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Questions and Answers
What does 𝑎. represent in the given notation?
What does 𝑎. represent in the given notation?
In the expression 𝑎.= 𝑓 𝑥; 𝜃, what does the notation ℎ.∘ ℎ.2$ ∘ ⋯ ∘ ℎ$ signify?
In the expression 𝑎.= 𝑓 𝑥; 𝜃, what does the notation ℎ.∘ ℎ.2$ ∘ ⋯ ∘ ℎ$ signify?
What is implied by the term 'Feedforward architecture' in this context?
What is implied by the term 'Feedforward architecture' in this context?
Which parameter is involved in the l-th layer of the neural network architecture?
Which parameter is involved in the l-th layer of the neural network architecture?
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What do the functions ℎ3 represent in the context of the neural network?
What do the functions ℎ3 represent in the context of the neural network?
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What is often more beneficial than using a stronger model with fewer training iterations?
What is often more beneficial than using a stronger model with fewer training iterations?
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What characteristic of ReLUs contributes to their state-of-the-art performance?
What characteristic of ReLUs contributes to their state-of-the-art performance?
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What is the main constraint when utilizing model parameters in deep learning?
What is the main constraint when utilizing model parameters in deep learning?
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Which statement is true regarding model complexity and hierarchies in deep learning?
Which statement is true regarding model complexity and hierarchies in deep learning?
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What trade-off is highlighted in the context of data efficient modules?
What trade-off is highlighted in the context of data efficient modules?
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What is the purpose of the loss or cost function in gradient-based learning?
What is the purpose of the loss or cost function in gradient-based learning?
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What does the notation $w^* = ext{arg max} , p(y|x; w)$ indicate in the context of maximum likelihood estimation?
What does the notation $w^* = ext{arg max} , p(y|x; w)$ indicate in the context of maximum likelihood estimation?
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Which statement best describes the output from the last layer, $p(y|x)$, in a neural network?
Which statement best describes the output from the last layer, $p(y|x)$, in a neural network?
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What is often the basis for the cost function used in training deep learning models?
What is often the basis for the cost function used in training deep learning models?
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What is the primary goal of maximum likelihood estimation in this context?
What is the primary goal of maximum likelihood estimation in this context?
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What is the primary purpose of the negative log-likelihood in relation to activation functions?
What is the primary purpose of the negative log-likelihood in relation to activation functions?
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How can activation functions be described when they limit the output range?
How can activation functions be described when they limit the output range?
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What effect does the choice of activation function have on a neural network?
What effect does the choice of activation function have on a neural network?
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Which statement accurately reflects a consequence of activation function saturation?
Which statement accurately reflects a consequence of activation function saturation?
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What role do squashing functions play in a neural network?
What role do squashing functions play in a neural network?
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What is the characteristic of linear models mentioned?
What is the characteristic of linear models mentioned?
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In the context of extending to nonlinear models, what is the role of φ?
In the context of extending to nonlinear models, what is the role of φ?
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What is a feature of the kernel trick mentioned?
What is a feature of the kernel trick mentioned?
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What does the notation y = f(x; θ, w) represent in the context of deep learning?
What does the notation y = f(x; θ, w) represent in the context of deep learning?
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Which of the following is NOT a characteristic of logistic and linear regression models?
Which of the following is NOT a characteristic of logistic and linear regression models?
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What is indicated by the term 'hidden layer' in the context of deep learning?
What is indicated by the term 'hidden layer' in the context of deep learning?
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Which technique is used for nonlinear dimension reduction?
Which technique is used for nonlinear dimension reduction?
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Why is it important to find the correct parameters θ?
Why is it important to find the correct parameters θ?
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Which statement is true about the capacity of linear models?
Which statement is true about the capacity of linear models?
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What does applying the linear model to transformed input φ(x) achieve?
What does applying the linear model to transformed input φ(x) achieve?
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What is the role of the scaling factor in the Gaussian distribution?
What is the role of the scaling factor in the Gaussian distribution?
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Why can the constant term in the Gaussian distribution be discarded?
Why can the constant term in the Gaussian distribution be discarded?
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What does the equivalence between maximum likelihood estimation and mean squared error imply?
What does the equivalence between maximum likelihood estimation and mean squared error imply?
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Which statement best describes the relationship between the Gaussian distribution and the parameter θ?
Which statement best describes the relationship between the Gaussian distribution and the parameter θ?
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In the context of predictive modeling using Gaussian distributions, what is a crucial observation?
In the context of predictive modeling using Gaussian distributions, what is a crucial observation?
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What implications does not needing to parametrize the Gaussian distribution have on modeling?
What implications does not needing to parametrize the Gaussian distribution have on modeling?
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Which aspect of the Gaussian distribution remains constant regardless of the functions used for prediction?
Which aspect of the Gaussian distribution remains constant regardless of the functions used for prediction?
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What does minimizing mean squared error in relation to the Gaussian distribution achieve?
What does minimizing mean squared error in relation to the Gaussian distribution achieve?
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Study Notes
Lecture 2: Deep Feedforward Networks
- Deep learning uses modular components
- Deep learning uses nonlinearities to generate complex mappings
- Deep learning utilizes gradient-based learning methods
- Backpropagation utilizes the chain rule for efficient computation
Lecture Overview
- Deep learning involves modularity
- Deep learning uses nonlinearities
- Gradient-based learning is a core technique
- The chain rule is fundamental
- Backpropagation is a key technique
Last Time
- Neural networks transform inputs to outputs
- The input is weighted and summed
- This sum triggers an activation function
- The output is a result of an activation function
From Linear Functions to Nonlinear Functions
- Linear functions (f=ReLU(Ax)) are based on matrix multiplication and the ReLU function
- Non-linear architectures are required for complex mappings
How Deep Neural Networks Do It
- Deep networks utilize multiple layers
- ReLU (Rectified Linear Unit): a non-linear activation function
- ReLU(x) = max(0, x)
- For example ReLU(3) = 3
- For example ReLU(-3) = 0
Deep Feedforward Networks
- Feedforward neural networks are a type of neural network architecture
- They are also called multi-layer perceptrons (MLPs)
- The goal is to approximate a function, f
- The network defines a mapping y=f(x; θ)
- The network learns parameters to best approximate a function
- No feedback connections
Deep Feedforward Networks as a Composite Function
- A deep network is a series of composable functions
- y=f(x; θ) = h₁∘h₂∘…∘h_(l)∘x (where θₗ is the parameter for the l-th layer)
Neural Networks in Blocks
- Feedforward networks can be visualized as blocks
- Inputs flow through hidden layers to produce an output
- Each hidden layer transforms the input from the previous layer
What is a Module?
- A module is a building block for function transformation
- It takes input data or the output of other modules
- Modules use an activation function with or without trainable parameters (w).
- Examples: f = Ax, f = exp(x)
Requirements
- Activation functions need to be differentiable in most contexts
- Be careful with cycles
Feedforward Model
- The majority of models are feedforward
- Almost all CNNs/Transformers
- Very simple architecture
Non-Linear Feature Learning Perspective
- Linear models include logistic regression and linear regression
- They are convex with a closed-form solution
- They can be fitted reliably and efficiently
- They have limited capacity
Non-linear Feature Learning Perspective
- Deep learning aims to find good feature representations
- Deep learning involves finding the best parameters to achieve accurate feature representation from x
- No longer a convex training problem
- Design families for q(x; θ)
- Utilize human knowledge about the problem domain for better generalization
Directed Acyclic Graph Models
- Mix the network architectures to match the task domain
- This method makes sense for problems with multiple inputs or modalities (e.g., RGB and LIDAR)
- Interweaved & skip connections
Hierarchies of Modules
- Data-efficient modules and hierarchies are used for better model efficiency
- Trade-off between model complexity and efficiency, more training iterations with a 'weaker' model often better
- ReLUs are often the activation function of choice as they help train faster
- GPU memory is a practical constraint
- Modules need to be computed in the correct order
Loopy Connections
- Past outputs can affect future inputs
- Such cycles are common in recurrent networks
- Loops must be unfolded to train the model properly
How to get w? Gradient-Based Learning
- The non-linearity results in a non-convex loss function
- Optimizers are used with iterative gradient calculations for complex function mappings
Cost Function
- Maximum likelihood is common in training neural functions
- Taking the logarithm leads to minimizing negative log-likelihood, equivalent to cross-entropy
Cost Functions
- Euclidean loss suitable for regression problems
- Sensitive to outliers, magnifying errors quadratically
- Other functions cross-entropy, KL-divergence
Cost Functions
- Cost functions define what the model should learn
- The gradient must be sufficiently large and predictable
- Functions that saturate may become predictable with poor results
- The negative log-likelihood helps to avoid the problem of saturating functions due to output exponentiation (eg. softmax)
Deep Learning Modules
- Study of deep learning modules
Activation Functions
- Activation functions transform weighted input sums to outputs
- Outputs with a limited range are called "squashing" functions
- The activation function choice significantly impacts a network's performance and capabilities
- Often a single function is utilized for all layers in a given network
- Functions need to be differentiable at most points
- Linear and ReLU are commonly used
Linear Units
- Identity function with no activation function saturation
- Strong and stable gradients
- Reliable learning with modules
Rectified Linear Unit (ReLU)
- ReLU = max(0,x)
- Sparse activation
- Better gradient propagation
- Efficient computation (addition and multiplication)
- Scale invariant
Rectified Linear Unit (ReLU) Potential Problems
- Non-differentiable at zero,
- Not zero-centered
- Unbounded
Leaky ReLU
- Leaky ReLU is differentiable at zero.
- Allows a small, positive gradient when the unit isn't active
- Parametric ReLU includes a learnable parameter (a)
Exponential Linear Unit (ELU)
- A smooth approximation to the rectifier
- Non-monotonic (gradient change)
- Default activation function for many models such as BERT
Gaussian Error Linear Unit (GELU)
- Similar to ELU but not monotonic with a change in the gradient
- A default function for Vision Transformers and state of the art models
Sigmoid and Tanh
- Tanh (x) has output range [-1, +1]
- Data centered around zero
- Less positive bias
- Saturates at the extremes (resulting in zero gradients)
- Easy to become overconfident at the extreme values
- Gradients are less than optimal leading to problems with middle layers
Softmax
- Outputs probability distributions
- Normalizes output, avoids large or small values for better stability
- Useful at the output layer
How to Choose an Activation Function
- ReLU or GELU are typical choices for hidden layers
- Linear, sigmoid, and tanh are common output layers (regression, binary classification, multiclass classification)
- Choose appropriate functionality based on the type of classification task
New Modules
- Any function that is differentiable is a valid module
- Modules of modules can be effectively implemented
- They should be implemented as cascades of simple modules
Architecture Design
- The structure of a neural network and how units connect to each other
- Networks are composed of organized units, called layers
- The chain layer is how successive layers are utilized to map input into output layers
- Each layer's calculation utilizes a given function with specific network parameters
Width and Depth
- Universal approximation theorem
- Large MLPs can represent any function, providing adequate hidden units
- Deeper networks often generalize better and reduce the number of units required to accurately model a given function
Width and Depth
- The number of parameters in layers of convolutions without an increase of depth isn't the most effective strategy to model better results
- Deeper networks often generalize better and reduce the amount of generalization errors in learning
Deeper Networks: Hierarchical Pattern Recognition
- Deeper networks show a division of labor between layers
- Layers learn different features, resulting in a hierarchical pattern understanding of input data
A Neural Network Jungle
- Detailed list of neural network architectures (e.g., Perceptrons, MLPs, RNNs, LSTMs, GRUs, Autoencoders, Convolutional Nets, Transformers, Generative Adversarial Nets, Deep Residual Nets, Neural Turing Machines)
Intermezzo: Chain Rule
- The chain rule is a fundamental concept in calculus used to calculate derivatives of functions formed by composing other functions
- It's used in deep learning for finding gradients of the cost function by utilizing calculated derivative values
- Useful technique for complex non-linear functions and calculations
Computational Graph
- Shows operations to calculate outputs with nodes representing variables
- A simple calculation function or variable represents an operation or a node within the graph
Example
- Show example calculations and graphs utilizing the chain rule in derivative calculations
Example
- Shows example calculations with diagrams to highlight the importance of differentiability in neural networks
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Description
Test your understanding of deep learning architectures and key concepts. This quiz covers topics such as feedforward architecture, neural network layers, and activation functions. Perfect for students studying machine learning or computer science.
