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What does the ReLU function output when the input is less than or equal to zero?

Negative input value

Zero (correct)

The input value itself

The maximum of the input and zero

If $f = ReLU(Ax)$ and $y = ReLU(B ReLU(Ax))$, what can be deduced about the relationship between $f$ and $y$?

y is influenced by f through a nonlinear transformation (correct)

y is always greater than f

y is independent of f

f is always greater than y

In the context provided, what scenario best describes the operation of the function $y = ReLU(B ReLU(Ax))$?

It applies two separate ReLU functions in isolation.

It constrains the output to only positive values.

It uses only the original input without transformations.

It compounds the output of multiple linear transformations before applying ReLU. (correct)

Why are inputs resulting in non-zero outputs significant in deep learning models using ReLU activations?

They allow for the activation of more neurons, leading to higher representation capacity. Signup and view all the answers

Which statement about deep learning and the XOR problem is true?

XOR requires a non-linear approach using multiple layers or neurons. Signup and view all the answers

What does the ReLU function output when the input is negative?

Zero Signup and view all the answers

Which of the following statements best represents the transition from linear to nonlinear functions in deep learning?

Nonlinear functions are more complex and can model a wider variety of data. Signup and view all the answers

Given the linear function notation f = ReLU(Ax), what does 'A' represent?

The weight matrix Signup and view all the answers

What is the purpose of applying a ReLU activation function in deep learning?

To introduce non-linearity into the model Signup and view all the answers

What limitation does a linear function have in the context of model complexity?

It can represent only linear relationships Signup and view all the answers

How is the term 'deep' typically understood in deep learning?

Describing the number of layers in the network Signup and view all the answers

What is the output of the function ReLU(−1)?

0 Signup and view all the answers

Which characteristic defines a nonlinear function compared to a linear function?

It cannot be represented with a simple equation. Signup and view all the answers

What is one disadvantage of the Euclidean loss function?

It magnifies errors quadratically, making it sensitive to outliers. Signup and view all the answers

Which of the following is NOT a characteristic of cost functions?

They must have a gradient that is unpredictable. Signup and view all the answers

What happens when the cost function becomes very flat?

It undermines the objective of guiding learning algorithms. Signup and view all the answers

In which scenario is the use of the cross-entropy cost function preferable?

For classification problems with discrete outcomes. Signup and view all the answers

Which of the following correctly describes the Euclidean loss function?

It computes the average distance between predicted and actual values. Signup and view all the answers

Which statement best defines the main purpose of cost functions?

To evaluate how well a model's predictions match the actual output. Signup and view all the answers

The KL-divergence cost function is primarily used for which type of tasks?

Comparing probability distributions. Signup and view all the answers

When blending different cost functions, what is a key consideration?

The gradients of combined functions must be consistent. Signup and view all the answers

What is the primary design focus when developing families φ(x; θ)?

To encode human knowledge for better generalization Signup and view all the answers

What characteristic defines a good classification problem?

The data must be linearly separable Signup and view all the answers

Why is sensitivity to outliers a significant concern with certain cost functions?

It can skew the model's learning process. Signup and view all the answers

In the context of feature learning, what does the XOR problem illustrate?

The inefficiency of linear models Signup and view all the answers

What does transforming the input space into a learned feature space allow a model to do?

View data as linearly separable Signup and view all the answers

What role does the learned representation play in solving the XOR problem?

It allows for linear classification in transformed space Signup and view all the answers

How are thresholds used in the XOR problem representation?

To determine separation between classes Signup and view all the answers

Why do neural networks utilize feature extraction for problems like XOR?

To learn complex boundary representations Signup and view all the answers

Which characteristic is essential for a representation to successfully solve the XOR problem?

Should create a linear boundary in transformed space Signup and view all the answers

What is an essential characteristic of a module in deep feedforward networks?

A module may or may not have trainable parameters. Signup and view all the answers

Which statement about the hidden layer in the provided feedforward network is correct?

The hidden layer contains two units. Signup and view all the answers

What does the notation 'f = exp(x)' in the context of deep feedforward networks represent?

An activation function that can be applied to network outputs. Signup and view all the answers

In the structure of a feedforward network, how are inputs typically represented?

As nodes connected to the hidden layer. Signup and view all the answers

Which feature distinguishes the depiction of the feedforward network in the provided content?

All units are represented as nodes in the graph. Signup and view all the answers

Why is the XOR example specifically mentioned in relation to the feedforward network?

It serves to illustrate the network's ability to handle non-linearly separable problems. Signup and view all the answers

What role do the weights (w) play in a feedforward network?

They determine how inputs are transformed and affect the network's learning process. Signup and view all the answers

What does the term 'deep' refer to in deep feedforward networks?

The addition of multiple hidden layers to the network. Signup and view all the answers

What is the main purpose of back-propagation in neural networks?

To compute the gradient of the loss function with respect to the module output Signup and view all the answers

Which of the following statements about back-propagation is true?

It calculates the chain rule for efficient computation of gradients Signup and view all the answers

In back-propagation, what does the Jacobian matrix represent?

The partial derivatives of the output with respect to its inputs Signup and view all the answers

What do we mean by 'back-propagating gradients'?

Recursively computing gradients from the output to the input layer Signup and view all the answers

Which equation correctly represents the relationship between gradients in back-propagation?

$rac{d\mathcal{L}}{d w_3} = \frac{d h}{d w_3} \cdot \frac{d\mathcal{L}}{d h}$ Signup and view all the answers

What does the term 'efficient order of operations' refer to in back-propagation?

The optimization of computing gradients across layers Signup and view all the answers

Which of the following is NOT a key component in the back-propagation process?

Feature scaling of input data Signup and view all the answers

How does back-propagation relate to the chain rule in calculus?

It uses the chain rule to combine partial derivatives for gradient computation Signup and view all the answers

Study Notes

Lecture 2: Deep Feedforward Networks

A robot wrote this entire article. Are you scared yet, human?
A powerful language model, GPT-3, was asked to write an essay from scratch.
The assignment was to write from scratch.

Lecture Overview

Modularity in deep learning
Deep learning nonlinearities
Gradient-based learning
Chain rule
Backpropagation

Last Time

Neural networks go from simple linear functions to more complex non-linear functions.
Deep neural networks employ non-linear functions.

From linear functions to nonlinear = from shallow to deep

The focus is on ReLU(Ax), nonlinear functions from shallow to deep.

From linear functions to nonlinear = from shallow to deep

The study includes an analysis of ReLU(Bf) equal to ReLU(B ReLU(Ax)).

We've learned XOR.

Neural networks learn the XOR function in the original x space.
This example demonstrates how deep networks can learn non-linear relationships.

Deep feedforward networks

Feedforward neural networks are also known as multi-layer perceptrons (MLPs).
The objective is to approximate a function f.
A feedforward network defines y = f(x; θ).
The network learns the parameter θ to get the best approximation of the function f.
Feedforward networks do not have feedback connections.
Recurrent neural networks have feedback connections.
Brains have many feedback connections.

Deep feedforward networks

Deep feedforward networks are comprised of functions in a composite function.
y = f (x; θ) = aₗ (x; θ₁, .... θₗ).
θₗ is a parameter in the l-th layer.
aₗ can be simplified to: aₗ = f(x; θ) = h₁° h₁-1°…。 h₁° x
Where each function hₗ is parameterized by θₗ.

Neural networks in blocks

The functions in the composite function can be visualized as a cascade of blocks.
The network is comprised of forward connections.
The structure includes an input, hidden layers, and an output.

What is a module?

A module is a building block/transformation/function.
Modules can receive input data or the output from another module.
Modules return output based on their activation function.
Modules may or may not have trainable parameters.
Examples include f = Ax and f = exp(x).

Requirements

Activation functions must be differentiable almost everywhere.
Take special care with cycles in the architecture of blocks.
No other requirements.

Feedforward model

Almost all CNNs/Transformers are feedforward models.
The model is as simple as possible.

Non-linear feature learning perspective

Linear models, including logistic regression and linear regression, are convex and efficiently fit with closed-form solutions.
However, linear models have limited capacity and reliability.
Neural networks allow you to extend to nonlinear models.
Applying a linear model to a transformed input, q(x), is a method. Kernel trick, e.g., the RBF kernel, is an application of a nonlinear dimension reduction.

Non-linear feature learning perspective

Deep learning aims to learn a function q where y = f(x;θ, w).
The learned θ corresponds to a good or linearly separable representation in the case of classification.
A good representation involves choosing the appropriate θ.

Neural networks in blocks

Modular design allows you to combine hierarchies to build complex architectures.
This offers greater data efficiency by using good knowledge of the problem domain to create diverse inputs.
Combining multiple inputs and modalities like RGB and LIDAR is an illustration.

Hierarchies of modules

Data efficient modules and hierarchies for efficient performance.
Often, increasing the number of iterations is better than having a stronger model.
ReLUs are efficient due to only using addition and multiplication.

Loopy connections

The past output of a module can be utilized as the future input of that same module.
Cycles in the architecture, also called recurrent neural networks.
Usually not used anymore.

How to get w? gradient-based learning

The non-linearity of neural networks results in a non-convex loss function.
Use iterative gradient based optimizers to train the network.
Stochastic gradient descent is a common approach but the parameter initialization is critical.

Cost function

Primarily, maximum likelihood estimation is used on the training set to maximize the likelihood of the model parameters.
Minimum negative log-likelihood of the model and actual distribution is found for cost function, effectively equivalent to minimizing cross entropy.
The Gaussian distribution, which is not parameterized, can be discarded for the cost function evaluation.

Cost functions

Euclidean loss, useful for regression problems, but sensitive to outliers.
Other cost functions like cross-entropy are useful for tasks like classification.

Cost functions

Cost functions describe the behavior of the model.
The gradient should be large and predictable to guide learning algorithms.
Functions that saturate become less useful for guide learning algorithms.
Negative log likelihood is helpful to ensure good performance.

Activation functions

Activation functions transform the weighted sum of inputs into an output from a node.
Squashing functions have a limited range.
Activation function choices greatly impact neural networks.
Should be differentiable to the greatest extent possible.

Linear Units

Identity activation function
No activation saturation.
Strong and stable gradients.
Reliable learning using linear modules.

Rectified Linear Unit (ReLU)

ReLU activation function.
The output of ReLU is the maximum of zero and the input.
The graph of ReLU is a straight line with slope 0, rising up at x=0 with the slope of 1.
Advantages of using ReLU for sparse activation, greater gradient propagation. Also, efficient computation since it simplifies to comparing the input with zero.
ReLU is also scale-invariant.
Potential problems include a lack of differentiability at zero, not being zero-centered, and unbounded.

Rectified Linear Unit (ReLU)

Potential problems include non-differentiability at zero, not being zero-centered, and unbounded output values.
Dead neurons problem - neurons can potentially become inactive.
Increasing learning rates might help deal with this problem.
In current deep networks ReLU is the common default.

Leaky ReLU

Allows a small, positive gradient when a unit is not active.
Parametric ReLU or PReLU treats a as a learnable parameter.

Exponential Linear Unit (ELU)

Smooth approximation to the rectifier.
Has a non-monotonic bump when x < 0.
Default activation function for many models, including BERT.

Gaussian Error Linear Unit (GELU)

Similar to ELU, but non-monotonic.
Default for Vision Transformaters.

Sigmoid and Tanh

Tanh has a better output range of -1 to 1.
Tanh is centered around 0 and not 0.5.
Sigmoids and tanh saturate at extreme values, leading to zero gradients.
They can be problematic for middle layers.

Softmax

Outputs probability distribution - summing the values from the outputs to one is commonly done.
Avoids exponentiation of large/small numbers for stability.

How to Choose an Activation Function

Default recommendation for hidden layers is ReLU or GELU
For outputs, use different types of activation depending on the task.

New modules

Any differentiable function is valuable.
Modules of modules can be as easy as tanh(ReLU(x)).

Architecture Design

Networks are composed of layers of interconnected units.
The first layer is defined by h(1) = g(1) (W(1) Tx+b(1)).

Quiz

If nonlinearities are removed from a deep network, then only a single layer is learned.
Adding a nonlinearity at the end is not effective.

Width and Depth

Universal approximation theorem shows any function can be represented using MLPs.
Depth can improve generalization error as networks with more layers may provide better representations of the information in the data, reducing the generalization error by improving the representation of the data and reducing redundancy.

Deeper networks: hierarchical pattern recognition

Deeper architectures in neural networks exhibit a division of labor where different layers specialize in recognizing different levels of features, and allows for a bottom-up processing of the input information.
Raw data input is progressively transformed into successively higher level features, like low level, mid level, and high level features.

Width and Depth

Increasing the number of parameters in convolutional layers without increasing depth is generally less effective than increasing depth.

A neural network jungle

Encompasses a range of neural network types, including perceptrons, MLPs, RNNs, LSTMs, GRUs, autoencoders, and various other types.
--Most important ones are MLPs, Variational Autoencoders, Convolutional Nets, and LSTMs/Transformers.

Intermezzo: Chain rule

Chain rule applied to compute derivatives of functions formed by composing other functions.

Computational graph

Graphs show computation sequences of variables in the form of nodes.
Each node in the graph denotes a variable or a simple function of the other variables.

Example

Example uses computational graphs to illustrate the chain rule in backpropagation.

Example

Illustrates the chain rule using a diagram.

How research gets done part II

Suggests different approaches for reading research papers.
The "pass" approach simplifies the process.

Backpropagation

Algorithm for training neural networks.

Backprop: even former head of Tesla AI thinks it's important

Backpropagation is a key concept in deep learning.

Backpropagation <=> Chain rule

Backpropagation is the application of the chain rule.

Backpropagation <=> Chain rule!!!

Backpropagation implementation involves recursively computing gradients.

Backpropagation <=> Chain rule but as an algorithm

Backpropagation computes chain rule, efficient ordering.

But you know this already from ML 1

Review of backpropagation as previously covered.

But why do we actually use Backprop?

To determine benefits of backpropagation.

Regarding point 4:

A 3x3x3 MLP can be trained easily without backpropagation magic.

Re: point 2:

The brain uses random synaptic weights to propagate errors.

Computational feasibility

Storing the Jacobian for large data sets is computationally intensive.

Chain rule visualized

Illustration of chain rule in multi-step operations.

What if the output is a scalar?

The Jacobian in this case is relatively small.

Chain rule visualized

The computation is simplified if the output is a scalar.

Chain rule visualized

The calculation in a chain rule can be visualized again as it is computationally simpler when the output is a scalar.

But we still need the Jacobian?

Sparse Jacobians are common in neural network operations.

Computational graphs: Forward graph

Computational graph for a feedforward network.

Computational graphs: Reverse graph

Backwards flow for computation.

Backpropagation in summary

Steps for applying backpropagation.

Backpropagation visualization

Visual representation and steps for backpropagation.

What's the big deal?

Reasons for using backpropagation.

Summary

Summary of deep feedforward networks, neural network modules, chain rule, and backpropagation
Reading material, including Deep Learning book, chapter 6 and Efficient Backprop, are suggested for further study.

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Questions and Answers

What does the ReLU function output when the input is less than or equal to zero?

If $f = ReLU(Ax)$ and $y = ReLU(B ReLU(Ax))$, what can be deduced about the relationship between $f$ and $y$?

In the context provided, what scenario best describes the operation of the function $y = ReLU(B ReLU(Ax))$?

Why are inputs resulting in non-zero outputs significant in deep learning models using ReLU activations?

Which statement about deep learning and the XOR problem is true?

What does the ReLU function output when the input is negative?

Which of the following statements best represents the transition from linear to nonlinear functions in deep learning?

Given the linear function notation f = ReLU(Ax), what does 'A' represent?

What is the purpose of applying a ReLU activation function in deep learning?

What limitation does a linear function have in the context of model complexity?

How is the term 'deep' typically understood in deep learning?

What is the output of the function ReLU(−1)?

Which characteristic defines a nonlinear function compared to a linear function?

What is one disadvantage of the Euclidean loss function?

Which of the following is NOT a characteristic of cost functions?

What happens when the cost function becomes very flat?

In which scenario is the use of the cross-entropy cost function preferable?

Which of the following correctly describes the Euclidean loss function?

Which statement best defines the main purpose of cost functions?

The KL-divergence cost function is primarily used for which type of tasks?

When blending different cost functions, what is a key consideration?

What is the primary design focus when developing families φ(x; θ)?

What characteristic defines a good classification problem?

Why is sensitivity to outliers a significant concern with certain cost functions?

In the context of feature learning, what does the XOR problem illustrate?

What does transforming the input space into a learned feature space allow a model to do?

What role does the learned representation play in solving the XOR problem?

How are thresholds used in the XOR problem representation?

Why do neural networks utilize feature extraction for problems like XOR?

Which characteristic is essential for a representation to successfully solve the XOR problem?

What is an essential characteristic of a module in deep feedforward networks?

Which statement about the hidden layer in the provided feedforward network is correct?

What does the notation 'f = exp(x)' in the context of deep feedforward networks represent?

In the structure of a feedforward network, how are inputs typically represented?

Which feature distinguishes the depiction of the feedforward network in the provided content?

Why is the XOR example specifically mentioned in relation to the feedforward network?

What role do the weights (w) play in a feedforward network?

What does the term 'deep' refer to in deep feedforward networks?

What is the main purpose of back-propagation in neural networks?

Which of the following statements about back-propagation is true?

In back-propagation, what does the Jacobian matrix represent?

What do we mean by 'back-propagating gradients'?

Which equation correctly represents the relationship between gradients in back-propagation?

What does the term 'efficient order of operations' refer to in back-propagation?

Which of the following is NOT a key component in the back-propagation process?

How does back-propagation relate to the chain rule in calculus?

Study Notes

Lecture 2: Deep Feedforward Networks

Lecture Overview

Last Time

From linear functions to nonlinear = from shallow to deep

From linear functions to nonlinear = from shallow to deep

We've learned XOR.

Deep feedforward networks

Deep feedforward networks

Neural networks in blocks

What is a module?

Requirements

Feedforward model

Non-linear feature learning perspective

Non-linear feature learning perspective

Neural networks in blocks

Hierarchies of modules

Loopy connections

How to get w? gradient-based learning

Cost function

Cost functions

Cost functions

Activation functions

Linear Units

Rectified Linear Unit (ReLU)

Rectified Linear Unit (ReLU)

Leaky ReLU

Exponential Linear Unit (ELU)

Gaussian Error Linear Unit (GELU)

Sigmoid and Tanh