Learn optimization for machine learning, including gradients, convex optimization, and gradient descent. Explore constrained optimization and advanced methods using NumPy and SciPy.

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In this course, you will learn about optimization, one of the fundamental pillars of mathematics and machine learning. Machine learning depends heavily on optimization because it allows the model to learn from data and generate precise predictions.

You will begin by introducing optimization. Then, you will learn about optimization basics, including gradients and integrals. Next, you will cover convex optimization. You will then learn how to compute gradient descent for non-convex optimization. Next, you’ll learn how to perform constrained optimization. You will finish the course by studying the miscellaneous methods of optimization, like Newton’s methods, quasi-Newton methods, and conjugate gradient descent.

After completing this course, you’ll have the practical skills to formulate, analyze, and implement optimization algorithms for machine learning using the NumPy and SciPy libraries. This will help you become a highly proficient data scientist or machine learning engineer.

Optimization for Machine Learning with NumPy and SciPy

$-\cos(\cos(log(x)))\sin(\log(x))\frac{1}{x}$

$\cos(\cos(log(x)))\sin(\log(x))\frac{1}{x}$

$-\cos(\cos(log(x)))\cos(\log(x))\frac{1}{x}$

Which is the correct derivative of $f(x) = \sin(cos(\log(x)))$ with respect to $x$?

Which is the correct derivative of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>c</mi><mi>o</mi><mi>s</mi><mo stretchy="false">(</mo><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = \sin(cos(\log(x)))</annotation></semantics></math>f(x)=sin(cos(log(x))) with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>x?

$\nabla f(x,y) = [y\cdot x^{y-1}, x^y \ln(x)]$

$\nabla f(x,y) = [x^y \ln(x), y\cdot x^{y-1}]$

$\nabla f(x,y) = [y\cdot x^{y-1}, x^{y-1}\ln(x)]$

$\nabla f(x,y) = [x^y \ln(x), x^y \ln(y)]$

Which is the correct gradient of $f(x, y) = x^y$ with respect to $(x,y)$?

Which is the correct gradient of <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>x</mi><mi>y</mi></msup></mrow><annotation encoding="application/x-tex">f(x, y) = x^y</annotation></semantics></math>f(x,y)=xy with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math>(x,y)?

$\nabla f(x, y) = \begin{bmatrix} y \cdot e^{xy} & x \cdot e^{xy} \\ y \cdot \cos(xy) & x \cdot \cos(xy) \end{bmatrix}$

$\nabla f(x, y) = \begin{bmatrix} y \cdot e^{xy} & y \cdot \sin(xy) \\ x \cdot e^{xy}  & x \cdot \sin(xy) \end{bmatrix}$

$\nabla f(x, y) = \begin{bmatrix} y \cdot e^{xy} & y \cdot \cos(xy) \\ x \cdot e^{xy}  & x \cdot \cos(xy) \end{bmatrix}$

Which is the correct gradient of the following vector-valued function $f(x, y) = [e^{xy}, \sin(xy)]$ with respect to $(x,y)$?

Which is the correct gradient of the following vector-valued function <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">[</mo><msup><mi>e</mi><mrow><mi>x</mi><mi>y</mi></mrow></msup><mo separator="true">,</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">f(x, y) = [e^{xy}, \sin(xy)]</annotation></semantics></math>f(x,y)=[exy,sin(xy)] with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math>(x,y)?

$H = \begin{bmatrix} -\sin(x) & 0 \\ 0 & -\cos(x)\end{bmatrix}$

$H = \begin{bmatrix} -\cos(x) & 0 \\ 0 & -\sin(x)\end{bmatrix}$

$H = \begin{bmatrix} 0  &-\sin(x) \\ -\cos(x) & 0\end{bmatrix}$

$H = \begin{bmatrix} 0  &-\cos(x) \\ -\sin(x) & 0\end{bmatrix}$

Which is the correct Hessian of the function $f(x, y) = \sin(x) + \cos(y)$ with respect to $(x,y)$?

Which is the correct Hessian of the function <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>sin</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x, y) = \sin(x) + \cos(y)</annotation></semantics></math>f(x,y)=sin(x)+cos(y) with respect to <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x,y)</annotation></semantics></math>(x,y)?

$(p \times q \times r) \times (m \times n)$

$(m \times n) \times (p \times q \times r)$

$(m \times p) \times (n \times q \times r)$

$(n \times m) \times (r \times q \times p)$

Which is the correct shape of the Jacobian of the following function $f: \R^{m \times n} \rightarrow \R^{p \times q \times r}$ ?

Which is the correct shape of the Jacobian of the following function <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo>:</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>m</mi><mo>×</mo><mi>n</mi></mrow></msup><mo>→</mo><msup><mi mathvariant="double-struck">R</mi><mrow><mi>p</mi><mo>×</mo><mi>q</mi><mo>×</mo><mi>r</mi></mrow></msup></mrow><annotation encoding="application/x-tex">f: \R^{m \times n} \rightarrow \R^{p \times q \times r}</annotation></semantics></math>f:Rm×n→Rp×q×r ?

$f(x,y) = \cos(8) - 4\sin(8)(x-2) -8\sin(8)(y-2)$

$f(x,y) = \cos(8) - 8\sin(8)(x-2) -4\sin(8)(y-2)$

$f(x,y) = \cos(8) + 4\sin(8)(x-2) +8\sin(8)(y-2)$

$f(x,y) = \cos(8) - 8\sin(8)(x-2) -8\sin(8)(y-2)$

Which is the correct linear approximation of the function $f(x,y) = \cos(xy^2)$ around the point $(x_0, y_0) = (2,2)$ according to the Taylor expansion?

Which is the correct linear approximation of the function <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>cos</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mi>y</mi><mn>2</mn></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x,y) = \cos(xy^2)</annotation></semantics></math>f(x,y)=cos(xy2) around the point <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>x</mi><mn>0</mn></msub><mo separator="true">,</mo><msub><mi>y</mi><mn>0</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mo separator="true">,</mo><mn>2</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x_0, y_0) = (2,2)</annotation></semantics></math>(x0​,y0​)=(2,2) according to the Taylor expansion?

Evaluate yourself on the concepts learned throughout this chapter.

Introduction to Optimization

Vector Calculus

Convex Optimization

Gradient Descent for Non-Convex Optimization

Use Particle Swarm Optimizer to Optimize a Non-convex Function

Constrained Optimization

Miscellaneous Methods

Course Conclusion

Test Your Concepts of Optimization

Training Support Vector Machines (SVMs)

Quiz Yourself on Vector Calculus