Neural Networks Demystified [Part 5: Numerical Gradient Checking]

Summary

This script discusses the importance of accurately calculating gradients in neural networks to avoid subtle errors that degrade performance. It suggests using numerical gradient checking as a testing method, akin to unit testing in software development. The script explains derivatives, their calculation using the definition, and how to apply this to a simple function and a neural network. It also introduces a method to quantify the similarity between computed and numerical gradients, ensuring gradient computations are correct before training the network.

Takeaways

• 🧮 Calculus is used to find the rate of change of cost with respect to parameters in neural networks.
• 🚩 Errors in gradient implementation can be subtle and hard to detect, leading to significant issues.
• 🔍 Testing gradient computation is crucial, similar to unit testing in software development.
• 📚 Derivatives are fundamental to understanding the slope or steepness of a function.
• 📐 The definition of a derivative involves the change in y values over an infinitely small change in x.
• 💻 Computers handle numerical estimates of derivatives effectively, despite not dealing well with infinitesimals.
• 📈 Numerical gradient checking involves comparing computed gradients with those estimated using a small epsilon.
• 🔢 For a simple function like x^2, numerical gradient checking can validate the correctness of symbolic derivatives.
• 🤖 The same numerical approach can be applied to neural networks to evaluate the gradient of the cost function.
• 📊 Comparing the numerical and official gradient vectors can help ensure gradient calculations are correct, with results typically around 10^-8 or less.

Q & A

What is the main concern when implementing calculus for finding the rate of change of cost with respect to parameters in a neural network?

The main concern is the potential for making mistakes in the calculus steps, which can be hard to detect since the code may still appear to function correctly even with incorrect gradient implementations.

Why are big, in-your-face errors considered less problematic than subtle errors when building complex systems?

Big errors are immediately noticeable and it's clear they need to be fixed, whereas subtle errors can hide in the code, causing slow degradation of performance and wasting hours of debugging time.

What is the suggested method to test the gradient computation part of neural network code?

The suggested method is to perform numerical gradient checking, similar to unit testing in software development, to ensure the gradients are computed and coded correctly.

What is the definition of a derivative and how does it relate to the slope of a function?

The derivative is defined as the slope of a function at a specific point, calculated as the limit of the ratio of the infinitesimal change in the function's output over the change in its input.

Why is it important to understand the definition of a derivative even after learning calculus shortcuts like the power rule?

Understanding the definition of a derivative provides a deeper insight into what a derivative represents and is crucial for solving problems where numerical methods are required, such as gradient checking.

How does the concept of 'epsilon' play a role in numerical gradient checking?

Epsilon is a small value used to perturb the input around a test point to estimate the derivative numerically. It helps in calculating the slope of the function at that point by finding the function values just above and below the test point.

What is the symbolic derivative of the function x squared?

The symbolic derivative of x squared is 2x, which is derived using the power rule of differentiation.

How can numerical gradient checking be applied to evaluate the gradient of a neural network's cost function?

By perturbing each weight with a small epsilon, computing the cost function for the perturbed values, and calculating the slope between these values, one can numerically estimate the gradient of the cost function.

What is meant by 'perturb' in the context of gradient checking?

To 'perturb' in gradient checking means to slightly alter the weights by adding or subtracting a small epsilon to compute the cost function and estimate the gradient.

How can you quantify the similarity between the numerical gradient vector and the official gradient calculation?

The similarity can be quantified by dividing the norm of the difference between the two vectors by the norm of their sum. A result on the order of 10^-8 or less indicates a correct gradient computation.

What is the next step after verifying the gradient computations in a neural network?

The next step after verifying the gradient computations is to train the neural network using the confirmed correct gradients.

Outlines

00:00

📐 Calculus and Gradient Checking

The paragraph discusses the importance of accurately calculating the rate of change of cost with respect to parameters in a neural network using calculus. It highlights the potential for errors in calculus steps and the difficulty in detecting these errors, as incorrect gradient implementations can appear to function correctly. The paragraph suggests testing the gradient computation part of the code to ensure accuracy, similar to unit testing in software development. It introduces the concept of numerical gradient checking as a method to verify the correctness of gradient calculations. The process involves using the definition of a derivative to compute the slope of a function at a given point, which is then compared to the symbolic derivative. The paragraph also explains the practical approach of using a small, reasonable value for epsilon to approximate derivatives numerically, which is key to gradient checking.

Keywords

💡Rate of change

The rate of change refers to how quickly a quantity changes with respect to another quantity. In the context of the video, it pertains to the derivative of the cost function, J, with respect to the parameters, W, which is crucial in optimization algorithms like gradient descent. The video emphasizes the importance of accurately calculating this rate of change to ensure the proper functioning of neural networks.

💡Parameters (W)

Parameters, often denoted as W, are the weights and biases within a neural network that are adjusted during the training process. The video discusses the importance of correctly calculating the rate of change of the cost function with respect to these parameters, as errors in this calculation can lead to suboptimal network performance.

💡Cost function (J)

The cost function, or loss function, is a measure of how well the neural network is performing. It quantifies the difference between the predicted outputs and the actual targets. In the video, the focus is on finding the rate of change of this cost function with respect to the network's parameters to minimize it effectively.

💡Gradient

A gradient in the context of neural networks is a vector of partial derivatives of the cost function with respect to each parameter. It points in the direction of the steepest increase of the cost function. The video explains the process of computing gradients to update the parameters and the need to verify their correctness to avoid subtle errors.

💡Numerical gradient checking

Numerical gradient checking is a method used to verify the correctness of an analytically computed gradient. It involves approximating the gradient numerically and comparing it to the analytical gradient. The video suggests using this technique to ensure that the gradients computed in the neural network's code are accurate.

💡Derivative

A derivative in calculus represents the rate at which a function changes with respect to its input. The video reviews derivatives as a foundational concept for understanding how changes in input variables (like weights in a neural network) affect the output (like the cost function).

💡Slope

Slope is a term used to describe the steepness or incline of a line, which in the context of derivatives, represents how quickly a function is changing. The video uses the concept of slope to explain the definition of a derivative and how it can be approximated numerically for gradient checking.

💡Epsilon

Epsilon, often used as a small positive number, is a term in the video that represents a small change made to a parameter when performing numerical gradient checking. By perturbing each weight by epsilon, the video demonstrates how to approximate the derivative of the cost function with respect to that weight.

💡Perturb

To perturb in this context means to slightly alter a value, such as a weight in a neural network, to observe the effect on the cost function. The video describes perturbing weights by adding and subtracting epsilon to compute the numerical gradient and compare it with the analytical gradient.

💡Vector

A vector in mathematics is a quantity with both magnitude and direction, which can be represented as an array of numbers. In the video, the gradient is described as a vector because it contains multiple components, each representing the partial derivative of the cost function with respect to a specific parameter.

💡Norm

The norm of a vector is a measure of its length or magnitude. In the context of the video, the norm is used to quantify the difference between the numerically computed gradient and the analytical gradient. Comparing these norms helps determine the accuracy of the gradient computation.

Highlights

Calculus is used to find the rate of change of cost with respect to parameters in neural networks.

Mistakes in calculus steps can lead to incorrect gradient implementations.

Networks lack a mechanism to indicate incorrect gradient computations.

Subtle errors in gradient computation can degrade performance over time.

Testing gradient computation is akin to unit testing in software development.

Derivatives provide the slope or steepness of a function.

The power rule is a common method for calculating derivatives.

Computing derivatives using the definition offers a deeper understanding.

The derivative definition is a slope formula applied to an infinitely small region.

Computers struggle with infinitely small numbers in calculations.

Reasonably small epsilon values can provide good numerical estimates of derivatives.

Numerical gradient checking compares computed derivatives to symbolic ones.

Testing one gradient at a time simplifies the process.

Perturbing weights and computing the cost function helps estimate gradients.

A numerical gradient vector is created by comparing cost function values with perturbed weights.

Comparing the numerical gradient vector to the official calculation verifies correctness.

The norm of the difference divided by the norm of the sum quantifies similarity between vectors.

Results should be on the order of 10^-8 or less for correct gradient computation.

Gradient errors can be eliminated before they impact neural network training.

Transcripts

Last time, we did a bunch of calculus to find the rate of change of our cost, J,

with respect to our parameters, W.

Although each calculus step was pretty straight forward, it’s still relatively easy to make mistakes.

What’s worse, is that our network doesn’t have a good way to tell us that it’s broken

– code with incorrectly implemented gradients may appear to be functioning just fine.

This is the most nefarious kind of error when building complex systems.

Big, in-your-face errors suck initially

but it’s clear that you must fix this error for your work to succeed.

More subtle errors can be more troublesome because they hide in your code and steal hours of your time,

slowly degrading performance, while you wonder what the problem is.

A good solution here is to test the gradient computation part of our code,

just as developer would unit test new portions of their code.

We’ll combine a simple understanding of the derivative with some mild cleverness to perform numerical gradient checking.

If our code passes this test,

we can be quite confident that we have computed and coded up our gradients correctly.

To get started, let’s quickly review derivatives.

Derivatives tell us the slope, or how steep a function is.

Once you’re familiar with calculus,

it’s easy to take for granted the inner workings of the derivative

we just accept that the derivative of x^2 is 2x by the power rule.

However, depending on how mean your calculus teacher was,

you may have spent months not being taught the power rule, and instead required to compute derivatives using the definition.

Taking derivatives this way is a bit tedious,

but still important

it provides us a deeper understanding of what a derivative is,

and it’s going to help us solve our current problem.

The definition of the derivative is really a glorified slope formula.

The numerator gives us the change in y values,

while the denominator is convenient way to express the change in x values.

By including the limit, we are applying the slope formula across an infinitely small region

it’s like zooming in on our function, until it becomes linear.

The definition tells us to zoom in until our x distance is infinitely small,

but computers can’t really handle infinitely small numbers,

especially when they’re in the bottom parts of fractions

if we try to plug in something too small,

we will quickly lose precision.

The good news here is that if we plug in something reasonable small,

we can still get surprisingly good numerical estimates of the derivative.

We’ll modify our approach slightly by picking a point in the middle of the interval we would like to test,

and call the distance we move in each direction epsilon.

Let’s test our method with a simple function, x squared.

We’ll choose a reasonable small value for epsilon,

and compute the slope of x^2 at a given point by finding the function value just above and just below our test point.

We can then compare our result to our symbolic derivative 2x,

at the test point

If the numbers match, we’re in business!

We can use the same approach to numerically evaluate the gradient of our neural network.

It’s a little more complicated this time, since we have 9 gradient values,

and we’re interested in the gradient of our cost function.

We’ll make things simpler by testing one gradient at a time.

We’ll “perturb” each weight

adding epsilon to the current value and computing the cost function,

subtracting epsilon from the current value and computing the cost function,

and then computing the slope between these two values.

We’ll repeat this process across all our weights,

and when we’re done we’ll have a numerical gradient vector,

with the same number of values as we have weights.

It’s this vector we would like to compare to our official gradient calculation.

We see that our vectors appear very similar,

which is a good sign, but we need to quantify just how similar they are.

A nice way to do this is to divide the norm of the difference

by the norm of the sum of the vectors we would like to compare.

Typical results should be on the order of 10^-8 or less if you’ve computed your gradient correctly.

And that’s it, we can now check our computations and eliminate gradient errors before they become a problem.

Next time we’ll train our Neural Network.