Machine Learning and Neural Networks - page 53
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Lecture 1.5 — Three types of learning
Lecture 1.5 — Three types of learning [Neural Networks for Machine Learning]
In this video, I will discuss three main types of machine learning: supervised learning, reinforcement learning, and unsupervised learning. The course will primarily focus on supervised learning in the first half and unsupervised learning in the second half. Unfortunately, due to time constraints, we will not cover reinforcement learning.
Supervised learning involves predicting an output given an input vector. The goal is to accurately predict a real number or a class label. Regression deals with real numbers, such as predicting stock prices, while classification involves assigning labels, like distinguishing between positive and negative cases or recognizing handwritten digits. Supervised learning relies on a model class, which is a set of candidate models represented by functions mapping inputs to outputs using numerical parameters (W). These parameters are adjusted to minimize the discrepancy between the predicted output (Y) and the correct output (t).
Reinforcement learning focuses on selecting actions or action sequences to maximize the rewards received. Actions are chosen based on occasional rewards, and the objective is to maximize the expected sum of future rewards. A discount factor is typically employed to prioritize immediate rewards over distant ones. Reinforcement learning presents challenges due to delayed rewards and the limited information conveyed by scalar rewards.
Unsupervised learning, which will be covered extensively in the course's second half, involves discovering useful internal representations of input data. For many years, unsupervised learning was overlooked in favor of clustering, as it was challenging to define the objectives of unsupervised learning. However, unsupervised learning serves various purposes, including creating internal representations beneficial for subsequent supervised or reinforcement learning. It aims to generate compact, low-dimensional representations of high-dimensional inputs, such as images, by identifying underlying manifolds. Unsupervised learning can also provide economical representations using learned features, where inputs can be expressed in binary or sparse codes. Additionally, unsupervised learning encompasses clustering, which can be viewed as an extreme case of finding sparse features, with one feature per cluster.
This video covers the three main types of machine learning: supervised learning, reinforcement learning, and unsupervised learning. While supervised learning focuses on predicting outputs, reinforcement learning centers around maximizing rewards through action selection. Unsupervised learning aims to discover useful internal representations, such as low-dimensional representations or learned features, and includes the identification of underlying clusters.
Lecture 2.1 — Types of neural network architectures
Lecture 2.1 — Types of neural network architectures [Neural Networks for Machine Learning]
Neural networks can have different types of architectures, which refer to how the neurons are connected. The most common architecture in practical applications is a feed-forward neural network, where information flows from input units through hidden layers to output units. On the other hand, recurrent neural networks are more interesting as they allow information to flow in cycles, enabling long-term memory and complex dynamics. Training recurrent networks is challenging due to their complexity, but recent progress has made them more trainable and capable of impressive tasks.
Another type of architecture is symmetrically connected networks, where the weights between units are the same in both directions. These networks follow an energy function and are easier to analyze compared to recurrent networks. However, they are more restricted in their capabilities and cannot model cycles.
In feed-forward neural networks, each layer computes transformations between the input and output, resulting in new representations at each layer. Non-linear functions are applied to the activities of neurons in each layer to capture similarity and dissimilarity between inputs. In contrast, recurrent neural networks utilize directed cycles in their connection graph, allowing for complex dynamics and sequential data modeling. The same weights are used at every time step, and the hidden units' states determine the states of the next time step.
Recurrent neural networks have the ability to remember information for a long time in their hidden states, but training them to utilize this ability is challenging. However, recent algorithms have made significant progress in training recurrent nets. These networks can be used for tasks like predicting the next character in a sequence, generating text, or modeling sequential data.
Overall, neural network architectures can vary in their connections and capabilities, ranging from feed-forward networks for straightforward computations to recurrent networks for memory and complex dynamics.
Lecture 2.2 — Perceptrons: first-generation neural networks
Lecture 2.2 — Perceptrons: first-generation neural networks [Neural Networks for Machine Learning]
Perceptrons, a class of algorithms for machine learning, were first investigated in the early 1960s. Initially, they showed great promise as learning devices, but their limitations were later revealed by Minsky and Papert, leading to a decline in their popularity. Minsky and Papert demonstrated that perceptrons were rather restricted in their ability to learn complex patterns in statistical pattern recognition tasks.
In the field of statistical pattern recognition, a standard approach is followed to recognize patterns. First, the raw input data is processed and converted into a set or vector of feature activations. This conversion is done using predefined programs based on common sense, where human expertise determines what features are relevant for the task at hand. It is important to note that this preprocessing stage does not involve learning. The selection of appropriate features is a crucial step and often involves trial and error. Different features are tried, and their effectiveness is evaluated. Through this iterative process, a set of features is eventually identified that allows the subsequent learning stage to effectively solve the problem.
The learning stage in statistical pattern recognition involves determining the weights associated with each feature activation. These weights represent the strength of evidence that each feature provides in favor of or against the hypothesis that the current input belongs to a particular pattern or class. By summing the weighted feature activations, a total evidence score is obtained, which is compared to a threshold. If the evidence surpasses the threshold, the input vector is classified as a positive example of the pattern being recognized.
Perceptrons are a specific type of statistical pattern recognition system. While there are various types of perceptrons, the standard form, referred to as an alpha perceptron by Rosenblatt, consists of input units that are transformed into feature activations. This transformation may resemble the behavior of neurons, but it is important to note that this stage of the system does not involve learning. Once the feature activations are obtained, the weights are learned using a learning algorithm.
Perceptrons gained prominence in the 1960s through the work of Frank Rosenblatt, who extensively studied and described them in his book "Principles of Neurodynamics." The book presented different kinds of perceptrons and was filled with innovative ideas. One of the most notable contributions was a powerful learning algorithm associated with perceptrons, which generated high expectations for their capabilities.
However, the initial enthusiasm surrounding perceptrons was met with skepticism when it was discovered that their learning algorithm had limitations. For instance, exaggerated claims were made about their ability to differentiate between partially obscured pictures of tanks and trucks. These claims were debunked when it was revealed that the perceptrons were simply measuring the total intensity of the pixels, a task humans perform with greater sensitivity. This kind of misunderstanding tarnished the perceptron's reputation and led to doubts about the effectiveness of neural network models as a whole.
In 1969, Minsky and Papert published a seminal book titled "Perceptrons" that critically analyzed the capabilities of perceptrons and highlighted their limitations. However, the broader field of artificial intelligence mistakenly extrapolated these limitations to all neural network models. The prevailing belief became that Minsky and Papert had proven neural network models to be impractical and incapable of learning complex tasks. In reality, Minsky and Papert's findings were specific to the perceptrons they studied and did not invalidate the potential of neural networks as a whole.
It is worth noting that the perceptron convergence procedure, which we will explore shortly, continues to be widely used today for tasks involving large feature vectors. In fact, major companies like Google employ perceptron-based algorithms to predict outcomes based on vast sets of features.
The decision unit in a perceptron is a binary threshold neuron, a type of neuron that has been encountered before in neural network models. To refresh our understanding, these neurons compute a weighted sum of inputs received from other neurons, add a bias term, and generate an output of one if the sum exceeds zero, otherwise producing an output of zero.
To simplify the learning process, biases can be treated as weights by augmenting each input vector with an additional input of constant value one. By doing so, the bias is incorporated as a weight on this extra input line, eliminating the need for a separate learning rule for biases. In essence, the bias becomes equivalent to a weight, with its value being the negative of the threshold.
Now, let's explore the perceptron's learning procedure, which is surprisingly powerful and guaranteed to converge to a solution. However, it is important to consider some caveats regarding its guarantee, which will be discussed later.
To begin, we include an extra component with a value of one in every input vector. We can then focus on the weights and disregard the biases since they are now treated as weights on the additional input line. Training cases are selected according to any policy that ensures each case is chosen within a reasonable time frame, although the precise definition of "reasonable time" may vary depending on the context.
After selecting a training case, we evaluate the output generated by the perceptron and compare it to the expected output. If the output is correct, indicating that the perceptron's decision aligns with the desired classification, we leave the weights unchanged. However, if the output is incorrect, we adjust the weights based on the following rules:
Remarkably, this simple learning procedure is guaranteed to find a set of weights that produce the correct output for every training case. However, an important condition must be satisfied: there must exist a feasible set of weights that can correctly classify all the training cases. Unfortunately, for many interesting problems, such a feasible set of weights may not exist.
The existence of a feasible set of weights heavily depends on the choice of features used. For many problems, the critical challenge lies in determining the most appropriate features to capture the relevant patterns. If the right features are selected, the learning process becomes feasible and effective. On the other hand, if inadequate features are chosen, learning becomes impossible, and the primary focus shifts to feature selection.
In conclusion, perceptrons played a significant role in the early development of neural network models. While their limitations were revealed by Minsky and Papert, it is important to note that their findings were specific to the perceptrons they examined and did not invalidate the broader potential of neural networks. The perceptron convergence procedure remains a valuable tool, particularly for tasks involving large feature vectors. However, the key to successful pattern recognition lies in selecting appropriate features, as using the right features greatly facilitates the learning process, while using inadequate features can render learning impossible.
Lecture 2.3 — A geometrical view of perceptrons
Lecture 2.3 — A geometrical view of perceptrons [Neural Networks for Machine Learning]
In this video, we will delve into the geometrical understanding of how perceptrons learn. To do this, we need to think in terms of weight space, a high-dimensional space where each point corresponds to a specific configuration of all the weights. In this space, we can represent training cases as planes, and the learning process involves positioning the weight vector on the correct side of all the training planes.
For those who are not well-versed in mathematics, this may be more challenging than previous material. It may require dedicating ample time to comprehend the upcoming content, particularly if you are unfamiliar with thinking about hyperplanes in high-dimensional spaces. You'll need to become comfortable visualizing a 14-dimensional space by first visualizing a three-dimensional space and then audibly reminding yourself of the dimensionality. It may seem peculiar, but it's a common practice to aid comprehension.
When dealing with hyperplanes in a 14-dimensional space, the complexity increases significantly, similar to transitioning from a 2D space to a 3D space. It's crucial to understand that a 14-dimensional space is vast and intricate. With that in mind, let's begin by focusing on weight space.
Weight space is a space that has one dimension for each weight in the perceptron. A point in weight space represents a specific configuration of all the weights, assuming we have eliminated the threshold. Every training case can be represented as a hyperplane passing through the origin in weight space. Consequently, points in this space correspond to weight vectors, while training cases correspond to planes.
For a particular training case, the weights must lie on one side of the hyperplane to produce the correct output. Let's visualize this concept through an example. Consider a training case where the correct answer is one. The weight vector needs to be on the same side of the hyperplane as the direction indicated by the training vector. Any weight vector on that side will have an angle with the input vector of less than 90 degrees, resulting in a positive scalar product. As we have eliminated the threshold, the perceptron will output one, providing the correct answer.
Conversely, if a weight vector lies on the wrong side of the plane, its angle with the input vector will exceed 90 degrees, yielding a negative scalar product. Consequently, the perceptron will output zero, leading to an incorrect answer.
To summarize, weight vectors on one side of the plane yield the correct answer, while those on the other side produce the wrong answer. Now, let's examine a different training case where the correct answer is zero.
In this case, any weight vector making an angle of less than 90 degrees with the input vector will result in a positive scalar product, causing the perceptron to output one, leading to an incorrect answer. Conversely, weight vectors on the other side of the plane, with an angle exceeding 90 degrees, will yield a scalar product less than zero, and the perceptron will output zero, correctly providing the answer.
Let's combine these two training cases in a single picture of weight space. The weight space becomes crowded, and a cone of possible weight vectors emerges. Any weight vector within this cone will yield the correct answer for both training cases. It's worth noting that the existence of such a cone is not guaranteed. There may be scenarios where no weight vectors provide the correct answers for all training cases. However, if such weight vectors exist, they will form a cone.
The learning algorithm considers training cases one by one, adjusting the weight vector to eventually lie within this cone. It's important to observe that if we have two good weight vectors that work for all training cases, their average will also lie within the cone. This implies that the problem is convex, and the average of two solutions is itself a solution. Convex learning problems simplify the process in machine learning.
Understanding weight space and the relationship between weight vectors and training cases provides a geometrical insight into how perceptrons learn. The goal is to find a weight vector that lies within the cone of possible solutions, which ensures correct classification for all training cases.
Lecture 2.4 — Why the learning works
Lecture 2.4 — Why the learning works [Neural Networks for Machine Learning]
In this video, we aim to present a proof that the perceptron learning procedure will ultimately lead the weights to converge within the cone of feasible solutions. However, I want to emphasize that this course primarily focuses on engineering aspects rather than rigorous proofs. There will be few proofs throughout the course. Nevertheless, understanding how perceptrons eventually find the correct answer provides valuable insights.
To construct our proof, we will leverage our geometric understanding of weight space and the perceptron learning process. We assume the existence of a feasible weight vector that yields the correct answer for all training cases. In the diagram, this is represented by the green dot.
The key idea in our proof is that every time the perceptron misclassifies a training case, the current weight vector will be updated in a manner that brings it closer to all feasible weight vectors. We can measure the squared distance between the current weight vector and a feasible weight vector as the sum of a squared distance along the input vector's line (the training case's definition) and another squared distance orthogonal to that line. The orthogonal squared distance remains constant, while the distance along the input vector's line decreases.
While this claim seems promising, we encounter an issue illustrated by the gold feasible weight vector in the diagram. It lies just on the right side of the plane defined by one of the training cases, whereas the current weight vector is on the wrong side. Additionally, the input vector is relatively large, causing the addition of the input vector to move the current weight vector further away from the gold feasible weight vector. As a result, our initial claim fails.
However, we can rectify this by introducing the concept of a generously feasible weight vector. These weight vectors not only classify all training cases correctly but also do so with a margin equal to or greater than the length of the input vector for each case. Inside the cone of feasible solutions, we have another cone of generously feasible solutions.
With this adjustment, our proof becomes valid. We can now claim that every time the perceptron misclassifies a case, the squared distance to all generously feasible weight vectors decreases by at least the squared length of the input vector. This update ensures that the weight vector moves closer to the generously feasible solutions.
While we won't provide a formal proof here, this informal sketch demonstrates the convergence process. If the input vectors are not infinitesimally small, the squared distance to all generously feasible weight vectors decreases by at least the squared length of the input vector after a finite number of mistakes. Consequently, the weight vector must eventually reside within the feasible region, assuming it exists. It doesn't necessarily have to lie in the generously feasible region, but at least within the feasible region to avoid making further mistakes.
To summarize, this is an informal overview of the proof demonstrating that the perceptron convergence procedure works. However, it's essential to note that the entire proof relies on the assumption that a generously feasible weight vector exists. If such a vector doesn't exist, the proof collapses.
Lecture 2.5 — What perceptrons can't do
Lecture 2.5 — What perceptrons can't do [Neural Networks for Machine Learning]
In this video, we will explore the limitations of perceptrons, which arise from the types of features used. The effectiveness of perceptrons heavily depends on the choice of features. With the right features, perceptrons can be incredibly versatile, but with the wrong features, their learning capabilities are severely restricted. This limitation led to perceptrons falling out of favor in the past. It highlights the challenge of learning the appropriate features, which is the crucial aspect of the learning process.
However, even without learning features, perceptrons can still achieve a lot. For instance, in tasks like determining the plausibility of an English sentence, one could manually define numerous features and learn their weights to decide the likelihood of a sentence being grammatically correct. Nevertheless, in the long run, learning features becomes necessary.
Perceptron research faced setbacks in the late 1960s and early 1970s when it was discovered that perceptrons had significant limitations. By choosing features manually and incorporating enough features, perceptrons can accomplish almost any task. For example, if we consider binary input vectors and create separate feature units that activate based on specific binary input vectors, we can achieve any discrimination on binary input vectors. However, this approach is impractical for real-world problem-solving as it requires an excessive number of feature units, hindering generalization. Attempting to generalize from a subset of cases while ignoring others is futile because new feature units would be required for the remaining cases, and determining the weights for those new feature units is challenging once the manual selection is complete.
There are stringent constraints on what perceptrons can learn once the feature units and their weights have been established. Let's examine a classic example to understand these limitations. We want to determine if a binary threshold decision unit can learn to identify whether two features have the same value. We have two positive cases and two negative cases, each defined by single-bit features with values of either 1 or 0. The positive cases occur when both features are on (1) or when both features are off (0), while the negative cases occur when one feature is on (1) and the other is off (0). The task seems straightforward, but algebraically, it can be proven impossible to satisfy all four inequalities formed by these input-output pairs. Consequently, it is not possible to find weights that allow the perceptron to provide the correct output for all four cases.
This limitation can also be understood geometrically. We imagine a data space where each point represents a data point, and weight vectors define planes perpendicular to the data points. To discriminate correctly, the weight plane should separate the positive cases from the negative cases. However, there are sets of training cases that are not linearly separable, meaning no hyperplane can correctly separate the cases where the output should be 1 from those where the output should be 0. This inability to separate cases correctly is known as a "set of training cases that is not linearly separable."
Another devastating example for perceptrons involves recognizing patterns that retain their identity even when translated with wraparound. Perceptrons fail to discriminate between patterns with the same number of pixels if the discrimination needs to work with translated and wrapped patterns. This limitation becomes apparent when considering patterns A and B. Pattern A has four "on" pixels arranged in a barcode-like shape, and pattern B also has four "on" pixels arranged differently. When translated with wraparound, perceptrons cannot learn to discriminate between these patterns. Minsky and Papert's Group Invariance theorem states that perceptrons cannot recognize patterns under translation if wraparound is allowed. This theorem was particularly significant in the history of perceptrons because pattern recognition aims to identify patterns despite transformations such as translation.
The theorem revealed that perceptrons, as they were originally formulated, are unable to handle pattern recognition tasks that require translation invariance with wraparound. This limitation greatly restricted their practical applications and led to a decline in interest and research on perceptrons during the late 1960s and early 1970s. However, it's important to note that these limitations only apply to single-layer perceptrons with binary threshold units. The field of artificial neural networks continued to evolve and overcome these limitations with the development of more advanced models, such as multilayer perceptrons (MLPs) and convolutional neural networks (CNNs). MLPs introduce hidden layers between the input and output layers, allowing for more complex and flexible representations of features. By incorporating nonlinear activation functions and using techniques like backpropagation for weight adjustment, MLPs can overcome the linear separability limitation of single-layer perceptrons.
CNNs, on the other hand, were specifically designed to address the problem of pattern recognition and image classification. They employ a hierarchical structure with convolutional layers that extract local features and pooling layers that capture spatial invariance. CNNs have demonstrated remarkable success in tasks like image recognition, object detection, and natural language processing.
The limitations of perceptrons highlighted the importance of feature learning, nonlinearity, and hierarchical representations in neural networks. Subsequent advancements in the field have led to the development of more sophisticated models with improved learning capabilities and broader applications.
While perceptrons have limitations in their ability to learn complex features and handle certain pattern recognition tasks, these limitations have been addressed through the development of more advanced neural network architectures. MLPs and CNNs, among other models, have overcome the restrictions of single-layer perceptrons and have become powerful tools in various domains of artificial intelligence.
Lecture 3.1 — Learning the weights of a linear neuron
Lecture 3.1 — Learning the weights of a linear neuron [Neural Networks for Machine Learning]
his video introduces the learning algorithm for a linear neuron, which achieves something different from the learning algorithm for a perceptron. In a perceptron, the weights always get closer to a good set of weights, while in a linear neuron, the outputs are always getting closer to the target outputs.
The perceptron convergence procedure ensures that changing the weights brings us closer to a good set of weights. However, this guarantee cannot be extended to more complex networks because averaging two good sets of weights may result in a bad set of weights. Therefore, for multi-layer neural networks, we don't use the perceptron learning procedure, and the proof of improvement during learning is different as well.
Multi-layer neural networks, often called multi-layer perceptrons (MLPs), require a different approach to show progress. Instead of showing that the weights get closer to a good set of weights, we demonstrate that the actual output values get closer to the target output values. This holds true even for non-convex problems, where averaging the weights of two good solutions does not yield a good solution.
The learning algorithm for a linear neuron is illustrated through a toy example. It involves starting with random guesses for the prices of portions and then adjusting these guesses iteratively to fit the observed prices. The iterative approach uses the delta rule to update the weights based on the learning rate, the number of portions, and the residual error.
The delta rule is derived by differentiating the error measure with respect to one of the weights. The learning rule states that the change in a weight is equal to the learning rate multiplied by the input value and the difference between the target and actual outputs. By iteratively applying the delta rule, the weights can be adjusted to minimize the error.
The learning procedure using the delta rule does not guarantee that individual weights will always improve. However, the difference between the target and estimated outputs tends to improve. The learning rate determines the speed of learning, and with a small enough learning rate, the weights can approach the best approximation for the given training cases.
It's important to note that even for linear systems, the learning process can be slow, especially when input dimensions are highly correlated. Determining how much weight should be attributed to each input dimension becomes challenging in such cases. Additionally, there is a similarity between the online version of the delta rule and the perceptron learning rule, where the weight vector is adjusted based on the input vector and the error. However, the delta rule incorporates the learning rate and the residual error. Choosing an appropriate learning rate is crucial for stable and efficient learning.
The iterative learning process described for the linear neuron can converge to a solution that minimizes the error measure. However, it's important to note that there may not be a perfect solution that exactly matches the desired outputs for all training cases. Instead, the goal is to find a set of weights that provides the best approximation and minimizes the error measure across all training cases. By making the learning rate sufficiently small and allowing the learning process to continue for a long enough time, we can approach this best approximation.
The speed of learning can vary, even for linear systems. When two input dimensions are highly correlated, it becomes difficult to determine how much weight should be attributed to each input dimension. For example, if the number of portions of ketchup and chips is always the same, it can take a long time for the learning process to correctly assign the price to each component.
Interestingly, there is a relationship between the delta rule and the learning rule for perceptrons. The online version of the delta rule, where weights are updated after each training case, bears similarities to the perceptron learning rule. In perceptron learning, the weight vector is incremented or decremented by the input vector, but only when an error occurs. In the online version of the delta rule, the weight vector is also adjusted by the input vector, but scaled by both the residual error and the learning rate.
One challenge in using the delta rule is selecting an appropriate learning rate. If the learning rate is too large, the system may become unstable, making it difficult to converge to a solution. On the other hand, if the learning rate is too small, the learning process may take an unnecessarily long time to reach a sensible set of weights.
The learning algorithm for a linear neuron aims to minimize the error between the target outputs and the actual outputs. It iteratively adjusts the weights using the delta rule, which incorporates the learning rate, the input values, and the difference between the target and actual outputs. Although the learning process can be slow and the weights may not individually improve, the overall goal is to approach the best approximation for the given training cases.
Lecture 3.2 — The error surface for a linear neuron
Lecture 3.2 — The error surface for a linear neuron [Neural Networks for Machine Learning]
In this video, we will explore the error surface of a linear neuron, which provides insights into the learning process. By visualizing this surface, we can gain a geometric understanding of how weights are learned in a linear neuron. The space we consider is similar to the weight space used in perceptrons but with an additional dimension.
Imagine a space where the horizontal dimensions represent the weights, and the vertical dimension represents the error. In this space, different weight settings are represented as points on the horizontal plane, and the height of each point corresponds to the error associated with that weight setting, summed over all training cases. For a linear neuron, the errors for each weight setting define an error surface, which takes the form of a quadratic bowl. A vertical cross-section of the error surface always yields a parabola, while a horizontal cross-section forms an ellipse. It's important to note that this behavior holds true only for linear systems with squared error. As we move to multi-layer nonlinear neural networks, the error surface becomes more complex.
While the weights remain within a reasonable range, the error surface remains smooth but can have numerous local minima. To facilitate learning, we employ the delta rule, which involves computing the derivative of the error with respect to the weights. Adjusting the weights proportionally to this derivative is equivalent to performing steepest descent on the error surface. Viewing the error surface from above reveals elliptical contour lines. The delta rule guides us perpendicular to these contour lines. In batch learning, where the gradient is computed over all training cases, the delta rule leads us in the right direction. However, we can also employ online learning, where the weights are updated after each training case, similar to perceptrons. In this case, the weight change moves us towards the constraint planes formed by the training cases.
By alternating between training cases, we can zigzag towards the solution point where the constraint lines intersect, indicating the weights that satisfy both cases. Additionally, examining the error surface enables us to understand the conditions that result in slow learning. If the ellipse representing the contour lines is highly elongated, which occurs when the lines corresponding to two training cases are nearly parallel, the gradient exhibits an unfavorable property. The gradient becomes large in the direction where we don't want to move far and small in the direction where we want to make significant progress. This mismatch impedes efficient learning and makes it challenging to traverse the ravine-like structure of the error surface along its elongated axis.
Visualizing the error surface of a linear neuron provides valuable insights into the learning process. Understanding the geometry of the surface helps us grasp the behavior of the delta rule and its implications for learning speed.
Lecture 3.3 — Learning weights of logistic output neuron
Lecture 3.3 — Learning weights of logistic output neuron [Neural Networks for Machine Learning]
To extend the learning rule from a linear neuron to multi-layer networks of nonlinear neurons, we need to take two steps. First, we need to generalize the learning rule for a single nonlinear neuron, specifically a logistic neuron. While logistic neurons are used as an example, other types of nonlinear neurons could be employed as well.
A logistic neuron computes its logit, denoted as z, which is the sum of its bias and the weighted sum of its input lines. The output, denoted as y, is a smooth nonlinear function of the logit. In the graph, it can be observed that the function approaches zero when z is large and negative, approaches one when z is large and positive, and exhibits smooth and nonlinear changes in between. The continuity of the logistic function provides convenient derivatives for learning. To obtain the derivatives of a logistic neuron with respect to the weights (which is crucial for learning), we first compute the derivative of the logit itself with respect to a weight. This derivative simplifies to the value on the input line, denoted as xi. Similarly, the derivative of the logit with respect to xi is the weight wi.
The derivative of the output with respect to the logit can be expressed in terms of the output itself. Specifically, if the output is represented as y, then dy/dz is given by y * (1 - y). The mathematical derivation of this result is provided on the next slide, and it involves tedious yet straightforward calculations. Having obtained the derivative of the output with respect to the logit and the derivative of the logit with respect to the weight, we can now determine the derivative of the output with respect to the weight. By applying the chain rule, we have dz/dw as xi and dy/dz as y * (1 - y). Consequently, we arrive at a learning rule for a logistic neuron that closely resembles the delta rule.
The change in the error, denoted as de/dwi, as we modify a weight, is obtained by summing over all training cases (n) the product of the value on an input line (xin) and the residual, which is the difference between the target output and the actual output of the neuron. However, there is an additional term stemming from the slope of the logistic function, namely yn * (1 - yn). With this slight modification of the delta rule, we arrive at the gradient descent learning rule for training a logistic neuron.
By applying the gradient descent learning rule to a logistic neuron, we can effectively train multi-layer networks of nonlinear neurons. This extends the learning rule beyond linear systems and enables us to tackle more complex tasks. To understand how this learning rule works in the context of multi-layer networks, let's consider a simple two-layer network as an example. We have an input layer with multiple neurons and an output layer with a single logistic neuron. The weights between the layers are denoted as W and the biases as b.
The learning process involves two steps. First, we compute the output of the network for a given input. This is done by propagating the inputs forward through the network, applying the logistic function to each neuron's total input (logit), and obtaining the final output. Next, we compute the gradients of the error with respect to the weights using the chain rule. Starting from the output layer, we calculate the derivative of the error with respect to the output, which is simply the difference between the target output and the actual output of the network. We then propagate this error gradient backward through the network, multiplying it by the derivative of the logistic function at each neuron to obtain the gradients at the hidden layer. Finally, we update the weights using the computed gradients and a learning rate. The learning rate determines the step size in the weight update and can be adjusted to control the speed of learning. The weight update follows the equation: ΔW = learning_rate * error_gradient * input, where ΔW represents the change in the weights.
This process of forward propagation, error backpropagation, and weight update is repeated iteratively for a set number of epochs or until the network reaches a desired level of performance. By adjusting the weights iteratively, the network gradually learns to make better predictions or classify inputs more accurately. It's important to note that the learning rule we discussed for logistic neurons can be generalized to other types of nonlinear activation functions as well. The key is to compute the derivatives of the activation functions accurately to propagate the error gradients effectively through the network.
By extending the learning rule from linear neurons to logistic neurons and applying it to multi-layer networks, we can train complex nonlinear models. This allows us to solve a wide range of tasks, including pattern recognition, classification, and regression, by iteratively adjusting the weights based on the error gradients.
Lecture 3.4 — The backpropagation algorithm
Lecture 3.4 — The backpropagation algorithm [Neural Networks for Machine Learning]
Now that we have covered the preliminaries, let's delve into the central issue of learning multiple layers of features. In this video, we will finally describe the backpropagation algorithm, which emerged in the 1980s and sparked a surge of interest in neural networks.
Before we dive into backpropagation, let's first discuss another algorithm that is not as effective but worth mentioning. Once we know how to learn the weights of a logistic unit, the next challenge is learning the weights of hidden units. Neural networks without hidden units are limited in the input-output mappings they can model. While a layer of hand-coded features, such as in a perceptron, enhances the network's power, designing those features for a new task remains a difficult and manual process.
Ideally, we would like to automate the feature design loop, allowing the computer to find good features without requiring human insights or repeated trial and error. This is where the concept of learning by perturbing the weights comes to mind. Randomly perturbing a weight is akin to a mutation, and we can observe if it improves the network's performance. If the performance improves, we save that weight change, resembling a form of reinforcement learning.
However, this method is highly inefficient. To evaluate whether changing a weight improves the network, multiple forward passes on a representative set of training cases are required. Assessing the impact of a weight change based on a single training case is inadequate. Additionally, as learning progresses, large changes in weights tend to make things worse since the proper relative values between weights are crucial. Towards the end of learning, not only is it time-consuming to evaluate each weight change, but the changes themselves must be small.
There are slightly better ways to use perturbations for learning. One approach is to perturb all the weights in parallel and then correlate the performance gain with the weight changes. However, this method does not provide significant improvement. The challenge lies in the need for numerous trials with different random perturbations of all the weights to isolate the effect of changing one weight amid the noise caused by changing the others.
An alternative approach that shows some improvement is to randomly perturb the activities of hidden units instead of the weights. Once it is determined that perturbing the activity of a hidden unit on a specific training case improves performance, the weight changes can be computed. Since there are fewer activities than weights, the algorithm becomes more efficient. However, backpropagation still surpasses these methods in efficiency, with a factor equal to the number of neurons in the network.
The core idea behind backpropagation is that while we may not know what the hidden units should be doing (hence the term "hidden"), we can compute how fast the error changes when we modify a hidden unit's activity on a particular training case. Instead of using the activities of hidden units as desired states, we utilize the error derivatives with respect to the activities. Since each hidden unit can influence multiple output units, its effects need to be combined, which can be done efficiently.
To summarize the backpropagation algorithm for a single training case, we start by defining the error as the squared difference between the target values of the output units and the actual values produced by the network. By differentiating this error, we obtain the expression for how the error changes with respect to the output of an output unit. We can then compute the error derivatives for the hidden units by summing the effects of all outgoing connections from the hidden unit using the previously computed error derivatives of the layer above.
The backpropagation algorithm allows us to propagate the error derivatives from one layer to the previous layer efficiently. Once we have the error derivatives for the hidden units, we can easily compute the error derivatives for the weights coming into a hidden unit. This is done by multiplying the error derivative with respect to the total input received by the unit by the activity of the unit in the layer below. The computed error derivatives for the weights represent how the error changes as we modify a particular weight.
Now, let's outline the steps involved in the backpropagation algorithm for a single training case:
Define the error as the squared difference between the target values and the actual output values of the network.
Compute the error derivative with respect to the output of each output unit by applying the chain rule, which involves multiplying the derivative of the output with respect to the total input by the error derivative with respect to the output.
Calculate the error derivative with respect to the output of each hidden unit by summing the effects of all outgoing connections from the hidden unit. This involves multiplying the weight on each connection by the error derivative computed in the layer above.
Repeat step 3 for all hidden layers, propagating the error derivatives backward through the network.
Compute the error derivatives for the weights coming into each unit. This is done by multiplying the error derivative with respect to the total input received by the unit by the activity of the unit in the layer below.
By following these steps, we can efficiently compute the error derivatives for all weights in the network based on a single training case. The backpropagation algorithm enables us to understand how modifying each weight affects the overall error and facilitates the learning process. Understanding and implementing the backpropagation algorithm is crucial for training deep neural networks with multiple layers of features. While this explanation may require careful study, grasping the underlying concepts and computations will be essential for effectively utilizing backpropagation in neural network training.
Once we have computed the error derivatives for all the weights in the network using the backpropagation algorithm, we can use this information to update the weights and improve the network's performance. This process is known as weight adjustment or weight updating. Weight updating typically involves the use of an optimization algorithm, such as gradient descent, to iteratively adjust the weights in the direction that minimizes the error. The basic idea is to update each weight by subtracting a small fraction of its corresponding error derivative, multiplied by a learning rate parameter.
The learning rate determines the step size in weight space and affects the convergence speed of the training process. Choosing an appropriate learning rate is important to ensure stable and efficient learning. A learning rate that is too large can cause the weights to diverge, while a learning rate that is too small can result in slow convergence. During the weight updating process, it's common to update the weights in small batches or even on a single training example at a time. This approach is known as stochastic gradient descent or mini-batch gradient descent, and it helps speed up the learning process and avoid getting stuck in local optima.
The weight updating process is typically repeated for multiple epochs, where each epoch consists of going through the entire training dataset. This allows the network to gradually adjust the weights based on the accumulated errors from multiple training examples, improving its generalization performance. It's important to note that backpropagation and weight updating are performed during the training phase of a neural network. Once the network is trained, it can be used for making predictions on new, unseen data by simply feeding the input through the network and obtaining the output.
Backpropagation and the ability to learn multiple layers of features have been crucial in the success of deep learning. With the backpropagation algorithm, neural networks can automatically learn complex representations of data, enabling them to solve a wide range of tasks, including image and speech recognition, natural language processing, and more.
The backpropagation algorithm is a powerful technique for efficiently computing error derivatives in neural networks with multiple layers. By propagating the error information backward through the network, we can determine how changes in the weights affect the overall error. This information is then used to update the weights iteratively, allowing the network to learn and improve its performance over time.