Machine Learning and Neural Networks - page 55
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Lecture 6.1 — Overview of mini batch gradient descent
Lecture 6.1 — Overview of mini batch gradient descent [Neural Networks for Machine Learning]
In this video, we will discuss stochastic gradient descent learning for neural networks, focusing on the mini-batch version, which is widely used in large neural networks. The error surface of a linear neuron forms a quadratic bowl, where the horizontal axes represent the weights and the vertical axis represents the error. For multi-layer non-linear networks, the error surface is more complex but locally approximated by a fraction of a quadratic bowl.
When using full batch learning, descending along the steepest gradient direction may not lead to the desired destination. The direction of steepest descent often runs almost perpendicular to the desired direction, causing convergence issues. This problem persists in non-linear multi-layer networks, as the error surfaces tend to be highly curved in some directions and less curved in others.
To address this, stochastic gradient descent (SGD) is employed. Instead of computing the gradient on the entire dataset, SGD computes the gradient on subsets or mini-batches of the data. This approach offers several advantages, such as reduced computation for weight updates and the ability to parallelize gradient computations for multiple training cases.
Using mini-batches helps avoid unnecessary weight sloshing. It is important to have mini-batches that are representative of the entire dataset and avoid ones that are uncharacteristic, such as having all examples from a single class. While there are full gradient algorithms available, mini-batch learning is generally preferred for large and redundant training sets due to its computational efficiency.
The basic mini-batch gradient descent learning algorithm involves guessing an initial learning rate and monitoring the network's performance. If the error worsens or oscillates, the learning rate is reduced. If the error falls too slowly, the learning rate can be increased. Automating the adjustment of the learning rate based on these observations is beneficial. Towards the end of learning, it is often helpful to decrease the learning rate to smooth out fluctuations in the weights caused by mini-batch gradients. Validation sets are used to assess when to decrease the learning rate and determine when the error stops decreasing consistently.
By carefully tuning the learning rate and adjusting it throughout the training process, mini-batch gradient descent provides an effective approach to train large neural networks on redundant datasets.
Additionally, it is worth noting that there are two main types of learning algorithms for neural networks: full gradient algorithms and mini-batch algorithms. Full gradient algorithms compute the gradient using all training cases, allowing for various optimization techniques to speed up learning. However, these methods developed for smooth nonlinear functions may require modifications to work effectively with multi-layer neural networks.
On the other hand, mini-batch learning is advantageous for highly redundant and large training sets. While mini-batches may need to be relatively large, they offer computational efficiency. The use of mini-batches allows for parallel computation of gradients for multiple training cases simultaneously, leveraging the capabilities of modern processors, such as graphics processing units (GPUs).
Throughout the training process, it is important to strike a balance with the learning rate. Adjustments to the learning rate should be made based on the network's performance and the observed behavior of the error. Towards the end of learning, reducing the learning rate can help achieve a final set of weights that is a good compromise, smoothing out fluctuations caused by mini-batch gradients.
Validation sets play a crucial role in monitoring the network's progress. By measuring the error on a separate validation set, one can assess if the error consistently stops decreasing and determine the appropriate time to adjust the learning rate. These validation examples are not used for training or final testing.
The mini-batch gradient descent learning algorithm provides a practical and efficient approach to training large neural networks on redundant datasets. Careful adjustment of the learning rate, monitoring of the network's performance using validation sets, and the use of representative mini-batches contribute to successful training outcomes.
Lecture 6.2 — A bag of tricks for mini batch gradient descent
Lecture 6.2 — A bag of tricks for mini batch gradient descent [Neural Networks for Machine Learning]
In this video, we'll discuss several issues that arise when using stochastic gradient descent with mini-batches. There are numerous tricks that can significantly improve performance, often referred to as the "black art" of neural networks. I'll cover some of the key tricks in this video.
Let's start with the first issue: weight initialization in a neural network. If two hidden units have the same weights and biases, they will always receive the same gradient and never differentiate from each other. To enable them to learn different feature detectors, we need to initialize them with different weights. This is typically done by using small random weights to break the symmetry. It's important to note that the size of the initial weights should not be the same for all units. Hidden units with larger fan-in (incoming connections) tend to saturate with larger weights, so smaller weights are preferred in such cases. On the other hand, hidden units with smaller fan-in benefit from larger weights. The size of the initial weights should be proportional to the square root of the fan-in.
Another important factor is shifting the inputs. Adding a constant value to each input component can have a significant impact on the learning speed. Shifting the inputs is particularly effective when using steepest descent. It is recommended to shift each input component so that, on average across the training data, it has a mean value of zero.
Next, let's consider the error surface and how it relates to the weights and training cases. In some cases, when the weights satisfy different training cases, the error surface can become elongated, making learning difficult. However, by subtracting a constant value from each input component, we can transform the error surface into a circular shape, which makes learning easier.
Another factor to consider is the activation function for hidden units. Hyperbolic tangents, which range between -1 and 1, are often preferred because they lead to hidden unit activities with a mean value close to zero. This can facilitate faster learning in subsequent layers. However, logistic activation functions have advantages as they provide an output of zero for small negative inputs, allowing the network to ignore fluctuations in those inputs. Hyperbolic tangents require larger inputs to ignore such fluctuations.
Scaling the inputs is also crucial for effective learning. By transforming the input components to have unit variance across the entire training set, with a typical value of one or minus one, we can improve the error surface. Scaling the inputs ensures that changes in weights have similar effects on all input components, leading to a more balanced learning process.
To speed up mini-batch learning, there are four main methods to consider:
Momentum: Instead of directly changing the weights based on the gradient, momentum uses the gradient to accelerate the weights' velocity. This approach allows the weights to retain information from previous gradients.
Adaptive learning rates: Using a separate adaptive learning rate for each parameter and adjusting it based on empirical measurements can improve learning. If the gradient keeps changing sign, the learning rate is reduced. If the gradient remains consistent, the learning rate is increased.
RMSprop: This method divides the learning rate by a running average of the recent gradient magnitudes. It effectively handles a wide range of gradients by scaling them appropriately.
Full batch learning: This approach involves using the entire training set for learning and employing advanced optimization techniques that consider curvature information. While it can be effective, it may require further adaptation to work with mini-batches.
These are just some of the techniques that can significantly enhance the performance of stochastic gradient descent with mini-batches.
In summary, we have discussed several important issues and techniques related to stochastic gradient descent with mini-batches in neural networks. These techniques serve as the "black art" of neural networks and can greatly improve their performance. Let's recap the main points:
Weight Initialization: To allow hidden units to learn different feature detectors, it is crucial to initialize their weights differently. By using small random weights that break symmetry, we can ensure that each unit starts off distinct from one another. The size of the initial weights should be proportional to the square root of the fan-in, which helps in achieving a good starting point.
Shifting Inputs: Shifting the inputs by adding a constant to each component can have a significant impact on the speed of learning. It is beneficial to shift each component so that, on average, the input has a value of zero. This can be achieved by subtracting the mean of the input values across the training data.
Scaling Inputs: Scaling the input values is another useful technique in stochastic gradient descent. Transforming the inputs such that each component has unit variance over the entire training set simplifies the learning process. Rescaling the inputs helps in creating a circular error surface, which makes gradient descent more efficient.
Decorrelating Input Components: Decorrelating the components of input vectors improves learning by removing correlations between features. Principal components analysis is a valuable method for achieving this. By removing components with small eigenvalues and scaling the remaining components, we can obtain a circular error surface, making gradient descent easier.
Common Problems: Some common problems encountered in neural network training include large initial learning rates leading to hidden units becoming stuck, slow learning when starting with small weights, and prematurely reducing the learning rate. It is important to strike a balance in adjusting the learning rate to ensure optimal learning.
Speeding up Mini-Batch Learning: There are four main methods for speeding up mini-batch learning explicitly designed for faster learning: momentum, adaptive learning rates, RMSprop, and full batch learning with curvature information. These techniques leverage various mechanisms, such as momentum, adaptive adjustments, and gradient magnitudes, to accelerate the learning process.
While these techniques significantly enhance the performance of stochastic gradient descent with mini-batches in neural networks, it is essential to consider the specific problem at hand and experiment with different approaches to achieve the best results. The field of optimization offers further advanced methods worth exploring for even more efficient learning in neural networks.
That concludes the main points covered in this video regarding the challenges and techniques related to stochastic gradient descent with mini-batches in neural networks.
Lecture 6.3 — The momentum method
Lecture 6.3 — The momentum method [Neural Networks for Machine Learning]
In this video, we will discuss several issues related to using stochastic gradient descent with mini-batches and explore some techniques to enhance its effectiveness. These techniques are often considered the "black art" of neural networks, and we will cover some of the key ones.
Firstly, let's address the problem of weight initialization in a neural network. If two hidden units have identical weights and biases, they will always receive the same gradient and cannot learn different feature detectors. To enable them to learn distinct features, it is crucial to initialize their weights differently. We achieve this by using small random weights that break the symmetry. Moreover, it is beneficial for the initial weights to have different magnitudes, considering the fan-in of each hidden unit. For instance, larger weights may lead to saturation in units with a significant fan-in, while smaller weights are suitable for units with a small fan-in. As a rule of thumb, the size of initial weights should be proportional to the square root of the fan-in.
Shifting the inputs, or adding a constant to each input component, can surprisingly have a significant impact on the speed of learning. This shift helps with steepest descent as it can substantially alter the error surface. It is often recommended to shift each input component to have an average value of zero across the entire training set.
Scaling the inputs is another important consideration when using steepest descent. Scaling involves transforming the input values to ensure that each component has unit variance over the entire training set, making them typical values of one or minus one. This scaling allows for better error surface behavior, avoiding high curvature for large input components and low curvature for small ones.
Decorrelating the input components can greatly facilitate learning. By removing correlations between input components, we reduce redundancy and improve learning efficiency. Principal components analysis (PCA) is a commonly used technique for achieving decorrelation. It involves removing components with small eigenvalues and scaling the remaining components by dividing them by the square roots of their eigenvalues. This process leads to a circular error surface for linear systems, which simplifies learning.
Now let's address some common problems encountered in neural network training. Starting with an excessively large learning rate can drive hidden units to extreme states, where their derivatives become close to zero. This causes learning to stop, often giving the impression of being stuck in a local minimum. In reality, the network is likely trapped on a plateau.
Another challenge arises when classifying using squared error or cross-entropy error. Initially, the network quickly learns the best guessing strategy, which is to make the output unit equal to the expected proportion of being one. However, further improvement beyond this guessing strategy can be slow, especially with deep networks and small initial weights.
When tuning the learning rate, it is essential to find the right balance. Decreasing the learning rate too soon or too much may hinder progress, while reducing it towards the end of training can help stabilize the network's performance.
Now let's explore four specific methods for significantly speeding up mini-batch learning:
Momentum: Instead of directly changing the weights based on the gradient, momentum involves using the gradient to update the velocity of the weight updates. This momentum allows the network to remember previous gradients and helps accelerate learning.
Adaptive learning rates: Assigning a separate learning rate for each parameter and slowly adjusting it based on empirical measurements can enhance learning. If the sign of the gradient keeps changing, indicating oscillation, the learning rate is decreased. If the sign remains consistent, indicating progress, the learning rate is increased.
RMSprop: This method involves dividing the weight updates by a running average of the recent gradient magnitudes. It dynamically adjusts the weight updates based on the magnitude of the gradients, allowing for effective handling of various gradient ranges. RMSprop is a mini-batch adaptation of the Rprop.
Lecture 6.4 — Adaptive learning rates for each connection
Lecture 6.4 — Adaptive learning rates for each connection [Neural Networks for Machine Learning]
In this video, we will explore a method known as adaptive learning rates, which was initially developed by Robbie Jacobs in the late 1980s and subsequently improved by other researchers. The concept behind adaptive learning rates is to assign a unique learning rate to each connection in a neural network, based on empirical observations of how the weight on that connection behaves during updates. This approach allows for fine-tuning the learning process by decreasing the learning rate when the weight's gradient keeps reversing and increasing it when the gradient remains consistent.
Having separate adaptive learning rates for each connection is advantageous, especially in deep multi-layer networks. In such networks, the learning rates can vary significantly between different weights, particularly between weights in different layers. Additionally, the fanning of a unit, which determines the size of overshoot effects when adjusting multiple incoming weights to correct an error, also calls for different learning rates. Larger fan-ins can lead to more significant overshoot effects, making it necessary to adapt learning rates accordingly.
To implement adaptive learning rates, a global learning rate is manually set and multiplied by a local gain specific to each weight. Initially, the local gain is set to 1 for every weight. The weight update is then determined by multiplying the learning rate by the local gain and the error derivative for that weight. The local gain is adapted by increasing it if the gradient for the weight does not change sign and decreasing it if the gradients have opposite signs. Additive increases and multiplicative decreases are employed, with the goal of rapidly dampening large gains if oscillations occur.
It is interesting to consider the behavior of adaptive learning rates when the gradients are random. In such cases, there will be an equal number of increases and decreases in the gains, resulting in hovering around a gain of 1. If the gradients consistently have the same direction, the gain can become much larger than 1, while consistently opposite gradients can make the gain much smaller than 1, indicating oscillation across a ravine.
To improve the effectiveness of adaptive learning rates, it is important to limit the size of the gains within a reasonable range, such as 0.1 to 10 or 0.01 to 100. Excessive gains can lead to instability and prevent weights from converging. While adaptive learning rates were initially designed for full batch learning, they can also be applied with mini-batches. However, larger mini-batches are preferred to minimize the influence of sampling errors and ensure that sign changes in gradients reflect traversing a ravine.
It is possible to combine adaptive learning rates with momentum, as suggested by Jacobs. Instead of comparing the current gradient with the previous gradient, the agreement in sign is determined between the current gradient and the velocity (accumulated gradient) for that weight. This combination harnesses the benefits of both momentum and adaptive learning rates. While adaptive learning rates handle axis-aligned effects, momentum can handle diagonal ellipses and quickly navigate in diagonal directions, which adaptive learning rates alone cannot achieve.
There are a few additional considerations and techniques to enhance the performance of adaptive learning rates. It is crucial to strike a balance and prevent the gains from becoming excessively large. If the gains become too large, they can lead to instability and fail to decrease rapidly enough, potentially causing damage to the weights.
Furthermore, it's worth noting that adaptive learning rates were primarily designed for full batch learning, where all training examples are processed in a single iteration. However, they can also be applied to mini-batch learning, where subsets of training examples are processed at a time. When using mini-batches, it is important to ensure that the mini-batch size is relatively large to mitigate the influence of sampling errors. This helps ensure that sign changes in gradients are indicative of traversing a ravine rather than being solely due to the sampling variability of the mini-batch.
It is also possible to combine adaptive learning rates with momentum, which can further enhance the optimization process. Instead of comparing the current gradient with the previous gradient, the agreement in sign can be assessed between the current gradient and the velocity (i.e., the accumulated gradient) for that weight. By incorporating momentum, a synergy between the advantages of momentum and adaptive learning rates can be achieved. Adaptive learning rates focus on handling axis-aligned effects, while momentum is capable of effectively dealing with diagonal ellipses and swiftly navigating in diagonal directions that may be challenging for adaptive learning rates alone.
Adaptive learning rates offer a way to fine-tune the learning process in neural networks by assigning individual learning rates to each connection based on empirical observations. They address the challenges of varying learning rates across different weights in deep multi-layer networks and consider the fanning of units. Techniques such as limiting the size of gains, appropriate mini-batch selection, and combining adaptive learning rates with momentum can further optimize the training process, resulting in improved performance and convergence.
Lecture 6.5 — Rmsprop: normalize the gradient
Lecture 6.5 — Rmsprop: normalize the gradient [Neural Networks for Machine Learning]
The video introduces a method called Rprop (Resilient Backpropagation), initially designed for full batch learning. It shares similarities with the popular backpropagation algorithm but with some differences. The speaker then discusses how to extend Rprop to work with mini-batches, which is essential for large redundant datasets. The resulting method, called iRprop (Improved Rprop), combines the advantages of Rprop with the efficiency of mini-batch learning.
The main motivation behind Rprop is to address the challenge of varying gradient magnitudes. Gradients in neural networks can range from tiny to huge, making it difficult to choose a single global learning rate. Rprop tackles this issue by considering only the sign of the gradient, ensuring that all weight updates are of the same size. This technique is particularly useful for escaping plateaus with small gradients, as even with tiny gradients, the weight updates can be substantial.
Rprop combines the sign of the gradient with adaptive step sizes based on the weight being updated. Instead of considering the magnitude of the gradient, it focuses on the step size previously determined for that weight. The step size adapts over time, increasing multiplicatively if the signs of the last two gradients agree and decreasing multiplicatively if they disagree. By limiting the step sizes, it prevents them from becoming too large or too small.
While Rprop works well for full batch learning, it faces challenges when applied to mini-batches. It violates the central idea behind stochastic gradient descent, which relies on averaging gradients over successive mini-batches when using a small learning rate. The speaker explains that Rprop increments the weight multiple times with its current step size and only decrements it once, leading to an undesired increase in weight magnitude.
To overcome this limitation, the speaker introduces RMSprop (Root Mean Square propagation) as a mini-batch version of Rprop. RMSprop ensures that the number used to divide the gradient remains consistent across nearby mini-batches. It achieves this by maintaining a moving average of the squared gradients for each weight. The squared gradients are weighted using a decay factor (e.g., 0.9) and combined with the previous mean square to compute the updated mean square. The square root of the mean square is then used to normalize the gradient, allowing for more effective learning.
The speaker mentions that further developments can be made to RMSprop, such as combining it with momentum or adaptive learning rates on each connection. Additionally, they refer to related methods, such as Nesterov momentum and a method proposed by Yann LeCun's group, which shares similarities with RMSprop.
In summary, the speaker recommends different learning methods based on the characteristics of the dataset. For small datasets or large datasets without much redundancy, full batch methods like nonlinear conjugate gradient, LBFGS, or L-BFGS-B are suitable. Adaptive learning rates or Rprop can also be used for neural networks. In the case of large redundant datasets, mini-batch methods are essential. The first option to try is standard gradient descent with momentum. RMSprop is another effective method to consider, as it combines the benefits of Rprop and mini-batch learning. The speaker suggests exploring further enhancements, but there is currently no simple recipe for training neural networks due to the diverse nature of networks and tasks.
Lecture 7.1 — Modeling sequences: a brief overview
Lecture 7.1 — Modeling sequences: a brief overview [Neural Networks for Machine Learning]
In this video, the speaker provides an overview of different types of models used for sequences. They start by discussing autoregressive models, which predict the next term in a sequence based on previous terms. They then mention more complex variations of autoregressive models that incorporate hidden units. The speaker goes on to introduce models with hidden state and dynamics, such as linear dynamical systems and hidden Markov models. Although these models are complex, their purpose is to show the relationship between recurrent neural networks and these types of models in the context of sequence modeling.
When modeling sequences using machine learning, the goal is often to transform one sequence into another. For example, converting English words to French words or turning sound pressures into word identities for speech recognition. In some cases, there may not be a separate target sequence, so the next term in the input sequence can serve as a teaching signal. This approach is more natural for temporal sequences, as there is a natural order for predictions. However, it can also be applied to images, blurring the distinction between supervised and unsupervised learning.
The speaker then provides a review of other sequence models before diving into recurrent neural networks (RNNs). They explain that autoregressive models without memory can be extended by adding hidden units in a feed-forward neural network. However, they emphasize that memory-less models are just one subclass of models for sequences. Another approach is to use models with hidden states and internal dynamics, which can store information for a longer time. These models, such as linear dynamical systems and hidden Markov models, involve probabilistic inference and learning algorithms.
Linear dynamical systems are widely used in engineering and have real-valued hidden states with linear dynamics and Gaussian noise. Hidden Markov models, on the other hand, use discrete distributions and probabilistic state transitions. They are commonly used in speech recognition and have efficient learning algorithms based on dynamic programming.
The speaker explains the limitations of hidden Markov models when it comes to conveying large amounts of information between the first and second halves of an utterance. This limitation is due to the limited memory capacity of the hidden state. This leads to the introduction of recurrent neural networks, which have distributed hidden states and non-linear dynamics, making them more efficient at remembering information.
Recurrent neural networks can exhibit various behaviors, including oscillation, settling to point attractors (useful for memory retrieval), and chaotic behavior (useful in certain circumstances). The idea that an RNN can learn to implement multiple programs using different subsets of its hidden state was initially thought to make them very powerful. However, RNNs are computationally challenging to train, and exploiting their full potential has been a difficult task.
The video provides an overview of different sequence models, introduces the concept of recurrent neural networks, and highlights their computational power and challenges in training.
Lecture 7.2 — Training RNNs with back propagation
Lecture 7.2 — Training RNNs with back propagation [Neural Networks for Machine Learning]
In this video, I will discuss the backpropagation through time algorithm, which is a common method for training recurrent neural networks. The algorithm is straightforward once you understand the relationship between a recurrent neural network and a feed-forward neural network with multiple layers representing different time steps. I will also cover various approaches for providing input and desired outputs to recurrent neural networks.
The diagram illustrates a simple recurrent network with three interconnected neurons. Each connection has a time delay of one, and the network operates in discrete time with integer ticks. To train a recurrent network, we need to recognize that it is essentially an expanded version of a feed-forward network in time. The recurrent network begins in an initial state at time zero and uses its connection weights to generate a new state at time one. It repeats this process, utilizing the same weights to produce subsequent new states.
Backpropagation is effective in learning when weight constraints are present. This has been observed in convolutional networks as well. To incorporate weight constraints, we compute gradients as usual, ignoring the constraints, and then modify the gradients to maintain the constraints. For example, if we want w1 to be equal to w2, we ensure that the change in w1 is equal to the change in w2 by taking the derivatives with respect to w1 and w2, adding or averaging them, and applying the same quantity to update both weights. As long as the weights satisfy the constraints initially, they will continue to do so.
The backpropagation through time algorithm is simply a term used to describe the process of treating a recurrent network as a feed-forward network with shared weights and training it using backpropagation. In the time domain, the forward pass accumulates activities at each time slice, while the backward pass extracts activities from the stack and computes error derivatives for each time step. This backward pass at each time step gives the algorithm its name—backpropagation through time.
After the backward pass, we sum or average the derivatives from all time steps for each weight. We then update all instances of that weight by the same amount, proportional to the sum or average of the derivatives. An additional consideration arises if we don't specify the initial state of all units, such as hidden or output units. In this case, we need to start them off in a particular state. One approach is to set default values, like 0.5, but it may not yield optimal results. Alternatively, we can learn the initial states by treating them as parameters and adjusting them based on the gradient of the error function with respect to the initial state.
There are various methods to provide input to a recurrent neural network. We can specify the initial state of all units, a subset of the units, or the states at each time step for a subset of the units. The latter is often used when dealing with sequential data. Similarly, there are different ways to specify the targets for a recurrent network. We can specify the desired final states for all units or for multiple time steps if we want to train it to settle at a particular attractor. By including derivatives from each time step during backpropagation, we can easily incorporate these specifications and encourage learning of attractors.
The backpropagation through time algorithm is a straightforward extension of backpropagation for training recurrent neural networks. By treating the recurrent network as an expanded feed-forward network with shared weights, we can apply backpropagation and adjust the initial states and weights to optimize training. Various methods exist to provide input and desired outputs to recurrent networks, allowing for flexibility in handling sequential data and training for specific objectives.
Lecture 7.3 — A toy example of training an RNN
Lecture 7.3 — A toy example of training an RNN [Neural Networks for Machine Learning]
In this video, I will explain how a recurrent neural network (RNN) solves a toy problem. A toy problem is chosen to showcase the capabilities of RNNs that are not easily achievable with feed-forward neural networks. The problem demonstrated here is binary addition. After the RNN learns to solve the problem, we can examine its hidden states and compare them to the hidden states of a finite state automaton that solves the same problem.
To illustrate the problem of adding two binary numbers, we could train a feed-forward neural network. However, there are limitations with this approach. We must determine the maximum number of digits for both input numbers and the output number in advance. Moreover, the processing applied to different bits of the input numbers does not generalize. As a result, the knowledge of adding the last two digits and dealing with carries resides in specific weights. When dealing with different parts of a long binary number, the knowledge needs to be encoded in different weights, leading to a lack of automatic generalization.
The algorithm for binary addition is depicted in the image. The states in the algorithm resemble those in a hidden Markov model, except they are not truly hidden. The system operates in one state at a time, performing an action upon entering a state (either printing 1 or 0). When in a state, it receives input, which consists of two numbers from the next column, causing a transition to a new state. For example, if it is in the carry state and just printed a 1, upon encountering a 1 1, it remains in the same state and prints another 1. However, if it encounters a 1 0 or 0 1, it transitions to the carry state but prints a 0. Similarly, a 0 0 leads it to the no carry state, where it prints a 1. This process continues for each time step.
A recurrent neural network for binary addition requires two input units and one output unit. It receives two input digits at each time step and produces an output corresponding to the column it encountered two time steps ago. To account for the time delay, the network needs a delay of two time steps, where the first step updates the hidden units based on the inputs, and the second step generates the output from the hidden state. The network architecture consists of three interconnected hidden units, although more hidden units could be used for faster learning. These hidden units have bidirectional connections with varying weights. The connections between hidden units allow the activity pattern at one time step to influence the hidden activity pattern at the next time step. The input units have feed-forward connections to the hidden units, allowing the network to observe the two digits in a column. Similarly, the hidden units have feed-forward connections to the output unit, enabling the production of an output.
It is fascinating to analyze what the recurrent neural network learns. It learns four distinct activity patterns in its three hidden units, which correspond to the nodes in the finite state automaton for binary addition. It is crucial not to confuse the units in a neural network with the nodes in a finite state automaton. The nodes in the automaton align with the activity vectors of the recurrent neural network. The automaton is restricted to one state at each time, just like the hidden units in the RNN, which have precisely one activity vector at each time step. While an RNN can emulate a finite state automaton, it is exponentially more powerful in representation. With N hidden neurons, it can have two to the power of N possible binary activity vectors. Although it only has N squared weights, it may not fully exploit the entire representational power. If the bottleneck lies in the representation, an RNN can outperform a finite state automaton.
This is particularly important when the input stream contains two separate processes occurring simultaneously. A finite state automaton needs to exponentially increase its number of states to handle the parallel processes. In contrast, a recurrent neural network only needs to double the number of hidden units, thereby doubling the number of units and quadrupling the number of binary vector states it can represent.
Lecture 7.4 — Why it is difficult to train an RNN?
Lecture 7.4 — Why it is difficult to train an RNN? [Neural Networks for Machine Learning]
In this video, I will discuss the problem of exploding and vanishing gradients that makes training recurrent neural networks (RNNs) difficult. For many years, researchers believed that modeling long-term dependencies with RNNs was nearly impossible. However, there are now four effective approaches to address this issue.
To understand why training RNNs is challenging, we need to recognize a crucial difference between the forward and backward passes in an RNN. In the forward pass, squashing functions like the logistic function are used to prevent the activity vectors from exploding. Each neuron in the RNN employs a logistic unit, which constrains the output between 0 and 1. This prevents the activity levels from growing uncontrollably.
In contrast, the backward pass is completely linear. Surprisingly, if we double the error derivatives at the final layer, all the error derivatives will also double during backpropagation. The gradients are determined by the slopes of the logistic curves at specific points (marked by red dots in the video). Once the forward pass is complete, the slopes of these tangents are fixed. During backpropagation, the gradients propagate through a linear system where the non-linearity's slope has been fixed. However, linear systems tend to suffer from the problem of exploding or dying gradients as they iterate. If the weights are small, the gradients shrink exponentially and become negligible. Conversely, if the weights are large, the gradients explode and overpower the learning process. These issues are more severe in RNNs compared to feed-forward neural networks, especially when dealing with long sequences.
Even with careful weight initialization, it remains challenging to capture dependencies between the current output and events that occurred many time steps ago. RNNs struggle with handling long-range dependencies. The video provides an example of how gradients can either vanish or explode when training a recurrent neural network to learn attractor states. Small differences in initial states lead to no change in the final state (vanishing gradients), while slight variations near the boundaries result in significant divergence (exploding gradients).
To address these challenges, there are four effective methods for training RNNs. The first method is Long Short-Term Memory (LSTM), which alters the network's architecture to improve memory capabilities. The second approach involves using advanced optimizers that can handle small gradients effectively. Hessian-free optimization, tailored for neural networks, excels at detecting small gradients with low curvature. The third method involves careful initialization of weights and creating a reservoir of weakly coupled oscillators within the hidden state. This allows the network to reverberate and remember input sequences. The connections between hidden units and outputs are then trained, while the recurrent connections remain fixed. The fourth method utilizes momentum and combines it with the initialization technique used in echo state networks. This modification improves the dynamics of the network, making it even more effective.
The ability to train RNNs has improved with these approaches, overcoming the challenges posed by exploding and vanishing gradients.
Lecture 7.5 — Long term Short term memory
Lecture 7.5 — Long term Short term memory [Neural Networks for Machine Learning]
In this video, I will explain the approach known as "Long Short-Term Memory" (LSTM) for training recurrent neural networks. LSTM aims to create a long-lasting short-term memory in a neural network by employing specialized modules that facilitate the gating of information.
The memory cell in LSTM is designed to retain information for an extended period. It consists of logistic and linear units with multiplicative interactions. When a logistic "write" gate is activated, information enters the memory cell from the rest of the recurrent network. The state of the "write" gate is determined by the recurrent network. The information remains in the memory cell as long as the "keep" gate is on, which is controlled by the rest of the system. To read the information from the memory cell, a logistic "read" gate is activated, and the stored value is retrieved and influences the future states of the recurrent neural network.
LSTM utilizes logistic units because they have differentiable properties, enabling backpropagation through them. This allows the network to learn and optimize the memory cell over multiple time steps. The backpropagation through the memory cell involves updating the weights based on the error derivatives, which can be propagated back through hundreds of time steps.
LSTM has been particularly successful in tasks like handwriting recognition. It can store and retrieve information effectively, even in the presence of cursive handwriting. It has shown superior performance compared to other systems in reading and writing tasks, and Canada Post has begun using LSTM-based systems for such purposes.
In the video, a demonstration of a handwriting recognition system based on LSTM is shown. The system takes in pen coordinates as input and produces recognized characters as output. The top row displays the recognized characters, the second row shows the states of selected memory cells, the third row visualizes the actual writing with pen coordinates, and the fourth row illustrates the gradient backpropagated to the XY locations, indicating the impact of past events on character recognition decisions.
LSTM has proven to be a powerful approach for training recurrent neural networks, enabling the capture and utilization of long-term dependencies in sequential data.