Neural Networks Demystified [Part 6: Training]

Summary

This script outlines the process of training a neural network using Python, including computing a cost function, gradient descent, and numerical validation of gradient computations. It introduces the BFGS algorithm, a sophisticated variant of gradient descent that estimates the second derivative to optimize the cost function surface. The script also highlights the importance of avoiding overfitting by ensuring the model's performance on unseen data. The training process is depicted, showing a decrease in cost and the model's ability to predict test scores based on sleep and study hours, with a surprising revelation that sleep has a more significant impact on grades than study time.

Takeaways

• 🤖 **Building a Neural Network**: The script discusses constructing a neural network in Python.
• 📊 **Cost Function**: It mentions the computation of a cost function to evaluate the network's performance.
• 🔍 **Gradient Computation**: The script explains computing the gradient of the cost function for training purposes.
• 🧭 **Numerical Validation**: It highlights the importance of numerically validating gradient computations.
• 🏋️‍♂️ **Gradient Descent**: The script introduces training the network using gradient descent.
• 🔄 **Challenges with Gradient Descent**: It points out potential issues with gradient descent like getting stuck in local minima or moving too quickly/slowly.
• 📚 **Optimization Field**: The script refers to the broader field of mathematical optimization for improving neural network training.
• 🔍 **BFGS Algorithm**: It introduces the BFGS algorithm, a sophisticated variant of gradient descent, for more efficient training.
• 🛠️ **Implementation with SciPy**: The script describes using SciPy's BFGS implementation within the minimize function for neural network training.
• 📈 **Monitoring Training**: It suggests implementing a callback function to track the cost function during training.
• 📉 **Overfitting Concerns**: The script concludes with a caution about overfitting, even when the network performs well on training data.

Q & A

What is the primary purpose of a cost function in a neural network?

The primary purpose of a cost function in a neural network is to measure how well the network is performing by quantifying the difference between the predicted outputs and the actual outputs.

Why is gradient computation important in training a neural network?

Gradient computation is important because it tells us the direction in which the cost function decreases the fastest, allowing us to adjust the network's parameters to minimize the cost and improve the network's performance.

What is the potential issue with using consistent step sizes in gradient descent?

Using consistent step sizes in gradient descent can lead to issues such as getting stuck in a local minimum or flat spot, moving too slowly and never reaching the minimum, or moving too quickly and overshooting the minimum.

What is mathematical optimization and how does it relate to training neural networks?

Mathematical optimization is a field dedicated to finding the best combination of inputs to minimize the output of an objective function. It relates to training neural networks as it deals with optimizing the network's parameters to minimize the cost function.

What is the BFGS algorithm and how does it improve upon standard gradient descent?

The BFGS algorithm is a sophisticated variant of gradient descent that estimates the second derivative or curvature of the cost function surface. It uses this information to make more informed movements towards the minimum, overcoming some limitations of plain gradient descent.

Why is it necessary to use a wrapper function when applying the BFGS algorithm to a neural network?

It is necessary to use a wrapper function when applying the BFGS algorithm to a neural network because the neural network implementation may not follow the required input and output semantics of the minimize function in the scipy.optimize package.

What role does the callback function play during the training of the neural network?

The callback function allows us to track the cost function value as the network trains, providing insights into the training process and helping to monitor the network's performance over iterations.

How does the training process affect the cost function value?

As the network is trained, the cost function value should ideally decrease monotonically, indicating that the network is learning and improving its predictions with each iteration.

What does it mean for the gradient to have very small values at the solution?

Having very small gradient values at the solution indicates that the cost function is flat at the minimum, suggesting that the network has found a stable point where further changes to the parameters would not significantly reduce the cost.

How does the trained network help in predicting test scores based on sleep and study hours?

Once trained, the network can predict test scores by inputting the number of hours slept and studied, providing insights into how these factors might influence performance.

What is the danger of overfitting in machine learning and how does it relate to the trained network?

Overfitting occurs when a model performs exceptionally well on training data but fails to generalize to new, unseen data. In the context of the trained network, it means that even though it performs well on the training data, it may not accurately predict test scores in real-world scenarios.

Outlines

00:00

🤖 Training a Neural Network with Gradient Descent

This paragraph discusses the process of training a neural network using gradient descent. It starts by summarizing the previous steps taken to build the network, compute the cost function, and validate the gradient computations. The challenges of implementing gradient descent, such as getting stuck in local minima or moving too slowly or quickly, are highlighted. The paragraph then introduces the broader field of mathematical optimization and mentions the BFGS algorithm as a sophisticated variant of gradient descent that estimates the second derivative to make more informed updates. The use of the BFGS algorithm through the scipy.optimize package is described, including setting up the objective function, initial parameters, and callback function to track the cost. The success of the training is shown by plotting the cost against the number of iterations, and the trained parameters are used to make predictions. The paragraph concludes by cautioning against overfitting, noting that good performance on training data does not guarantee real-world applicability.

Keywords

💡Neural Network

A neural network is a computational model that is inspired by the structure and function of biological neural networks in the brain. In the video, the neural network is designed to predict test scores based on input data like sleep and study hours. It is trained using techniques like gradient descent to minimize error in predictions.

💡Cost Function

The cost function measures how well a neural network performs by calculating the error between predicted outputs and actual outputs. In the script, the cost function is used to evaluate the performance of the network and guide optimization. Lowering the cost means the model is improving its accuracy.

💡Gradient Descent

Gradient descent is an optimization algorithm used to minimize the cost function by adjusting the network's parameters in small steps. In the video, it is mentioned as a method to 'march downhill' towards a minimum, although it can face challenges like getting stuck in local minima or moving too slowly.

💡BFGS Algorithm

The BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm is a more advanced optimization technique than standard gradient descent. It estimates the curvature of the cost function to make more informed adjustments to the network's parameters. The video highlights its use to improve the convergence of training the neural network.

💡Second Derivative (Curvature)

The second derivative or curvature of the cost function describes how the slope changes, giving more insight into the 'shape' of the cost landscape. By estimating this, BFGS improves optimization by taking more efficient steps, avoiding some issues that arise with simple gradient descent, like slow progress or getting stuck in flat regions.

💡Overfitting

Overfitting occurs when a model performs exceptionally well on training data but fails to generalize to new, unseen data. In the video, the neural network fits the training data perfectly, but the speaker warns that this doesn't guarantee the model will work in real-world scenarios. It's a common issue in machine learning.

💡Optimization

Optimization refers to the process of adjusting the model's parameters to minimize the cost function. The video discusses various optimization methods, from basic gradient descent to more advanced techniques like the BFGS algorithm, which can find better solutions faster.

💡Jacobian

The Jacobian matrix contains the partial derivatives of a function and helps in calculating the gradient. In the video, the 'Jacobian' parameter is set to True to indicate that the neural network is computing its gradients internally during training.

💡Training Data

Training data consists of the inputs and corresponding outputs that are used to teach the neural network how to make predictions. In this video, the training data involves the number of hours spent sleeping and studying, with the goal of predicting a test score. The network's performance is judged based on how well it fits this data.

💡Callback Function

A callback function is a function that is executed after certain events or steps during training. In the video, a callback function is used to track the value of the cost function as the network trains, helping visualize its progress and convergence.

Highlights

Building a neural network in Python

Computing a cost function to evaluate network performance

Computing the gradient of the cost function for training

Numerically validating gradient computations

Deciding to train the network using gradient descent

Challenges with gradient descent implementation

Risks of getting stuck in local minimum or flat spot

Potential issues with step size in gradient descent

The complexity of gradient descent in high-dimensional space

No guarantee of convergence with gradient descent

Optimization techniques beyond neural networks

Yann LeCun's 1998 publication on optimization techniques

Using the BFGS algorithm for more sophisticated gradient descent

BFGS algorithm estimates second derivative for better movements

Using SciPy's minimize function with BFGS

Wrapping the neural network for compatibility with minimize function

Implementing a callback function to track cost

Observing a monotonically decreasing cost function

Fewer function evaluations with BFGS compared to brute force

Evaluating the gradient at the solution

Training a network to predict test scores based on sleep and study hours

Exploring the input space for optimal sleep and study combinations

Potential overfitting despite excellent training performance

Transcripts

so far we've built a neural network in

Python computed a cost function to let

us know how well our network is

performing computed the gradient of our

cost function so we can train our

Network and last time we numerically

validated our gradient

computations after all that work it's

finally time to train our neural

network back in part three we decided to

train our network using gradient descent

while gradient descent is conceptually

pretty straightforward its

implementation can actually be quite

complete

especially as we increase the size and

number of layers in our

network if we just March downhill with

consistent step sizes we may get stuck

in a local minimum or flat spot we may

move too slowly and never reach our

minimum or we may move too quickly and

bounce out of our minimum and remember

all this must happen in high dimensional

space making things significant more

complex gradient descent is a

wonderfully clever method but provides

no guarantee that we will converge to a

good solution that we will converge to a

solution in a certain amount of time or

that we will converge to a solution at

all the good and bad news here is that

this problem is not unique to neural

networks there's an entire field

dedicated to finding the best

combination of inputs to minimize the

output of an objective function the

field of mathematical

optimization the bad news is that

optimization can be a bit overwhelming

there are many different techniques that

could apply to to our problem part of

what makes optimization challenging is

the broad range of approaches covered

from very rigorous theoretical methods

to Hands-On more heuristics driven

methods Yan laon's 1998 publication

efficient back propop presents an

excellent review of various optimization

techniques as applied to neural

networks here we're going to use a more

sophisticated variant on gradient

descent the popular Bren Fletcher gold

farb shano numerical optimization

algorithm the bfgs algorithm overcomes

some of the limitations of plain

gradient descent by estimating the

second derivative or curvature of the

cost function surface and using this

information to make more informed

movements

downhill bfgs will allow us to find

Solutions more often and more quickly

we'll use the bfgs implementation built

into the scipi optimized package

specifically within the minimize

function to use bfgs the minimize

function requires us to pass in an

objective function that accepts a vector

of parameters input and output data and

returns both the cost and gradients our

neural network implementation doesn't

quite follow these semantics so we'll

use a wrapper function to give it this

Behavior we'll also pass in initial

parameters set the Jacobian parameter to

True since we're Computing the gradient

within our neural network class set the

method to bfgs pass in our input and

output data and some options finally

we'll Implement a call back function

that allows us to track the cost

function value as we train the network

once the network is trained we'll

replace the original random parameters

with the trained parameters if we plot

the cost against the number of

iterations through training we see a

nice monotonically decreasing function

further we see that the number of

function evaluations required to find a

solution is less than 100 and far less

than the 10 to the 27th function

evaluations that would have been

required to find a solution by Brute

Force as shown in part

three finally we can evaluate our

gradient at our solution and see very

small values this makes sense as our

minimum should be quite

flat the more exciting thing here is

that we have finally trained a network

that can predict your score on a test

based on how many hours you sleep and

how many hours you study the night

before if we run our training data

through our forward method now we see

that our predictions are excellent we

can go one step further and explore the

input space for various combinations of

hour sleeping and hour studying and

maybe we can figure out an optimal

combination of the two for your next

test our results look pretty reasonable

and we see that for our model sleep

actually has a bigger impact on your

grade than study something I wish I had

realized when I was in school so we're

done right nope we've made possibly the

most dangerous and tempting error in

machine learning overfitting Although

our network is performing incredibly

well in our training data that doesn't

mean that our model is a good fit for

the real world and that's what we'll

work on next

time