The Emergence of the Sigmoid Function from Entropy Minimization

Posted by Dr Bouarfa Mahi on 07 Feb, 2025

Abstract

The sigmoid function plays a central role in artificial intelligence, machine learning, and decision theory. While it has been widely used as an activation function, its origins have remained largely empirical. In this article, we demonstrate that the sigmoid function emerges naturally from the fundamental definition of entropy and structured knowledge accumulation. By deriving the sigmoid function directly from entropy minimization, we provide a first-principles explanation for its effectiveness in learning systems. This discovery offers profound insights into the relationship between entropy reduction, knowledge structuration, and decision-making in AI.

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1. Introduction

Entropy is a measure of uncertainty in a system, fundamental to information theory and AI learning processes. Traditional approaches view the sigmoid function as a convenient non-linearity, but its deep connection to entropy minimization has remained unexplored.

This article develops a rigorous derivation of the sigmoid function from first principles, showing that it naturally arises when modeling how intelligence structures knowledge over probabilistic decision-making. Importantly, the sigmoid function emerges from entropy because we have redefined entropy beyond Shannon entropy, aligning it with structured knowledge evolution. However, it is crucial to note that while we used the sigmoid function to redefine entropy, this does not mean that entropy is derived from the sigmoid function.

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2. The Fundamental Entropy Equation

We begin with the fundamental entropy equation, which expresses entropy as an integral of structured knowledge accumulation over probabilistic decision $D$:

$$ H = -\frac{1}{\ln 2} \int z \, dD $$

where:

• $H$ represents entropy (uncertainty in decision-making),
• $z$ is structured knowledge,
• $D$ is the probability of making the correct decision.

Differentiating Shannon entropy with respect to $D$, we obtain:

$$ \frac{dH}{dD} = \log_2 \left(\frac{1 - D}{D}\right) $$

which describes the rate of entropy change as decision probability evolves.

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3. Derivation of the Sigmoid Function from Entropy

We express structured knowledge $z$ in terms of entropy:

$$ z = -\frac{dH}{dD} \ln 2 $$

Substituting our entropy derivative:

$$ z = -\ln 2 \cdot \log_2 \left(\frac{1 - D}{D}\right) $$

Using the logarithm identity $\log_2 x = \displaystyle \frac{\ln x}{\ln 2}$, we simplify:

$$ z = -\ln \left(\frac{1 - D}{D}\right) $$

which can be rewritten as:

$$ \frac{1 - D}{D} = e^{-z} $$

Rearranging for $D$:

$$ D = \frac{1}{1 + e^{-z}} $$

This is exactly the sigmoid function:

$$ \sigma(z) = \frac{1}{1 + e^{-z}} $$

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4. Interpretation: Why the Sigmoid Function Emerges from Entropy

This derivation proves that the sigmoid function is not an arbitrary activation function—it emerges directly from entropy minimization.

Key Insights:

• Entropy reduction is the driver of structured knowledge formation.
• Decision probability follows a logistic trajectory as knowledge accumulates.
• The sigmoid function represents an optimal path for entropy minimization.

This provides the first theoretical foundation for why sigmoid functions work so effectively in AI—they naturally model the evolution of structured intelligence.

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5. Implications for AI and Learning Systems

This discovery has far-reaching implications for how we design and understand AI models:

• Machine Learning Optimization: AI training should focus not just on minimizing loss, but on structuring entropy reduction.
• Neural Network Design: Activation functions should be viewed as entropy structuration mechanisms rather than arbitrary non-linearities.
• Human Learning Models: This theory aligns with cognitive science, where intelligence evolves by reducing uncertainty and structuring knowledge efficiently.

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6. Conclusion: The Entropy-Sigmoid Theorem

Formal Theorem:

Probabilistic decision-making follows a sigmoid function as a direct consequence of entropy reduction and structured knowledge accumulation.

This provides a first-principles derivation of the sigmoid function and a new foundation for AI learning theory. Future work should explore entropy-aware optimization techniques and expand this framework to multi-class decision systems such as the softmax function.

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7. Future Directions

• Experimental Validation: Can we track entropy reduction in AI models and verify sigmoid emergence?
• Entropy-Aware Learning Algorithms: Can we design AI that explicitly follows structured entropy minimization instead of brute-force optimization?
• Softmax and Multi-Class Extensions: Does the same principle generalize to multi-class decision-making?

This work redefines the mathematical foundations of AI learning. Instead of arbitrary optimization, AI can now be seen as an entropy-structuring system that follows a well-defined mathematical trajectory.

ENTROPY SIGMOID   

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