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📗 (Fall 2017 Final Q7, Fall 2014 Midterm Q17, Fall 2013 Final Q10) We use gradient descent to find the minimum of the function \(f\left(x\right)\) = with step size \(\eta > 0\). If we start from the point \(x_{0}\) = , how small should \(\eta\) be so we make progress in each iteration? Check all values of \(\eta\) that make progress.
📗 Hint: the minimum is 0, so "making progress" means getting closer to 0 in at least the first iteration.
📗 Choices:
📗 (Fall 2017 Final Q15, Fall 2010 Final Q5) Let \(x = \left(x_{1}, x_{2}, x_{3}\right)\). We want to minimize the objective function \(f\left(x\right)\) = using gradient descent. Let the stepsize \(\eta\) = . If we start at the vector \(x^{\left(0\right)}\) = , what is the next vector \(x^{\left(1\right)}\) produced by gradient descent?
📗 Answer (comma separated vector): .
📗 (Fall 2017 Final Q23) Consider a rectified linear unit (ReLU) with input \(x\) and a bias term. The output can be written as \(y\) = . Here, the weight is and the bias is . Write down the input value \(x\) that produces a specific output \(y\) = .
📗 Answer: .
📗 (Spring 2017 Final Q3, Spring 2018 Final Q7) Consider a Linear Threshold Unit (LTU) perceptron with initial weights and bias . Given a new input and . Let the learning rate be , compute the updated weights, :
📗 Answer (comma separated vector): .
📗 (Fall 2016 Final Q15, Fall 2011 Midterm Q11) Let \(f\left(z\right) = \dfrac{1}{1 + \exp\left(-z\right)}, z = w^\top x = w_{1} x_{1} + w_{2} x_{2} + ... + w_{d} x_{d}\), \(d\) = be a sigmoid perceptron with inputs \(x_{1} = ... = x_{d}\) = and weights \(w_{1} = ... = w_{d}\) = . There is no constant bias input of one. If the desired output is \(y\) = , and the sigmoid perceptron update rule has a learning rate of , what will happen after one step of update? Each \(w_{i}\) will change by (enter a number):
📗 Hint: the change should be \(-\alpha \left(a - y\right) x\), do not forget the minus sign in front.
📗 Answer: .
📗 (Fall 2014 Final Q4) Which functions are (weakly) convex on \(\mathbb{R}\)?
📗 Hint: either plot the functions or find the ones with non-negative second derivative (i.e. positive semi-definite Hessian matrix in higher dimensions).
📗 Choices:
📗 (Fall 2012 Final Q8) Consider a single sigmoid perceptron with bias weight \(w_{0}\) = , a single input \(x_{1}\) with weight \(w_{1}\) = , and the sigmoid activation function \(g\left(z\right) = \dfrac{1}{1 + \exp\left(-z\right)}\). For what input \(x_{1}\) does the perceptron output value \(a\) = .
📗 Minor update to the notation: \(z = w_{0} + w_{1} x_{1}\) is the linear part, and \(a\) is the target output (or activation in the lectures).
📗 Answer: .
📗 (Fall 2011 Midterm Q10, Spring 2018 Final Q4) With a linear threshold unit perceptron, implement the following function. That is, you should write down the weights \(w_{0}, w_{A}, w_{B}\). Enter the bias first, then the weights on A and B.
📗 Play the "which face is real" game: Link a few times and enter your score as a percentage. Discuss how to distinguish real and fake faces on Piazza: Link.
📗 My accuracy is and I have participated in the discussion on Piazza.
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📗 Answer: .
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