Neural Networks Demystified [Part 6: Training]
Summary
TLDRThis 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
🤖 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.
Mindmap
Keywords
💡Neural Network
💡Cost Function
💡Gradient Descent
💡BFGS Algorithm
💡Second Derivative (Curvature)
💡Overfitting
💡Optimization
💡Jacobian
💡Training Data
💡Callback Function
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
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