Training Neural Networks and Backpropagation

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Your Name

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January 28, 2025

Introduction

This lecture focuses on the fundamental concepts and techniques involved in training neural networks. The primary objective is to equip students with a comprehensive understanding of how neural networks learn from data. We will delve into the core principles of gradient descent, a widely used optimization algorithm, and explore its application in adjusting network parameters to minimize the loss function. Additionally, we will examine the backpropagation algorithm, which efficiently computes the gradients necessary for gradient descent in the context of neural networks.

Key Concepts

The key concepts covered in this lecture include:

The role of the loss function in quantifying the performance of a neural network.
The relationship between training data and the cost function landscape.
The iterative nature of gradient descent and its reliance on local information.
The challenges posed by local minima and plateaus in the optimization process.
The mathematical formulation of gradient descent and the significance of the learning rate.
The application of backpropagation for efficient gradient computation in neural networks.
The importance of monitoring the training process and diagnosing potential issues.

Learning Outcomes

By the end of this lecture, students will be able to:

Understand the core principles of training neural networks.
Explain the mechanics of gradient descent and its role in optimization.
Describe the backpropagation algorithm and its significance in neural network training.
Analyze the impact of training data and network architecture on the cost function.
Identify and address common challenges encountered during the training process.

Training Neural Networks

Objective: Minimizing the Loss Function

The primary goal in training neural networks is to find the optimal set of weights and biases that minimize the discrepancy between the network’s predictions and the actual target values. This discrepancy is quantified by a **loss function**, which serves as a measure of the network’s performance. The objective, therefore, is to minimize this loss function.

Cost Function and Training Data

Defining the Cost Function

Given a training dataset with $n$ samples, the cost function $J$ is typically defined as the average of the individual loss functions over all training examples. For instance, if $L(y^{(i)}, a^{(i)})$ represents the loss for the $i$-th training example, where $y^{(i)}$ is the target value and $a^{(i)}$ is the network’s prediction, the cost function can be expressed as: \[J(W, b) = \frac{1}{n} \sum_{i=1}^{n} L(y^{(i)}, a^{(i)}) \label{eq:cost_function}\] Here, $W$ and $b$ represent all the weights and biases in the network, respectively.

Impact of Training Data on the Cost Function Landscape

It is crucial to understand that the cost function is intrinsically linked to both the network’s architecture (including weights and biases) and the training data.

**Dependency on Network Parameters:** For a fixed training dataset, changing the network’s weights and biases will alter the network’s predictions and consequently change the value of the cost function.
**Dependency on Training Data:** Conversely, for a fixed network architecture and parameters, changing the training data (e.g., using different target values) will result in a different cost function landscape.

In essence, each unique combination of network architecture and training data yields a unique cost function landscape.

Gradient Descent: An Iterative Optimization Approach

Using Local Information for Parameter Updates

Gradient descent is an iterative optimization algorithm that aims to find the minimum of a function (in this case, the cost function) by iteratively adjusting the parameters (weights and biases) based on the local gradient of the function. The gradient provides information about the direction of the steepest ascent, and by moving in the opposite direction, we can iteratively approach a minimum.

Visualizing the Cost Function as a Landscape

The cost function can be visualized as a landscape where the height represents the value of the cost function, and the coordinates represent the values of the network’s parameters. The goal of gradient descent is to navigate this landscape and find the lowest point, which corresponds to the minimum of the cost function.

Limitations of Exhaustive Search Methods

An exhaustive search, such as a grid search over the parameter space, is computationally infeasible for high-dimensional problems like training neural networks. Gradient descent offers a more efficient alternative by iteratively refining the parameters based on local gradient information.

The Gradient Descent Algorithm

Random Initialization of Network Parameters

The gradient descent algorithm typically starts with a random initialization of the network’s weights and biases. This provides a starting point for the iterative optimization process.

Iterative Parameter Adjustment

The core of gradient descent involves iteratively updating the network’s parameters by moving in the direction opposite to the gradient of the cost function. This process can be visualized as a person descending a hill by taking steps in the direction of the steepest downward slope.

Challenges: Local Minima and Plateaus

**Local Minima:** The cost function landscape may contain multiple local minima, which are points that are lower than their immediate surroundings but not the global minimum. Gradient descent can get trapped in local minima, preventing it from reaching the optimal solution.
**Plateaus:** Plateaus are regions of the cost function landscape where the gradient is very close to zero. This can significantly slow down the convergence of gradient descent, as the parameter updates become very small.

Note: In practice, the cost function landscape can have large regions where the gradient is close to zero, making it challenging to find the global minimum. This is especially true for complex neural networks with many parameters.

Exploration Strategies to Escape Local Minima

Various techniques have been developed to mitigate the issue of local minima, such as:

**Stochastic Gradient Descent:** Introducing randomness into the parameter updates can help the algorithm escape local minima.
**Momentum:** Incorporating momentum into the update rule can help the algorithm overcome small local minima and accelerate convergence.

Mathematical Formulation of Gradient Descent

Requirement for Differentiable Loss Functions

Gradient descent relies on the computation of gradients, which requires the cost function to be differentiable with respect to the network’s parameters.

The Parameter Update Rule

The update rule for a parameter $\theta$ (which can be a weight or a bias) in gradient descent is given by: \[\theta_{new} = \theta_{old} - \alpha \frac{\partial J}{\partial \theta} \label{eq:gradient_descent}\] where $\alpha$ is the learning rate, and $\frac{\partial J}{\partial \theta}$ is the partial derivative of the cost function with respect to $\theta$.

The Role of the Learning Rate ($\alpha$)

The learning rate $\alpha$ controls the step size taken in the direction opposite to the gradient.

**Large Learning Rate:** A large learning rate can lead to faster convergence but may also cause overshooting, where the algorithm oscillates around the minimum or even diverges.
**Small Learning Rate:** A small learning rate ensures smaller steps, reducing the risk of overshooting, but it can lead to slow convergence.

Note: In practice, the learning rate is often tested using a grid search approach to find a suitable value. It’s also common to use adaptive learning rate methods that automatically adjust the learning rate during training.

Simultaneous Updates of Network Parameters

It is crucial to update all network parameters simultaneously based on the gradients computed at the current parameter values. Updating parameters sequentially can lead to incorrect updates, as the cost function landscape changes after each parameter update.

Intuitive Understanding of Gradient Descent

Illustrative Example with a Simple Cost Function

Consider a simple cost function $J(\theta) = (\theta - 1)^2$. Let’s analyze the behavior of gradient descent for different initial values of $\theta$.

**Case 1: $\theta_{old} = 3.8$** The derivative $\frac{dJ}{d\theta} = 2(\theta - 1)$. At $\theta = 3.8$, the derivative is positive ($2(3.8-1) = 5.6$). Therefore, the update rule becomes $\theta_{new} = 3.8 - \alpha * 5.6$. Since $\alpha$ is positive, $\theta_{new}$ will be less than $3.8$, moving towards the minimum at $\theta = 1$.
**Case 2: $\theta_{old} = 0$** At $\theta = 0$, the derivative is negative ($2(0-1) = -2$). The update rule becomes $\theta_{new} = 0 - \alpha * (-2) = 0 + 2\alpha$. Since $\alpha$ is positive, $\theta_{new}$ will be greater than $0$, again moving towards the minimum at $\theta = 1$.

Derivatives as Indicators of Descent Direction

The derivative of the cost function with respect to a parameter indicates the direction of the steepest ascent. By taking the negative of the derivative, we obtain the direction of the steepest descent, which guides the parameter updates towards the minimum.

Monitoring the Training Process

Importance of Tracking the Cost Function’s Value

While the parameter updates in gradient descent do not explicitly require the value of the cost function (only its derivative), it is crucial to monitor the cost function’s value during training. This provides insights into the convergence behavior and helps diagnose potential issues.

Periodic Evaluation of the Cost Function

A common practice is to evaluate the cost function after a certain number of iterations (e.g., every 10 steps). This allows us to track the progress of the optimization process and observe whether the cost function is decreasing as expected.

Diagnosing Issues with the Learning Rate

By plotting the cost function’s value over iterations, we can identify potential problems with the learning rate:

**Oscillations or Divergence:** If the cost function oscillates or increases, it may indicate that the learning rate is too large.
**Slow Convergence:** If the cost function decreases very slowly, it may suggest that the learning rate is too small.

Adaptive Learning Rate Techniques

Adaptive learning rate methods, such as Adam or RMSprop, automatically adjust the learning rate during training based on the observed gradients. These techniques can often improve convergence speed and stability.

Backpropagation: Applying Gradient Descent to Neural Networks

A Simple Neural Network Example

Network Architecture: Inputs, Weights, Bias, Sigmoid Activation

Consider a simple neural network with two inputs ($x_1$, $x_2$), two weights ($w_1$, $w_2$), a bias ($b$), and a sigmoid activation function ($\sigma$). The network’s output is given by: \[a = \sigma(z) = \sigma(w_1x_1 + w_2x_2 + b) \label{eq:neural_network_output}\]

Binary Cross-Entropy Loss for Classification

For binary classification tasks, the binary cross-entropy loss function is commonly used: \[L(y, a) = -[y \log(a) + (1-y) \log(1-a)] \label{eq:binary_cross_entropy}\] where $y$ is the true label (0 or 1) and $a$ is the network’s predicted probability.

Computational Graph Representation

Forward Pass: Calculating the Weighted Sum (Z)

The computation in the network can be represented as a computational graph. The first step is to calculate the weighted sum of inputs plus the bias: \[z = w_1x_1 + w_2x_2 + b \label{eq:weighted_sum}\]

Applying the Activation Function

The next step is to apply the sigmoid activation function to $z$: \[a = \sigma(z) = \frac{1}{1 + e^{-z}} \label{eq:sigmoid}\]

Calculating the Loss

Finally, the loss function is computed using the network’s output $a$ and the true label $y$: \[L(y, a) = -[y \log(a) + (1-y) \log(1-a)]\]

Derivatives and the Chain Rule

Review of Essential Derivative Rules

Derivative of $\ln(x)$: $\frac{d}{dx} \ln(x) = \frac{1}{x}$
Derivative of a constant: $\frac{d}{dx} c = 0$
Derivative of a sum: $\frac{d}{dx} (f(x) + g(x)) = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)$
Chain rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Derivative of the Sigmoid Activation Function

The derivative of the sigmoid function is given by: \[\frac{d\sigma(z)}{dz} = \sigma(z)(1 - \sigma(z)) = a(1-a) \label{eq:sigmoid_derivative}\]

Calculating Gradients using Backpropagation

Derivative of the Loss with Respect to the Activation (A)

To apply gradient descent, we need to compute the derivatives of the loss function with respect to the network’s parameters. We start by computing the derivative of the loss with respect to $a$: \[\frac{\partial L}{\partial a} = -\frac{y}{a} + \frac{1-y}{1-a} \label{eq:dL_da}\]

Derivative of the Loss with Respect to the Weighted Sum (Z)

Next, we use the chain rule to compute the derivative of the loss with respect to $z$:

\[\frac{\partial L}{\partial z} = \frac{\partial L}{\partial a} \frac{\partial a}{\partial z} = \left(-\frac{y}{a} + \frac{1-y}{1-a}\right) a(1-a) = a - y \label{eq:dL_dz}\]

Derivatives with Respect to Weights and Bias Parameters

Finally, we compute the derivatives with respect to the weights and bias:

\[\begin{aligned} \frac{\partial L}{\partial w_1} &= \frac{\partial L}{\partial z} \frac{\partial z}{\partial w_1} = (a-y)x_1 \label{eq:dL_dw1} \\ \frac{\partial L}{\partial w_2} &= \frac{\partial L}{\partial z} \frac{\partial z}{\partial w_2} = (a-y)x_2 \label{eq:dL_dw2} \\ \frac{\partial L}{\partial b} &= \frac{\partial L}{\partial z} \frac{\partial z}{\partial b} = (a-y) \label{eq:dL_db} \end{aligned}\]

Gradient Descent with Multiple Training Samples

Averaging the Loss Over a Batch of Samples

When training with multiple samples, the cost function is the average of the individual losses: \[J(W, b) = \frac{1}{n} \sum_{i=1}^{n} L(y^{(i)}, a^{(i)})\]

Calculating Gradients for the Batch Cost Function

The derivative of the cost function with respect to a parameter is the average of the derivatives of the individual losses:

\[\frac{\partial J}{\partial w_1} = \frac{1}{n} \sum_{i=1}^{n} \frac{\partial L^{(i)}}{\partial w_1} = \frac{1}{n} \sum_{i=1}^{n} (a^{(i)} - y^{(i)})x_1^{(i)} \label{eq:dJ_dw1}\] Similarly for $w_2$ and $b$: \[\begin{aligned} \frac{\partial J}{\partial w_2} = \frac{1}{n} \sum_{i=1}^{n} (a^{(i)} - y^{(i)})x_2^{(i)} \label{eq:dJ_dw2} \\ \frac{\partial J}{\partial b} = \frac{1}{n} \sum_{i=1}^{n} (a^{(i)} - y^{(i)}) \label{eq:dJ_db} \end{aligned}\]

Initialize weights $w_1$, $w_2$ and bias $b$ randomly Set learning rate $\alpha$ Initialize $\frac{\partial J}{\partial w_1} = 0$, $\frac{\partial J}{\partial w_2} = 0$, $\frac{\partial J}{\partial b} = 0$ $z^{(i)} = w_1x_1^{(i)} + w_2x_2^{(i)} + b$ $a^{(i)} = \sigma(z^{(i)})$ $\frac{\partial L^{(i)}}{\partial a} = -\frac{y^{(i)}}{a^{(i)}} + \frac{1-y^{(i)}}{1-a^{(i)}}$ $\frac{\partial L^{(i)}}{\partial z} = a^{(i)} - y^{(i)}$ $\frac{\partial L^{(i)}}{\partial w_1} = (a^{(i)} - y^{(i)})x_1^{(i)}$ $\frac{\partial L^{(i)}}{\partial w_2} = (a^{(i)} - y^{(i)})x_2^{(i)}$ $\frac{\partial L^{(i)}}{\partial b} = (a^{(i)} - y^{(i)})$ $\frac{\partial J}{\partial w_1} += \frac{\partial L^{(i)}}{\partial w_1}$ $\frac{\partial J}{\partial w_2} += \frac{\partial L^{(i)}}{\partial w_2}$ $\frac{\partial J}{\partial b} += \frac{\partial L^{(i)}}{\partial b}$ $\frac{\partial J}{\partial w_1} = \frac{1}{n}\frac{\partial J}{\partial w_1}$ $\frac{\partial J}{\partial w_2} = \frac{1}{n}\frac{\partial J}{\partial w_2}$ $\frac{\partial J}{\partial b} = \frac{1}{n}\frac{\partial J}{\partial b}$ $w_1 = w_1 - \alpha \frac{\partial J}{\partial w_1}$ $w_2 = w_2 - \alpha \frac{\partial J}{\partial w_2}$ $b = b - \alpha \frac{\partial J}{\partial b}$

The complexity of each iteration of Algorithm [alg:gradient_descent_backpropagation] is $O(n)$, where $n$ is the number of training samples. This is because we need to iterate through each training sample to compute the gradients. The number of iterations required for convergence depends on various factors, including the learning rate, the complexity of the cost function landscape, and the desired level of accuracy.

Conclusion

In this lecture, we explored the fundamental concepts of training neural networks, focusing on gradient descent and backpropagation. We learned that the goal of training is to minimize the cost function, which measures the discrepancy between the network’s predictions and the true target values. Gradient descent is an iterative algorithm that achieves this by updating the network’s parameters in the direction opposite to the gradient of the cost function.

We discussed the importance of the learning rate, which controls the step size in gradient descent, and how to monitor the training process by tracking the cost function’s value. We also examined the challenges posed by local minima and plateaus and briefly touched upon techniques to address these issues.

Furthermore, we delved into the backpropagation algorithm, which efficiently computes the gradients required for gradient descent in neural networks. By representing the network’s computations as a computational graph, we can apply the chain rule to calculate the derivatives of the loss function with respect to each parameter.

Key Takeaways

Key takeaways from this lecture include:

The cost function is a crucial component in training neural networks, and its landscape is influenced by both the network architecture and the training data.
Gradient descent is a powerful optimization algorithm that iteratively adjusts parameters to minimize the cost function, but it can be sensitive to the learning rate and may get trapped in local minima.
Backpropagation provides an efficient way to compute the gradients needed for gradient descent by propagating derivatives backward through the computational graph.
Monitoring the cost function during training is essential for diagnosing issues and ensuring convergence.

Future Directions

In the next lecture, we will build upon these concepts and explore more advanced techniques for training neural networks, including different activation functions, regularization methods, and optimization algorithms. We will also discuss how to handle larger and more complex neural network architectures.

Preparation for Next Lecture

To prepare for the next lecture, consider the following questions:

How might the choice of activation function affect the training process and the performance of a neural network?
What are some strategies for preventing overfitting in neural networks, and how do they relate to the concepts discussed in this lecture?
How could we extend the backpropagation algorithm to handle networks with multiple hidden layers and different activation functions?

--- title: "Training Neural Networks and Backpropagation" author: "Your Name" date: "2025-01-28" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This lecture focuses on the fundamental concepts and techniques involved in training neural networks. The primary objective is to equip students with a comprehensive understanding of how neural networks learn from data. We will delve into the core principles of gradient descent, a widely used optimization algorithm, and explore its application in adjusting network parameters to minimize the loss function. Additionally, we will examine the backpropagation algorithm, which efficiently computes the gradients necessary for gradient descent in the context of neural networks. ## Key Concepts The key concepts covered in this lecture include: - The role of the loss function in quantifying the performance of a neural network. - The relationship between training data and the cost function landscape. - The iterative nature of gradient descent and its reliance on local information. - The challenges posed by local minima and plateaus in the optimization process. - The mathematical formulation of gradient descent and the significance of the learning rate. - The application of backpropagation for efficient gradient computation in neural networks. - The importance of monitoring the training process and diagnosing potential issues. ## Learning Outcomes By the end of this lecture, students will be able to: - Understand the core principles of training neural networks. - Explain the mechanics of gradient descent and its role in optimization. - Describe the backpropagation algorithm and its significance in neural network training. - Analyze the impact of training data and network architecture on the cost function. - Identify and address common challenges encountered during the training process. # Training Neural Networks ## Objective: Minimizing the Loss Function The primary goal in training neural networks is to find the optimal set of weights and biases that minimize the discrepancy between the network's predictions and the actual target values. This discrepancy is quantified by a \*\*loss function\*\*, which serves as a measure of the network's performance. The objective, therefore, is to minimize this loss function. ## Cost Function and Training Data ### Defining the Cost Function Given a training dataset with $n$ samples, the cost function $J$ is typically defined as the average of the individual loss functions over all training examples. For instance, if $L(y^{(i)}, a^{(i)})$ represents the loss for the $i$-th training example, where $y^{(i)}$ is the target value and $a^{(i)}$ is the network's prediction, the cost function can be expressed as: $$J(W, b) = \frac{1}{n} \sum_{i=1}^{n} L(y^{(i)}, a^{(i)}) \label{eq:cost_function}$$ Here, $W$ and $b$ represent all the weights and biases in the network, respectively. ### Impact of Training Data on the Cost Function Landscape It is crucial to understand that the cost function is intrinsically linked to both the network's architecture (including weights and biases) and the training data. - \*\*Dependency on Network Parameters:\*\* For a fixed training dataset, changing the network's weights and biases will alter the network's predictions and consequently change the value of the cost function. - \*\*Dependency on Training Data:\*\* Conversely, for a fixed network architecture and parameters, changing the training data (e.g., using different target values) will result in a different cost function landscape. In essence, each unique combination of network architecture and training data yields a unique cost function landscape. ## Gradient Descent: An Iterative Optimization Approach ### Using Local Information for Parameter Updates Gradient descent is an iterative optimization algorithm that aims to find the minimum of a function (in this case, the cost function) by iteratively adjusting the parameters (weights and biases) based on the local gradient of the function. The gradient provides information about the direction of the steepest ascent, and by moving in the opposite direction, we can iteratively approach a minimum. ### Visualizing the Cost Function as a Landscape The cost function can be visualized as a landscape where the height represents the value of the cost function, and the coordinates represent the values of the network's parameters. The goal of gradient descent is to navigate this landscape and find the lowest point, which corresponds to the minimum of the cost function. ::: center ::: ### Limitations of Exhaustive Search Methods An exhaustive search, such as a grid search over the parameter space, is computationally infeasible for high-dimensional problems like training neural networks. Gradient descent offers a more efficient alternative by iteratively refining the parameters based on local gradient information. ## The Gradient Descent Algorithm ### Random Initialization of Network Parameters The gradient descent algorithm typically starts with a random initialization of the network's weights and biases. This provides a starting point for the iterative optimization process. ### Iterative Parameter Adjustment The core of gradient descent involves iteratively updating the network's parameters by moving in the direction opposite to the gradient of the cost function. This process can be visualized as a person descending a hill by taking steps in the direction of the steepest downward slope. ### Challenges: Local Minima and Plateaus - \*\*Local Minima:\*\* The cost function landscape may contain multiple local minima, which are points that are lower than their immediate surroundings but not the global minimum. Gradient descent can get trapped in local minima, preventing it from reaching the optimal solution. - \*\*Plateaus:\*\* Plateaus are regions of the cost function landscape where the gradient is very close to zero. This can significantly slow down the convergence of gradient descent, as the parameter updates become very small. ::: tcolorbox **Note:** In practice, the cost function landscape can have large regions where the gradient is close to zero, making it challenging to find the global minimum. This is especially true for complex neural networks with many parameters. ::: ### Exploration Strategies to Escape Local Minima Various techniques have been developed to mitigate the issue of local minima, such as: - \*\*Stochastic Gradient Descent:\*\* Introducing randomness into the parameter updates can help the algorithm escape local minima. - \*\*Momentum:\*\* Incorporating momentum into the update rule can help the algorithm overcome small local minima and accelerate convergence. ## Mathematical Formulation of Gradient Descent ### Requirement for Differentiable Loss Functions Gradient descent relies on the computation of gradients, which requires the cost function to be differentiable with respect to the network's parameters. ### The Parameter Update Rule The update rule for a parameter $\theta$ (which can be a weight or a bias) in gradient descent is given by: $$\theta_{new} = \theta_{old} - \alpha \frac{\partial J}{\partial \theta} \label{eq:gradient_descent}$$ where $\alpha$ is the learning rate, and $\frac{\partial J}{\partial \theta}$ is the partial derivative of the cost function with respect to $\theta$. ### The Role of the Learning Rate ($\alpha$) The learning rate $\alpha$ controls the step size taken in the direction opposite to the gradient. - \*\*Large Learning Rate:\*\* A large learning rate can lead to faster convergence but may also cause overshooting, where the algorithm oscillates around the minimum or even diverges. - \*\*Small Learning Rate:\*\* A small learning rate ensures smaller steps, reducing the risk of overshooting, but it can lead to slow convergence. ::: tcolorbox **Note:** In practice, the learning rate is often tested using a grid search approach to find a suitable value. It's also common to use adaptive learning rate methods that automatically adjust the learning rate during training. ::: ### Simultaneous Updates of Network Parameters It is crucial to update all network parameters simultaneously based on the gradients computed at the current parameter values. Updating parameters sequentially can lead to incorrect updates, as the cost function landscape changes after each parameter update. ## Intuitive Understanding of Gradient Descent ### Illustrative Example with a Simple Cost Function Consider a simple cost function $J(\theta) = (\theta - 1)^2$. Let's analyze the behavior of gradient descent for different initial values of $\theta$. - \*\*Case 1: $\theta_{old} = 3.8$\*\* The derivative $\frac{dJ}{d\theta} = 2(\theta - 1)$. At $\theta = 3.8$, the derivative is positive ($2(3.8-1) = 5.6$). Therefore, the update rule becomes $\theta_{new} = 3.8 - \alpha * 5.6$. Since $\alpha$ is positive, $\theta_{new}$ will be less than $3.8$, moving towards the minimum at $\theta = 1$. - \*\*Case 2: $\theta_{old} = 0$\*\* At $\theta = 0$, the derivative is negative ($2(0-1) = -2$). The update rule becomes $\theta_{new} = 0 - \alpha * (-2) = 0 + 2\alpha$. Since $\alpha$ is positive, $\theta_{new}$ will be greater than $0$, again moving towards the minimum at $\theta = 1$. ::: center ::: ### Derivatives as Indicators of Descent Direction The derivative of the cost function with respect to a parameter indicates the direction of the steepest ascent. By taking the negative of the derivative, we obtain the direction of the steepest descent, which guides the parameter updates towards the minimum. ## Monitoring the Training Process ### Importance of Tracking the Cost Function's Value While the parameter updates in gradient descent do not explicitly require the value of the cost function (only its derivative), it is crucial to monitor the cost function's value during training. This provides insights into the convergence behavior and helps diagnose potential issues. ### Periodic Evaluation of the Cost Function A common practice is to evaluate the cost function after a certain number of iterations (e.g., every 10 steps). This allows us to track the progress of the optimization process and observe whether the cost function is decreasing as expected. ### Diagnosing Issues with the Learning Rate By plotting the cost function's value over iterations, we can identify potential problems with the learning rate: - \*\*Oscillations or Divergence:\*\* If the cost function oscillates or increases, it may indicate that the learning rate is too large. - \*\*Slow Convergence:\*\* If the cost function decreases very slowly, it may suggest that the learning rate is too small. ::: center ::: ### Adaptive Learning Rate Techniques Adaptive learning rate methods, such as Adam or RMSprop, automatically adjust the learning rate during training based on the observed gradients. These techniques can often improve convergence speed and stability. # Backpropagation: Applying Gradient Descent to Neural Networks ## A Simple Neural Network Example ### Network Architecture: Inputs, Weights, Bias, Sigmoid Activation Consider a simple neural network with two inputs ($x_1$, $x_2$), two weights ($w_1$, $w_2$), a bias ($b$), and a sigmoid activation function ($\sigma$). The network's output is given by: $$a = \sigma(z) = \sigma(w_1x_1 + w_2x_2 + b) \label{eq:neural_network_output}$$ ::: center ::: ### Binary Cross-Entropy Loss for Classification For binary classification tasks, the binary cross-entropy loss function is commonly used: $$L(y, a) = -[y \log(a) + (1-y) \log(1-a)] \label{eq:binary_cross_entropy}$$ where $y$ is the true label (0 or 1) and $a$ is the network's predicted probability. ## Computational Graph Representation ### Forward Pass: Calculating the Weighted Sum (Z) The computation in the network can be represented as a computational graph. The first step is to calculate the weighted sum of inputs plus the bias: $$z = w_1x_1 + w_2x_2 + b \label{eq:weighted_sum}$$ ### Applying the Activation Function The next step is to apply the sigmoid activation function to $z$: $$a = \sigma(z) = \frac{1}{1 + e^{-z}} \label{eq:sigmoid}$$ ### Calculating the Loss Finally, the loss function is computed using the network's output $a$ and the true label $y$: $$L(y, a) = -[y \log(a) + (1-y) \log(1-a)]$$ ::: center ::: ## Derivatives and the Chain Rule ### Review of Essential Derivative Rules - Derivative of $\ln(x)$: $\frac{d}{dx} \ln(x) = \frac{1}{x}$ - Derivative of a constant: $\frac{d}{dx} c = 0$ - Derivative of a sum: $\frac{d}{dx} (f(x) + g(x)) = \frac{d}{dx} f(x) + \frac{d}{dx} g(x)$ - Chain rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$ ### Derivative of the Sigmoid Activation Function ::: tcolorbox The derivative of the sigmoid function is given by: $$\frac{d\sigma(z)}{dz} = \sigma(z)(1 - \sigma(z)) = a(1-a) \label{eq:sigmoid_derivative}$$ ::: ## Calculating Gradients using Backpropagation ### Derivative of the Loss with Respect to the Activation (A) To apply gradient descent, we need to compute the derivatives of the loss function with respect to the network's parameters. We start by computing the derivative of the loss with respect to $a$: $$\frac{\partial L}{\partial a} = -\frac{y}{a} + \frac{1-y}{1-a} \label{eq:dL_da}$$ ### Derivative of the Loss with Respect to the Weighted Sum (Z) Next, we use the chain rule to compute the derivative of the loss with respect to $z$: ::: tcolorbox $$\frac{\partial L}{\partial z} = \frac{\partial L}{\partial a} \frac{\partial a}{\partial z} = \left(-\frac{y}{a} + \frac{1-y}{1-a}\right) a(1-a) = a - y \label{eq:dL_dz}$$ ::: ### Derivatives with Respect to Weights and Bias Parameters Finally, we compute the derivatives with respect to the weights and bias: ::: tcolorbox $$\begin{aligned} \frac{\partial L}{\partial w_1} &= \frac{\partial L}{\partial z} \frac{\partial z}{\partial w_1} = (a-y)x_1 \label{eq:dL_dw1} \\ \frac{\partial L}{\partial w_2} &= \frac{\partial L}{\partial z} \frac{\partial z}{\partial w_2} = (a-y)x_2 \label{eq:dL_dw2} \\ \frac{\partial L}{\partial b} &= \frac{\partial L}{\partial z} \frac{\partial z}{\partial b} = (a-y) \label{eq:dL_db} \end{aligned}$$ ::: ## Gradient Descent with Multiple Training Samples ### Averaging the Loss Over a Batch of Samples When training with multiple samples, the cost function is the average of the individual losses: $$J(W, b) = \frac{1}{n} \sum_{i=1}^{n} L(y^{(i)}, a^{(i)})$$ ### Calculating Gradients for the Batch Cost Function The derivative of the cost function with respect to a parameter is the average of the derivatives of the individual losses: ::: tcolorbox $$\frac{\partial J}{\partial w_1} = \frac{1}{n} \sum_{i=1}^{n} \frac{\partial L^{(i)}}{\partial w_1} = \frac{1}{n} \sum_{i=1}^{n} (a^{(i)} - y^{(i)})x_1^{(i)} \label{eq:dJ_dw1}$$ Similarly for $w_2$ and $b$: $$\begin{aligned} \frac{\partial J}{\partial w_2} = \frac{1}{n} \sum_{i=1}^{n} (a^{(i)} - y^{(i)})x_2^{(i)} \label{eq:dJ_dw2} \\ \frac{\partial J}{\partial b} = \frac{1}{n} \sum_{i=1}^{n} (a^{(i)} - y^{(i)}) \label{eq:dJ_db} \end{aligned}$$ ::: ::::: tcolorbox :::: algorithm ::: algorithmic Initialize weights $w_1$, $w_2$ and bias $b$ randomly Set learning rate $\alpha$ Initialize $\frac{\partial J}{\partial w_1} = 0$, $\frac{\partial J}{\partial w_2} = 0$, $\frac{\partial J}{\partial b} = 0$ $z^{(i)} = w_1x_1^{(i)} + w_2x_2^{(i)} + b$ $a^{(i)} = \sigma(z^{(i)})$ $\frac{\partial L^{(i)}}{\partial a} = -\frac{y^{(i)}}{a^{(i)}} + \frac{1-y^{(i)}}{1-a^{(i)}}$ $\frac{\partial L^{(i)}}{\partial z} = a^{(i)} - y^{(i)}$ $\frac{\partial L^{(i)}}{\partial w_1} = (a^{(i)} - y^{(i)})x_1^{(i)}$ $\frac{\partial L^{(i)}}{\partial w_2} = (a^{(i)} - y^{(i)})x_2^{(i)}$ $\frac{\partial L^{(i)}}{\partial b} = (a^{(i)} - y^{(i)})$ $\frac{\partial J}{\partial w_1} += \frac{\partial L^{(i)}}{\partial w_1}$ $\frac{\partial J}{\partial w_2} += \frac{\partial L^{(i)}}{\partial w_2}$ $\frac{\partial J}{\partial b} += \frac{\partial L^{(i)}}{\partial b}$ $\frac{\partial J}{\partial w_1} = \frac{1}{n}\frac{\partial J}{\partial w_1}$ $\frac{\partial J}{\partial w_2} = \frac{1}{n}\frac{\partial J}{\partial w_2}$ $\frac{\partial J}{\partial b} = \frac{1}{n}\frac{\partial J}{\partial b}$ $w_1 = w_1 - \alpha \frac{\partial J}{\partial w_1}$ $w_2 = w_2 - \alpha \frac{\partial J}{\partial w_2}$ $b = b - \alpha \frac{\partial J}{\partial b}$ ::: []{#alg:gradient_descent_backpropagation label="alg:gradient_descent_backpropagation"} :::: ::::: The complexity of each iteration of Algorithm [\[alg:gradient_descent_backpropagation\]](#alg:gradient_descent_backpropagation){reference-type="ref" reference="alg:gradient_descent_backpropagation"} is $O(n)$, where $n$ is the number of training samples. This is because we need to iterate through each training sample to compute the gradients. The number of iterations required for convergence depends on various factors, including the learning rate, the complexity of the cost function landscape, and the desired level of accuracy. # Conclusion In this lecture, we explored the fundamental concepts of training neural networks, focusing on gradient descent and backpropagation. We learned that the goal of training is to minimize the cost function, which measures the discrepancy between the network's predictions and the true target values. Gradient descent is an iterative algorithm that achieves this by updating the network's parameters in the direction opposite to the gradient of the cost function. We discussed the importance of the learning rate, which controls the step size in gradient descent, and how to monitor the training process by tracking the cost function's value. We also examined the challenges posed by local minima and plateaus and briefly touched upon techniques to address these issues. Furthermore, we delved into the backpropagation algorithm, which efficiently computes the gradients required for gradient descent in neural networks. By representing the network's computations as a computational graph, we can apply the chain rule to calculate the derivatives of the loss function with respect to each parameter. ## Key Takeaways Key takeaways from this lecture include: - The cost function is a crucial component in training neural networks, and its landscape is influenced by both the network architecture and the training data. - Gradient descent is a powerful optimization algorithm that iteratively adjusts parameters to minimize the cost function, but it can be sensitive to the learning rate and may get trapped in local minima. - Backpropagation provides an efficient way to compute the gradients needed for gradient descent by propagating derivatives backward through the computational graph. - Monitoring the cost function during training is essential for diagnosing issues and ensuring convergence. ## Future Directions In the next lecture, we will build upon these concepts and explore more advanced techniques for training neural networks, including different activation functions, regularization methods, and optimization algorithms. We will also discuss how to handle larger and more complex neural network architectures. ## Preparation for Next Lecture To prepare for the next lecture, consider the following questions: - How might the choice of activation function affect the training process and the performance of a neural network? - What are some strategies for preventing overfitting in neural networks, and how do they relate to the concepts discussed in this lecture? - How could we extend the backpropagation algorithm to handle networks with multiple hidden layers and different activation functions?