding Transformers' approximation capabilities at the level of fundamental operations also helps explain the performance and limitations of current large models on tasks such as mathematical reasoning and logical judgment. For example, if we know that approximating multiplication requires a specific number of attention layers, we can better understand why smaller models tend to make errors on arithmetic tasks.

Looking ahead, this research direction may develop along the following paths: extending the conversion method to more complex function classes, such as trigonometric and exponential functions; combining it with actual training dynamics to verify the consistency between theoretical predictions and experimental results; and applying resource bound analysis to practical problems such as model compression and efficient architecture search.

Overall, this work adds an important piece to the theoretical foundation of Transformers, taking us one step further on the path to "understanding why AI works."