Machine Learning and Neural Networks - page 52
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The Adam Optimizer (DL 21)
The Adam Optimizer (DL 21)
The goal of this lecture is to demonstrate how we can address the weaknesses of stochastic gradient descent (SGD) and introduce the Adam optimizer, which has been widely used for training neural networks.
In SGD, we update each weight in the neural network by subtracting the product of the learning rate and the partial derivative of the loss with respect to that weight for each batch we train on. This process takes steps in the direction of the negative gradient of the loss, gradually minimizing it. However, SGD can encounter several failure modes that hinder its effectiveness.
To illustrate these issues, we start with a one-dimensional example where we aim to minimize the loss. In this case, SGD takes us downhill by following the gradient's direction, gradually approaching the minimum. However, there are scenarios where SGD fails to be helpful. If we reach a local minimum, where the gradient is zero, we won't move away from that point. Similarly, being on a plateau of the loss function, where the gradient is also zero, prevents progress. Although stochasticity from training on different batches can sometimes help escape local minima or plateaus, it doesn't completely solve these problems.
In higher-dimensional optimization problems, such as training neural networks with multiple weight dimensions, additional challenges arise. Saddle points, where there are local minima and maxima in different directions, can cause the gradient to be zero, slowing down learning. Additionally, discrepancies in the scales of partial derivatives across different dimensions can lead to zigzagging behavior during gradient descent, causing overshooting and slower convergence.
The Adam optimizer addresses these issues with two key ideas. The first is momentum, which favors continuing in the same direction when several steps have been taken in that direction. Momentum helps overcome plateaus and can guide the algorithm away from saddle points. The second idea involves maintaining a moving average of the partial derivatives and the squared partial derivatives. By estimating the second moment (variance) of the partial derivatives, we can normalize the steps taken in each dimension, reducing zigzagging and overshooting.
The Adam update rule replaces the partial derivative term in the weight update with the moving average of the partial derivative (v) divided by the square root of the moving average of the squared partial derivative (s), along with a small constant (epsilon) to avoid division by zero. This update rule ensures more balanced steps across dimensions and has proven to be more effective than vanilla SGD for training neural networks.
The Adam optimizer addresses the limitations of SGD by incorporating momentum and maintaining moving averages of the partial derivatives. These modifications enhance the optimizer's ability to navigate local minima, plateaus, and saddle points while reducing zigzagging behavior and overshooting. As a result, Adam has become a popular choice for training neural networks in practice.
Auto-Encoders (DL 22)
Auto-Encoders (DL 22)
In many deep learning scenarios, we often encounter the idea of training a neural network on one dataset and using a hidden layer from that network to encode data that can be applied to other problems or datasets. This concept is known as transfer learning. For example, with residual networks, pre-training involves learning useful image processing techniques that can be later applied to different datasets. By discarding the output layers of the pre-trained model and adding new output layers for a new task, we essentially create a new network that processes the encoded data produced by the pre-trained model.
Word embeddings also serve the purpose of data encoding, where the goal is to learn a representation that captures meaningful information in the hidden layers. This idea extends to various other contexts as well. One notable model that embraces this concept is the autoencoder. An autoencoder is trained on unsupervised data, where the input data and the output data are the same. While it may seem trivial to solve this regression problem using linear regression, the primary objective of an autoencoder is to learn a more compact representation in its hidden layers.
By progressively reducing the size of the hidden layers, the autoencoder forces the network to learn compressed representations of the input data. If the network can consistently reproduce the original input from this compressed representation, it effectively learns data compression. For instance, if we have a 200x200 pixel image as input and reduce it to a hidden layer of 1000 neurons, which can then expand back to reproduce a close approximation of the original image, we achieve a 20:1 compression ratio.
However, using a neural network solely for data compression is not particularly useful since there are more efficient non-learning-based compression algorithms available. Instead, the value of an autoencoder lies in using either the first half of the network to generate a representation for transfer learning in other deep learning tasks or the second half as a decoder to generate data examples from the input set.
The initial application of an autoencoder for transfer learning was prominent in the early days of deep learning. However, better approaches to transfer learning have since been developed. On the other hand, utilizing the decoder part of the network to generate data samples became the foundation for many other deep learning algorithms.
The simplest approach involves achieving the maximum possible compression by minimizing the size of the hidden layer. In this scenario, any reasonable input vector given to the decoder should generate data resembling the distribution of the input data. However, determining the optimal size of the hidden layer is challenging. It can be either too small, making it incapable of reproducing the inputs accurately, or too large, resulting in the generation of unrealistic data that does not resemble the original dataset.
To address this issue, we can modify the architecture to encourage the autoencoder to learn representations that resemble randomly sampled vectors. This modification leads us to the variational autoencoder. In a variational autoencoder, the middle hidden vector is replaced with two vectors representing the mean and variance. The training process involves generating a random vector using a normal distribution, which is then combined with the hidden encoding vectors to create the input for the decoder. Additionally, the loss for the encoder network includes a divergence term that encourages the mean and variance to stay close to a normal distribution. This helps cluster the representations around the center of the space, making it more reliable for random sampling. Thus, the variational autoencoder allows us to generate samples that closely resemble the data distribution learned by the network.
In summary, the concept of using a neural network's hidden layer as a data encoding has evolved into the ability to sample from a learned probability distribution. This opens the doors to generative adversarial networks and the generation of diverse and interesting data.
Generative Adversarial Networks (DL 23)
Generative Adversarial Networks (DL 23)
In the last lecture, we covered variational autoencoders, which are a type of generative modeling approach. The main goal of the autoencoder is to learn latent variables that can be used for sampling from the generative distribution. Another way to think about generating samples from a distribution is through computational random number generators.
When using a random library in programming, samples from random distributions are generated based on a sequence of random or pseudo-random bits. The random number generator performs computations to transform this sequence of bits into samples from a different distribution. Many distributions are built on top of uniform distributions in these generators.
This alternative approach to generative modeling involves training a generator neural network. The generator takes random noise as input and transforms it into a random sample from a data distribution. For example, if the dataset consists of images of puppies, the goal is to train the neural network to generate random pictures of puppies given any input noise.
To train the generator network, an additional neural network called the discriminator is used. The discriminator takes inputs from either real training data or the output of the generator network and determines whether the input is real or fake. The generator network aims to produce samples that can deceive the discriminator, while the discriminator aims to distinguish real data from fake data. This creates an adversarial relationship between the two networks.
The training process involves training the discriminator first, allowing it to learn the distribution of real data. Then, the generator is trained to produce outputs that resemble real data and can fool the discriminator. Training alternates between the discriminator and the generator to improve their performance.
The loss function for the generator network can be the opposite of the discriminator's loss or a different loss function altogether. Gradients can be propagated back through the discriminator into the generator network to update its weights based on the loss function. This allows the generator to learn how to improve its objective function.
Different loss functions can be used for the generator and discriminator networks, especially when the goal is to generate samples for specific categories within the data distribution. The generator can be conditioned on additional information such as labels to produce samples that fool the discriminator into outputting specific categories.
While training the adversarial networks, there are potential failure modes to consider. One failure mode is the generator network simply producing samples from the real data distribution, which would not be minimizing its objective function. Overfitting is another concern, as the generator could memorize the actual data instead of generating diverse samples.
To avoid overfitting, it is important to limit the generator's exposure to the real data and ensure it does not have too many opportunities to memorize it. The real data set is not directly input into the generator network. The generator learns from the real data indirectly when it influences the weights in the discriminator network, which then affects the loss passed back to the generator.
If successful, a trained generator network can generate samples that resemble the real data but go beyond it. This can be useful for data augmentation in training other neural networks and for artistic purposes. Examples of generative adversarial networks being used for art and data augmentation were discussed in the lecture.
Additionally, trained generator networks can be valuable in various applications beyond data augmentation and art. One such application is in generating synthetic data to improve the training of neural networks for solving other important problems.
By leveraging the generator network, we can generate data samples that are specifically tailored to enhance the training of neural networks. For example, if we have a classification problem where the classes are imbalanced, we can use the generator to generate additional samples for the underrepresented class. This can help balance the dataset and improve the model's ability to learn the minority class.
Furthermore, generative adversarial networks have the potential to generate samples that explore the space between different categories or feature combinations. For instance, if we provide the generator with a combination of features like 0.5 dog and 0.5 cat, it can produce a sample that combines characteristics of both dogs and cats. This ability to interpolate between different categories or features opens up possibilities for creative and novel outputs.
Generative adversarial networks have found applications in various domains. In the field of computer vision, they have been used to generate realistic images, enhance image quality, and even create deep fakes. In natural language processing, they have been employed to generate realistic text, translate between languages, and even create chatbots.
It's important to note that training generative adversarial networks can be a challenging task. It requires careful tuning of hyperparameters, selecting appropriate loss functions, and managing the trade-off between the generator and discriminator networks. Additionally, ensuring the stability of training and avoiding mode collapse, where the generator only produces a limited set of samples, are important considerations.
Despite these challenges, generative adversarial networks have demonstrated impressive capabilities in generating realistic and diverse samples. Ongoing research continues to advance the field, exploring new architectures, loss functions, and training techniques to further improve the performance and reliability of these networks.
In conclusion, generative adversarial networks offer a powerful framework for generative modeling. By training a generator and discriminator network in an adversarial manner, we can learn to generate samples that resemble the real data distribution. This opens up exciting possibilities in data augmentation, creative applications, and improving training for various machine learning tasks.
AlphaGo & AlphaGo Zero (DL 24)
AlphaGo & AlphaGo Zero (DL 24)
AlphaGo and AlphaGo Zero are two go-playing agents developed by Google subsidiary DeepMind. These systems combine deep convolutional neural networks with self-play reinforcement learning to achieve significant advancements in go-playing algorithms. In 2016, AlphaGo became the first go AI to defeat a human world champion. In this video, we will explore how DeepMind created these systems and discuss key findings from the research papers published on both AlphaGo and AlphaGo Zero.
Go is a two-player game with simple rules: players take turns placing black and white stones on an empty intersection of the board. Stones or groups of stones surrounded by the opponent's pieces are captured and removed from the board. The game ends when both players pass, and the score is determined by the number of stones and empty intersections surrounded.
Developing an AI algorithm for go requires planning multiple moves ahead. Chess engines like Deep Blue achieved this by considering all possible move sequences and evaluating resulting board positions. However, evaluating the quality of a go board position is more challenging due to the game's complexity and higher branching factor. Restricting the search space to promising moves and determining board position quality were significant problems that AlphaGo addressed using deep learning.
AlphaGo solved these problems by training deep neural networks to estimate the value and policy of board states. The value network predicts the probability of winning from a given state, while the policy network estimates move quality. These models guide the planning algorithm by restricting attention to promising moves and providing quality estimates.
The architecture of AlphaGo and AlphaGo Zero differs. The original AlphaGo used separate policy and value networks, while AlphaGo Zero employed a single network with separate heads for policy and value outputs. Both architectures incorporate residual blocks to extract important board state information. The training data for AlphaGo included games played by high-level amateurs, while AlphaGo Zero used data solely from self-play.
Training the value network is relatively simple, using board state representations and win/loss labels. Training the policy network is more complex, as it requires predicting move quality. AlphaGo Zero improved on this by training the policy network on move quality estimates generated by the search algorithm during rollouts. Over time, the policy network learns to estimate move quality several steps into the future.
Both AlphaGo and AlphaGo Zero use Monte Carlo Tree Search (MCTS) as their planning algorithm. MCTS performs rollouts to gather information about move sequence values and uncertainty. The search algorithm uses the policy and value networks to evaluate board states and estimate move quality. Through self-play reinforcement learning, both systems improve their networks' quality estimates and become stronger go players.
Overall, the development of AlphaGo and AlphaGo Zero represents a significant milestone in go-playing AI. These systems combine deep learning with reinforcement learning and planning algorithms to achieve remarkable performance and strategic play in the game of go.
Computation Graphs (DL 25)
Computation Graphs (DL 25)
This lecture focuses on computational graphs, which are visual representations of the data flow and sequence of computations in a program. While computational graphs are commonly used for understanding forward and backward propagation in neural networks, they can be applied to any program. By making the implicit operations in a neural network explicit, computational graphs provide a clearer understanding of the computations involved.
In a computational graph, each node represents a computation, such as multiplying weights by activations, summing weighted inputs, computing activation functions, or calculating loss. The connections between nodes represent dependencies between variables in the program. By knowing how to take the derivative of any node in the graph, we can represent both forward and backpropagation steps in a neural network.
To compute the partial derivatives needed for gradient descent in a neural network, we propagate the derivatives backward through the network using the chain rule. At each step, we multiply the derivative of the current operation by the derivative of the previous node. When a node has multiple outputs, we sum the derivatives coming from each output.
The computational graph allows us to compute the outputs of a neural network and calculate the partial derivatives of the loss with respect to each weight. By working backward through a topological sort of the graph and propagating the derivatives, we can determine the partial derivatives for any parameter in the network.
The lecture also provides examples of computational graphs, illustrating how intermediate values and derivatives are computed. By breaking down functions into smaller computations and assigning names to intermediate values, we can compute both function outputs and their partial derivatives using the computational graph.
Furthermore, computational graphs can handle not only scalar variables but also variables representing vectors, matrices, or tensors. By using variables that correspond to higher-dimensional objects, such as activation vectors and weight matrices, we can apply computational graphs to densely connected neural networks and other complex computations.
To extend the computational graph for a densely connected neural network, we can introduce variables that correspond to vectors of activations or matrices of weights. Let's name the vector of activations at this entire layer of the network as "a1," denoted by a vector hat symbol (^). Similarly, we can represent the weights as a matrix W1.
In this extended graph, the input to each node in the layer is the dot product of the activations vector (a1) and the corresponding weights matrix (W1). We can represent this operation as a matrix multiplication: a1 * W1.
Furthermore, we can introduce a bias vector (b1) associated with each node in the layer. The bias term is added element-wise to the dot product of activations and weights before applying an activation function.
Next, we apply an activation function (such as a sigmoid or ReLU) element-wise to the resulting vector. Let's denote this vector as "a2" (with a hat symbol), representing the activations of the next layer.
We can repeat this process for subsequent layers in the neural network, connecting the nodes with edges and propagating the activations and weights through the graph.
To calculate the forward pass in this extended computational graph, we would start with the input values (such as pixel intensities for an image) and propagate them forward through the graph, applying matrix multiplications, element-wise additions, and activation functions at each node until we obtain the final output.
When it comes to backpropagation, the goal is to calculate the partial derivatives of the loss function with respect to each weight in the network. By extending the computational graph, we can trace the flow of gradients backward through the network, enabling us to efficiently compute these partial derivatives using the chain rule.
During backpropagation, we start with the derivative of the loss function with respect to the final output and use the chain rule to propagate it backward through the graph. At each node, we multiply the incoming derivative by the derivative of the corresponding operation (activation function, matrix multiplication, etc.) with respect to its inputs.
By following this process, we can compute the gradients for each weight in the network, which allows us to update the weights using optimization algorithms like gradient descent and its variants.
In summary, extending the computational graph to represent a densely connected neural network allows us to visualize and compute the forward and backward propagation steps. It enables efficient computation of the gradients and facilitates the optimization of the network through weight updates.
Automatic Differentiation (DL 26)
Automatic Differentiation (DL 26)
Reverse mode automatic differentiation (AD) is a technique used to compute the gradients of functions. In Julia, the Zygote library provides automatic differentiation capabilities. When working on large-scale machine learning projects in Julia, the Flux deep learning library, built on top of Zygote, is commonly used.
Zygote offers a "gradient" function that takes another function and input arguments, and it automatically computes the gradient at those points. For example, given a function and the input (1, 2, -1), Zygote can compute the gradient as (22, 4, -12). This feature is convenient but similar to what you might have implemented in Project Zero.
The power of automatic differentiation lies in its ability to compute gradients for more complex functions. For instance, let's consider a function to compute the nth element of the Fibonacci sequence. Using Zygote's gradient function, we can compute the gradient of this Fibonacci function. At inputs (0, 1, 12), the partial derivative with respect to "a" is 89, and with respect to "b" is 144. However, there is no partial derivative with respect to "n" since it's not a continuous variable in this function.
To understand how Zygote computes the gradient for such functions, we can look into reverse mode automatic differentiation. In reverse mode AD, a computation graph is built as the function is executed, and partial derivatives are backpropagated through the graph. To achieve this, the numerical values of variables are replaced with objects that store both the value and additional information for derivative computation.
Two types of information can be stored in these objects: (1) the value of the variable and its partial derivatives with respect to each input (forward mode AD), or (2) the value of the variable and the preceding variables in the computation and the function used to compute its value (reverse mode AD). For deep learning, reverse mode AD is more useful as it scales better in the number of outputs (typically one) rather than inputs (e.g., weight parameters in a neural network).
By creating these reverse mode auto-diff objects and building a computation graph during function evaluation, we can perform backpropagation later. The intermediate variables store the results of computations, and the parent edges in the objects indicate dependencies between nodes in the graph. The computational graph, including function nodes and dependencies, is constructed implicitly. Applying the chain rule to each node, the derivatives can be propagated backward through the graph.
This collection of reverse mode auto-diff objects, with their values, parents, and functions, is commonly stored in a gradient tape. With this approach, even functions with more complex intermediate computations can be differentiated, as long as the derivatives of the components are known. The values of these variables can be scalars, vectors, matrices, or tensors, enabling differentiation of functions with various data types.
In summary, reverse mode automatic differentiation, supported by libraries like Zygote in Julia, allows us to compute gradients for functions efficiently. By building a computation graph and propagating derivatives through it, we can automate the process of computing gradients, making it suitable for deep learning and other complex applications.
Coursera Neural Networks for Machine Learning — Geoffrey Hinton - Lecture 1.1 — Why do we need machine learning
Lecture 1.1 — Why do we need machine learning [Neural Networks for Machine Learning]
Welcome to the Coursera course on neural networks for machine learning! In this course, we will explore the fascinating field of neural networks and their applications in machine learning. Before diving into the intricacies of neural network learning algorithms, let's take a moment to discuss the importance of machine learning, its uses, and provide some examples to illustrate its capabilities.
Machine learning is necessary for solving complex problems that are difficult to address with traditional programming approaches. For instance, recognizing a three-dimensional object from different viewpoints, under varying lighting conditions, and in cluttered scenes is a challenging task. The complexity lies in the fact that we don't fully understand how our brains perform such recognition, making it challenging to write explicit programs to solve these problems. Even if we did uncover the underlying program, it might be extremely complicated to implement effectively.
Another example is detecting fraudulent credit card transactions. Traditional rule-based systems struggle to capture the intricacies of fraud patterns, as they require combining numerous unreliable rules that constantly change over time. Machine learning offers an alternative approach by leveraging a large number of examples that specify correct outputs for given inputs. A learning algorithm processes these examples to produce a program that effectively tackles the task. The resulting program may look different from traditional handcrafted programs, potentially containing millions of weighted numbers. However, if implemented correctly, it can generalize well to new cases and adapt to changing data by retraining on updated information.
Machine learning excels in recognizing patterns, such as objects in real scenes, facial expressions, or spoken words. It is also powerful in identifying anomalies, such as unusual sequences of credit card transactions or abnormal sensor readings in a nuclear power plant. Additionally, machine learning is valuable in prediction tasks, like forecasting stock prices or predicting user preferences based on their past choices and the behavior of others.
Throughout this course, we will use the MNIST database of handwritten digits as a standard example to explain many machine learning algorithms. This database is widely used and allows for effective comparison of different methods. By using such tasks, we can better grasp the underlying concepts and principles of machine learning.
These examples only scratch the surface of the remarkable capabilities of machine learning and neural networks. With technological advancements and readily available computational resources, complex machine learning models can be trained and deployed efficiently. These models have the potential to tackle increasingly complex tasks, pushing the boundaries of what we can achieve with machine learning.
In this course, we will delve into the intricacies of neural networks, discussing their architectures, training algorithms, and practical implementation techniques. By the end of the course, you will have a strong foundation in neural networks and be equipped to apply them to a wide range of problems.
Join us on this exciting journey into the world of neural networks for machine learning. Get ready to expand your knowledge, enhance your skills, and unlock the potential of this transformative technology!
Lecture 1.2 — What are neural networks
Lecture 1.2 — What are neural networks [Neural Networks for Machine Learning]
In this video, I will discuss real neurons in the brain, which serve as the foundation for artificial neural networks that we will explore in this course. Although we won't focus much on real neurons throughout most of the course, I wanted to provide a brief overview initially.
There are several reasons to study how networks of neurons can compute. Firstly, it helps us understand the functioning of the brain. While conducting experiments directly on the brain seems logical, it is a complex and delicate organ that doesn't withstand manipulation well. Therefore, computer simulations are essential for comprehending empirical findings.
Secondly, studying neural networks allows us to grasp the concept of parallel computation, inspired by the brain's ability to compute through a vast network of relatively slow neurons. Understanding this style of parallel computation could lead to advancements in parallel computers, which differ significantly from conventional serial processors. It is particularly effective for tasks in which the brain excels, such as vision, but not well-suited for tasks like multiplication.
The third reason, relevant to this course, involves solving practical problems using innovative learning algorithms inspired by the brain. These algorithms can be highly valuable even if they don't precisely mimic the brain's operations. Thus, while we won't delve deeply into how the brain functions, it serves as a source of inspiration, indicating that large parallel networks of neurons can perform complex computations.
In this video, I will provide more insights into the workings of the brain. A typical cortical neuron consists of a cell body, an axon for sending messages to other neurons, and a dendritic tree for receiving messages from other neurons. At the point where one neuron's axon connects with another neuron's dendritic tree, we find a synapse. When a spike of activity travels along the axon, it injects charge into the postsynaptic neuron.
A neuron generates spikes when the charge received in its dendritic tree depolarizes a region called the axon hillock. Once depolarized, the neuron transmits a spike along its axon, which is essentially a wave of depolarization.
Synapses themselves have an interesting structure. They contain vesicles filled with transmitter chemicals. When a spike reaches the axon, it triggers the migration and release of these vesicles into the synaptic cleft. Transmitter molecules diffuse across the synaptic cleft and bind to receptor molecules on the membrane of the postsynaptic neuron. This binding alters the shape of the molecules, creating holes in the membrane that allow specific ions to flow in or out of the postsynaptic neuron, thereby changing its depolarization state.
Synapses are relatively slow compared to computer memory, but they possess advantages over random access memory in computers. They are small, low-power, and adaptable. Adaptability is crucial as it enables synapses to change their strengths by utilizing locally available signals. This adaptability facilitates learning and the ability to perform intricate computations.
The question then arises: How do synapses decide how to change their strength? What are the rules for their adaptation? These are essential considerations.
To summarize, the brain functions through neurons that receive inputs from other neurons. Only a small fraction of neurons receive inputs from sensory receptors. Neurons communicate within the cortex by transmitting spikes of activity. The effect of an input on a neuron depends on its synaptic weight, which can be positive or negative. These synaptic weights adapt, allowing the entire network to learn and perform various computations, such as object recognition, language comprehension, planning, and motor control.
The brain is composed of approximately 10^11 neurons, each with around 10^4 synaptic weights. Consequently, the brain contains an immense number of synaptic weights, many of which contribute to ongoing computations within milliseconds. This provides the brain with superior bandwidth for storing knowledge compared to modern workstations.
Another intriguing aspect of the brain is its modularity. Different regions of the cortex end up specializing in different functions. Inputs from the senses are directed to specific regions genetically, influencing their eventual functionality. Local damage to the brain results in specific effects, such as the loss of language comprehension or object recognition. The brain's flexibility is evident in the fact that functions can relocate to other parts of the brain in response to early damage. This suggests that the cortex contains a flexible, universal learning algorithm that can adapt to particular tasks based on experience.
In conclusion, the brain performs rapid parallel computation once it has learned, combined with remarkable flexibility. It is akin to an FPGA, where standard parallel hardware is built, and subsequent information determines the specific parallel computation to be performed. Conventional computers achieve flexibility through sequential programming, but this necessitates fast central processes to access program lines and perform lengthy sequential computations.
Lecture 1.3 — Some simple models of neurons
Lecture 1.3 — Some simple models of neurons [Neural Networks for Machine Learning]
I will describe some simple models of neurons, including linear neurons, threshold neurons, and more complex models. These models are simpler than real neurons but still allow us to create neural networks for machine learning. When understanding complex systems, we need to simplify and idealize them to grasp their workings. This involves removing non-essential details and applying mathematics and analogies. While it's important not to overlook essential properties, it can be valuable to study models that are known to be incorrect but still useful in practice. For instance, neural networks often use neurons that communicate real values instead of discrete spikes, even though real cortical neurons behave differently.
The simplest type of neuron is the linear neuron, which has computational limitations but provides insights into more complex neurons. Its output is determined by a bias and the weighted sum of input activities. A plot of the bias plus weighted activities forms a straight line. In contrast, binary threshold neurons, introduced by McCulloch and Pitts, send a spike of activity if the weighted sum exceeds a threshold. These spikes represent truth values that neurons combine to produce their own truth value. While logic was once seen as the main paradigm for understanding the mind, the brain is now thought to combine various unreliable evidence sources, making logic less suitable.
Binary threshold neurons can be described by two equivalent equations. The total input is the sum of input activities multiplied by weights, and the output is one if the total input is above the threshold. Alternatively, the total input includes a bias term, and the output is one if the total input is above zero. A rectified linear neuron combines properties of linear neurons and binary threshold neurons. It computes a linear weighted sum but applies a non-linear function to determine the output. The output is zero if the sum is below zero and equal to the sum if it's above zero, resulting in a non-linear but linear above-zero output.
Sigmoid neurons are commonly used in artificial neural networks. They provide a real-valued output that is a smooth and bounded function of the total input. The logistic function is often used, where the output is one divided by one plus the negative exponent of the total input. For large positive inputs, the output is one, while for large negative inputs, the output is zero. The sigmoid function has smooth derivatives, facilitating learning in neural networks.
Stochastic binary neurons use the same equations as logistic units, but instead of outputting the probability as a real number, they make a probabilistic decision and output either one or zero. The probability represents the likelihood of producing a spike. If the input is very positive, they will likely produce a one, while a very negative input will likely result in a zero. Rectified linear units follow a similar principle but introduce randomness in spike production. The output of a rectified linear unit represents the rate of producing spikes, and the actual spike times are determined by a random Poisson process within the unit.
These stochastic behaviors in binary neurons and rectified linear units introduce intrinsic randomness to the neural network. While the rate of spike production is deterministic, the actual timing of spikes becomes a random process. This randomness adds variability and stochasticity to the system.
Understanding these different neuron models provides us with a range of computational capabilities. Linear neurons are computationally limited but can offer insights into more complex systems. Binary threshold neurons allow for decision-making based on threshold comparisons. Rectified linear neurons combine linearity and non-linearity, enabling decision-making and linear processing simultaneously. Sigmoid neurons provide smooth, bounded outputs and are commonly used in neural networks due to their differentiable nature. Stochastic binary neurons and rectified linear units introduce randomness into the system, allowing for probabilistic decision-making and introducing variability.
By combining different types of neurons in neural networks, we can create powerful models for machine learning tasks. These networks can learn from data, adapt their weights and biases, and make predictions or classifications based on learned patterns. Understanding the principles and behaviors of these neuron models helps us design and train effective neural networks.
However, it's essential to remember that these neuron models are simplified abstractions of real neurons in the brain. The brain is an incredibly complex and dynamic system, and these models serve as approximations to capture certain aspects of neural processing. While they may not capture the full complexity of real neurons, they provide useful tools for building computational models and achieving impressive machine learning capabilities.
Studying different models of neurons, including linear neurons, threshold neurons, rectified linear neurons, sigmoid neurons, and stochastic binary neurons, allows us to understand various computational properties and behaviors. These models form the foundation for constructing neural networks and enable us to perform diverse machine learning tasks. While simplified, they offer valuable insights into the functioning of neural systems.
Lecture 1.4 — A simple example of learning
Lecture 1.4 — A simple example of learning [Neural Networks for Machine Learning]
In this example of machine learning, we will explore a simple neural network that learns to recognize digits. Throughout the process, you will witness the evolution of weights using a basic learning algorithm.
Our focus is on training a straightforward network to identify handwritten shapes. The network consists of two layers: input neurons representing pixel intensities and output neurons representing classes. The objective is for the output neuron corresponding to a specific shape to become active when that shape is presented.
Each active pixel "votes" for the shapes it is part of, and these votes have varying intensities. The shape with the most votes wins, assuming there is competition among the output units. We will delve into this competitive aspect in a later lecture.
To visualize the weights, we need a display that can accommodate thousands of weights. Instead of writing the weights on individual connections between input and output units, we will create small maps for each output unit. These maps represent the strength of connections from input pixels by using black and white blobs. The area of each blob indicates the magnitude, while the color represents the sign of the connection.
Initially, the weights are assigned small random values. To improve the weights, we will present the network with data and train it to adjust the weights accordingly. When an image is shown, we increment the weights from the active pixels to the correct class. However, to prevent the weights from becoming excessively large, we also decrement the weights from the active pixels to the class the network guesses. This training approach guides the network to make the right decisions rather than sticking to its initial tendencies.
After showing the network several hundred training examples, we observe the weights again. They begin to form regular patterns. With further training examples, the weights continue to change and eventually stabilize. At this point, the weights resemble templates for the shapes. For instance, the weights going into the "one" unit serve as a template for identifying ones. Similarly, the weights going into the "nine" unit focus on discriminating between nines and sevens based on the presence or absence of specific features.
It is worth noting that this learning algorithm, due to the simplicity of the network, can only achieve a limited ability to discriminate shapes. The learned weights effectively function as templates, and the network determines the winner based on the overlap between the template and the ink. However, this approach falls short when faced with the complexity of variations in handwritten digits. To address this, we need to extract features and analyze their arrangements, as simple template matching of whole shapes cannot solve the problem adequately.
In summary, the example demonstrates the training of a simple neural network to recognize digits. While the network's weights evolve and resemble templates for the shapes, the limitations of this approach become apparent when faced with the intricate variations in handwritten digits.