3 - The Bottom of the Bowl Flashcards
Who was Bernard Widrow?
A young academic at Stanford University in the autumn of 1959.
What was the focus of Widrow’s work?
Adaptive filters and the use of calculus to optimize them.
Who is Marcian ‘Ted’ Hoff?
A graduate student who approached Widrow for discussion.
What significant algorithm did Widrow and Hoff invent?
The least mean squares (LMS) algorithm.
What is the LMS algorithm foundational for?
Training artificial neural networks.
Where did Widrow grow up?
A small town in Connecticut.
What did Widrow’s father do for a living?
Ran an ice-manufacturing plant.
What did Widrow initially want to be when he grew up?
An electrician.
What subtle course correction did Widrow’s father suggest?
To become an electrical engineer instead of an electrician.
Where did Widrow obtain his degrees?
MIT.
What workshop did Widrow attend in the summer of 1956?
A workshop on artificial intelligence at Dartmouth College.
Who is credited with coining the term ‘artificial intelligence’?
John McCarthy.
What was the main goal of the Dartmouth Summer Research Project?
To explore how machines can simulate aspects of learning and intelligence.
What did Widrow conclude after six months of thinking about thinking?
It would take twenty-five years to build a thinking machine with the technology of that time.
What did Widrow turn his attention to after abandoning plans for a thinking machine?
Adaptive filters that could learn to remove noise from signals.
Who developed the theory that Widrow was particularly interested in?
Norbert Wiener.
What is the goal of an adaptive filter?
To learn from its mistakes and improve over time.
What does the mean squared error (MSE) measure?
The average of the squares of the errors made by the filter.
What mathematical method is used to minimize the mean squared error?
The method of steepest descent.
What does the term ‘gradient’ refer to in calculus?
The slope of a function at a given point.
What is the derivative of the function y = x^2?
2x.
What is the purpose of differential calculus?
To calculate the slope of a continuous function.
At what point is the slope of a function typically zero?
At the minimum of the function.
What is the method of steepest descent also known as?
The method of gradient descent.
What must be calculated to take a step toward the minimum of a curve?
The slope or gradient at the current location.
What is the significance of the step size in the gradient descent method?
It must be small to avoid overshooting the minimum.
True or False: The steps in gradient descent become larger as you approach the minimum.
False.
What happens to the step size during gradient descent as you approach the minimum?
The jumps along the curve become smaller as you near the bottom.
This is because the gradient is getting smaller.
What type of functions have a single, well-defined minimum?
Convex functions.
The global minimum is the bottom of the bowl-shaped graph.
What is a saddle point in the context of optimization?
An unstable point where the gradient is zero but is not a minimum.
The function does not have a global or local minimum at a saddle point.
What is the gradient in a multi-variable function?
A vector composed of partial derivatives with respect to each variable.
The gradient points away from the minimum.
How do you calculate the gradient for a function with multiple variables?
By taking partial derivatives of the function with respect to each variable.
The notation used includes ∂ for partial derivatives.
What is the significance of the gradient vector in optimization?
It indicates the direction of steepest descent.
To move towards the minimum, one must follow the negative of the gradient.
What is an adaptive filter in signal processing?
A filter that adjusts its parameters to minimize the error between the desired and actual output signals.
It is essential in applications like digital communications.
What equation describes the error in an adaptive filter?
en = dn - yn.
Here, dn is the desired signal and yn is the output signal.
What is the function of the adaptive filter during a modem handshake?
It learns the characteristics of noise to create an error-free communication channel.
This is crucial for digital devices transmitting over noisy analog lines.
What does it mean for a function to be differentiable?
It means that the function can be differentiated to find its derivatives.
Differentiability allows for the calculation of gradients.
What does the notation ‘z = x^2 + y^2’ represent in optimization?
An elliptic paraboloid surface in three-dimensional space.
This represents a function with two variables.
What is the role of partial derivatives in finding the gradient?
They provide the components of the gradient vector.
Each component corresponds to a variable’s contribution to the slope.
How can you express the output of an adaptive filter mathematically?
yn = w.xn, where w is the weight vector and xn is the input vector.
This represents the linear combination of inputs adjusted by weights.
What field of study does multi-variate calculus belong to?
Calculus involving functions with multiple variables.
It is essential for understanding gradient descent in machine learning.
What is the primary purpose of an adaptive filter?
To adapt to varying noise conditions and minimize output error.
This is crucial for maintaining signal integrity in communication systems.
What happens when you start from a different location while descending a gradient?
You may veer away from the saddle point.
The starting point can dictate the convergence path in optimization.
What is the relationship between functions and vectors in optimization?
The gradient is a vector derived from the function’s partial derivatives.
This illustrates the interplay between different mathematical domains.
What is the formula for the output of an adaptive filter?
yn = w.xn
Where xn = [ xn, xn1 , …] and w = [ w0, w1, …]
What is the expression for the error made by the filter at the n-th time step?
en = dn - yn
This can be rewritten as en = dn - w.xn
What is the goal of an adaptive filter?
To minimize the error between the generated output and the desired signal
How do we calculate the average error in adaptive filtering?
Using mean absolute error (MAE) or mean squared error (MSE)
MSE is preferred due to its statistical properties and differentiability
What is the mathematical representation of the value to be minimized in adaptive filtering?
J = E (( dn - yn )^2)
This represents the expected value of the squared errors.
What type of function is formed when relating J to the filter parameter w?
A quadratic function
What method can be used to minimize J if the correlation between inputs and outputs is unknown?
Method of steepest descent
What does stochastic gradient descent (SGD) refer to?
A method where the direction of each step in descent is slightly random
What is the output of an adaptive neuron designed by Widrow and Hoff?
y = w0 x0 + w1 x1 + w2 x2
What does the term ‘bias’ refer to in the context of adaptive neurons?
w0, which is the coefficient associated with input x0 set to 1
What is the update rule for the weights in the LMS algorithm?
w new = w old + 2 με x
Where μ = step size, ε = error, and x = vector of a single data point.
What is the error in the context of an adaptive neuron?
ε = d - w^T x
Where d is the desired output.
What is the significance of the LMS algorithm?
It is widely used in adaptive filters and is the first algorithm for training artificial neurons using gradient descent principles
What was the original context in which Widrow and Hoff discovered their algorithm?
They were working on adaptive filters and neural elements at Stanford
What was the result of running the algorithm on the analog computer?
It verified that the algorithm worked
What was the first task after confirming the algorithm worked?
Building a single adaptive neuron
What is the problem with calculating the optimal values for filter parameters?
It requires more samples of input and desired output, making calculations time-consuming
What does the gradient represent in the context of optimization?
A vector of partial derivatives of the mean squared error with respect to each weight
What is the nature of the function representing the expectation value of squared errors?
Bowl-shaped function in higher-dimensional space
What is the method used to estimate the gradient without full calculations?
Using an estimate based on just one data point
Who were the key figures behind the development of the LMS algorithm?
Widrow and Hoff
What is ADALINE?
ADALINE stands for ‘adaptive linear neuron’ and is an adaptive neuron that learns to be a good neuron.
What algorithm does ADALINE use?
ADALINE uses the LMS algorithm.
What does the LMS algorithm do in the context of ADALINE?
It separates an input space into two regions, helping to find the weights that represent the linearly separating hyperplane.
What are the dimensions of the input space used for representing letters in ADALINE?
The input space is a 16-dimensional space defined by 4×4 pixels.
How are letters represented in the 4×4 pixel space?
Each letter is represented by 16 binary digits, which can be either 0 or 1.
What is the main difference between ADALINE and Rosenblatt’s perceptron?
ADALINE uses the LMS algorithm while the perceptron uses a different algorithm to find the linearly separating hyperplane.
What did Widrow discover about the LMS algorithm while waiting for a flight?
He discovered that the LMS algorithm is an unbiased estimate and that taking extremely small steps leads to the optimal value for the weights.
What are the two types of neural architectures mentioned in the text?
- ADALINE (single layer of adaptive neurons)
- MADALINE (multiple layers: input, hidden, output)
What was the challenge with training MADALINE?
It was hard to train MADALINE.
What is the significance of the LMS algorithm in relation to backpropagation?
The LMS algorithm is the foundation of backpropagation, which is essential for modern AI.
What key role did Hoff play in the development of Intel?
Hoff was one of the key people behind the development of the Intel 4004, the company’s first general-purpose microprocessor.
What was the title of the 1963 episode of Science in Action featuring MADALINE?
‘Computers that Learn.’
What was the public perception of MADALINE as described in the Science in Action episode?
It was described as a machine that can learn to balance a broom, which was presented as a remarkable feat.
Who were the two key figures mentioned in laying the foundation for modern deep neural networks?
- Frank Rosenblatt
- Bernard Widrow
True or False: The assessment of neural network limitations by Minsky and Papert greatly affected research in the field.
True
Fill in the blank: The LMS algorithm helps ADALINE find the weights representing the _______.
linearly separating hyperplane
What was Widrow’s response to the question about the name ‘ADALINE’?
He stated that it spells ‘Adaptive Linear Neuron.’
