#### Lukasz Kaisar (Google Brain Research Scientist) – “Deep Learning (Aug 2018)

#### Chapters

#### Abstract

Abstract

The Evolution and Challenges of Neural Networks: From Basic Concepts to Advanced Architectures

—

Introduction

Neural networks, a cornerstone of modern artificial intelligence, have evolved significantly over the years. Starting with simple models and progressing to complex architectures like recurrent neural networks (RNNs) and Neural Turing Machines (NTMs), the journey reflects a continuous quest for computational efficiency and adaptability. This article delves into this evolution, highlighting key concepts like TensorFlow’s role, the intricacies of neural network training, and the advances in architectures, including RNNs and NTMs. It also addresses practical considerations and theoretical complexities, offering insights into the future trajectory of neural network development.

—

Evolution of Neural Networks and TensorFlow’s Role

TensorFlow, a prominent framework in the field, plays a pivotal role in advancing neural networks. It enables the creation and execution of complex models, such as those trained on the MNIST dataset. The ease of integrating TensorFlow with platforms like Google Colab, which provides GPU support, has democratized access to powerful computational resources. The article starts by exploring TensorFlow’s graph creation and execution capabilities, illustrating its utility in running basic neural network models, including a simple model with one hidden and one output layer trained on MNIST data. The discussion then extends to various optimizers and the concept of distributed tuning, a common practice in deep learning, emphasizing the practical aspects of neural network implementation.__Potentially Universal Computation:__

TensorFlow’s computational graph is not restricted to fixed circuits. This flexibility allows neural networks to perform tasks requiring recurrence and conditional computation, including evaluating arbitrary functions.__Long Multiplication Computation:__

With n steps of recurrent computation, the network can calculate the result of long multiplication of two binary numbers written in memory.__Parallel Operations:__

The computations are highly parallelized, enabling the network to efficiently perform a large number of operations simultaneously.__Memory Storage:__

The numbers to be multiplied are stored in the network’s memory, eliminating the need for intermediate storage space.__Limited Computation Steps:__

In contrast to traditional multiplication algorithms, which require n squared steps, this approach requires only n steps for multiplication.__Number Length:__

The length of the binary numbers involved in the multiplication can be arbitrary, with n representing the length of both numbers.__Parallel Addition:__

Although the convolution operation can only access one element to the left and right, addition can be performed in n steps with parallel processing.__Parallel Accumulation:__

Efficient accumulation of parallel additions is necessary to avoid requiring n squared space.__Historical Research:__

Previous research in the 1980s demonstrated the capacity of parallel models to perform complex computations effectively.__Algorithm Simplicity:__

The algorithm’s simplicity allows for efficient computation, unlike traditional methods that require extensive intermediate storage.

—

From Basic to Advanced Neural Network Architectures

The journey from basic neural networks to advanced architectures encompasses a significant leap in computational capabilities. This section examines the shift from single-layer networks, limited to polynomial time computations, to more sophisticated models like RNNs and NTMs. The limitations of threshold circuits and their evolution into more powerful computational models are discussed, emphasizing the increasing complexity and capabilities of modern neural networks.

Threshold Circuits and Complexity Classes

Threshold circuits are a more powerful class than AC circuits because they lack the parity problem that AC circuits face. Threshold circuits do not have an equivalent concept of parity, making them a more general class. Few separation results are known for threshold circuits. For AC circuits, a hierarchy can be proven, but for threshold circuits, the depth hierarchy remains open. All related questions are essentially unsolved. Circuits have an input of fixed size, so the question arises of how to handle variable-sized inputs. Circuits are typically constructed in a uniform way, with a polynomial-time program generating the circuit or formula for each input length.__Architecture Overview__:

The architecture consists of a single column, containing a vector of floats representing the input.

– Convolutional layers with gating mechanisms (CGRU layers) are applied sequentially to the input vector.

– The output is the final result of the network.__Long Binary Multiplication__:

The network learns to perform long binary multiplication on 20-bit numbers, generalizing to 2,000-bit numbers with no errors.__Proof of Learning__:

Proving that the network has learned an algorithm formally is challenging.

– The network is tested on a set 100 times longer than the training set.

– While it performs well on this set, it fails on an even larger set, indicating that it has not fully learned the multiplication algorithm.__Visualization__:

The state of the network is represented as a 2D grid, with the input on the height axis and the activations on the width axis (channel dimension).

– In the case of duplication, the network learns to shift down a pattern in the input over time.

– In the case of reversing, the network learns a “tapey” movement, moving parts of the input up and down to achieve the reversal.__Adding and Multiplying__:

The network can also learn to add and multiply binary numbers.

– In the case of addition, the network spreads out the numbers to align the digits before combining them.

– In the case of multiplication, the spreading and combining of numbers is more complex and difficult to visualize.__Training Challenges__:

Training the network is challenging.

– Curriculum learning (starting with short numbers and gradually increasing length) and relaxation (starting with different sets of parameters and gradually bringing them together) are used to improve training results.

– Dropout on the recurrent connection and noise added to gradients are also used to aid training.__Recent Developments__:

Recent work by colleagues from Riga has shown that most of the tricks used to train the network are not necessary.

– With improved neural GPU, their models train successfully almost every time.

Complexity Increase with Recurrence in Neural Networks

In standard connections like residual connections, the overall network complexity remains relatively unchanged, even with multiple layers. Recurrence, however, significantly alters the complexity landscape. With recurrence, the complexity increases rapidly, particularly if it is executed numerous times (n times or 2 to the n times). Recurrent networks can potentially simulate Turing machines, as they can manipulate information left, right, and internally. Networks with a limited number of layers (e.g., three layers) may be unable to compute specific tasks. Although not mathematically proven, this limitation is widely accepted. A task requiring the network to perform long multiplication illustrates the need for expressiveness and learning capabilities. Gradient descent, the primary training method for neural networks, may be inadequate for finding optimal solutions for such complex tasks. RNNs, a type of neural network with recurrence, have input, state, and output components. Each cell iterates through an input sequence, producing an output at each step based on its state and input. The simplest RNN cell involves a non-linearity applied to a linear combination of the current state and input. Training RNNs becomes increasingly difficult as the sequence length increases due to vanishing gradients. Tricks like adding a residual connection help alleviate this problem, allowing for better gradient propagation.

Vanishing Gradients and Residual Connections

Recurrent neural networks (RNNs) with multiple non-linear layers can suffer from vanishing gradients during backpropagation. This occurs because the gradients of the ReLU activation function, which is commonly used in RNNs, can be 0 for some inputs, causing information to vanish. To address the vanishing gradient problem, residual connections were introduced. Residual connections add the input of a layer to its output, allowing gradients to flow more easily through the network. This helps prevent the vanishing gradient problem and enables the network to learn long-term dependencies more effectively. To prevent the accumulation of large values in residual connections, layer normalization or special gates can be used. Layer normalization normalizes the activations of a layer before adding them to the output. Gated mechanisms, such as the Gated Recurrent Unit (GRU), allow the network to control which parts of the information are retained or forgotten at each time step.

Advanced RNN Architectures: Neural Turing Machines and ConvGRUs

Neural Turing Machines (NTMs) and Convolutional GRUs (ConvGRUs) represent significant advancements in RNN architectures. This section explains the capabilities of these models, including their computational universality and ability to perform complex computations like long multiplication. The challenges in training NTMs, such as the need for curriculum learning and parameter relaxation, are discussed, along with the strategies employed to overcome these obstacles.__Navigating Tricks and Tweaks in Neural GPU Architectures and Convolutional RNNs__:

Convolutional RNNs, such as convolutional GRUs and LSTMs, replace fully connected layers with convolutions.__Neural GPU Improvements:__

Using Adamax optimizer is more effective than Adam for algorithmic tasks.

– Slight modifications to the loss function can help achieve saturation without using additional tricks.__Sequence Repetition Task:__

A toy problem of repeating bits at even positions in a sequence is introduced.

– Convolutional layers should theoretically solve this task easily.__Data Preprocessing:__

Data sets are created from Python generators.

– Tensors are reshaped and converted to one-hot embeddings for integers.__Word Embeddings:__

Word embeddings are used in NLP tasks to represent words as dense vectors.

– In this example, one-hot embeddings are multiplied by a matrix to create word embeddings.

—

Challenges in Training Neural Networks: The Case of RNNs

RNNs, despite their advanced capabilities, face significant training challenges, particularly the vanishing gradient problem. This part of the article explores techniques to mitigate these challenges, such as residual connections and sophisticated optimization algorithms. It also touches upon the limitations of RNNs, like their constant memory size, and the emergence of more powerful architectures like NTMs and neural GPU architectures.

—

Recurrent Neural Networks and Vanishing Gradients

The concept of vanishing gradients in RNNs is a critical challenge in neural network training. This section delves into the technical aspects of this problem, exploring how ReLU activations and residual connections impact the training process. It also discusses the role of layer normalization and special gates in preventing vector norm escalation, providing a comprehensive overview of the methods employed to address these challenges.

—

Practical Applications and Future Prospects

The article concludes with a look at the practical applications of neural networks and their future prospects. It covers topics like the introduction of positional information in networks, the use of 2D convolution for 1D tasks, and the limitations of fixed positional information. The final part previews upcoming exercises aimed at tackling tasks without fixed positional embedding and explores further advancements in RNNs.

—

Conclusion

The evolution of neural networks from simple to complex architectures underscores the dynamic nature of this field. As these systems become more sophisticated, they offer new possibilities and challenges, reflecting the ever-expanding horizons of artificial intelligence and computational capabilities. This article provides a comprehensive overview of this journey, from the basics of TensorFlow and neural networks to the advanced architectures of RNNs and NTMs, offering a glimpse into the future of neural network development and application.

Notes by: Alkaid