Why Adversarial Models, Such as the One-Pixel Attack, Are Effective Against Neural Networks

Adversarial attacks, especially one-pixel attacks, are notorious for their ability to trick neural networks into making incorrect predictions. Eric Jang's insights into the two primary reasons for their success—the weird geometry of high-dimensional spaces and the excessive curvature of neural network decision boundaries—provide a crucial perspective. However, in this essay, we will delve deeper into the former, emphasizing why it is a more fundamental and unavoidable issue.

Introduction to Adversarial Attacks in High-Dimensional Spaces

Adversarial examples are not unique to neural networks. They can be constructed for any supervised machine learning algorithm. This essay will explore the mathematical underpinnings of why these attacks are almost inevitable in high-dimensional classification tasks, such as image and audio recognition, where neural networks have proven to be the most effective.

The Geometry of High-Dimensional Spaces

The existence of adversarial examples can be explained through the concentration of measure phenomenon, a well-studied topic in probability theory. This phenomenon is closely related to the 'weird' geometry of high-dimensional spaces, where the volume or measure of a set can increase dramatically with even a small expansion.

Concentration of Measure

Consider a simple classification task where the data points ( x in mathcal{X} ) are drawn from a probability measure (mu). Let (mathcal{E} subseteq mathcal{X}) be the set of data points where the algorithm (e.g., a neural network) makes an error. In the presence of an adversary with a budget ( b ), the algorithm can now make an error on any data point ( x' ) that is no farther than distance ( b ) from (mathcal{E}), i.e., the distance between ( x ) and ( x' ) is less than ( b ). Let (mathcal{E}_b) denote the ( b )-expansion of the error set (mathcal{E}).

Visualization: b-Expansion of the Error Set

Figure 1: b-expansion of the error set (mathcal{E}) in the domain (mathcal{X})

Adversarial examples occur when the set (mathcal{E}_b) is significantly larger than the set (mathcal{E}). In other words, most data points outside the error set (mathcal{E}) should be within a small distance ( b ) of the error set. If the distance of ( x ) to the error set is smaller than ( b ), it can be pushed into the expanded error set (mathcal{E}_b) in the presence of an adversary.

Blow-Up in High-Dimensional Settings

In low-dimensional settings, the expansion of the error set does not increase its volume significantly if the adversarial budget ( b ) is small enough. However, this intuition breaks down spectacularly in high-dimensional settings. For example, consider a 1000-dimensional unit sphere with an error set (mathcal{E}) that sums to 50% of its total surface area. Expanding this set by just 0.1 of its diameter results in the error set covering 99% of the sphere's surface area! This kind of blow-up is a general phenomenon that does not depend on the specific geometry or measure used.

Visualizing the Expansion

Figure: 0.1-expansion of a 50% error set on a 1000-dimensional sphere covers 99% of the sphere!

Intensity of Concentration

To further illustrate the concept, consider Figures showing the intensity of concentration. As the concentration of the measure on the space becomes sharper, the expanded error set (mathcal{E}_b) increasingly covers a larger portion of the domain (mathcal{X}).

Mathematical Formulation

The usual error rate of the algorithm is (mu(mathcal{E})) (the measure of the error set), while the adversarial error rate is (mu(mathcal{E}_b)) (the measure of the b-expansion of the error set). No matter what the algorithm is, adversarial attack success is inevitable if (mu(mathcal{E}_b) gg mu(mathcal{E})), which is exactly what happens when the measure (mu) exhibits sharp concentration.

Conclusion and Future Research

Understanding the geometry of high-dimensional spaces and the concentration of measure phenomenon is crucial for developing more robust machine learning models. While the theory behind adversarial examples is an active research area, this essay provides a foundational perspective on why these attacks are a fundamental challenge to neural networks. Future work should focus on leveraging this understanding to improve the robustness of machine learning algorithms.

Bonus: For those interested in exploring the weird properties of high-dimensional spaces further, I recommend reading the following essays: [Providing links/urls here].