Neural Networks Demystified [Part 3: Gradient Descent]

Summary

This script discusses improving neural network predictions by introducing the concept of a cost function to quantify prediction errors. It explains the necessity of minimizing this cost function, which is a function of the network's weights and examples. The script highlights the impracticality of brute force optimization due to the curse of dimensionality and introduces gradient descent as a more efficient method. It also touches on the importance of choosing a convex cost function to avoid local minima and mentions stochastic gradient descent as a variant that can work with non-convex functions.

Takeaways

• 🧠 **Inaccuracy in Predictions**: The initial neural network predictions were inaccurate, indicating a need for improvement.
• 🔍 **Cost Function Introduction**: A cost function was introduced to quantify the error in predictions, which is essential for model improvement.
• 📈 **Cost Function Calculation**: The cost function sums the squares of the errors, multiplied by one half for simplification.
• 🔧 **Minimizing the Cost**: The process of training a network is essentially minimizing this cost function.
• 🔗 **Cost Function Dependency**: The cost function depends on the examples and the weights of the synapses.
• 🚀 **Brute Force Inefficiency**: Trying all possible weights (brute force) is inefficient due to the curse of dimensionality.
• ⏱️ **Time Complexity**: As the number of weights increases, the time required to find the optimal weights grows exponentially.
• 📉 **Gradient Descent**: Gradient descent is introduced as a method to minimize the cost function by iteratively moving in the direction that reduces the cost.
• 📐 **Convexity of Cost Function**: The choice of a quadratic cost function ensures its convexity, which aids gradient descent in finding the global minimum.
• 🔄 **Stochastic Gradient Descent**: Using examples one at a time (stochastic gradient descent) can sometimes bypass the issue of non-convex loss functions.
• 🛠️ **Batch Gradient Descent**: The script concludes with the intention to implement batch gradient descent, using all examples at once to keep the function convex.

Q & A

What was the initial problem with the neural network's predictions?

The initial problem was that the neural network made really bad predictions of a test score based on the number of hours slept and studied the night before.

What is a cost function in the context of neural networks?

A cost function is a measure of how wrong or costly our model's predictions are given our examples. It quantifies the error between the predicted values and the actual values.

How is the overall cost computed in the script?

The overall cost is computed by squaring each error value and adding these values together, then multiplying by one half to simplify future calculations.

What is meant by minimizing the cost function?

Minimizing the cost function means adjusting the weights of the neural network to reduce the error between the predicted and actual values to the smallest possible amount.

Why is it not feasible to try all possible weights to find the best combination?

Trying all possible weights is not feasible due to the curse of dimensionality, which exponentially increases the number of evaluations required as the number of weights increases.

What is the curse of dimensionality?

The curse of dimensionality refers to the phenomenon where the volume of the space increases so fast with the addition of each dimension that even a large number of samples provides little information about the overall space.

How long would it take to evaluate all possible combinations of nine weights?

Evaluating all possible combinations of nine weights would take one quadrillion, 268 trillion, 391 billion, 679 million, 350 thousand, 583 years and a half.

What is numerical estimation and why is it used?

Numerical estimation is a method used to approximate the value of a mathematical function. It's used to determine the direction in which the cost function is decreasing, which helps in adjusting the weights.

What is gradient descent and how does it help in minimizing the cost function?

Gradient descent is an optimization algorithm used to find the minimum of a function by iteratively moving in the direction of the steepest descent as defined by the negative of the gradient. It helps in minimizing the cost function by taking steps in the direction that reduces the cost the most.

Why is the cost function chosen to be the sum of squared errors?

The cost function is chosen to be the sum of squared errors to exploit the convex nature of quadratic equations, which ensures that the function has a single global minimum and no local minima.

What is the difference between batch gradient descent and stochastic gradient descent?

Batch gradient descent uses all examples at once to compute the gradient and update the weights, whereas stochastic gradient descent uses one example at a time, which can lead to a noisy but potentially faster optimization process.