Day 2-Forward Propogation, Loss Functions, Chain Rule Of Derivatives|Deep Learning Live

Name: Day 2-Forward Propogation, Loss Functions, Chain Rule Of Derivatives|Deep Learning Live
Uploaded: 2022-05-03T00:00:00.000Z
Duration: 3 h 28 min 45 s

Introduction and Session Overview

Welcome and Confirmation

The speaker begins by checking if the audience can hear them, encouraging interaction through likes and subscriptions.

They express excitement for the upcoming session, indicating it will be informative and engaging.

Recap of Previous Session

The speaker invites attendees to share what they learned in the previous live session, emphasizing community engagement.

They mention that today's session will cover various deep learning concepts, building on prior knowledge.

Agenda for Today's Session

Key Topics to Cover

The agenda includes understanding forward propagation as a foundational concept in neural networks.

Before discussing loss functions, the speaker plans to explain the chain rule of derivatives, which is crucial for understanding backpropagation.

Additional Concepts

The session will address the vanishing gradient problem, an important issue in training deep neural networks.

Different types of loss functions will also be explored to provide a comprehensive understanding of model evaluation metrics.

Deep Learning Fundamentals

Neural Network Structure

A basic structure of a neural network is introduced: three inputs leading to one hidden neuron and one output neuron. This setup illustrates how data flows through layers.

Bias is explained as an essential component added at each hidden layer to improve model performance.

Forward Propagation Process

The process starts with assigning random weights to inputs; these weights are critical for determining output values during forward propagation.

Understanding Forward and Backward Propagation in Neural Networks

Overview of Forward Propagation

The output y can be represented as w^T x + b , indicating a linear regression model. However, the activation function introduces non-linear properties, enabling the solution of non-linear problems.

Forward propagation occurs in every layer of the neural network, leading to an output denoted as haty . Following this, a loss function is calculated to evaluate performance.

Transitioning to Backward Propagation

To minimize the loss function, weight updates are necessary. This process occurs during backward propagation.

Two key topics will be covered: the weight updation formula and the chain rule of differentiation.

Weight Update Mechanism

In a simple setup with three inputs, one hidden neuron, and one output neuron, weights are interconnected. The goal is to update these weights based on calculated losses.

The weight update formula involves adjusting previous weights by subtracting a product of learning rate and the derivative of loss concerning old weights.

Understanding Derivatives in Weight Updates

The generic weight update formula is expressed as:

w_textnew = w_textold - textlearning rate times partial L/partial w_textold

Here, the derivative represents the slope calculation essential for determining how much to adjust each weight.

Gradient Descent Concept

The concept of gradient descent emerges from calculating slopes similar to those used in linear regression. It visualizes how weights relate to loss functions.

A graph representing this relationship shows points where global minima occur—indicating optimal training conditions for updating weights effectively.

Visualizing Loss Function Dynamics

When visualizing this process in 3D (like an inverted mountain), our objective is reaching specific points that represent minimal loss.

As we apply derivatives at various points along this curve, tangent lines help determine whether slopes are positive or negative—critical for effective weight adjustments.

Identifying Slope Directions

Understanding Weight Updates in Gradient Descent

The Concept of Slope and Weight Adjustment

A negative slope indicates the need to increase weights to reach the global minima. The goal is to adjust weights effectively during training.

The weight update formula for a negative slope becomes: w_textnew = w_textold - textlearning rate times (textnegative value) .

Multiplying two negatives results in a positive, meaning that increasing weights is possible when starting from a negative slope.

Thus, with a negative slope, w_textnew will always be greater than w_textold , confirming that we can successfully increase weights.

Positive Slope and Its Implications

In contrast, a positive slope requires decreasing weights to move towards the desired outcome (global minima).

For a positive slope, the weight update formula changes to: w_textnew = w_textold - textlearning rate times (textpositive value) .

This leads to w_textnew < w_textold , demonstrating that we are indeed reducing the weights as needed.

Gradient Descent Overview

The process described is part of gradient descent, which relies on loss functions for effective weight updates.

An example loss function used in linear regression is (y - haty)^2 , which helps visualize how gradients affect weight adjustments.

Importance of Learning Rate

The learning rate plays a crucial role; smaller values lead to gradual convergence towards global minima.

A larger learning rate may cause erratic jumps in weight adjustments, potentially preventing convergence altogether.

Practical Considerations and Examples

Recommended learning rates typically range from 0.001 to 0.01 for stable convergence during training.

Understanding these principles lays the groundwork for grasping backpropagation and its associated processes.

Chain Rule of Differentiation in Backpropagation

Introduction to Chain Rule

Understanding Weight Update in Neural Networks

Forward Propagation and Weight Update Formula

The weight update formula is introduced: w_textnew = w_textold - textlearning rate times fracpartial textlosspartial w_textold . This formula is crucial for understanding how weights are adjusted during training.

Derivative of Loss with Respect to Weights

The focus shifts to calculating the derivative of loss concerning a specific weight, denoted as w_4 . The updated formula for w_4 is presented: w_4,textnew = w_4,textold - textlearning rate times fracpartial textlosspartial w_4,textold .

Chain Rule Application

To find the derivative of loss with respect to w_4, the chain rule will be applied. This method allows for breaking down complex derivatives into simpler parts.

Understanding Neuron Outputs

The output from neurons, such as o_1 and o_2, is discussed. These outputs are derived from inputs multiplied by weights, followed by an activation function.

Updating Weights Using Chain Rule

The process of updating weights using the chain rule involves recognizing that loss depends on neuron outputs. Specifically, it starts with finding the derivative of loss concerning o_2, which then leads to further calculations involving derivatives related to weights.

Bias Update Mechanism

Bias Updation Formula

Similar to weight updates, bias also has an updation formula: b_2,textnew = b_2,textold - textlearning rate times fracpartial textlosspartial b_2,textold. This highlights that biases undergo similar adjustments as weights during training.

Applying Chain Rule for Other Weights

Updating Weight w_1

A new example focuses on updating another weight, specifically w_1. The corresponding formula is provided: w_1,textnew = w_1,textold - textlearning rate × fracpartial textlosspartial w_1,mathrmold.

Exploring Dependencies in Chain Rule

Understanding Chain Rule in Neural Networks

Dependency of Outputs on Weights

The output o2 is dependent on o1 , leading to the formulation of the derivative of o2 with respect to o1 .

The next step involves recognizing that o1 is dependent on weight w1 , allowing for the calculation of the derivative of o1 concerning w1 .

Weight Update Formulas

The weight update formula for w2 is introduced: new weight equals old weight minus learning rate times the derivative of loss concerning old weight.

Acknowledgment that derivatives can be expanded using chain rule, even when considering additional weights like w4 .

Example Neural Network Structure

An example neural network structure is described, consisting of an input layer, hidden neurons, and an output neuron.

Each component's outputs are labeled (e.g., outputs as o11, o21, o22, o31 ), culminating in a final predicted output ( yhat ).

Updating Weights Using Chain Rule

A prompt encourages viewers to pause and attempt updating weight w1 using chain rule before revealing the solution.

The correct formula for updating weights is reiterated: new weight equals old weight minus learning rate times the derivative of loss concerning old weight.

Paths in Chain Rule Calculation

Two distinct paths for calculating derivatives are discussed; one path follows through dependencies from loss to output neuron back to weights.

The first route emphasizes how loss depends on various outputs sequentially linked back to their respective weights.

Combining Derivative Paths

The first part of the calculation involves multiplying derivatives along one path leading from loss through multiple layers down to a specific weight.

A second path is introduced that also leads back to the same weight but through different outputs, emphasizing how both paths contribute cumulatively.

Understanding the Chain Rule of Differentiation

Introduction to Chain Rule

The chain rule of differentiation is introduced as a fundamental concept in calculus, essential for understanding how derivatives work in complex functions.

The speaker emphasizes the importance of grasping this concept thoroughly, indicating that they will provide detailed explanations and visual aids.

Transition to Vanishing Gradient Problem

After discussing the chain rule, the focus shifts to the vanishing gradient problem, which is highlighted as a significant issue in deep learning.

The speaker notes that understanding this problem is crucial for interviews and practical applications in neural networks.

Exploring the Vanishing Gradient Problem

Deep Neural Networks Overview

A deep neural network structure is described, including layers and weights (w1, w2, w3, etc.), leading to an output (y hat).

The mean squared error (MSE) loss function is introduced with its formula: 1/2(y - haty)^2 .

Activation Functions

The sigmoid activation function is discussed as a key component of neural networks; it outputs values between 0 and 1.

Biases are added at each hidden layer to enhance model performance.

Weight Update Mechanism

Weight Update Formula

The weight update formula for w1 is presented: w_1,textnew = w_1,textold - textlearning rate times partial L/partial w_1 .

Application of Chain Rule in Weight Updates

The application of the chain rule for updating weights involves multiple derivatives linked through each layer's output.

Sigmoid Activation Function Characteristics

Sigmoid Function Formula

The formula for the sigmoid activation function is given: S(x) = 1/1 + e^-x , where outputs are classified based on a threshold (0.5).

Derivative Properties

It’s noted that the derivative of the sigmoid function ranges between 0 and 0.25, which affects backpropagation during training.

Understanding the Derivative and Vanishing Gradient Problem in Neural Networks

Derivative of Sigmoid Function

The derivative curve of the sigmoid function ranges between 0 to 0.25, indicating that the output of the sigmoid function is constrained within this range.

In various calculations involving derivatives, the sigmoid function is consistently applied, affecting all values derived from it.

During backpropagation, when calculating derivatives with respect to outputs (e.g., o4_1), the value will always be between 0 to 0.25 due to the properties of the sigmoid activation function.

Impact on Weight Updates

As derivatives are calculated through chain rule processes, they tend to decrease over iterations (e.g., from 0.10 to smaller values).

This reduction leads to very small derivative values during weight updates, which can significantly impact learning rates.

When small values are multiplied by a small learning rate during weight updates (w_new = w_old - learning_rate * small_value), changes in weights become negligible.

Vanishing Gradient Problem

The situation where weights do not update effectively due to minimal changes is termed as the vanishing gradient problem.

This issue arises because w_new becomes approximately equal to w_old, leading to stagnation in weight adjustments.

Solutions and Alternative Activation Functions

To address the vanishing gradient problem, alternative activation functions should be utilized instead of solely relying on sigmoid functions.

Various activation functions will be explored including:

Sigmoid

Tanh

ReLU

Leaky ReLU

PReLU

Exploring Activation Functions

A detailed discussion on different activation functions will follow, focusing on their functionalities and effectiveness in mitigating issues like vanishing gradients.

Confirmation from participants indicates understanding before proceeding with further explanations about these activation functions.

Key Characteristics of Sigmoid Function

The sigmoid function's output ranges between 0 and 1; its derivative also remains limited between 0 and 0.25 which contributes directly to the vanishing gradient problem.

Understanding the Sigmoid Activation Function and Its Limitations

Properties of the Sigmoid Activation Function

The sigmoid activation function has a smooth gradient, which helps prevent jumps in output values, facilitating quicker convergence during training.

However, the output is not centered around zero, which can hinder efficient weight updates in neural networks.

A zero-centered curve allows for easier weight updates as it includes both negative and positive values. The sigmoid function does not pass through zero, making it non-zero-centered.

Disadvantages of the Sigmoid Activation Function

When inputs are slightly away from the origin, the gradient becomes very small (almost zero), leading to issues like vanishing gradients during backpropagation.

The sigmoid function's exponential operation is computationally expensive due to its formula: sigma(x) = 1/1 + e^-x , increasing time complexity.

Major disadvantages include susceptibility to vanishing gradients, non-zero-centered outputs, and slow computation due to power operations.

Transitioning from Sigmoid to Tanh Activation Function

To address these limitations, researchers developed alternative activation functions like Tanh (hyperbolic tangent).

The Tanh function ranges between -1 and 1 with its formula: textTanh(x) = frace^x - e^-xe^x + e^-x .

Advantages of Tanh Over Sigmoid

Unlike sigmoid, Tanh has a derivative that ranges between 0 and 1. This characteristic improves performance in deep networks by mitigating some vanishing gradient issues.

Although Tanh also faces challenges with deep networks regarding vanishing gradients, it is still preferred over sigmoid due to its better properties.

Summary of Key Differences Between Sigmoid and Tanh

Activation Functions in Neural Networks

Overview of Activation Functions

The discussion begins with the importance of activation functions in neural networks, highlighting that the output interval is crucial for weight updates. Zero-centric functions are noted to be better than sigmoid functions for binary classification.

Acknowledgment of ongoing research in activation functions, leading to the emergence of new equations and methods. The ReLU (Rectified Linear Unit) function is introduced as a popular choice among researchers.

Rectified Linear Unit (ReLU)

ReLU is defined as an activation function where the output is either 0 or the input value x . If x is negative, the output becomes 0; if positive, it retains its value.

The derivative of ReLU can only be 0 or 1, which raises concerns about "dead neurons" during backpropagation when weights become zero.

When a neuron’s derivative becomes zero during backpropagation, it leads to no weight updates, effectively rendering that neuron inactive.

Advantages and Disadvantages of ReLU

Despite its drawbacks, ReLU performs exceptionally well due to its simplicity and speed compared to sigmoid and tanh functions. It addresses issues like vanishing gradients effectively.

However, one major disadvantage remains: ReLU is not zero-centric since it does not produce negative values.

Leaky ReLU Activation Function

To address dead neurons in standard ReLU, Leaky ReLU introduces a small slope (e.g., 0.01 times x ) for negative inputs instead of outputting zero.

This modification ensures that even when inputs are negative, there will always be some gradient available during backpropagation.

Further Developments: Exponential Linear Units

The discussion transitions towards Exponential Linear Units (ELUs), hinting at their unique formula and potential advantages over traditional activation functions like ReLU.

Activation Functions in Neural Networks

Overview of Activation Functions

The discussion begins with the importance of activation functions, highlighting that when the value is greater than 0, we have 'x'; otherwise, a specific operation is applied. This results in an exponential curve ranging between 0 and 1.

Mention of small variations in activation functions like softmax and pre-relu, indicating their significance in neural network performance.

Types of Activation Functions

Explanation of different types:

ReLU (Rectified Linear Unit): If input a_i is 0, it behaves as ReLU; if greater than 0, it acts as Leaky ReLU; if it's a learnable parameter, it becomes Pre-ReLU.

Introduction to Swish, a self-gated function developed by Google. It has unique properties but poses challenges in finding derivatives at zero.

Choosing the Right Activation Function

Emphasis on understanding which activation function to use based on the problem type.

Warning against using sigmoid or tanh due to the vanishing gradient problem.

Recommendations for Binary Classification

For binary classification tasks:

Use ReLU activation functions in hidden layers for efficiency.

In output layers, employ sigmoid activation functions to ensure proper classification outcomes.

Adjustments When Convergence Fails

If convergence issues arise with ReLU:

Consider switching to Pre-ReLU or ELU (Exponential Linear Unit).

Multi-Class Classification Strategies

In multi-class classification problems:

Utilize softmax activation functions in output layers while maintaining ReLU or its variants in hidden layers for optimal performance.

Regression Problem Guidelines

For regression tasks:

Use ReLU or its variations in hidden layers.

Apply linear activation functions in output layers since regression predicts continuous values.

Transitioning to Loss Functions

The session transitions towards discussing loss functions after establishing foundational knowledge about activation functions.

The speaker checks for understanding among participants before moving forward into deeper discussions about loss and cost functions.

Understanding Loss Functions in Deep Learning

Introduction to Loss Functions

The speaker emphasizes the importance of loss functions in deep learning, stating they are "super important" and plans to cover them thoroughly.

A brief overview is provided on how loss functions relate to neural networks, highlighting that loss is calculated after forward propagation and needs to be minimized.

Types of Problems Addressed by Loss Functions

The discussion distinguishes between two main types of problems solved by artificial neural networks: regression and classification.

Regression Problems

In regression, the dataset typically includes continuous output features. An example given involves predicting salary based on years of experience and degree level.

The output feature in regression is a continuous value, making it essential for tasks like salary prediction.

Classification Problems

For classification problems, an example involving study hours and passing or failing an exam illustrates binary classification scenarios.

Multi-class classification examples are also discussed, where different combinations of playing and studying hours affect the likelihood of passing.

Overview of Loss Functions for Regression

The speaker introduces three primary loss functions used in regression:

Mean Squared Error (MSE)

Mean Absolute Error (MAE)

Huber Loss

Distinction Between Loss Function and Cost Function

A new term, cost function, is introduced alongside loss function. The difference lies in their application during training:

Loss Function: Calculated for individual data points.

Cost Function: Calculated over a batch of data points during training epochs.

Forward Propagation Process Explained

During forward propagation with multiple records (e.g., 100), the process involves calculating predictions (y hat), followed by computing losses using the formula y - haty .

To improve efficiency, instead of processing one record at a time, batches can be processed together leading to a more efficient calculation method.

Understanding Mean Squared Error (MSE)

MSE is defined as 1/2(y - haty)^2 . This formula applies both to individual losses and aggregated costs across batches.

Understanding Mean Squared Error and Its Implications

Introduction to Loss Functions

The mean squared error (MSE) is defined as the average of the squares of the differences between actual values (y) and predicted values (y hat). This formula represents a common loss function in regression analysis.

Quadratic Equation Overview

The MSE can be viewed as a quadratic equation, which follows the form a^2 - 2ab + b^2. A general definition of a quadratic equation is expressed as ax^2 + bx + c.

Characteristics of Quadratic Equations

When plotted, quadratic equations create a parabolic curve. This curve is significant for understanding gradient descent in optimization problems.

Advantages of Quadratic Equations

Differentiability: The curve formed by the quadratic equation is differentiable, allowing for effective weight updates during optimization.

Single Minimum: It has only one global or local minimum, simplifying convergence during training.

Faster Convergence: The nature of this curve allows for quicker convergence towards optimal solutions.

Disadvantages of Quadratic Equations

Sensitivity to Outliers: A major drawback is its lack of robustness against outliers. Outliers can significantly distort the regression line due to squaring errors in MSE calculations.

Impact of Outliers on Regression Lines

When outliers are present, they heavily influence the position of the regression line. Removing outliers leads to a more accurate fit compared to when they are included.

Transitioning to Mean Absolute Error (MAE)

To address issues with MSE, researchers developed mean absolute error (MAE), which uses absolute differences instead of squared differences.

Benefits and Characteristics of MAE

Robustness to Outliers: MAE does not square errors; thus, it remains less affected by outlier data points.

Visual Representation and Calculation Challenges

The graph representing MAE differs from that of MSE; it appears linear rather than parabolic. Calculating derivatives requires sub-gradient methods due to its piecewise nature, making it slightly more complex but manageable.

Understanding Loss Functions in Regression and Classification

Huber Loss in Linear Regression

Huber loss is a combination of mean squared error (MSE) and mean absolute error (MAE), providing a robust alternative for regression tasks.

The formula for Huber loss includes a condition based on the difference between actual values (y) and predicted values (y hat). It uses MSE when there are no outliers.

A hyperparameter, denoted as delta, determines whether to apply MSE or MAE based on the magnitude of the error. If the error is less than delta, MSE is used; otherwise, MAE applies.

This approach allows for effective handling of outliers by switching between two different loss calculations depending on their presence.

Transition to Classification Loss Functions

The discussion shifts towards classification loss functions, specifically focusing on cross entropy as a primary method for evaluating model performance in classification tasks.

Cross entropy can be categorized into binary cross entropy for binary classification problems and categorical cross entropy for multi-class classification scenarios.

Binary Cross Entropy

Binary cross entropy is utilized specifically for binary classification tasks. It measures how well the predicted probabilities align with actual outcomes.

The log loss function used in logistic regression serves as the basis for binary cross entropy:

-y log(y hat) - (1-y)log(1-y hat).

When y equals 0 or 1, this function simplifies to either -log(1-y hat) or -log(y hat), respectively, demonstrating its adaptability based on true labels.

Categorical Cross Entropy

Categorical cross entropy addresses multi-class classification problems by evaluating how well predictions match one-hot encoded true labels.

Understanding Multi-Class Classification and Loss Functions

Initial Steps in Multi-Class Classification

The process begins by categorizing outputs into three columns: good, bad, and neutral. Each category is represented as a binary value (1 or 0), where the presence of a class becomes 1 and all others become 0.

Loss Function for Multi-Class Classification

The loss function used is categorical cross-entropy, which quantifies the difference between predicted probabilities and actual classes. The formula involves summing over categories to calculate loss.

Formula Breakdown

The loss function can be expressed mathematically as:

[

textLoss = sum_j=1^c y_ij cdot log_e(y_ij^hat)

]

where c represents the number of categories.

Understanding Output Representation

In this context, y_i denotes the output for each row. For example, y_i1, y_i2, y_i3 represent different classes for the first row.

Class Indicator Values

When determining class membership:

If an element belongs to a class (e.g., good), it is marked as 1; otherwise, it is marked as 0.

Activation Function: Softmax

To derive predicted probabilities ( y^hat_ij ), a softmax activation function is applied at the output layer. This function transforms raw scores into probabilities that sum to one.

Softmax Calculation Example

For instance, if raw scores are [10, 20, ..., n], softmax calculates probabilities using:

[

P(j|i) = frace^z_jsum_k e^z_k

]

resulting in values like [0.4, 0.5,...].

Importance of Probability Summation

It’s crucial that all calculated probabilities sum up to one; this ensures valid probability distribution across classes.

Distinction Between Binary and Multi-Class Problems

For multi-class classification problems, softmax is used in the output layer while sigmoid activation functions are reserved for binary classification tasks.

Summary of Key Concepts Learned

Using ReLU in hidden layers with softmax in output indicates multi-class classification.

Using ReLU with sigmoid indicates binary classification.

Different loss functions apply based on activation types (sigmoid for binary).

Understanding Loss Functions and Activation in Neural Networks

Overview of Loss Functions

The output layer utilizes a linear activation function, with various loss functions applicable depending on the task:

Mean Squared Error (MSE) or Mean Absolute Error (MAE) for regression tasks.

Huber loss can also be applied as an alternative.

Categorical Cross Entropy is used for multi-class classification problems.

Binary Cross Entropy is appropriate for binary classification tasks.

Session Recap and Engagement

The speaker acknowledges the lengthy session duration of 1 hour and 40 minutes, expressing gratitude to participants.

Participants are encouraged to show their understanding by liking the session and sharing it widely.

Key Learnings from the Session

A summary of topics covered includes:

Forward propagation techniques.

Chain rule of derivatives.

Issues related to vanishing gradients.

Understanding different activation functions.

Resources and Community Engagement

Participants are directed to visit ineuron.ai for community classes in deep learning, where they can access course materials and videos.

Resources from previous sessions are available for download, including today's materials which will be uploaded shortly after class.

Upcoming Topics and Encouragement

Tomorrow's discussion will focus on optimizers, which promises to be engaging.

The speaker encourages more participation in future sessions, emphasizing the importance of community support through likes, subscriptions, and sharing content across platforms like LinkedIn and WhatsApp.

Closing Remarks

The speaker expresses hope that attendees enjoyed the session while wishing everyone a happy Eid.

Emphasis is placed on continuous learning and helping others within the community as part of personal growth.

Future Sessions