Backpropagation — Topic 79 of Machine Learning Foundations

Jon Krohn
8 Dec 202106:05
EducationalLearning
32 Likes 10 Comments

TLDRThe video script explains the fundamental relationship between partial derivatives and the back propagation algorithm, which is widely used in training artificial neural networks, including deep learning models. Back propagation is essentially an application of the chain rule of partial derivatives to optimize model parameters. The script clarifies that the theory behind back propagation is the same whether applied to a simple regression model or a complex deep neural network with millions of parameters across thousands of layers. It also introduces the concept of mini-batches for handling large datasets in deep learning, where a forward pass through the network generates an estimated output, followed by a backward pass that employs automatic differentiation to calculate the partial derivatives of the cost function with respect to each model parameter. Finally, the script outlines the use of gradient descent to update the model's parameters, emphasizing the iterative process of forward pass, cost comparison, automatic differentiation, and parameter adjustment.

Takeaways
  • 📚 Back propagation is a concept used in training artificial neural networks, particularly deep learning networks.
  • 🔗 The underlying theory of back propagation is the chain rule of partial derivatives of cost, which is fundamental in calculus.
  • 📈 The chain rule applies to neural networks with thousands of layers and millions or billions of model parameters.
  • 🔑 The partial derivatives of cost with respect to model parameters, a concept introduced through simpler models, extends to complex deep neural networks.
  • 📉 In machine learning, the cost function often used is mean squared error, which compares the estimated output (y-hat) with the true output (y).
  • 🔁 Back propagation involves a forward pass through the network to estimate y-hat, followed by a backward pass for automatic differentiation.
  • 📊 Training deep learning models involves large datasets that are divided into mini-batches for efficiency.
  • 🧠 Each neuron in a neural network has its own parameters, contributing to the complexity of the model.
  • 🛠️ Tools like PyTorch and TensorFlow use autograd to calculate the partial derivatives of cost with respect to all model parameters.
  • 🔍 After calculating the partial derivatives, gradient descent is used to update the model parameters, including weights and biases.
  • 📌 The process of training a neural network includes four main steps: forward pass, cost comparison, automatic differentiation, and gradient descent.
Q & A

  • What is the relationship between partial derivatives and back propagation?

    -The relationship between partial derivatives and back propagation is that back propagation is essentially the application of the chain rule of partial derivatives to calculate the gradients of the cost function with respect to the model parameters in a neural network. This is crucial for updating the parameters to minimize the cost during training.

  • Why is back propagation widely used in training artificial neural networks?

    -Back propagation is widely used because it efficiently computes the gradient of the cost function for all the parameters in a neural network, which is essential for gradient-based optimization algorithms like gradient descent to adjust the parameters and improve the model's performance.

  • What is the role of the chain rule of partial derivatives in back propagation?

    -The chain rule of partial derivatives is fundamental to back propagation as it allows the computation of the gradient of the cost function through the nested functions or layers in a neural network, enabling the update of each parameter based on its contribution to the overall cost.

  • How does the concept of mini-batches come into play in the context of back propagation?

    -Mini-batches are smaller subsets of the large training dataset. By using mini-batches, the model performs a forward pass on a smaller set of data, calculates the cost and gradients, and updates the parameters. This approach is more memory-efficient and can lead to better generalization compared to using the entire dataset at once.

  • What is a forward pass in the context of neural networks?

    -A forward pass is the process of inputting data into a neural network and propagating it through the layers until an output (y-hat) is produced. It is the first step in both the training and inference phases, where the network makes its predictions.

  • What is the purpose of comparing y-hat to y using a cost function?

    -The comparison of y-hat (predicted output) to y (true output) using a cost function measures the error or difference between the prediction and the actual value. This cost is used to guide the optimization process and update the model's parameters to reduce the error in future predictions.

  • How does automatic differentiation facilitate back propagation?

    -Automatic differentiation, often implemented via libraries like Autograd in PyTorch or TensorFlow, calculates the partial derivatives of the cost function with respect to each model parameter. This simplifies and speeds up the back propagation process by eliminating the need for manual derivative calculations.

  • What are the two different kinds of parameters in a neural network?

    -The two different kinds of parameters in a neural network are weights and biases. Weights are the coefficients that are multiplied by inputs, and biases are the additional terms added to the weighted sum before applying an activation function.

  • What is the final step after performing automatic differentiation in back propagation?

    -The final step after performing automatic differentiation is to use an optimization algorithm, such as gradient descent, to adjust the parameters of the model in the direction that minimizes the cost function.

  • Why is understanding partial derivatives important for studying deep learning?

    -Understanding partial derivatives is important for deep learning because it forms the mathematical foundation for understanding how changes in model parameters affect the cost function. This knowledge is essential for the optimization process, which is a core part of training deep neural networks.

  • How does the process of back propagation relate to the concept of forward propagation?

    -Back propagation is essentially the reverse of forward propagation. While forward propagation is the process of computing the output of the neural network for given inputs, back propagation is the process of computing the gradients of the cost function with respect to the inputs, allowing for the update of the network's parameters to reduce the cost.

  • What is the significance of the number of layers in a neural network when discussing back propagation?

    -The number of layers in a neural network is significant because it affects the complexity of the back propagation process. Deeper networks with more layers require the computation of more gradients, which can be more computationally intensive but also allow for the modeling of more complex patterns in the data.

Outlines
00:00
🧠 Understanding Backpropagation in Neural Networks

This paragraph introduces the concept of backpropagation, a fundamental technique for training artificial neural networks, including deep learning models. Backpropagation leverages the chain rule of partial derivatives to calculate the cost function's gradient with respect to the model's parameters. The paragraph emphasizes that the theory behind backpropagation is the same regardless of the neural network's complexity, whether it's a simple model with a few layers or a deep learning model with thousands of layers and billions of parameters. The process involves a forward pass through the network to generate an output estimate, followed by a comparison with the true output using a cost function. The key takeaway is that the understanding of partial derivatives from simpler models directly applies to the more complex deep neural networks.

05:00
🔢 Automatic Differentiation and Gradient Descent in Neural Networks

The second paragraph delves into the specifics of the backpropagation process, highlighting the role of automatic differentiation in calculating the partial derivatives of the cost function with respect to each model parameter. It outlines the steps involved in training a neural network: performing a forward pass to obtain an estimated output, comparing this estimate to the actual output using a cost function, using automatic differentiation to find the gradient of the cost, and finally, employing gradient descent to update the model's parameters. The distinction between two types of parameters in neural networks, weights and biases, is mentioned, though not elaborated upon due to the foundational nature of the course. The paragraph concludes by reiterating the process steps and segueing into the next topic of higher order partial derivatives.

Mindmap
Keywords
💡Back Propagation
Back propagation is a method used for training artificial neural networks, particularly deep learning networks. It involves the calculation of the gradient of the loss function with respect to the model's parameters. In the context of the video, back propagation utilizes the chain rule of partial derivatives to update the network's weights and biases, which is a crucial step in the learning process. The script explains that this process is an extension of the same theory used for simpler models, now applied to networks with potentially millions of parameters.
💡Partial Derivatives
Partial derivatives are a mathematical concept that describes the rate at which a function changes with respect to a variable while keeping other variables constant. In the video, partial derivatives are essential for understanding how the cost function changes with respect to the model parameters. They are used in the chain rule to compute the gradients needed for back propagation, which is a fundamental aspect of training neural networks as highlighted in the script.
💡Deep Learning Networks
Deep learning networks refer to neural networks with multiple layers, capable of learning complex patterns in data. The video emphasizes that the principles of back propagation and partial derivatives apply to these networks, which can have thousands of layers and billions of parameters. These networks are a central theme of the video as they represent the advanced applications of the concepts being taught.
💡Chain Rule
The chain rule is a fundamental principle in calculus that allows the computation of the derivative of a composite function. In the video, the chain rule is applied to compute the partial derivatives of the cost function with respect to the model parameters across multiple layers of a neural network. It is illustrated as a key mechanism that enables the back propagation of errors in deep learning.
💡Cost Function
A cost function, also known as a loss function, measures the difference between the predicted output of a model and the actual output. In the context of the video, the cost function, such as mean squared error, is critical for guiding the learning process. The partial derivatives of the cost function with respect to the model parameters are computed during back propagation to minimize this cost and improve the model's predictions.
💡Mean Squared Error
Mean squared error (MSE) is a common cost function used in machine learning to measure the average squared difference between the estimated values and the actual value. It is mentioned in the video as an example of a cost function that can be used to compare the model's estimated output (y-hat) with the true output (y). The goal during training is to minimize this error through the process of back propagation and gradient descent.
💡Neural Networks
Neural networks are a set of algorithms designed to recognize patterns. They are inspired by the human brain's neural networks and are composed of interconnected neurons. The video discusses how neural networks, especially deep neural networks, use back propagation to adjust their parameters. The script provides an example of a simple neural network with four layers to illustrate the concept.
💡Gradient Descent
Gradient descent is an optimization algorithm used to minimize a function by iteratively moving in the direction of the steepest descent, as defined by the negative of the gradient. In the video, gradient descent is used to update the weights and biases of the neural network based on the gradients computed during back propagation. It is a fundamental step in the training process of neural networks to find the set of parameters that minimize the cost function.
💡Weights and Biases
In neural networks, weights and biases are the parameters that are adjusted during the learning process. Weights determine the strength of the connection between neurons, while biases help to shift the activation function, allowing the network to fit more complex patterns. The video script mentions that both weights and biases are updated using gradient descent as part of the back propagation process.
💡Mini-Batches
Mini-batches are subsets of the training dataset that are used in the training process of neural networks. The video script explains that large datasets are divided into mini-batches to facilitate more manageable and efficient training. This technique, known as mini-batch gradient descent, is part of the optimization process discussed in the context of the machine learning foundation series.
💡Forward Pass
A forward pass in neural networks involves the process of inputting data through the network to obtain an output. The video script describes how, during a forward pass, the input data (x values) are processed through the layers of the neural network to produce an estimated output (y-hat). This step is essential before the calculation of the cost and the subsequent back propagation can occur.
💡Automatic Differentiation
Automatic differentiation is a set of techniques to compute derivatives of numerical functions with high efficiency and accuracy. In the context of the video, it is used during the back propagation phase to calculate the partial derivatives of the cost function with respect to the model parameters. The script emphasizes that this method is crucial for updating the parameters of deep learning models with potentially millions of parameters.
Highlights

Back propagation is a fundamental approach used in training artificial neural networks, including deep learning networks.

The underlying theory of back propagation is the chain rule of partial derivatives of cost, such as mean squared error cost.

The same chain rule of partial derivatives applies to neural networks with thousands of layers and billions of model parameters.

The concept of partial derivatives with respect to model parameters extends from simple regression models to complex deep neural networks.

In machine learning, fitting a line to data points involves calculating partial derivatives, a concept that scales to more complex models.

Deep learning models often involve millions of parameters spread across thousands of layers.

The forward pass in a neural network is the process of inputting data through the model to estimate an output.

Back propagation involves automatic differentiation to calculate the partial derivative of cost with respect to each model parameter.

Large datasets are typically split into mini-batches for training deep learning models.

A forward pass produces an estimated output (y-hat), which is compared to the true output using a cost function.

Gradient descent is used to adjust the parameters in the model based on the results of the back propagation.

There are two types of parameters in a neural network: weights and biases.

All parameters in a neural network, including weights and biases, are updated using gradient descent.

The process of training a neural network involves a forward pass, cost comparison, automatic differentiation, and gradient descent.

The theory of partial derivatives and linear algebra underpins the operation of deep learning models.

The machine learning foundation series covers the basics that are essential for understanding more advanced topics in deep learning.

Higher order partial derivatives are a topic for further exploration in the machine learning foundation series.

Transcripts

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Related Tags
Partial DerivativesBackpropagationNeural NetworksDeep LearningMachine LearningChain RuleCost FunctionMean Squared ErrorGradient DescentAutomatic DifferentiationModel Parameters