Recurrent Neural Networks and Their Applications

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January 28, 2025

Introduction

This lecture covers Recurrent Neural Networks (RNNs), their architectures, training methods, and applications. We will delve into the intricacies of how RNNs handle sequential data, focusing on their ability to maintain a "memory" of past inputs. This "memory" is crucial for processing sequences where the order of elements matters, such as text, music, and speech. We will also explore various RNN architectures, including many-to-one, one-to-many, and many-to-many models. These different architectures allow us to apply RNNs to a wide range of tasks, such as:

Text classification: Determining the topic or sentiment of a text.
Image captioning: Generating a textual description of an image.
Machine translation: Translating text from one language to another.
Named Entity Recognition: Identifying and classifying named entities in text (e.g., persons, locations, organizations).

Additionally, we will discuss the limitations of standard RNNs. These limitations include difficulty in learning long-term dependencies and the lack of a mechanism for selectively storing and retrieving information. Finally, we will introduce advanced architectures like Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs) that address these limitations using gating mechanisms. These mechanisms allow for selective updating of the hidden state, enabling the network to retain important information over long periods while still adapting to new information.

Recurrent Neural Networks (RNNs) and their ability to process sequential data.
Different RNN architectures: many-to-one, one-to-many, and many-to-many.
Applications of RNNs in text classification, image captioning, machine translation, and named entity recognition.
Training RNNs using Backpropagation Through Time (BPTT).
Shared weights in RNNs and their implications.
Bidirectional RNNs for capturing context from both past and future.
Deep RNNs for learning complex representations.
Limitations of standard RNNs: difficulty with long-term dependencies and lack of selective information storage.
Advanced RNN architectures: GRUs and LSTMs with gating mechanisms.

Recurrent Neural Networks (RNNs)

Introduction to Sequential Data

Sequential data is a type of data where the order of elements matters. Examples include text, music, speech, and video. Unlike traditional data, where each data point is independent, sequential data points have dependencies on previous data points. For instance, in a sentence, the meaning of a word is often determined by the words that came before it. Similarly, in a piece of music, the next note depends on the sequence of notes that preceded it.

Text: "The cat sat on the mat." (The meaning of "sat" depends on "cat" and "on the mat" depends on "sat")
Music: A melody where each note is influenced by the preceding notes.
Speech: The sound of a phoneme can vary depending on the surrounding phonemes.
Video: Each frame in a video is related to the frames before and after it.

Word Embeddings: Representing Words as Vectors

To process text with neural networks, we need to convert words into numerical representations. Word embeddings are a way to represent words as vectors in a continuous vector space. This allows us to capture semantic relationships between words. Words with similar meanings will have similar vector representations.

A word embedding is a mapping from a discrete set of words to a continuous vector space $\mathbb{R}^n$, where each word $w$ is represented by a vector $v_w \in \mathbb{R}^n$. This mapping is learned from a large corpus of text data.

For example, the words "king" and "queen" would be close together in the vector space, while "king" and "apple" would be farther apart. The distance between vectors reflects the semantic similarity between the corresponding words. These embeddings can be pre-trained and made available, such as the ones from the Stanford Natural Language Processing Group.

Illustration of word embeddings in vector space. Words with similar meanings are closer together.

The Role of Hidden States in RNNs: Memory

RNNs are designed to process sequential data by maintaining a hidden state that acts as a "memory" of past inputs. At each time step, the RNN takes an input (e.g., a word embedding) and updates its hidden state based on the current input and the previous hidden state. This allows the network to accumulate information over time and make predictions based on the entire sequence seen so far.

The hidden state $h_t$ at time step $t$ in an RNN is a vector that represents the network’s memory of past inputs up to time $t$. It is updated recursively as follows: \[h_t = f(W_{xh} x_t + W_{hh} h_{t-1} + b_h) \label{eq:hidden_state}\] where $x_t$ is the input at time step $t$, $W_{xh}$ and $W_{hh}$ are weight matrices, $b_h$ is a bias vector, and $f$ is an activation function (e.g., $\tanh$ or ReLU).

Imagine reading a book. As you read each word, you update your understanding of the story based on the current word and your memory of the previous words. The hidden state in an RNN is like your understanding of the story at a particular point in time. It summarizes everything you’ve read so far.

Visualizing RNNs: Compact and Unfolded Representations

RNNs can be visualized in two main ways:

Compact representation: Shows the network with a feedback loop, indicating the recurrent connection from the hidden state to itself. This representation emphasizes the recursive nature of the network.
Unfolded representation: Shows the network unrolled across time, with a separate copy of the network for each time step. This representation makes it easier to see how the network processes a sequence of inputs.

Compact and unfolded representations of an RNN. The unfolded representation shows the network unrolled across three time steps.

Shared Weights in RNNs

In an RNN, the weight matrices $W_{xh}$ and $W_{hh}$ are shared across all time steps. This means that the same weights are used to process the input and update the hidden state at each time step. This is a crucial aspect of RNNs and has several important implications.

Outputs at Each Time Step

An RNN can produce an output at each time step. The output $y_t$ at time step $t$ is computed based on the hidden state $h_t$:

\[y_t = g(W_{hy} h_t + b_y) \label{eq:output}\]

where $W_{hy}$ is a weight matrix, $b_y$ is a bias vector, and $g$ is an activation function (e.g., softmax for classification). The specific activation function depends on the task. For example, if we are doing binary classification, we might use a sigmoid function to produce a probability between 0 and 1. If we are doing multi-class classification, we might use a softmax function to produce a probability distribution over multiple classes.

Initializing the Hidden State ($A_0$)

The initial hidden state $h_0$ (often denoted as $A_0$) is typically initialized to a vector of zeros or a random vector. This represents the network’s initial "memory" before it has processed any input. If prior knowledge about the sequence is available, it can be used to initialize $h_0$. For example, if we are processing text and we know the topic of the text, we might initialize $h_0$ with a vector that represents that topic. In many cases, however, we don’t have any prior knowledge, so we simply initialize $h_0$ to a vector of zeros.

Applications of RNNs

RNNs can be applied to a variety of tasks involving sequential data. Here are a few examples:

Text Completion

RNNs can be used to predict the next word in a sequence, given the previous words. This is useful for applications like autocomplete and predictive text. For example, if you type "The cat sat on the," an RNN can predict that the next word is likely to be "mat" or "couch."

Sentiment Analysis of Text Reviews

RNNs can analyze text reviews to determine the sentiment (positive, negative, or neutral) expressed in the review. This is useful for understanding customer opinions about products or services. For example, an RNN can process a review like "This product is amazing! I highly recommend it." and classify it as positive. The model achieves this by processing the review sequentially, word by word, and updating its hidden state to capture the overall sentiment expressed in the review.

Example: Sentiment Analysis with RNNs

Suppose we want to classify movie reviews as positive or negative. We can train an RNN on a dataset of movie reviews, where each review is labeled as positive or negative. The RNN will process each review word by word, updating its hidden state after each word. The final hidden state will then be used to predict the sentiment of the review.

Training Recurrent Neural Networks

Backpropagation Through Time (BPTT)

Backpropagation Through Time (BPTT) is the algorithm used to train RNNs. It is an extension of the standard backpropagation algorithm, adapted to handle the sequential nature of the data and the recurrent connections in the network.

Forward Pass

During the forward pass, the input sequence is fed to the network one time step at a time. At each time step $t$, the network computes the hidden state $h_t$ and the output $y_t$ based on the current input $x_t$ and the previous hidden state $h_{t-1}$. This process can be visualized as unfolding the RNN through time, as shown in Figure 2.

The hidden state is updated according to Equation [eq:hidden_state], and the output is computed using Equation [eq:output]. This process is repeated for each time step in the sequence, from $t=1$ to $t=T$, where $T$ is the length of the sequence.

Loss Calculation

After the forward pass is completed, the loss is computed by comparing the predicted outputs ($y_1, y_2, ..., y_T$) to the target outputs ($\hat{y}_1, \hat{y}_2, ..., \hat{y}_T$). The total loss is the sum of the losses at each time step.

Let $y_t$ be the predicted output at time step $t$ and $\hat{y}_t$ be the target output at time step $t$. The loss at time step $t$ is given by $L_t(y_t, \hat{y}_t)$, which measures the difference between the predicted and target outputs at that time step. The total loss $L$ is the sum of the losses over all time steps: \[L = \sum_{t=1}^T L_t(y_t, \hat{y}_t) \label{eq:loss_function}\]

The specific form of the loss function $L_t$ depends on the task. For example, for regression tasks, we might use the mean squared error:

\[L_t(y_t, \hat{y}_t) = (y_t - \hat{y}_t)^2\]

For binary classification tasks, we might use the binary cross-entropy:

\[L_t(y_t, \hat{y}_t) = -\hat{y}_t \log(y_t) - (1 - \hat{y}_t) \log(1 - y_t)\]

For multi-class classification tasks, we might use the categorical cross-entropy:

\[L_t(y_t, \hat{y}_t) = -\sum_{c=1}^C \hat{y}_{t,c} \log(y_{t,c})\]

where $C$ is the number of classes, and $\hat{y}_{t,c}$ and $y_{t,c}$ are the target and predicted probabilities for class $c$ at time step $t$, respectively.

Backward Pass

During the backward pass, the gradients of the loss with respect to the network parameters ($W_{xh}$, $W_{hh}$, $W_{hy}$, $b_h$, and $b_y$) are computed and used to update the parameters. The gradients are propagated backward through time, from the last time step $T$ to the first time step 1. This is done by applying the chain rule of calculus, taking into account the recurrent connections in the network.

The gradients are accumulated over all time steps, and then the parameters are updated using an optimization algorithm, such as stochastic gradient descent (SGD) or Adam.

Input: Input sequence $x = (x_1, x_2, ..., x_T)$, target sequence $\hat{y} = (\hat{y}_1, \hat{y}_2, ..., \hat{y}_T)$, initial hidden state $h_0$, learning rate $\eta$ Initialize: $W_{xh}$, $W_{hh}$, $W_{hy}$, $b_h$, $b_y$ randomly Forward Pass: $h_t = f(W_{xh} x_t + W_{hh} h_{t-1} + b_h)$ $y_t = g(W_{hy} h_t + b_y)$ Loss Calculation: $L = \sum_{t=1}^T L_t(y_t, \hat{y}_t)$ Backward Pass: Initialize gradients: $\frac{\partial L}{\partial W_{xh}} = 0$, $\frac{\partial L}{\partial W_{hh}} = 0$, $\frac{\partial L}{\partial W_{hy}} = 0$, $\frac{\partial L}{\partial b_h} = 0$, $\frac{\partial L}{\partial b_y} = 0$ Compute $\frac{\partial L_t}{\partial y_t}$ Compute $\frac{\partial L_t}{\partial h_t}$ (taking into account $\frac{\partial L}{\partial h_{t+1}}$ from the next time step) Accumulate gradients: $\frac{\partial L}{\partial W_{hy}} += \frac{\partial L_t}{\partial y_t} \frac{\partial y_t}{\partial W_{hy}}$, $\frac{\partial L}{\partial b_y} += \frac{\partial L_t}{\partial y_t} \frac{\partial y_t}{\partial b_y}$ Compute $\frac{\partial L_t}{\partial h_k}$ (taking into account $\frac{\partial L_t}{\partial h_{k+1}}$) Accumulate gradients: $\frac{\partial L}{\partial W_{xh}} += \frac{\partial L_t}{\partial h_k} \frac{\partial h_k}{\partial W_{xh}}$, $\frac{\partial L}{\partial W_{hh}} += \frac{\partial L_t}{\partial h_k} \frac{\partial h_k}{\partial W_{hh}}$, $\frac{\partial L}{\partial b_h} += \frac{\partial L_t}{\partial h_k} \frac{\partial h_k}{\partial b_h}$ Parameter Update: $W_{xh} = W_{xh} - \eta \frac{\partial L}{\partial W_{xh}}$ $W_{hh} = W_{hh} - \eta \frac{\partial L}{\partial W_{hh}}$ $W_{hy} = W_{hy} - \eta \frac{\partial L}{\partial W_{hy}}$ $b_h = b_h - \eta \frac{\partial L}{\partial b_h}$ $b_y = b_y - \eta \frac{\partial L}{\partial b_y}$

Connections and Weights for Output

To produce an output at each time step, the hidden state is connected to an output layer. The output layer has its own weight matrix $W_{hy}$ and bias vector $b_y$. The output $y_t$ at time step $t$ is computed by applying an activation function $g$ to the linear combination of the hidden state $h_t$ and the output weights and biases:

\[y_t = g(W_{hy} h_t + b_y)\]

The dimensionality of the output layer depends on the task. For example, for binary classification, the output layer would have a single unit with a sigmoid activation function to produce a probability between 0 and 1. For multi-class classification, the output layer would have multiple units, one for each class, with a softmax activation function to produce a probability distribution over the classes. The number of units in the output layer corresponds to the number of classes.

RNN Architectures for Different Tasks

Recurrent Neural Networks (RNNs) can be configured in various ways to handle different types of tasks. The architecture of an RNN is determined by how the inputs and outputs are arranged and how the hidden states are used. In this section, we will explore three main types of RNN architectures: many-to-one, one-to-many, and many-to-many.

Many-to-One RNNs

Many-to-one RNNs take a sequence of inputs and produce a single output. This architecture is useful for tasks where we need to summarize a sequence into a single value or label.

Many-to-one RNN architecture. The network takes a sequence of inputs ((x_1, x_2, x_3, x_4)) and produces a single output ((y)).

Text Classification (e.g., Spam Detection)

In text classification, the input is a sequence of words (represented as word embeddings), and the output is a class label (e.g., spam or not spam, positive or negative sentiment). The RNN processes the entire sequence word by word, updating its hidden state at each time step. The final hidden state, which encodes information from the entire sequence, is then used to produce a single output, which is the predicted class label.

Input: Sequence of word embeddings $x = (x_1, x_2, ..., x_T)$, where $x_i \in \mathbb{R}^d$ Output: Class label $y \in \{1, 2, ..., C\}$ Initialize hidden state $h_0 = \mathbf{0}$ $h_t = \text{RNN}(x_t, h_{t-1})$ $y = \text{softmax}(W h_T + b)$

For example, in spam detection, the input could be a sequence of word embeddings representing an email. The RNN processes the email word by word, and the final hidden state is used to predict whether the email is spam or not.

Consider the task of classifying movie reviews as positive or negative. The input is a sequence of word embeddings representing a review, such as "This movie was great!". The RNN processes the review word by word, and the final hidden state is used to predict the sentiment (positive or negative).

One-to-Many RNNs

One-to-many RNNs take a single input and produce a sequence of outputs. This architecture is useful for tasks where we need to generate a sequence from a single piece of information.

One-to-many RNN architecture. The network takes a single input ((x)) and produces a sequence of outputs ((y_1, y_2, y_3, y_4)).

Image Captioning

In image captioning, the input is an image, and the output is a sequence of words that describe the image. The image is typically processed by a convolutional neural network (CNN) to extract features, which are then fed as input to the RNN. The RNN generates the caption one word at a time, conditioning each word on the previously generated words and the image features.

Input: Image features $x \in \mathbb{R}^d$ (extracted from a CNN) Output: Sequence of words $y = (y_1, y_2, ..., y_T)$, where $y_i \in \text{Vocabulary}$ Initialize hidden state $h_0 = \mathbf{0}$ $h_1 = \text{RNN}(x, h_0)$ $y_1 = \text{softmax}(W h_1 + b)$ $h_t = \text{RNN}(y_{t-1}, h_{t-1})$ $y_t = \text{softmax}(W h_t + b)$

For example, given an image of a cat sitting on a mat, the CNN would extract features representing the cat, the mat, and their spatial relationship. The RNN would then take these features as input and generate a caption like "a cat sitting on a mat."

Another example of a one-to-many task is music generation. In this case, the input could be a single note or a musical style, and the output would be a sequence of notes or musical pieces. The RNN generates the music one note at a time, conditioning each note on the previously generated notes and the initial input.

Many-to-Many RNNs

Many-to-many RNNs take asequence of inputs and produce a sequence of outputs. This architecture is useful for tasks where both the input and the output are sequences, and the lengths of the input and output sequences may or may not be the same.

There are two main types of many-to-many RNNs: those where the input and output sequences have the same length, and those where they have different lengths.

Many-to-many RNN architectures. Left: The input and output sequences have the same length. Right: The input and output sequences have different lengths.

Machine Translation (Encoder-Decoder Model)

In machine translation, the input is a sentence in one language, and the output is the translation of the sentence in another language. The lengths of the input and output sentences may be different. This task is typically solved using an encoder-decoder model, which consists of two RNNs: an encoder and a decoder.

Input: Source sentence $x = (x_1, x_2, ..., x_T)$, where $x_i \in \text{Source Vocabulary}$ Output: Target sentence $y = (y_1, y_2, ..., y_{T'})$, where $y_i \in \text{Target Vocabulary}$ Encoder: Initialize encoder hidden state $h_0^e = \mathbf{0}$ $h_t^e = \text{RNN}_e(x_t, h_{t-1}^e)$ Decoder: Initialize decoder hidden state $h_0^d = h_T^e$ $h_t^d = \text{RNN}_d(y_{t-1}, h_{t-1}^d)$ $y_t = \text{softmax}(W h_t^d + b)$

The encoder processes the input sentence word by word and encodes it into a fixed-length vector called the context vector. The context vector represents the meaning of the entire input sentence. The decoder then takes the context vector as input and generates the translated sentence one word at a time.

The encoder’s job is to compress the meaning of the input sentence into a fixed-length vector. The decoder’s job is to unpack that vector and generate the corresponding output sentence.

Named Entity Recognition (NER)

In named entity recognition, the input is a sentence, and the output is a sequence of labels, one for each word in the sentence. Each label indicates whether the corresponding word is part of a named entity (e.g., person, location, organization) and, if so, what type of entity it is. In this case, the input and output sequences have the same length.

Input: Sequence of word embeddings $x = (x_1, x_2, ..., x_T)$, where $x_i \in \mathbb{R}^d$ Output: Sequence of labels $y = (y_1, y_2, ..., y_T)$, where $y_i \in \{1, 2, ..., C\}$ Initialize hidden state $h_0 = \mathbf{0}$ $h_t = \text{RNN}(x_t, h_{t-1})$ $y_t = \text{softmax}(W h_t + b)$

For example, given the sentence "Einstein was born in Ulm", the output might be "B-PER O O B-LOC", where "B-PER" indicates the beginning of a person’s name, "B-LOC" indicates the beginning of a location, and "O" indicates that the word is not part of a named entity.

Input sentence: "Apple Inc. is headquartered in Cupertino."

Output labels: "B-ORG I-ORG O O O B-LOC O"

Here, "B-ORG" and "I-ORG" indicate the beginning and inside of an organization name, "B-LOC" indicates the beginning of a location name, and "O" indicates other words.

Bidirectional Recurrent Neural Networks

Motivation: Capturing Context from Both Directions

In standard RNNs, the hidden state at each time step only encodes information from the past, i.e., from the inputs that came before the current time step. However, in many tasks, it is beneficial to also consider the future context, i.e., the inputs that come after the current time step. For example, when predicting a missing word in a sentence, it is helpful to know both the words that came before and the words that come after the missing word.

Consider the sentence "The cat sat on the ___." To predict the missing word, it is helpful to know that the next word is "mat" or "couch". A standard RNN, which only considers the past context, might not be able to make an accurate prediction.

Bidirectional RNNs (BRNNs) address this limitation by processing the input sequence in both directions: from left to right and from right to left. This allows the network to capture information from both past and future contexts.

Forward and Backward Networks

A BRNN consists of two separate RNNs:

A forward RNN that processes the sequence from left to right (from $t=1$ to $t=T$).
A backward RNN that processes the sequence from right to left (from $t=T$ to $t=1$).

Bidirectional RNN architecture. The network consists of a forward RNN (top) that processes the sequence from left to right and a backward RNN (bottom) that processes the sequence from right to left. The hidden states of the forward and backward RNNs are combined to make predictions.

The forward RNN computes a sequence of hidden states $(\overrightarrow{h_1}, \overrightarrow{h_2}, ..., \overrightarrow{h_T})$ by iterating through the input sequence from left to right:

\[\overrightarrow{h_t} = f(\overrightarrow{W_{xh}} x_t + \overrightarrow{W_{hh}} \overrightarrow{h_{t-1}} + \overrightarrow{b_h}) \label{eq:forward_hidden}\]

where $x_t$ is the input at time step $t$, $\overrightarrow{W_{xh}}$ and $\overrightarrow{W_{hh}}$ are the weight matrices for the forward RNN, $\overrightarrow{b_h}$ is the bias vector for the forward RNN, and $f$ is an activation function.

The backward RNN computes a sequence of hidden states $(\overleftarrow{h_1}, \overleftarrow{h_2}, ..., \overleftarrow{h_T})$ by iterating through the input sequence from right to left:

\[\overleftarrow{h_t} = f(\overleftarrow{W_{xh}} x_t + \overleftarrow{W_{hh}} \overleftarrow{h_{t+1}} + \overleftarrow{b_h}) \label{eq:backward_hidden}\]

where $\overleftarrow{W_{xh}}$ and $\overleftarrow{W_{hh}}$ are the weight matrices for the backward RNN, and $\overleftarrow{b_h}$ is the bias vector for the backward RNN.

Note that the forward and backward RNNs have different weight matrices and biases. They are independent of each other, except that they share the input sequence.

Combining Hidden States for Prediction

At each time step $t$, the hidden states of the forward and backward RNNs, $\overrightarrow{h_t}$ and $\overleftarrow{h_t}$, are combined to form a single hidden state $h_t$ that represents the context from both directions. The most common way to combine the hidden states is by concatenation:

\[h_t = [\overrightarrow{h_t}; \overleftarrow{h_t}] \label{eq:combined_hidden}\]

where $[\cdot ; \cdot]$ denotes concatenation. The combined hidden state $h_t$ is then used to make the prediction $y_t$ at that time step:

\[y_t = g(W_{hy} h_t + b_y) \label{eq:prediction}\]

where $W_{hy}$ is a weight matrix, $b_y$ is a bias vector, and $g$ is an activation function.

Input: Input sequence $x = (x_1, x_2, ..., x_T)$ Output: Output sequence $y = (y_1, y_2, ..., y_T)$ Forward Pass (Left to Right): Initialize $\overrightarrow{h_0} = \mathbf{0}$ $\overrightarrow{h_t} = f(\overrightarrow{W_{xh}} x_t + \overrightarrow{W_{hh}} \overrightarrow{h_{t-1}} + \overrightarrow{b_h})$ Backward Pass (Right to Left): Initialize $\overleftarrow{h_{T+1}} = \mathbf{0}$ $\overleftarrow{h_t} = f(\overleftarrow{W_{xh}} x_t + \overleftarrow{W_{hh}} \overleftarrow{h_{t+1}} + \overleftarrow{b_h})$ Prediction: $h_t = [\overrightarrow{h_t}; \overleftarrow{h_t}]$ $y_t = g(W_{hy} h_t + b_y)$

Capture both past and future context.
Improve performance on tasks where future context is important.
Provide a more complete representation of the sequence at each time step.

When implementing a bidirectional RNN, it’s important to note that the backward pass cannot be computed until the entire input sequence is available. This means that bidirectional RNNs cannot be used for real-time processing where the input sequence is received one element at a time.

Deep Recurrent Neural Networks

Stacking Multiple RNN Layers

Deep Recurrent Neural Networks (Deep RNNs) are formed by stacking multiple RNN layers on top of each other. In a deep RNN, the hidden state of one RNN layer is fed as input to the next RNN layer. This creates a hierarchical representation of the input sequence, where each layer learns to represent the sequence at a different level of abstraction.

Deep RNN with three layers. The hidden state of each layer is fed as input to the next layer. The superscript denotes the layer number.

The equations for a deep RNN with $L$ layers are as follows:

\[\begin{aligned} h_t^{(l)} &= f(W_{xh}^{(l)} h_t^{(l-1)} + W_{hh}^{(l)} h_{t-1}^{(l)} + b_h^{(l)}) \quad \text{for } l = 1, 2, ..., L \\ y_t &= g(W_{hy} h_t^{(L)} + b_y) \end{aligned}\]

where:

$h_t^{(l)}$ is the hidden state of layer $l$ at time step $t$.
$h_t^{(0)} = x_t$ is the input at time step $t$.
$W_{xh}^{(l)}$ and $W_{hh}^{(l)}$ are the weight matrices for layer $l$.
$b_h^{(l)}$ is the bias vector for layer $l$.
$f$ is an activation function.
$y_t$ is the output at time step $t$.
$W_{hy}$ is the weight matrix for the output layer.
$b_y$ is the bias vector for the output layer.
$g$ is an activation function for the output layer.

Input: Input sequence $x = (x_1, x_2, ..., x_T)$ Output: Output sequence $y = (y_1, y_2, ..., y_T)$ $h_t^{(0)} = x_t$ $h_t^{(l)} = f(W_{xh}^{(l)} h_t^{(l-1)} + W_{hh}^{(l)} h_{t-1}^{(l)} + b_h^{(l)})$ $y_t = g(W_{hy} h_t^{(L)} + b_y)$

Can learn more complex and abstract representations of the input sequence.
Can capture hierarchical dependencies in the data.
Can potentially achieve better performance than shallow RNNs on complex tasks.

Deep RNNs can be more difficult to train than shallow RNNs due to the vanishing and exploding gradient problems. The gradients can become very small or very large as they are propagated back through multiple layers and time steps. This can make it difficult for the network to learn long-range dependencies. In practice, it is uncommon to see deep RNNs with more than 2-3 layers, as the benefits of adding more layers often diminish beyond that point, and the training difficulties increase. Techniques like gradient clipping and specialized architectures like LSTMs and GRUs can help mitigate these issues.

Imagine a team of experts working together to analyze a complex document. Each expert specializes in a different aspect of the document, such as grammar, vocabulary, or topic. The first expert analyzes the raw text and passes their findings to the second expert, who builds upon the first expert’s analysis to gain a deeper understanding. This process continues with each expert adding their own expertise until the final expert produces a comprehensive analysis of the document. In a deep RNN, each layer is like an expert that specializes in a different level of abstraction. The first layer might learn low-level features like word patterns, while higher layers learn more abstract features like sentence structure and overall meaning.

Limitations of Standard RNNs

While standard RNNs are powerful models for sequence processing, they suffer from some limitations that can hinder their performance on certain tasks. In this section, we will discuss two key limitations: challenges with long-term dependencies and the need for selective information storage.

Challenges with Long-Term Dependencies

Standard RNNs have difficulty learning long-term dependencies, where the relevant information is separated by many time steps. This means that the network struggles to connect information that is far apart in the input sequence.

Consider the sentence: "The cat, which already ate a lot of food, was full." To understand that "was full" refers to the "cat" and not "food", the network needs to remember the subject of the sentence, which appeared many time steps earlier.

This difficulty arises due to the vanishing and exploding gradient problems. During backpropagation through time, the gradients are multiplied by the weight matrix $W_{hh}$ at each time step. If the largest singular value of $W_{hh}$ is less than 1, the gradients will shrink exponentially as they are propagated back through many time steps, leading to the vanishing gradient problem. Conversely, if the largest singular value of $W_{hh}$ is greater than 1, the gradients will grow exponentially, leading to the exploding gradient problem.

Illustration of the vanishing/exploding gradient problem in standard RNNs. The gradients can shrink or grow exponentially as they are propagated back through many time steps.

These problems make it difficult for the network to learn long-range dependencies, as the gradients that carry information about these dependencies either vanish or explode during training.

The Need for Selective Information Storage

In many tasks, some information in the input sequence is more important than other information. For example, when summarizing a document, some sentences are more important than others. Standard RNNs do not have a mechanism for selectively storing and retrieving information. They update their hidden state at every time step, regardless of the importance of the input.

This can lead to the loss of important information over time, as the hidden state may be overwritten byless important information. The hidden state has a limited capacity, and it tries to store information about the entire sequence seen so far. As the sequence gets longer, the hidden state may "forget" important information from the past.

Imagine you are packing for a trip and you have a backpack with limited capacity. You need to decide which items to keep and which to discard. A standard RNN is like a backpack that automatically replaces some items with new ones at every step, without considering their importance. Important items may be discarded to make room for less important ones.

Ideally, we would like the network to have a mechanism for selectively storing and retrieving information, so that it can focus on the most important parts of the input sequence and retain them in memory for as long as needed. This is a key motivation behind advanced RNN architectures like LSTMs and GRUs, which we will discuss in the next section.

Advanced RNN Architectures

To address the limitations of standard RNNs, researchers have developed more advanced RNN architectures that incorporate gating mechanisms. These gates control the flow of information within the network, allowing it to selectively update and retain information over time. In this section, we will introduce two popular advanced RNN architectures: Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs).

Gated Recurrent Units (GRUs)

Gated Recurrent Units (GRUs) are a type of RNN that use gating mechanisms to control the flow of information into and out of the hidden state. These gates allow the network to selectively update and retain information, addressing the limitations of standard RNNs.

A GRU has two gates:

An update gate ($z_t$) that determines how much of the past information (from previous time steps) needs to be passed along to the future.
A reset gate ($r_t$) that determines how much of the past information to forget.

GRU cell with update and reset gates. The gates control the flow of information from the previous hidden state (h_{t-1}) and the current input (x_t) to the new hidden state (h_t).

The update gate $z_t$ is computed as follows:

\[z_t = \sigma(W_z x_t + U_z h_{t-1} + b_z)\]

where $x_t$ is the input at time step $t$, $h_{t-1}$ is the previous hidden state, $W_z$ and $U_z$ are weight matrices, $b_z$ is a bias vector, and $\sigma$ is the sigmoid function.

The reset gate $r_t$ is computed similarly:

\[r_t = \sigma(W_r x_t + U_r h_{t-1} + b_r)\]

where $W_r$, $U_r$, and $b_r$ are weight matrices and a bias vector for the reset gate.

The candidate hidden state $\tilde{h}_t$ is computed as:

\[\tilde{h}_t = \tanh(W x_t + U (r_t \odot h_{t-1}) + b)\]

where $W$ and $U$ are weight matrices, $b$ is a bias vector, and $\odot$ denotes element-wise multiplication.

The new hidden state $h_t$ is a linear interpolation between the previous hidden state $h_{t-1}$ and the candidate hidden state $\tilde{h}_t$, controlled by the update gate $z_t$:

\[h_t = (1 - z_t) \odot h_{t-1} + z_t \odot \tilde{h}_t\]

Input: Input sequence $x = (x_1, x_2, ..., x_T)$, initial hidden state $h_0$ Output: Hidden states $h = (h_1, h_2, ..., h_T)$ $z_t = \sigma(W_z x_t + U_z h_{t-1} + b_z)$ $r_t = \sigma(W_r x_t + U_r h_{t-1} + b_r)$ $\tilde{h}_t = \tanh(W x_t + U (r_t \odot h_{t-1}) + b)$ $h_t = (1 - z_t) \odot h_{t-1} + z_t \odot \tilde{h}_t$

Long Short-Term Memory (LSTMs)

Long Short-Term Memory (LSTMs) are another type of RNN with gating mechanisms. LSTMs have a more complex architecture than GRUs, with separate memory cells and gates for input, output, and forgetting.

An LSTM cell has three gates:

An input gate ($i_t$) that controls how much of the new information to store in the cell state.
A forget gate ($f_t$) that controls how much of the old cell state to forget.
An output gate ($o_t$) that controls how much of the cell state to output to the hidden state.

LSTM cell with input, forget, and output gates. The gates control the flow of information into and out of the cell state (C_t) and the hidden state (h_t).

The gates are computed as follows:

\[\begin{aligned} i_t &= \sigma(W_i x_t + U_i h_{t-1} + b_i) \\ f_t &= \sigma(W_f x_t + U_f h_{t-1} + b_f) \\ o_t &= \sigma(W_o x_t + U_o h_{t-1} + b_o) \end{aligned}\]

where $x_t$ is the input at time step $t$, $h_{t-1}$ is the previous hidden state, $W_i$, $W_f$, $W_o$, $U_i$, $U_f$, $U_o$ are weight matrices, $b_i$, $b_f$, $b_o$ are bias vectors, and $\sigma$ is the sigmoid function.

The candidate cell state $\tilde{C}_t$ is computed as:

\[\tilde{C}_t = \tanh(W_C x_t + U_C h_{t-1} + b_C)\]

where $W_C$ and $U_C$ are weight matrices, and $b_C$ is a bias vector.

The new cell state $C_t$ is a combination of the old cell state $C_{t-1}$ and the candidate cell state $\tilde{C}_t$, controlled by the forget gate $f_t$ and the input gate $i_t$:

\[C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t\]

The new hidden state $h_t$ is the output of the cell state passed through an activation function and controlled by the output gate $o_t$: \[h_t = o_t \odot \tanh(C_t)\]

Input: Input sequence $x = (x_1, x_2, ..., x_T)$, initial hidden state $h_0$, initial cell state $C_0$ Output: Hidden states $h = (h_1, h_2, ..., h_T)$ $i_t = \sigma(W_i x_t + U_i h_{t-1} + b_i)$ $f_t = \sigma(W_f x_t + U_f h_{t-1} + b_f)$ $o_t = \sigma(W_o x_t + U_o h_{t-1} + b_o)$ $\tilde{C}_t = \tanh(W_C x_t + U_C h_{t-1} + b_C)$ $C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t$ $h_t = o_t \odot \tanh(C_t)$

Selective Updating of Hidden States

Both GRUs and LSTMs allow for selective updating of the hidden state. This means that the network can choose which parts of the hidden state to update and which parts to keep the same. This is achieved through the gating mechanisms.

In GRUs, the update gate $z_t$ controls how much of the new information to incorporate into the hidden state. If $z_t$ is close to 1, the new hidden state will be mostly based on the candidate hidden state $\tilde{h}_t$, and if $z_t$ is close to 0, the new hidden state will be mostly based on the previous hidden state $h_{t-1}$. The reset gate $r_t$ controls how much of the previous hidden state to use when computing the candidate hidden state.

In LSTMs, the input gate $i_t$ controls how much of the new information to store in the cell state, the forget gate $f_t$ controls how much of the old cell state to forget, and the output gate $o_t$ controls how much of the cell state to output to the hidden state.

Allow the network to learn long-term dependencies by selectively retaining information over many time steps.
Mitigate the vanishing gradient problem by providing a more direct path for the gradients to flow through during backpropagation.
Enable the network to focus on the most relevant parts of the input sequence and ignore irrelevant information.

LSTMs are generally more powerful than GRUs due to their more complex architecture and additional gating mechanisms. However, GRUs are computationally more efficient as they have fewer parameters. In practice, the choice between GRUs and LSTMs depends on the specific task and dataset, and it is often a good idea to try both architectures.

Conclusion

In this lecture, we explored Recurrent Neural Networks (RNNs) and their ability to process sequential data. We began by introducing the concept of sequential data and how it differs from traditional data. We then delved into the fundamental building blocks of RNNs, including word embeddings, hidden states, and shared weights. We discussed various RNN architectures, including many-to-one, one-to-many, and many-to-many models, and their applications in tasks like text classification, image captioning, machine translation, and named entity recognition. We also covered the training of RNNs using Backpropagation Through Time (BPTT) and the concept of shared weights, which allows RNNs to generalize across sequences of different lengths and recognize patterns regardless of their position in the sequence.

We examined Bidirectional RNNs, which process sequences in both directions (left-to-right and right-to-left), allowing them to capture both past and future context when making predictions. We also looked at Deep RNNs, which stack multiple RNN layers to learn hierarchical representations of the input sequence. We highlighted the limitations of standard RNNs, particularly their difficulty in handling long-term dependencies due to the vanishing and exploding gradient problems, and the need for selective information storage. Finally, we introduced advanced architectures like Gated Recurrent Units (GRUs) and Long Short-Term Memory networks (LSTMs), which address these limitations through gating mechanisms that allow for selective updating of the hidden state and cell state. These gating mechanisms enable GRUs and LSTMs to learn long-range dependencies and retain important information over many time steps.

RNNs are designed for processing sequential data, where the order of elements matters.
Word embeddings represent words as vectors, capturing semantic relationships.
Hidden states in RNNs act as a memory of past inputs.
Shared weights in RNNs enable generalization across time and reduce the number of parameters.
Many-to-one, one-to-many, and many-to-many architectures suit different types of tasks.
BPTT is used to train RNNs, adapting backpropagation for sequential data.
Bidirectional RNNs consider both past and future context.
Deep RNNs learn hierarchical representations by stacking multiple layers.
Standard RNNs struggle with long-term dependencies due to vanishing/exploding gradients.
GRUs and LSTMs use gating mechanisms to selectively update and retain information, addressing the limitations of standard RNNs.

For the next lecture, consider these questions:

How do GRUs and LSTMs differ in their gating mechanisms? (Hint: Consider the number of gates and how they control information flow.)
What are some practical applications where the ability to handle long-term dependencies is crucial? (Hint: Think about tasks involving natural language processing, speech recognition, or time-series analysis.)
How can we evaluate the performance of different RNN architectures on a given task? (Hint: Consider appropriate evaluation metrics and validation strategies.)
How does the concept of attention, which we will explore in the next lecture, further enhance the ability of RNNs to handle long-range dependencies and focus on relevant information?

In the next lecture, we will dive into attention mechanisms, a powerful technique that complements RNNs and further improves their ability to handle long sequences and complex tasks. Attention allows the network to focus on specific parts of the input sequence when making predictions, rather than relying solely on the fixed-length hidden state. This is particularly useful in tasks like machine translation and image captioning, where different parts of the input sequence may be relevant for generating different parts of the output sequence.

--- title: "Recurrent Neural Networks and Their Applications" author: "Your Name" date: "2025-01-28" format: html: toc: true # Table of Contents toc-depth: 2 code-tools: true theme: cosmo # Or "journal" for Distill-like minimalism --- # Introduction This lecture covers Recurrent Neural Networks (RNNs), their architectures, training methods, and applications. We will delve into the intricacies of how RNNs handle sequential data, focusing on their ability to maintain a \"memory\" of past inputs. This \"memory\" is crucial for processing sequences where the order of elements matters, such as text, music, and speech. We will also explore various RNN architectures, including many-to-one, one-to-many, and many-to-many models. These different architectures allow us to apply RNNs to a wide range of tasks, such as: - **Text classification:** Determining the topic or sentiment of a text. - **Image captioning:** Generating a textual description of an image. - **Machine translation:** Translating text from one language to another. - **Named Entity Recognition:** Identifying and classifying named entities in text (e.g., persons, locations, organizations). Additionally, we will discuss the limitations of standard RNNs. These limitations include difficulty in learning long-term dependencies and the lack of a mechanism for selectively storing and retrieving information. Finally, we will introduce advanced architectures like Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs) that address these limitations using gating mechanisms. These mechanisms allow for selective updating of the hidden state, enabling the network to retain important information over long periods while still adapting to new information. ::: tcolorbox - Recurrent Neural Networks (RNNs) and their ability to process sequential data. - Different RNN architectures: many-to-one, one-to-many, and many-to-many. - Applications of RNNs in text classification, image captioning, machine translation, and named entity recognition. - Training RNNs using Backpropagation Through Time (BPTT). - Shared weights in RNNs and their implications. - Bidirectional RNNs for capturing context from both past and future. - Deep RNNs for learning complex representations. - Limitations of standard RNNs: difficulty with long-term dependencies and lack of selective information storage. - Advanced RNN architectures: GRUs and LSTMs with gating mechanisms. ::: # Recurrent Neural Networks (RNNs) ## Introduction to Sequential Data Sequential data is a type of data where the order of elements matters. Examples include text, music, speech, and video. Unlike traditional data, where each data point is independent, sequential data points have dependencies on previous data points. For instance, in a sentence, the meaning of a word is often determined by the words that came before it. Similarly, in a piece of music, the next note depends on the sequence of notes that preceded it. ::: tcolorbox - **Text:** \"The cat sat on the mat.\" (The meaning of \"sat\" depends on \"cat\" and \"on the mat\" depends on \"sat\") - **Music:** A melody where each note is influenced by the preceding notes. - **Speech:** The sound of a phoneme can vary depending on the surrounding phonemes. - **Video:** Each frame in a video is related to the frames before and after it. ::: ## Word Embeddings: Representing Words as Vectors To process text with neural networks, we need to convert words into numerical representations. Word embeddings are a way to represent words as vectors in a continuous vector space. This allows us to capture semantic relationships between words. Words with similar meanings will have similar vector representations. ::: tcolorbox A word embedding is a mapping from a discrete set of words to a continuous vector space $\mathbb{R}^n$, where each word $w$ is represented by a vector $v_w \in \mathbb{R}^n$. This mapping is learned from a large corpus of text data. ::: For example, the words \"king\" and \"queen\" would be close together in the vector space, while \"king\" and \"apple\" would be farther apart. The distance between vectors reflects the semantic similarity between the corresponding words. These embeddings can be pre-trained and made available, such as the ones from the Stanford Natural Language Processing Group. <figure id="fig:word_embeddings"> <figcaption>Illustration of word embeddings in vector space. Words with similar meanings are closer together.</figcaption> </figure> ## The Role of Hidden States in RNNs: Memory RNNs are designed to process sequential data by maintaining a hidden state that acts as a \"memory\" of past inputs. At each time step, the RNN takes an input (e.g., a word embedding) and updates its hidden state based on the current input and the previous hidden state. This allows the network to accumulate information over time and make predictions based on the entire sequence seen so far. ::: tcolorbox The hidden state $h_t$ at time step $t$ in an RNN is a vector that represents the network's memory of past inputs up to time $t$. It is updated recursively as follows: $$h_t = f(W_{xh} x_t + W_{hh} h_{t-1} + b_h) \label{eq:hidden_state}$$ where $x_t$ is the input at time step $t$, $W_{xh}$ and $W_{hh}$ are weight matrices, $b_h$ is a bias vector, and $f$ is an activation function (e.g., $\tanh$ or ReLU). ::: ::: tcolorbox Imagine reading a book. As you read each word, you update your understanding of the story based on the current word and your memory of the previous words. The hidden state in an RNN is like your understanding of the story at a particular point in time. It summarizes everything you've read so far. ::: ## Visualizing RNNs: Compact and Unfolded Representations RNNs can be visualized in two main ways: - **Compact representation:** Shows the network with a feedback loop, indicating the recurrent connection from the hidden state to itself. This representation emphasizes the recursive nature of the network. - **Unfolded representation:** Shows the network unrolled across time, with a separate copy of the network for each time step. This representation makes it easier to see how the network processes a sequence of inputs. <figure id="fig:rnn_representations"> <figcaption>Compact and unfolded representations of an RNN. The unfolded representation shows the network unrolled across three time steps.</figcaption> </figure> ## Shared Weights in RNNs In an RNN, the weight matrices $W_{xh}$ and $W_{hh}$ are shared across all time steps. This means that the same weights are used to process the input and update the hidden state at each time step. This is a crucial aspect of RNNs and has several important implications. ### Motivation for Weight Sharing Sharing weights across time steps has several advantages: ::: tcolorbox Imagine you have a stamp with a specific pattern. You can use this same stamp to imprint the pattern at different locations on a piece of paper. The stamp represents the shared weights in an RNN, and the different locations on the paper represent different time steps. The pattern is the same regardless of where you stamp it. ::: ## Outputs at Each Time Step An RNN can produce an output at each time step. The output $y_t$ at time step $t$ is computed based on the hidden state $h_t$: $$y_t = g(W_{hy} h_t + b_y) \label{eq:output}$$ where $W_{hy}$ is a weight matrix, $b_y$ is a bias vector, and $g$ is an activation function (e.g., softmax for classification). The specific activation function depends on the task. For example, if we are doing binary classification, we might use a sigmoid function to produce a probability between 0 and 1. If we are doing multi-class classification, we might use a softmax function to produce a probability distribution over multiple classes. ## Initializing the Hidden State ($A_0$) The initial hidden state $h_0$ (often denoted as $A_0$) is typically initialized to a vector of zeros or a random vector. This represents the network's initial \"memory\" before it has processed any input. If prior knowledge about the sequence is available, it can be used to initialize $h_0$. For example, if we are processing text and we know the topic of the text, we might initialize $h_0$ with a vector that represents that topic. In many cases, however, we don't have any prior knowledge, so we simply initialize $h_0$ to a vector of zeros. ## Applications of RNNs RNNs can be applied to a variety of tasks involving sequential data. Here are a few examples: ### Text Completion RNNs can be used to predict the next word in a sequence, given the previous words. This is useful for applications like autocomplete and predictive text. For example, if you type \"The cat sat on the,\" an RNN can predict that the next word is likely to be \"mat\" or \"couch.\" ### Sentiment Analysis of Text Reviews RNNs can analyze text reviews to determine the sentiment (positive, negative, or neutral) expressed in the review. This is useful for understanding customer opinions about products or services. For example, an RNN can process a review like \"This product is amazing! I highly recommend it.\" and classify it as positive. The model achieves this by processing the review sequentially, word by word, and updating its hidden state to capture the overall sentiment expressed in the review. ::: mdframed **Example: Sentiment Analysis with RNNs** Suppose we want to classify movie reviews as positive or negative. We can train an RNN on a dataset of movie reviews, where each review is labeled as positive or negative. The RNN will process each review word by word, updating its hidden state after each word. The final hidden state will then be used to predict the sentiment of the review. ::: # Training Recurrent Neural Networks ## Backpropagation Through Time (BPTT) Backpropagation Through Time (BPTT) is the algorithm used to train RNNs. It is an extension of the standard backpropagation algorithm, adapted to handle the sequential nature of the data and the recurrent connections in the network. ### Forward Pass During the forward pass, the input sequence is fed to the network one time step at a time. At each time step $t$, the network computes the hidden state $h_t$ and the output $y_t$ based on the current input $x_t$ and the previous hidden state $h_{t-1}$. This process can be visualized as unfolding the RNN through time, as shown in Figure [2](#fig:rnn_representations){reference-type="ref" reference="fig:rnn_representations"}. The hidden state is updated according to Equation [\[eq:hidden_state\]](#eq:hidden_state){reference-type="ref" reference="eq:hidden_state"}, and the output is computed using Equation [\[eq:output\]](#eq:output){reference-type="ref" reference="eq:output"}. This process is repeated for each time step in the sequence, from $t=1$ to $t=T$, where $T$ is the length of the sequence. ### Loss Calculation After the forward pass is completed, the loss is computed by comparing the predicted outputs ($y_1, y_2, ..., y_T$) to the target outputs ($\hat{y}_1, \hat{y}_2, ..., \hat{y}_T$). The total loss is the sum of the losses at each time step. ::: tcolorbox Let $y_t$ be the predicted output at time step $t$ and $\hat{y}_t$ be the target output at time step $t$. The loss at time step $t$ is given by $L_t(y_t, \hat{y}_t)$, which measures the difference between the predicted and target outputs at that time step. The total loss $L$ is the sum of the losses over all time steps: $$L = \sum_{t=1}^T L_t(y_t, \hat{y}_t) \label{eq:loss_function}$$ ::: The specific form of the loss function $L_t$ depends on the task. For example, for regression tasks, we might use the mean squared error: $$L_t(y_t, \hat{y}_t) = (y_t - \hat{y}_t)^2$$ For binary classification tasks, we might use the binary cross-entropy: $$L_t(y_t, \hat{y}_t) = -\hat{y}_t \log(y_t) - (1 - \hat{y}_t) \log(1 - y_t)$$ For multi-class classification tasks, we might use the categorical cross-entropy: $$L_t(y_t, \hat{y}_t) = -\sum_{c=1}^C \hat{y}_{t,c} \log(y_{t,c})$$ where $C$ is the number of classes, and $\hat{y}_{t,c}$ and $y_{t,c}$ are the target and predicted probabilities for class $c$ at time step $t$, respectively. ### Backward Pass During the backward pass, the gradients of the loss with respect to the network parameters ($W_{xh}$, $W_{hh}$, $W_{hy}$, $b_h$, and $b_y$) are computed and used to update the parameters. The gradients are propagated backward through time, from the last time step $T$ to the first time step 1. This is done by applying the chain rule of calculus, taking into account the recurrent connections in the network. The gradients are accumulated over all time steps, and then the parameters are updated using an optimization algorithm, such as stochastic gradient descent (SGD) or Adam. :::: algorithm ::: algorithmic **Input:** Input sequence $x = (x_1, x_2, ..., x_T)$, target sequence $\hat{y} = (\hat{y}_1, \hat{y}_2, ..., \hat{y}_T)$, initial hidden state $h_0$, learning rate $\eta$ **Initialize:** $W_{xh}$, $W_{hh}$, $W_{hy}$, $b_h$, $b_y$ randomly **Forward Pass:** $h_t = f(W_{xh} x_t + W_{hh} h_{t-1} + b_h)$ $y_t = g(W_{hy} h_t + b_y)$ **Loss Calculation:** $L = \sum_{t=1}^T L_t(y_t, \hat{y}_t)$ **Backward Pass:** Initialize gradients: $\frac{\partial L}{\partial W_{xh}} = 0$, $\frac{\partial L}{\partial W_{hh}} = 0$, $\frac{\partial L}{\partial W_{hy}} = 0$, $\frac{\partial L}{\partial b_h} = 0$, $\frac{\partial L}{\partial b_y} = 0$ Compute $\frac{\partial L_t}{\partial y_t}$ Compute $\frac{\partial L_t}{\partial h_t}$ (taking into account $\frac{\partial L}{\partial h_{t+1}}$ from the next time step) Accumulate gradients: $\frac{\partial L}{\partial W_{hy}} += \frac{\partial L_t}{\partial y_t} \frac{\partial y_t}{\partial W_{hy}}$, $\frac{\partial L}{\partial b_y} += \frac{\partial L_t}{\partial y_t} \frac{\partial y_t}{\partial b_y}$ Compute $\frac{\partial L_t}{\partial h_k}$ (taking into account $\frac{\partial L_t}{\partial h_{k+1}}$) Accumulate gradients: $\frac{\partial L}{\partial W_{xh}} += \frac{\partial L_t}{\partial h_k} \frac{\partial h_k}{\partial W_{xh}}$, $\frac{\partial L}{\partial W_{hh}} += \frac{\partial L_t}{\partial h_k} \frac{\partial h_k}{\partial W_{hh}}$, $\frac{\partial L}{\partial b_h} += \frac{\partial L_t}{\partial h_k} \frac{\partial h_k}{\partial b_h}$ **Parameter Update:** $W_{xh} = W_{xh} - \eta \frac{\partial L}{\partial W_{xh}}$ $W_{hh} = W_{hh} - \eta \frac{\partial L}{\partial W_{hh}}$ $W_{hy} = W_{hy} - \eta \frac{\partial L}{\partial W_{hy}}$ $b_h = b_h - \eta \frac{\partial L}{\partial b_h}$ $b_y = b_y - \eta \frac{\partial L}{\partial b_y}$ ::: []{#alg:bptt label="alg:bptt"} :::: ## Connections and Weights for Output To produce an output at each time step, the hidden state is connected to an output layer. The output layer has its own weight matrix $W_{hy}$ and bias vector $b_y$. The output $y_t$ at time step $t$ is computed by applying an activation function $g$ to the linear combination of the hidden state $h_t$ and the output weights and biases: $$y_t = g(W_{hy} h_t + b_y)$$ ::: tcolorbox The dimensionality of the output layer depends on the task. For example, for binary classification, the output layer would have a single unit with a sigmoid activation function to produce a probability between 0 and 1. For multi-class classification, the output layer would have multiple units, one for each class, with a softmax activation function to produce a probability distribution over the classes. The number of units in the output layer corresponds to the number of classes. ::: # RNN Architectures for Different Tasks Recurrent Neural Networks (RNNs) can be configured in various ways to handle different types of tasks. The architecture of an RNN is determined by how the inputs and outputs are arranged and how the hidden states are used. In this section, we will explore three main types of RNN architectures: many-to-one, one-to-many, and many-to-many. ## Many-to-One RNNs Many-to-one RNNs take a sequence of inputs and produce a single output. This architecture is useful for tasks where we need to summarize a sequence into a single value or label. <figure id="fig:many-to-one"> <figcaption>Many-to-one RNN architecture. The network takes a sequence of inputs ($x_1, x_2, x_3, x_4$) and produces a single output ($y$).</figcaption> </figure> ### Text Classification (e.g., Spam Detection) In text classification, the input is a sequence of words (represented as word embeddings), and the output is a class label (e.g., spam or not spam, positive or negative sentiment). The RNN processes the entire sequence word by word, updating its hidden state at each time step. The final hidden state, which encodes information from the entire sequence, is then used to produce a single output, which is the predicted class label. :::: algorithm ::: algorithmic **Input:** Sequence of word embeddings $x = (x_1, x_2, ..., x_T)$, where $x_i \in \mathbb{R}^d$ **Output:** Class label $y \in \{1, 2, ..., C\}$ Initialize hidden state $h_0 = \mathbf{0}$ $h_t = \text{RNN}(x_t, h_{t-1})$ $y = \text{softmax}(W h_T + b)$ ::: :::: For example, in spam detection, the input could be a sequence of word embeddings representing an email. The RNN processes the email word by word, and the final hidden state is used to predict whether the email is spam or not. ::: tcolorbox Consider the task of classifying movie reviews as positive or negative. The input is a sequence of word embeddings representing a review, such as \"This movie was great!\". The RNN processes the review word by word, and the final hidden state is used to predict the sentiment (positive or negative). ::: ## One-to-Many RNNs One-to-many RNNs take a single input and produce a sequence of outputs. This architecture is useful for tasks where we need to generate a sequence from a single piece of information. <figure id="fig:one-to-many"> <figcaption>One-to-many RNN architecture. The network takes a single input ($x$) and produces a sequence of outputs ($y_1, y_2, y_3, y_4$).</figcaption> </figure> ### Image Captioning In image captioning, the input is an image, and the output is a sequence of words that describe the image. The image is typically processed by a convolutional neural network (CNN) to extract features, which are then fed as input to the RNN. The RNN generates the caption one word at a time, conditioning each word on the previously generated words and the image features. :::: algorithm ::: algorithmic **Input:** Image features $x \in \mathbb{R}^d$ (extracted from a CNN) **Output:** Sequence of words $y = (y_1, y_2, ..., y_T)$, where $y_i \in \text{Vocabulary}$ Initialize hidden state $h_0 = \mathbf{0}$ $h_1 = \text{RNN}(x, h_0)$ $y_1 = \text{softmax}(W h_1 + b)$ $h_t = \text{RNN}(y_{t-1}, h_{t-1})$ $y_t = \text{softmax}(W h_t + b)$ ::: :::: For example, given an image of a cat sitting on a mat, the CNN would extract features representing the cat, the mat, and their spatial relationship. The RNN would then take these features as input and generate a caption like \"a cat sitting on a mat.\" ::: tcolorbox Another example of a one-to-many task is music generation. In this case, the input could be a single note or a musical style, and the output would be a sequence of notes or musical pieces. The RNN generates the music one note at a time, conditioning each note on the previously generated notes and the initial input. ::: ## Many-to-Many RNNs Many-to-many RNNs take asequence of inputs and produce a sequence of outputs. This architecture is useful for tasks where both the input and the output are sequences, and the lengths of the input and output sequences may or may not be the same. There are two main types of many-to-many RNNs: those where the input and output sequences have the same length, and those where they have different lengths. <figure id="fig:many-to-many"> <figcaption>Many-to-many RNN architectures. Left: The input and output sequences have the same length. Right: The input and output sequences have different lengths.</figcaption> </figure> ### Machine Translation (Encoder-Decoder Model) In machine translation, the input is a sentence in one language, and the output is the translation of the sentence in another language. The lengths of the input and output sentences may be different. This task is typically solved using an encoder-decoder model, which consists of two RNNs: an encoder and a decoder. :::: algorithm ::: algorithmic **Input:** Source sentence $x = (x_1, x_2, ..., x_T)$, where $x_i \in \text{Source Vocabulary}$ **Output:** Target sentence $y = (y_1, y_2, ..., y_{T'})$, where $y_i \in \text{Target Vocabulary}$ **Encoder:** Initialize encoder hidden state $h_0^e = \mathbf{0}$ $h_t^e = \text{RNN}_e(x_t, h_{t-1}^e)$ **Decoder:** Initialize decoder hidden state $h_0^d = h_T^e$ $h_t^d = \text{RNN}_d(y_{t-1}, h_{t-1}^d)$ $y_t = \text{softmax}(W h_t^d + b)$ ::: :::: The encoder processes the input sentence word by word and encodes it into a fixed-length vector called the context vector. The context vector represents the meaning of the entire input sentence. The decoder then takes the context vector as input and generates the translated sentence one word at a time. ::: tcolorbox The encoder's job is to compress the meaning of the input sentence into a fixed-length vector. The decoder's job is to unpack that vector and generate the corresponding output sentence. ::: ### Named Entity Recognition (NER) In named entity recognition, the input is a sentence, and the output is a sequence of labels, one for each word in the sentence. Each label indicates whether the corresponding word is part of a named entity (e.g., person, location, organization) and, if so, what type of entity it is. In this case, the input and output sequences have the same length. :::: algorithm ::: algorithmic **Input:** Sequence of word embeddings $x = (x_1, x_2, ..., x_T)$, where $x_i \in \mathbb{R}^d$ **Output:** Sequence of labels $y = (y_1, y_2, ..., y_T)$, where $y_i \in \{1, 2, ..., C\}$ Initialize hidden state $h_0 = \mathbf{0}$ $h_t = \text{RNN}(x_t, h_{t-1})$ $y_t = \text{softmax}(W h_t + b)$ ::: :::: For example, given the sentence \"Einstein was born in Ulm\", the output might be \"B-PER O O B-LOC\", where \"B-PER\" indicates the beginning of a person's name, \"B-LOC\" indicates the beginning of a location, and \"O\" indicates that the word is not part of a named entity. ::: tcolorbox Input sentence: \"Apple Inc. is headquartered in Cupertino.\" Output labels: \"B-ORG I-ORG O O O B-LOC O\" Here, \"B-ORG\" and \"I-ORG\" indicate the beginning and inside of an organization name, \"B-LOC\" indicates the beginning of a location name, and \"O\" indicates other words. ::: # Bidirectional Recurrent Neural Networks ## Motivation: Capturing Context from Both Directions In standard RNNs, the hidden state at each time step only encodes information from the past, i.e., from the inputs that came before the current time step. However, in many tasks, it is beneficial to also consider the future context, i.e., the inputs that come after the current time step. For example, when predicting a missing word in a sentence, it is helpful to know both the words that came before and the words that come after the missing word. ::: tcolorbox Consider the sentence \"The cat sat on the \_\_\_.\" To predict the missing word, it is helpful to know that the next word is \"mat\" or \"couch\". A standard RNN, which only considers the past context, might not be able to make an accurate prediction. ::: Bidirectional RNNs (BRNNs) address this limitation by processing the input sequence in both directions: from left to right and from right to left. This allows the network to capture information from both past and future contexts. ## Forward and Backward Networks A BRNN consists of two separate RNNs: - A **forward RNN** that processes the sequence from left to right (from $t=1$ to $t=T$). - A **backward RNN** that processes the sequence from right to left (from $t=T$ to $t=1$). <figure id="fig:birnn"> <figcaption>Bidirectional RNN architecture. The network consists of a forward RNN (top) that processes the sequence from left to right and a backward RNN (bottom) that processes the sequence from right to left. The hidden states of the forward and backward RNNs are combined to make predictions.</figcaption> </figure> The forward RNN computes a sequence of hidden states $(\overrightarrow{h_1}, \overrightarrow{h_2}, ..., \overrightarrow{h_T})$ by iterating through the input sequence from left to right: $$\overrightarrow{h_t} = f(\overrightarrow{W_{xh}} x_t + \overrightarrow{W_{hh}} \overrightarrow{h_{t-1}} + \overrightarrow{b_h}) \label{eq:forward_hidden}$$ where $x_t$ is the input at time step $t$, $\overrightarrow{W_{xh}}$ and $\overrightarrow{W_{hh}}$ are the weight matrices for the forward RNN, $\overrightarrow{b_h}$ is the bias vector for the forward RNN, and $f$ is an activation function. The backward RNN computes a sequence of hidden states $(\overleftarrow{h_1}, \overleftarrow{h_2}, ..., \overleftarrow{h_T})$ by iterating through the input sequence from right to left: $$\overleftarrow{h_t} = f(\overleftarrow{W_{xh}} x_t + \overleftarrow{W_{hh}} \overleftarrow{h_{t+1}} + \overleftarrow{b_h}) \label{eq:backward_hidden}$$ where $\overleftarrow{W_{xh}}$ and $\overleftarrow{W_{hh}}$ are the weight matrices for the backward RNN, and $\overleftarrow{b_h}$ is the bias vector for the backward RNN. Note that the forward and backward RNNs have different weight matrices and biases. They are independent of each other, except that they share the input sequence. ## Combining Hidden States for Prediction At each time step $t$, the hidden states of the forward and backward RNNs, $\overrightarrow{h_t}$ and $\overleftarrow{h_t}$, are combined to form a single hidden state $h_t$ that represents the context from both directions. The most common way to combine the hidden states is by concatenation: $$h_t = [\overrightarrow{h_t}; \overleftarrow{h_t}] \label{eq:combined_hidden}$$ where $[\cdot ; \cdot]$ denotes concatenation. The combined hidden state $h_t$ is then used to make the prediction $y_t$ at that time step: $$y_t = g(W_{hy} h_t + b_y) \label{eq:prediction}$$ where $W_{hy}$ is a weight matrix, $b_y$ is a bias vector, and $g$ is an activation function. :::: algorithm ::: algorithmic **Input:** Input sequence $x = (x_1, x_2, ..., x_T)$ **Output:** Output sequence $y = (y_1, y_2, ..., y_T)$ **Forward Pass (Left to Right):** Initialize $\overrightarrow{h_0} = \mathbf{0}$ $\overrightarrow{h_t} = f(\overrightarrow{W_{xh}} x_t + \overrightarrow{W_{hh}} \overrightarrow{h_{t-1}} + \overrightarrow{b_h})$ **Backward Pass (Right to Left):** Initialize $\overleftarrow{h_{T+1}} = \mathbf{0}$ $\overleftarrow{h_t} = f(\overleftarrow{W_{xh}} x_t + \overleftarrow{W_{hh}} \overleftarrow{h_{t+1}} + \overleftarrow{b_h})$ **Prediction:** $h_t = [\overrightarrow{h_t}; \overleftarrow{h_t}]$ $y_t = g(W_{hy} h_t + b_y)$ ::: :::: ::: tcolorbox - Capture both past and future context. - Improve performance on tasks where future context is important. - Provide a more complete representation of the sequence at each time step. ::: ::: tcolorbox When implementing a bidirectional RNN, it's important to note that the backward pass cannot be computed until the entire input sequence is available. This means that bidirectional RNNs cannot be used for real-time processing where the input sequence is received one element at a time. ::: # Deep Recurrent Neural Networks ## Stacking Multiple RNN Layers Deep Recurrent Neural Networks (Deep RNNs) are formed by stacking multiple RNN layers on top of each other. In a deep RNN, the hidden state of one RNN layer is fed as input to the next RNN layer. This creates a hierarchical representation of the input sequence, where each layer learns to represent the sequence at a different level of abstraction. <figure id="fig:deep_rnn"> <figcaption>Deep RNN with three layers. The hidden state of each layer is fed as input to the next layer. The superscript denotes the layer number.</figcaption> </figure> The equations for a deep RNN with $L$ layers are as follows: $$\begin{aligned} h_t^{(l)} &= f(W_{xh}^{(l)} h_t^{(l-1)} + W_{hh}^{(l)} h_{t-1}^{(l)} + b_h^{(l)}) \quad \text{for } l = 1, 2, ..., L \\ y_t &= g(W_{hy} h_t^{(L)} + b_y) \end{aligned}$$ where: - $h_t^{(l)}$ is the hidden state of layer $l$ at time step $t$. - $h_t^{(0)} = x_t$ is the input at time step $t$. - $W_{xh}^{(l)}$ and $W_{hh}^{(l)}$ are the weight matrices for layer $l$. - $b_h^{(l)}$ is the bias vector for layer $l$. - $f$ is an activation function. - $y_t$ is the output at time step $t$. - $W_{hy}$ is the weight matrix for the output layer. - $b_y$ is the bias vector for the output layer. - $g$ is an activation function for the output layer. :::: algorithm ::: algorithmic **Input:** Input sequence $x = (x_1, x_2, ..., x_T)$ **Output:** Output sequence $y = (y_1, y_2, ..., y_T)$ $h_t^{(0)} = x_t$ $h_t^{(l)} = f(W_{xh}^{(l)} h_t^{(l-1)} + W_{hh}^{(l)} h_{t-1}^{(l)} + b_h^{(l)})$ $y_t = g(W_{hy} h_t^{(L)} + b_y)$ ::: :::: ::: tcolorbox - Can learn more complex and abstract representations of the input sequence. - Can capture hierarchical dependencies in the data. - Can potentially achieve better performance than shallow RNNs on complex tasks. ::: ::: tcolorbox Deep RNNs can be more difficult to train than shallow RNNs due to the vanishing and exploding gradient problems. The gradients can become very small or very large as they are propagated back through multiple layers and time steps. This can make it difficult for the network to learn long-range dependencies. In practice, it is uncommon to see deep RNNs with more than 2-3 layers, as the benefits of adding more layers often diminish beyond that point, and the training difficulties increase. Techniques like gradient clipping and specialized architectures like LSTMs and GRUs can help mitigate these issues. ::: ::: tcolorbox Imagine a team of experts working together to analyze a complex document. Each expert specializes in a different aspect of the document, such as grammar, vocabulary, or topic. The first expert analyzes the raw text and passes their findings to the second expert, who builds upon the first expert's analysis to gain a deeper understanding. This process continues with each expert adding their own expertise until the final expert produces a comprehensive analysis of the document. In a deep RNN, each layer is like an expert that specializes in a different level of abstraction. The first layer might learn low-level features like word patterns, while higher layers learn more abstract features like sentence structure and overall meaning. ::: # Limitations of Standard RNNs While standard RNNs are powerful models for sequence processing, they suffer from some limitations that can hinder their performance on certain tasks. In this section, we will discuss two key limitations: challenges with long-term dependencies and the need for selective information storage. ## Challenges with Long-Term Dependencies Standard RNNs have difficulty learning long-term dependencies, where the relevant information is separated by many time steps. This means that the network struggles to connect information that is far apart in the input sequence. ::: tcolorbox Consider the sentence: \"The cat, which already ate a lot of food, was full.\" To understand that \"was full\" refers to the \"cat\" and not \"food\", the network needs to remember the subject of the sentence, which appeared many time steps earlier. ::: This difficulty arises due to the **vanishing and exploding gradient problems**. During backpropagation through time, the gradients are multiplied by the weight matrix $W_{hh}$ at each time step. If the largest singular value of $W_{hh}$ is less than 1, the gradients will shrink exponentially as they are propagated back through many time steps, leading to the **vanishing gradient problem**. Conversely, if the largest singular value of $W_{hh}$ is greater than 1, the gradients will grow exponentially, leading to the **exploding gradient problem**. <figure id="fig:vanishing_gradient"> <figcaption>Illustration of the vanishing/exploding gradient problem in standard RNNs. The gradients can shrink or grow exponentially as they are propagated back through many time steps.</figcaption> </figure> These problems make it difficult for the network to learn long-range dependencies, as the gradients that carry information about these dependencies either vanish or explode during training. ## The Need for Selective Information Storage In many tasks, some information in the input sequence is more important than other information. For example, when summarizing a document, some sentences are more important than others. Standard RNNs do not have a mechanism for selectively storing and retrieving information. They update their hidden state at every time step, regardless of the importance of the input. This can lead to the loss of important information over time, as the hidden state may be overwritten byless important information. The hidden state has a limited capacity, and it tries to store information about the entire sequence seen so far. As the sequence gets longer, the hidden state may \"forget\" important information from the past. ::: tcolorbox Imagine you are packing for a trip and you have a backpack with limited capacity. You need to decide which items to keep and which to discard. A standard RNN is like a backpack that automatically replaces some items with new ones at every step, without considering their importance. Important items may be discarded to make room for less important ones. ::: Ideally, we would like the network to have a mechanism for selectively storing and retrieving information, so that it can focus on the most important parts of the input sequence and retain them in memory for as long as needed. This is a key motivation behind advanced RNN architectures like LSTMs and GRUs, which we will discuss in the next section. # Advanced RNN Architectures To address the limitations of standard RNNs, researchers have developed more advanced RNN architectures that incorporate gating mechanisms. These gates control the flow of information within the network, allowing it to selectively update and retain information over time. In this section, we will introduce two popular advanced RNN architectures: Gated Recurrent Units (GRUs) and Long Short-Term Memory (LSTMs). ## Gated Recurrent Units (GRUs) Gated Recurrent Units (GRUs) are a type of RNN that use gating mechanisms to control the flow of information into and out of the hidden state. These gates allow the network to selectively update and retain information, addressing the limitations of standard RNNs. A GRU has two gates: - An **update gate** ($z_t$) that determines how much of the past information (from previous time steps) needs to be passed along to the future. - A **reset gate** ($r_t$) that determines how much of the past information to forget. <figure id="fig:gru"> <figcaption>GRU cell with update and reset gates. The gates control the flow of information from the previous hidden state $h_{t-1}$ and the current input $x_t$ to the new hidden state $h_t$.</figcaption> </figure> The update gate $z_t$ is computed as follows: $$z_t = \sigma(W_z x_t + U_z h_{t-1} + b_z)$$ where $x_t$ is the input at time step $t$, $h_{t-1}$ is the previous hidden state, $W_z$ and $U_z$ are weight matrices, $b_z$ is a bias vector, and $\sigma$ is the sigmoid function. The reset gate $r_t$ is computed similarly: $$r_t = \sigma(W_r x_t + U_r h_{t-1} + b_r)$$ where $W_r$, $U_r$, and $b_r$ are weight matrices and a bias vector for the reset gate. The candidate hidden state $\tilde{h}_t$ is computed as: $$\tilde{h}_t = \tanh(W x_t + U (r_t \odot h_{t-1}) + b)$$ where $W$ and $U$ are weight matrices, $b$ is a bias vector, and $\odot$ denotes element-wise multiplication. The new hidden state $h_t$ is a linear interpolation between the previous hidden state $h_{t-1}$ and the candidate hidden state $\tilde{h}_t$, controlled by the update gate $z_t$: $$h_t = (1 - z_t) \odot h_{t-1} + z_t \odot \tilde{h}_t$$ :::: algorithm ::: algorithmic **Input:** Input sequence $x = (x_1, x_2, ..., x_T)$, initial hidden state $h_0$ **Output:** Hidden states $h = (h_1, h_2, ..., h_T)$ $z_t = \sigma(W_z x_t + U_z h_{t-1} + b_z)$ $r_t = \sigma(W_r x_t + U_r h_{t-1} + b_r)$ $\tilde{h}_t = \tanh(W x_t + U (r_t \odot h_{t-1}) + b)$ $h_t = (1 - z_t) \odot h_{t-1} + z_t \odot \tilde{h}_t$ ::: :::: ## Long Short-Term Memory (LSTMs) Long Short-Term Memory (LSTMs) are another type of RNN with gating mechanisms. LSTMs have a more complex architecture than GRUs, with separate memory cells and gates for input, output, and forgetting. An LSTM cell has three gates: - An **input gate** ($i_t$) that controls how much of the new information to store in the cell state. - A **forget gate** ($f_t$) that controls how much of the old cell state to forget. - An **output gate** ($o_t$) that controls how much of the cell state to output to the hidden state. <figure id="fig:lstm"> <figcaption>LSTM cell with input, forget, and output gates. The gates control the flow of information into and out of the cell state $C_t$ and the hidden state $h_t$.</figcaption> </figure> The gates are computed as follows: $$\begin{aligned} i_t &= \sigma(W_i x_t + U_i h_{t-1} + b_i) \\ f_t &= \sigma(W_f x_t + U_f h_{t-1} + b_f) \\ o_t &= \sigma(W_o x_t + U_o h_{t-1} + b_o) \end{aligned}$$ where $x_t$ is the input at time step $t$, $h_{t-1}$ is the previous hidden state, $W_i$, $W_f$, $W_o$, $U_i$, $U_f$, $U_o$ are weight matrices, $b_i$, $b_f$, $b_o$ are bias vectors, and $\sigma$ is the sigmoid function. The candidate cell state $\tilde{C}_t$ is computed as: $$\tilde{C}_t = \tanh(W_C x_t + U_C h_{t-1} + b_C)$$ where $W_C$ and $U_C$ are weight matrices, and $b_C$ is a bias vector. The new cell state $C_t$ is a combination of the old cell state $C_{t-1}$ and the candidate cell state $\tilde{C}_t$, controlled by the forget gate $f_t$ and the input gate $i_t$: $$C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t$$ The new hidden state $h_t$ is the output of the cell state passed through an activation function and controlled by the output gate $o_t$: $$h_t = o_t \odot \tanh(C_t)$$ :::: algorithm ::: algorithmic **Input:** Input sequence $x = (x_1, x_2, ..., x_T)$, initial hidden state $h_0$, initial cell state $C_0$ **Output:** Hidden states $h = (h_1, h_2, ..., h_T)$ $i_t = \sigma(W_i x_t + U_i h_{t-1} + b_i)$ $f_t = \sigma(W_f x_t + U_f h_{t-1} + b_f)$ $o_t = \sigma(W_o x_t + U_o h_{t-1} + b_o)$ $\tilde{C}_t = \tanh(W_C x_t + U_C h_{t-1} + b_C)$ $C_t = f_t \odot C_{t-1} + i_t \odot \tilde{C}_t$ $h_t = o_t \odot \tanh(C_t)$ ::: :::: ## Selective Updating of Hidden States Both GRUs and LSTMs allow for selective updating of the hidden state. This means that the network can choose which parts of the hidden state to update and which parts to keep the same. This is achieved through the gating mechanisms. In GRUs, the update gate $z_t$ controls how much of the new information to incorporate into the hidden state. If $z_t$ is close to 1, the new hidden state will be mostly based on the candidate hidden state $\tilde{h}_t$, and if $z_t$ is close to 0, the new hidden state will be mostly based on the previous hidden state $h_{t-1}$. The reset gate $r_t$ controls how much of the previous hidden state to use when computing the candidate hidden state. In LSTMs, the input gate $i_t$ controls how much of the new information to store in the cell state, the forget gate $f_t$ controls how much of the old cell state to forget, and the output gate $o_t$ controls how much of the cell state to output to the hidden state. ::: tcolorbox - Allow the network to learn long-term dependencies by selectively retaining information over many time steps. - Mitigate the vanishing gradient problem by providing a more direct path for the gradients to flow through during backpropagation. - Enable the network to focus on the most relevant parts of the input sequence and ignore irrelevant information. ::: ::: tcolorbox LSTMs are generally more powerful than GRUs due to their more complex architecture and additional gating mechanisms. However, GRUs are computationally more efficient as they have fewer parameters. In practice, the choice between GRUs and LSTMs depends on the specific task and dataset, and it is often a good idea to try both architectures. ::: # Conclusion In this lecture, we explored Recurrent Neural Networks (RNNs) and their ability to process sequential data. We began by introducing the concept of sequential data and how it differs from traditional data. We then delved into the fundamental building blocks of RNNs, including word embeddings, hidden states, and shared weights. We discussed various RNN architectures, including many-to-one, one-to-many, and many-to-many models, and their applications in tasks like text classification, image captioning, machine translation, and named entity recognition. We also covered the training of RNNs using Backpropagation Through Time (BPTT) and the concept of shared weights, which allows RNNs to generalize across sequences of different lengths and recognize patterns regardless of their position in the sequence. We examined Bidirectional RNNs, which process sequences in both directions (left-to-right and right-to-left), allowing them to capture both past and future context when making predictions. We also looked at Deep RNNs, which stack multiple RNN layers to learn hierarchical representations of the input sequence. We highlighted the limitations of standard RNNs, particularly their difficulty in handling long-term dependencies due to the vanishing and exploding gradient problems, and the need for selective information storage. Finally, we introduced advanced architectures like Gated Recurrent Units (GRUs) and Long Short-Term Memory networks (LSTMs), which address these limitations through gating mechanisms that allow for selective updating of the hidden state and cell state. These gating mechanisms enable GRUs and LSTMs to learn long-range dependencies and retain important information over many time steps. ::: tcolorbox - RNNs are designed for processing sequential data, where the order of elements matters. - Word embeddings represent words as vectors, capturing semantic relationships. - Hidden states in RNNs act as a memory of past inputs. - Shared weights in RNNs enable generalization across time and reduce the number of parameters. - Many-to-one, one-to-many, and many-to-many architectures suit different types of tasks. - BPTT is used to train RNNs, adapting backpropagation for sequential data. - Bidirectional RNNs consider both past and future context. - Deep RNNs learn hierarchical representations by stacking multiple layers. - Standard RNNs struggle with long-term dependencies due to vanishing/exploding gradients. - GRUs and LSTMs use gating mechanisms to selectively update and retain information, addressing the limitations of standard RNNs. ::: For the next lecture, consider these questions: - How do GRUs and LSTMs differ in their gating mechanisms? (Hint: Consider the number of gates and how they control information flow.) - What are some practical applications where the ability to handle long-term dependencies is crucial? (Hint: Think about tasks involving natural language processing, speech recognition, or time-series analysis.) - How can we evaluate the performance of different RNN architectures on a given task? (Hint: Consider appropriate evaluation metrics and validation strategies.) - How does the concept of attention, which we will explore in the next lecture, further enhance the ability of RNNs to handle long-range dependencies and focus on relevant information? ::: tcolorbox In the next lecture, we will dive into attention mechanisms, a powerful technique that complements RNNs and further improves their ability to handle long sequences and complex tasks. Attention allows the network to focus on specific parts of the input sequence when making predictions, rather than relying solely on the fixed-length hidden state. This is particularly useful in tasks like machine translation and image captioning, where different parts of the input sequence may be relevant for generating different parts of the output sequence. :::