Exercises

Section 5.9 Exercises

Exercise tags. The tags below identify the primary outcomes and competency types for each exercise group. Individual problems may also involve earlier outcomes.

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Subsection Directional derivatives and gradient descent

Primary outcomes. U5-LO2, U5-LO3. Competencies. P+C+T.

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OpenStax 4.6 #281
¹
openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient
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OpenStax 4.6 #283
²
openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient
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OpenStax 4.6 #285
³
openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient
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OpenStax 4.6 #297
⁴
openstax.org/books/calculus-volume-3/pages/4-6-directional-derivatives-and-the-gradient
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Subsection Symmetric matrices and quadratic forms

Primary outcomes. U5-LO4, U5-LO5. Competencies. P+C.

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Nicholson 8.2.5(b,d,f)
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math.libretexts.org/Bookshelves/Linear_Algebra/Linear_Algebra_with_Applications_(Nicholson)/08%3A_Orthogonality/8.02%3A_Orthogonal_Diagonalization/8.2E%3A_Orthogonal_Diagonalization_Exercises
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Subsection Hessians and the second derivative test

Primary outcomes. U5-LO1, U5-LO5. Competencies. P+C.

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OpenStax 4.7 #319
⁶
openstax.org/books/calculus-volume-3/pages/4-7-maxima-minima-problems
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OpenStax 4.7 #323
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openstax.org/books/calculus-volume-3/pages/4-7-maxima-minima-problems
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Subsection Additional applied and computational problems

These exercises connect the main Unit 5 ideas: critical points, directional derivatives, gradient descent, eigenvectors, Hessian eigenvalues, quadratic forms, least squares from gradients, fixed-hidden-layer training, and code interpretation. Solutions are collected in Appendix C.5.

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Checkpoint 5.9.1. Critical points are candidates.

Let

\begin{equation*} f(x,y)=x^2-y^2. \end{equation*}

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Find all critical points of \(f\text{.}\)
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Evaluate \(f(t,0)\) and \(f(0,t)\text{.}\)
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Explain why the critical point is not a local minimum or a local maximum.
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Tags. [U5-LO1, U5-LO5 | C+R | Core]

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Checkpoint 5.9.2. Directional derivative as a dot product.

Suppose

\begin{equation*} \nabla f(\mathbf{a})= \begin{bmatrix} 6\\ 8 \end{bmatrix}. \end{equation*}

Let

\begin{equation*} \mathbf{v}_1= \begin{bmatrix} 1\\ 0 \end{bmatrix}, \qquad \mathbf{v}_2= \begin{bmatrix} 0\\ 1 \end{bmatrix}, \qquad \mathbf{v}_3= \frac{1}{5} \begin{bmatrix} 3\\ 4 \end{bmatrix}, \qquad \mathbf{v}_4= -\frac{1}{5} \begin{bmatrix} 3\\ 4 \end{bmatrix}. \end{equation*}

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Compute \(D_{\mathbf{v}_i}f(\mathbf{a})\) for \(i=1,2,3,4\text{.}\)
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Which listed direction gives the largest directional derivative?
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Which listed direction gives the smallest directional derivative?
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Explain how this uses the Unit 1 dot product.
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Tags. [U5-LO2 | P+C+R | Core]

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Checkpoint 5.9.3. Reading a gradient descent loop.

Assume grad_f(x) computes \(\nabla f(\mathbf{x})\text{.}\) Consider the code:

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x = x0
for k in range(num_steps):
    x = x - alpha * grad_f(x)

What mathematical update rule is represented?
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Which quantity is the learning rate?
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What shape must grad_f(x) have?
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What would change if the minus sign were a plus sign?
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Does this code prove that a minimum has been found?
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Tags. [U5-LO2, U5-LO3 | C+T | Core]

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Checkpoint 5.9.4. Learning-rate diagnosis.

For \(f(x)=x^2\text{,}\) start at \(x_0=4\) and use

\begin{equation*} x_{k+1}=x_k-\alpha f'(x_k). \end{equation*}

The table shows the loss values \(f(x_k)\) for three learning rates.

\begin{equation*} \begin{array}{c|ccc} k \amp \alpha=0.05 \amp \alpha=0.20 \amp \alpha=1.05\\ \hline 0 \amp 16.000 \amp 16.000 \amp 16.000\\ 1 \amp 12.960 \amp 5.760 \amp 19.360\\ 2 \amp 10.498 \amp 2.074 \amp 23.426\\ 3 \amp 8.503 \amp 0.746 \amp 28.345 \end{array} \end{equation*}

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Which learning rate is making slow but steady progress?
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Which learning rate is making faster useful progress?
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Which learning rate appears unstable?
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Does a decreasing loss table prove that the global minimum has been found?
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Tags. [U5-LO3 | C+T | Core]

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Checkpoint 5.9.5. Eigenvectors as special directions.

Let

\begin{equation*} A= \begin{bmatrix} 3\amp1\\ 0\amp2 \end{bmatrix}, \qquad \mathbf{v}_1= \begin{bmatrix} 1\\ 0 \end{bmatrix}, \qquad \mathbf{v}_2= \begin{bmatrix} -1\\ 1 \end{bmatrix}. \end{equation*}

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Compute \(A\mathbf{v}_1\text{.}\)
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Compute \(A\mathbf{v}_2\text{.}\)
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Identify the eigenvalue for each eigenvector.
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Explain what the matrix map does to these two special directions.
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Tags. [U5-LO4 | P+C+R | Core]

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Checkpoint 5.9.6. Quadratic form and definiteness.

Let

\begin{equation*} H= \begin{bmatrix} 4\amp0\\ 0\amp1 \end{bmatrix}. \end{equation*}

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Compute \(\mathbf{h}^T H\mathbf{h}\) for

\begin{equation*} \mathbf{h}= \begin{bmatrix} 1\\ 0 \end{bmatrix}, \qquad \mathbf{h}= \begin{bmatrix} 0\\ 1 \end{bmatrix}, \qquad \mathbf{h}= \begin{bmatrix} 1\\ 1 \end{bmatrix}. \end{equation*}

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Write \(\mathbf{h}^T H\mathbf{h}\) for a general vector

\begin{equation*} \mathbf{h}= \begin{bmatrix} h_1\\ h_2 \end{bmatrix}. \end{equation*}

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Is \(H\) positive definite, negative definite, or indefinite?
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If \(H\) is the Hessian at a critical point, what does the second derivative test conclude?
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Tags. [U5-LO4, U5-LO5 | P+C | Core]

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Checkpoint 5.9.7. Hessian eigenvalue classification.

Suppose \(\mathbf{a}\) is a critical point of a scalar-valued function \(f\text{.}\) For each possible list of Hessian eigenvalues, classify the critical point as a local minimum, local maximum, saddle point, or inconclusive.

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\(\displaystyle 3,5\)
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\(\displaystyle -2,-7\)
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\(\displaystyle 4,-1\)
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\(\displaystyle 0,2\)
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Explain why the zero eigenvalue case is different.
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Tags. [U5-LO5 | C+R | Core]

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Checkpoint 5.9.8. Least squares from gradients.

Let

\begin{equation*} A= \begin{bmatrix} 1\amp0\\ 1\amp1\\ 1\amp2 \end{bmatrix}, \qquad \mathbf{b}= \begin{bmatrix} 1\\ 2\\ 2 \end{bmatrix}. \end{equation*}

Define

\begin{equation*} h(\mathbf{x})=\|A\mathbf{x}-\mathbf{b}\|^2. \end{equation*}

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Compute \(A^T A\) and \(A^T\mathbf{b}\text{.}\)
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Use \(\nabla h(\mathbf{x})=2A^T(A\mathbf{x}-\mathbf{b})\) to write the normal equations.
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Solve the normal equations.
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Compute the residual \(\mathbf{r}=\mathbf{b}-A\hat{\mathbf{x}}\text{.}\)
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Check that \(A^T\mathbf{r}=\mathbf{0}\text{.}\)
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Tags. [U5-LO6, U4-LO4 | P+C+R | Core]

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Checkpoint 5.9.9. Residual orthogonality in code.

Consider the code:

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xhat = np.linalg.lstsq(A, b, rcond=None)[0]
r = b - A @ xhat
A.T @ r

What is xhat?
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What is r?
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What condition is checked by A.T @ r?
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Does A.T @ r being close to zero mean that r is close to zero?
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Which earlier unit used this same condition?
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Tags. [U5-LO6, U4-LO4 | C+T | Core]

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Checkpoint 5.9.10. Fixed-hidden-layer design matrix.

Let \(\sigma(t)=\tanh(t)\text{,}\) and define

\begin{equation*} N_{\mathbf{c}}(t)=c_0+c_1\sigma(t)+c_2\sigma(t-1)+c_3\sigma(t+1). \end{equation*}

Use the input values \(t=-1,0,1\text{.}\)

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Write the design matrix \(A\) for the model \(A\mathbf{c}\approx \mathbf{y}\text{.}\)
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Which vector is trained?
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Why is this a least-squares problem?
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Is the model linear as a function of \(t\text{?}\)
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Is the model linear as a function of \(\mathbf{c}\text{?}\)
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Tags. [U5-LO6, U5-LO7 | C+M+R | Core]

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Checkpoint 5.9.11. Rank-one update.

Let

\begin{equation*} \mathbf{g}= \begin{bmatrix} 1\\ -2\\ 3 \end{bmatrix}, \qquad \mathbf{h}= \begin{bmatrix} 4\\ 0 \end{bmatrix}. \end{equation*}

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Compute \(\mathbf{g}\mathbf{h}^T\text{.}\)
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What is the shape of \(\mathbf{g}\mathbf{h}^T\text{?}\)
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Explain why \(\mathbf{g}\mathbf{h}^T\) has rank at most one.
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Interpret the update \(W_{\mathrm{new}}=W-\alpha \mathbf{g}\mathbf{h}^T\text{.}\)
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In NumPy, why is np.outer(g, h) appropriate when g and h are stored as one-dimensional arrays?
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Tags. [U5-LO7, U2-LO3 | P+C+T | Core]

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