Training Artificial Neural Networks (Part 1)

CSI 4106 - Fall 2024

Marcel Turcotte

Version: Oct 23, 2024 15:18

Preamble

Quote of the Day

Sir Demis Hassabis is the Co-founder and CEO of Google DeepMind, a leading company dedicated to addressing some of the most complex scientific and engineering challenges of our era to propel scientific advancement. A chess prodigy from the age of four, Hassabis achieved master-level proficiency by 13 and served as the captain for several England junior chess teams. In 2024, he was awarded the Nobel Prize in Chemistry for his contributions to the development of AlphaFold.

Learning objectives

• Explain the architecture and function of feedforward neural networks (FNNs).
• Describe the backpropagation algorithm and its role in training neural networks.
• Identify common activation functions and understand their impact on network performance.
• Understand the vanishing gradient problem and strategies to mitigate it.

As with Assignment 2, I have compiled the important concepts for the next assignment into a single lecture.

Summary

3Blue1Brown

It is highly recommended that you watch this video. While it covers the concepts we have already explored, it presents the material in a manner that is challenging to replicate in a classroom setting.

• Provides a clear explanation of the intuition behind the effectiveness of neural networks, detailing the hierarchy of concepts briefly mentioned in the last lecture.
• Offers a compelling rationale for the necessity of a bias term.
• Similarly, elucidates the concept of activation functions and the importance of a squashing function.
• The segment beginning at 13m 26s offers a visual explanation of the linear algebra involved: \(\sigma(W X^T + b)\).

In my opinion, this is an excellent and informative video.

Summary - DL

• Deep learning (DL) is a machine learning technique that can be applied to supervised learning (including regression and classification), unsupervised learning, and reinforcement learning.

• Inspired from the structure and function of biological neural networks found in animals.

• Comprises interconnected neurons (or units) arranged into layers.

Summary - FNN

Neural networks have inputs and outputs.

The network consists of three layers: input, hidden, and output. The input layer contains two nodes, the hidden layer comprises three nodes, and the output layer has two nodes. Additional hidden layers and nodes per layer can be added, which will be discussed later.

It is often useful to include explicit input nodes that do not perform calculations, known as input units or input neurons. These nodes act as placeholders to introduce input features into the network, passing data directly to the next layer without transformation. In the network diagram, these are the light blue nodes on the left. Typically, the number of input units corresponds to the number of features.

Information in this architecture flows unidirectionally—from left to right, moving from input to output. Consequently, it is termed a feedforward neural network (FNN).

Summary - FNN

Neural networks can have a significantly large number of input nodes, often in the hundreds or thousands, depending on the complexity of the data. Additionally, they may contain numerous hidden layers. For instance, ResNet, which won the ILSVRC 2015 image classification task, features 152 layers. The authors of ResNet have demonstrated results for networks with 100 and even 1000 layers (He et al. 2016). However, the number of output nodes tends to be relatively small. In regression problems, there is typically one output node, while in classification tasks (whether multiclass or multilabel), the number of output nodes corresponds to the number of classes.

The number of layers and nodes can vary based on the specific requirements.

Summary - units

In the diagram above, it is important to clarify that the inputs and output pertain specifically to this individual unit, rather than to the entire network’s global inputs and output.

The name activation originates from the function’s role in determining whether a neuron should be “activated” or “fired” based on its input.

Historically, the concept was inspired by biological neurons, where a neuron activates and transmits a signal to other neurons if its input exceeds a certain threshold. In artificial neural networks, the activation function serves a similar purpose by introducing non-linearity into the model. This non-linearity is crucial because it enables the network to learn complex patterns and representations in the data.

Introducing a fictitious input \(x^{(0)} = 1\) is a hack that simplifies the expression \(x^T\theta + b\).

Common Activation Functions

Consider the following observations:

• The sigmoid function produces outputs within the open interval \((0, 1)\).
• The hyperbolic tangent function (\(\tanh\)) has an image spanning the open interval \((-1, 1)\).
• The Rectified Linear Unit (ReLU) function outputs values in the interval \([0, \infty)\).

Additionally, note:

• The maximum derivative value of the sigmoid function is 0.25.
• The maximum derivative value of the \(\tanh\) function is 1.
• The derivative of the ReLU function is 0 for negative inputs and 1 for positive inputs.

Furthermore:

• A node employing ReLU as its activation function generates outputs within the range \([0, \infty)\). However, its derivative, utilized in gradient descent during backpropagation, is constant, taking values of either 0 or 1.

Géron (2022) – 10_neural_nets_with_keras.ipynb

Universal Approximation

The universal approximation theorem states that a feedforward neural network with a single hidden layer containing a finite number of neurons can approximate any continuous function on a compact subset of \(\mathbb{R}^n\), given appropriate weights and activation functions.

Cybenko (1989); Hornik, Stinchcombe, and White (1989)

Notation

Notation

A two-layer perceptron computes:

\[ y = \phi_2(\phi_1(X)) \]

where

\[ \phi_l(Z) = \phi(W_lZ_l + b_l) \]

Where \(\phi\) is an activation function, \(W\) a weight matrix, \(X\) an input matrix, and \(b\) a bias vector.

Notation

A 3-layer perceptron computes:

\[ y = \phi_3(\phi_2(\phi_1(X))) \]

where

\[ \phi_l(Z) = \phi(W_lZ_l + b_l) \]

Notation

A \(k\)-layer perceptron computes:

\[ y = \phi_k( \ldots \phi_2(\phi_1(X)) \ldots ) \]

where

\[ \phi_l(Z) = \phi(W_lZ_l + b_l) \]

A feedforward network exhibits a consistent structure, where each layer executes the same type of computation on varying inputs. Specifically, the input to layer \(l\) is the output from layer \(l-1\).

Back-propagation

3Blue1Brown

Back-propagation

Learning representations by back-propagating errors

David E. Rumelhart, Geoffrey E. Hinton & Ronald J. Williams

We describe a new learning procedure, back-propagation, for networks of neurone-like units. The procedure repeatedly adjusts the weights of the connections in the network so as to minimize a measure of the difference between the actual output vector of the net and the desired output vector. As a result of the weight adjustments, internal ‘hidden’ units which are not part of the input or output come to represent important features of the task domain, and the regularities in the task are captured by the interactions of these units. The ability to create useful new features distinguishes back-propagation from earlier, simpler methods such as the perceptron-convergence procedure.

I am presenting here the abstract from the seminal Nature publication where Hinton and colleagues introduced the backpropagation algorithm. This abstract is both elegant and informative, effectively capturing the core principles of modern neural networks: the concept of a loss function, the iterative adjustment of weights through the gradient descent algorithm, and the critical role of hidden layers in generating useful task-dependent features.

Nature is a prestigious journal, and it only occasionally publishes content related to computer science.

At the time of this publication, Hinton was affiliated with Carnegie Mellon University. As a reminder, Hinton received the Nobel Prize in Physics in 2024 for his contributions to developing foundational methods in modern machine learning.

The abstract highlights the rationale for using hidden layers in neural networks. The initial hidden layers learn simple representations directly from the input data, while subsequent layers identify associations among these representations. Each layer builds upon the knowledge of previous layers, culminating in the network’s final output.

Rumelhart, Hinton, and Williams (1986)

Before the back-propagation

• Limitations, such as the inability to solve the XOR classification task, essentially stalled research on neural networks.

• The perceptron was limited to a single layer, and there was no known method for training a multi-layer perceptron.

• Single-layer perceptrons are limited to solving classification tasks that are linearly separable.

Back-propagation: contributions

• The model employs mean squared error as its loss function.

• Gradient descent is used to minimize loss.

• A sigmoid activation function is used instead of a step function, as its derivative provides valuable information for gradient descent.

• Shows how updating internal weights using a two-pass algorithm consisting of a forward pass and a backward pass.

• Enables training multi-layer perceptrons.

Backpropagation: top level

1. Initialization

2. Forward Pass

3. Compute Loss

4. Backward Pass (Backpropagation)

5. Repeat 2 to 5.

The algorithm stops either after a predefined number of epochs or when convergence criteria are satisfied.

Backpropagation: 1. Initialization

Initialize the weights and biases of the neural network.

1. Zero Initialization
   • All weights are initialized to zero.
   • Symmetry problems, all neurons produce identical outputs, preventing effective learning.
2. Random Initialization
   • Weights are initialized randomly, often using a uniform or normal distribution.
   • Breaks the symmetry between neurons, allowing them to learn.
   • If not scaled properly, leads to slow convergence or vanishing/exploding gradients.

Initializing weights and biases to zero works for logistic regression because it is a linear model with a single layer. In logistic regression, each feature’s weight is independently adjusted during training, and the optimization process can converge correctly regardless of the initial weights, provided the data is linearly separable.

However, zero initialization does not work well for neural networks due to their multi-layered structure. Here’s why:

1. Symmetry Breaking: Neural networks require breaking symmetry between neurons in each layer so that they can learn different features. If all weights are initialized to zero, each neuron in a layer will compute the same output and receive the same gradient during backpropagation. This results in the neurons updating identically, preventing them from learning distinct features and effectively rendering multiple neurons redundant.

2. Non-Linearity: Neural networks rely on non-linear transformations between layers to model complex relationships in the data. Zero initialization inhibits the ability of neurons to activate differently, impeding the network’s capacity to capture non-linear patterns.

See also: Xavier/Glorot and He initialization (later)

Backpropagation: 2. Forward Pass

For each example in the training set (or in a mini-batch):

• Input Layer: Pass input features to first layer.

• Hidden Layers: For each hidden layer, compute the activations (output) by applying the weighted sum of inputs plus bias, followed by an activation function (e.g., sigmoid, ReLU).

• Output Layer: Same process as hidden layers. Output layer activations represent the predicted values.

In practice, it is the mini-batch version of this algorithm that is being used.

The forward pass is almost identical to applying the network for prediction (.predict()), with the exception that intermediate (activation) results are saved, as they are needed for the backward pass.

Backpropagation: 3. Compute Loss

Calculate the loss (error) using a suitable loss function by comparing the predicted values to the actual target values.

A smaller loss indicates that the predicted values are closer to the actual target values.

The value of the loss function can serve as a stopping criterion, with backpropagation halting when the loss is sufficiently small.

Crucially, the derivative of the loss function provides essential information for adjusting the network’s weights and bias terms.

More on the various loss functions coming later: mean squared error for regression tasks or cross-entropy loss for classification tasks.

Backpropagation: 4. Backward Pass

• Output Layer: Compute the gradient of the loss with respect to the output layer’s weights and biases using the chain rule of calculus.

• Hidden Layers: Propagate the error backward through the network, layer by layer. For each layer, compute the gradient of the loss with respect to the weights and biases. Use the derivative of the activation function to help calculate these gradients.

• Update Weights and Biases: Adjust the weights and biases using the calculated gradients and a learning rate, which determines the step size for each update.

At the end of the presentation, links are provided to a series of videos by Herman Kamper. These videos elucidate the intricacies of the backpropagation algorithm across various architectures, both with and without forks, utilizing function composition and graph computation approaches.

While the algorithm is complex due to the numerous cases it entails, its regular structure makes it suitable for automation. Specifically, algorithms like automatic differentiation (autodiff) facilitate this process.

In 1970, Seppo Ilmari Linnainmaa introduced the algorithm known as reverse mode automatic differentiation in his MSc thesis. Although he did not apply this algorithm to neural networks, it is more general than backpropagation.

Common optimization techniques like gradient descent or its variants (e.g., Adam) are employed.

Key Concepts

• Activation Functions: Functions like sigmoid, ReLU, and tanh introduce non-linearity, which allows the network to learn complex patterns.

• Learning Rate: A hyperparameter that controls how much to change the model in response to the estimated error each time the model weights are updated.

• Gradient Descent: An optimization algorithm used to minimize the loss function by iteratively moving towards the steepest descent as defined by the negative of the gradient.

Summary

Attribution: Angermueller et al. (2016)

Training

Vanishing gradients

• Vanishing gradient problem: Gradients become too small, hindering weight updates.

• Stalled neural network research (again) in early 2000s.

• Sigmoid and its derivative (range: 0 to 0.25) were key factors.

• Common initialization: Weights/biases from \(\mathcal{N}(0, 1)\) contributed to the issue.

Glorot and Bengio (2010) shed light on the problems.

The vanishing gradient problem often occurs with activation functions like the sigmoid and hyperbolic tangent (tanh), leading to difficulties in training deep neural networks due to diminishing gradients that slow down learning.

In contrast, the exploding gradient problem, which involves gradients growing excessively large, is typically observed in architectures such as recurrent neural networks (RNNs).

Both issues can significantly affect the stability and convergence of gradient-based optimization techniques, thereby hindering the effective training of deep models.

Vanishing gradients: solutions

• Alternative activation functions: Rectified Linear Unit (ReLU) and its variants (e.g., Leaky ReLU, Parametric ReLU, and Exponential Linear Unit).

• Weight Initialization: Xavier (Glorot) or He initialization.

Other techniques exists to mitigate the problem, including those:

• Batch Normalization: Implement batch normalization to standardize the inputs to each layer, which can help stabilize and accelerate training by reducing internal covariate shift and maintaining effective gradient flow.

• Residual Networks: Use residual connections, as seen in ResNet architectures, which allow gradients to flow more easily through the network by providing shortcut paths that bypass one or more layers.

Glorot and Bengio

Figure 6

Figure 7

The graphs presented illustrate the normalized histograms of activation values and back-propagated gradients associated with the hyperbolic tangent activation function.

The produce the top diagrams, Glorot and Bengio used an initialization method that was popular at the time, whereas the bottom diagrams have been produced using a new scheme (Glorot).

Glorot and Bengio (2010), page 254.

Glorot and Bengio

Objective: Mitigate the unstable gradients problem in deep neural networks.

Signal Flow:

• Forward Direction: Ensure stable signal propagation for accurate predictions.
• Reverse Direction: Maintain consistent gradient flow during backpropagation.

Glorot and Bengio (2010): pay attention to signal flow in both directions!

Glorot and Bengio

Variance Matching:

• Forward Pass: Ensure the output variance of each layer matches its input variance.

• Backward Pass: Maintain equal gradient variance before and after passing through each layer.

Keras employs Glorot initialization by default, which is well-suited for activation functions such as sigmoid, tanh, and softmax.

He initialization

A similar but slightly different initialization method design to work with ReLU, as well as Leaky ReLU, ELU, GELU, Swish, and Mish.

Ensure that the initialization method matches the chosen activation function.

import tensorflow as tf
from tensorflow.python.keras.layers import Dense

dense = Dense(50, activation="relu", kernel_initializer="he_normal")

• Glorot Initialization (Xavier Initialization): This method sets the initial weights based on the number of input and output units for each layer, aiming to keep the variance of activations consistent across layers. It is particularly effective for activation functions like sigmoid and tanh.

• He Initialization: This approach adjusts the weight initialization to be suitable for layers using ReLU and its variants, by scaling the variance according to the number of input units only.

AKA Kaiming initialization.

Note

Randomly initializing the weights¹ is sufficient to break symmetry in a neural network, allowing the bias terms to be set to zero without impacting the network’s ability to learn effectively.

1. Proper initialization of weights, such as using Xavier/Glorot or He initialization, is crucial and should be aligned with the choice of activation function to ensure optimal network performance.

Activation Function: Leaky ReLU

When the input to the ReLU activation function, the weighted sum plus bias, is negative for all the training examples, the output value of ReLU is zero. But also, its derivative is 0, which effectively deactivates the neuron. Leaky ReLU, or other variants, effectively mitigates the issue.

import tensorflow as tf
from tensorflow.python.keras.layers import Dense

leaky_relu = tf.keras.layers.LeakyReLU(negative_slope=0.2)
dense = tf.keras.layers.Dense(50, activation=leaky_relu, kernel_initializer="he_normal")

Keras proposes 18 layer activation functions at the time of writing.

The Leaky ReLU, a variant of the standard ReLU activation function, effectively mitigates the issue of dying ReLU nodes. For negative input values, it introduces a linear component with a slope governed by the parameter negative_slope.

Prologue

Summary

• Artificial Neural Networks (ANNs):
  • Inspired by biological neural networks.
  • Consist of interconnected neurons arranged in layers.
  • Applicable to supervised, unsupervised, and reinforcement learning.
• Feedforward Neural Networks (FNNs):
  • Information flows unidirectionally from input to output.
  • Comprised of input, hidden, and output layers.
  • Can vary in the number of layers and nodes per layer.
• Activation Functions:
  • Introduce non-linearity to enable learning complex patterns.
  • Common functions: Sigmoid, Tanh, ReLU, Leaky ReLU.
  • Choice of activation function affects gradient flow and network performance.
• Universal Approximation Theorem:
  • A neural network with a single hidden layer can approximate any continuous function.
• Backpropagation Algorithm:
  • Training involves forward pass, loss computation, backward pass, and weight updates.
  • Utilizes gradient descent to minimize the loss function.
  • Enables training of multi-layer perceptrons by adjusting internal weights.
• Vanishing Gradient Problem:
  • Gradients become too small during backpropagation, hindering training.
  • Mitigation strategies include using ReLU activation functions and proper weight initialization (Glorot or He initialization).
• Weight Initialization:
  • Random initialization breaks symmetry and allows effective learning.
  • Glorot initialization suits sigmoid and tanh activations.
  • He initialization is optimal for ReLU and its variants.
• Key Concepts:
  • Learning rate determines the step size during optimization.
  • Gradient descent is used to update weights in the direction of minimizing loss.
  • Proper selection of activation functions and initialization methods is crucial for effective training.

3Blue1Brown

A series of videos, with animations, providing the intuition behind the backpropagation algorithm.

• Neural networks (playlist)

  • What is backpropagation really doing? (12m 47s)
  • Backpropagation calculus (10m 18s)

Prerequisite: Gradient descent, how neural networks learn? (20m 33s)

StatQuest

• Neural Networks Pt. 2: Backpropagation Main Ideas (17m 34s)
• Backpropagation Details Pt. 1: Optimizing 3 parameters simultaneously (18m 32s)
• Backpropagation Details Pt. 2: Going bonkers with The Chain Rule (13m 9s)

Prerequisites: The Chain Rule (18m 24s) & Gradient Descent, Step-by-Step (23m 54s)

Herman Kamper

One of the most thorough series of videos on the backpropagation algorithm.

• Introduction to neural networks (playlist)

  • Backpropagation (without forks) (31m 1s)
  • Backprop for a multilayer feedforward neural network (4m 2s)
  • Computational graphs and automatic differentiation for neural networks (6m 56s)
  • Common derivatives for neural networks (7m 18s)
  • A general notation for derivatives (in neural networks) (7m 56s)
  • Forks in neural networks (13m 46s)
  • Backpropagation in general (now with forks) (3m 42s)

Next lecture

• We will talk about softmas, cross-entropy, and regularization.

References

Angermueller, Christof, Tanel Pärnamaa, Leopold Parts, and Oliver Stegle. 2016. “Deep Learning for Computational Biology.” Mol Syst Biol 12 (7): 878. https://doi.org/10.15252/msb.20156651.

Cybenko, George V. 1989. “Approximation by Superpositions of a Sigmoidal Function.” Mathematics of Control, Signals and Systems 2: 303–14. https://api.semanticscholar.org/CorpusID:3958369.

Géron, Aurélien. 2022. Hands-on Machine Learning with Scikit-Learn, Keras, and TensorFlow. 3rd ed. O’Reilly Media, Inc.

Glorot, Xavier, and Yoshua Bengio. 2010. “Understanding the Difficulty of Training Deep Feedforward Neural Networks.” In Proceedings of the Thirteenth International Conference on Artificial Intelligence and Statistics, edited by Yee Whye Teh and Mike Titterington, 9:249–56. Proceedings of Machine Learning Research. Chia Laguna Resort, Sardinia, Italy: PMLR. https://proceedings.mlr.press/v9/glorot10a.html.

He, Kaiming, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. 2016. “Deep Residual Learning for Image Recognition.” In 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 770–78. https://doi.org/10.1109/CVPR.2016.90.

Hornik, Kurt, Maxwell Stinchcombe, and Halbert White. 1989. “Multilayer Feedforward Networks Are Universal Approximators.” Neural Networks 2 (5): 359–66. https://doi.org/https://doi.org/10.1016/0893-6080(89)90020-8.

Rumelhart, David E., Geoffrey E. Hinton, and Ronald J. Williams. 1986. “Learning representations by back-propagating errors.” Nature 323 (6088): 533–36. https://doi.org/10.1038/323533a0.

Russell, Stuart, and Peter Norvig. 2020. Artificial Intelligence: A Modern Approach. 4th ed. Pearson. http://aima.cs.berkeley.edu/.

Marcel Turcotte

Marcel.Turcotte@uOttawa.ca

School of Electrical Engineering and Computer Science (EECS)

University of Ottawa