Fast Gradient Sign Method

⚡ FGSM Perturbation Calculator – Visualize Adversarial Noise Impact

How the FGSM Perturbation Calculator Works

This calculator helps you understand the effect of the Fast Gradient Sign Method (FGSM) by computing how a small perturbation with a given epsilon and gradient direction modifies an input value.

Enter the epsilon value to set the magnitude of the perturbation, choose the gradient sign to control the direction of change, and specify the original input value. The calculator will show the calculated perturbation amount, the perturbed input before clipping, and the clipped input constrained to the valid range of 0 to 1 or 0 to 255, depending on the original input scale.

If the chosen epsilon is too large, the calculator will warn you that the perturbation may cause noticeable changes leading to misclassification. Use this tool to experiment with adversarial noise levels and see how even small changes can impact model predictions.

How Fast Gradient Sign Method Works

Introduction to FGSM

The Fast Gradient Sign Method (FGSM) is a popular adversarial attack technique used in the field of machine learning and deep learning.
It perturbs the input data by adding small changes based on the gradients of the model’s loss function, creating adversarial examples that mislead the model.

Generating Adversarial Examples

FGSM calculates the gradient of the loss function with respect to the input data.
The perturbation is crafted by taking the sign of this gradient and scaling it with a predefined parameter (epsilon).
The perturbed input is then fed back into the model to test its vulnerability to adversarial attacks.

Applications

FGSM is widely used to evaluate and improve the robustness of machine learning models.
It is applied in tasks such as image classification, where adversarial examples are generated to reveal weaknesses in the model.
This technique is also used to develop defenses against adversarial attacks.

Advantages and Limitations

FGSM is computationally efficient and easy to implement, making it suitable for large-scale testing.
However, it creates adversarial examples with a single step, which might not always uncover the most complex vulnerabilities in robust models.

⚡ Fast Gradient Sign Method: Core Formulas and Concepts

1. Basic FGSM Formula

Given a model with loss function J(θ, x, y), the FGSM adversarial example is calculated as:

x_adv = x + ε * sign(∇_x J(θ, x, y))

Where:

• x is the original input
• y is the true label
• ε is the perturbation magnitude
• ∇_x J is the gradient of the loss with respect to the input
• sign() is the element-wise sign function

2. Sign Function Definition

sign(z) =
  +1 if z > 0
   0 if z = 0
  -1 if z < 0

3. Model Prediction Change

After adding the perturbation, the model may predict a different class:

f(x) = y
f(x_adv) ≠ y

4. Targeted FGSM Variant

For a targeted attack toward class y_target:

x_adv = x - ε * sign(∇_x J(θ, x, y_target))

The sign is flipped to move the input toward the target class.

Visualisation of FGSM

This diagram provides a visual explanation of how FGSM works, a method used in adversarial machine learning to generate adversarial examples that fool deep neural networks by adding small perturbations to input data.

1. Original Input (x)

The process begins with a clean input image x, which is initially fed into a model. This image represents the data that the model would normally classify correctly.

• Example: An image of a person.
• Input symbol: x

2. Gradient Computation

The model computes the gradient of the loss function J(θ, x, y) with respect to the input x, where:

• θ — model parameters
• y — true label

This gradient indicates the direction in which the loss increases most rapidly with respect to the input.

3. Perturbation Generation

The perturbation is calculated using the sign of the gradient and a small scalar η:

• η · sign(∇ₓJ(θ, x, y))

This creates a noise pattern that is intentionally designed to maximize the model’s prediction error, but is small enough to be imperceptible to humans.

4. Adversarial Input (x̄)

The adversarial example x̄ is constructed by adding the perturbation to the original input:

• x̄ = x + η · sign(∇ₓJ(θ, x, y))

This new image looks visually similar to x but can cause the model to misclassify it, demonstrating a vulnerability in the system.

Key Purpose

FGSM helps researchers understand and improve the robustness of AI models by exposing how small, calculated changes to input data can lead to incorrect predictions.

Types of FGSM

• Standard FGSM. The basic version of FGSM generates adversarial examples using a single step based on the gradient of the loss function.
• Iterative FGSM (I-FGSM). An extension of FGSM that applies the perturbation iteratively, creating stronger adversarial examples.
• Targeted FGSM. Generates adversarial examples to misclassify inputs as a specific target class, rather than any incorrect class.

Practical Use Cases for Businesses Using FGSM

• Fraud Detection Testing. Generates adversarial examples to expose vulnerabilities in transaction fraud detection systems, enabling improvements in AI model robustness.
• Medical Imaging Validation. Tests AI diagnostic tools by introducing adversarial perturbations to imaging data, ensuring accuracy in critical healthcare applications.
• Autonomous Navigation. Evaluates object detection and path planning algorithms in autonomous vehicles under adversarial conditions, improving safety and reliability.
• Product Recommendation Security. Enhances recommendation systems by ensuring resistance to adversarial inputs that could skew results or harm user experience.
• Intrusion Detection. Identifies potential security gaps in AI-based intrusion detection systems by simulating adversarial attacks, bolstering network security measures.

🧪 FGSM: Practical Examples

Example 1: Crafting an Adversarial Image

Original input image x is correctly classified as digit 7 by a model:

f(x) = 7

Gradient of loss w.r.t. input gives:

∇_x J = [0.1, -0.2, 0.3, ...]

Using ε = 0.01 and applying FGSM:

x_adv = x + 0.01 * sign(∇_x J)

The resulting image x_adv is misclassified as 3:

f(x_adv) = 3

Example 2: Targeted FGSM Attack

We want to fool the model into classifying input x as class 2:

x_adv = x - ε * sign(∇_x J(θ, x, y_target=2))

By using the negative gradient, the perturbation leads the model toward the desired target class.

Model output:

f(x) = 6
f(x_adv) = 2

Example 3: Visualizing the Perturbation

Let the perturbation vector be:

δ = ε * sign(∇_x J) = [0.01, -0.01, 0.01, ...]

We can visualize the difference between the original and adversarial image:

Difference = x_adv - x = δ

Even though the change is small and invisible to the human eye, it can drastically alter the model's prediction.

import torch

def fgsm_attack(image, epsilon, data_grad):
    # Generate adversarial image by adding sign of gradient
    sign_data_grad = data_grad.sign()
    perturbed_image = image + epsilon * sign_data_grad
    return torch.clamp(perturbed_image, 0, 1)
  

model.eval()
image.requires_grad = True

# Forward pass
output = model(image)
loss = loss_fn(output, target)

# Backward pass
model.zero_grad()
loss.backward()
data_grad = image.grad.data

# Generate adversarial example
epsilon = 0.03
adv_image = fgsm_attack(image, epsilon, data_grad)

# Evaluate model on adversarial input
output_adv = model(adv_image)
  

Conclusion

Fast Gradient Sign Method (FGSM) is a crucial technique for testing and improving the robustness of AI models against adversarial attacks.
As industries increasingly rely on AI, FGSM's role in enhancing model security and reliability will continue to grow, driving advancements in AI defense mechanisms.

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