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

Which is the global optimal solution of the function $f(x) = \frac{1}{x} + \sqrt{x}$?

Which is the global optimal solution 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 stretchy="false">)</mo><mo>=</mo><mfrac><mn>1</mn><mi>x</mi></mfrac><mo>+</mo><msqrt><mi>x</mi></msqrt></mrow><annotation encoding="application/x-tex">f(x) = \frac{1}{x} + \sqrt{x}</annotation></semantics></math>f(x)=x1​+x<svg xmlns="http://www.w3.org/2000/svg" width='400em' height='1.08em' viewBox='0 0 400000 1080' preserveAspectRatio='xMinYMin slice'><path d='M95,702
c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14
c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54
c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10
s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429
c69,-144,104.5,-217.7,106.5,-221
l0 -0
c5.3,-9.3,12,-14,20,-14
H400000v40H845.2724
s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7
c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z
M834 80h400000v40h-400000z'/></svg>​?

**(Select all that apply.)** Which functions are convex functions? 

(Select all that apply.) Which functions are convex functions?

Which is the global optimal solution of the function $f(x) = -0.3
\log(1-x) - 0.7\log(x)$?

Which is the global optimal solution 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 stretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>0.3</mn><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>x</mi><mo stretchy="false">)</mo><mo>−</mo><mn>0.7</mn><mi>log</mi><mo>⁡</mo><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(x) = -0.3
\log(1-x) - 0.7\log(x)</annotation></semantics></math>f(x)=−0.3log(1−x)−0.7log(x)?

**(Select all that apply.)** Which options are a valid subgradient of the function $f(x) = |x-2| + |0.5x-5| + |x+1|$ at $x = 2$? 

(Select all that apply.) Which options are a valid subgradient 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 stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mo>−</mo><mn>2</mn><mi mathvariant="normal">∣</mi><mo>+</mo><mi mathvariant="normal">∣</mi><mn>0.5</mn><mi>x</mi><mo>−</mo><mn>5</mn><mi mathvariant="normal">∣</mi><mo>+</mo><mi mathvariant="normal">∣</mi><mi>x</mi><mo>+</mo><mn>1</mn><mi mathvariant="normal">∣</mi></mrow><annotation encoding="application/x-tex">f(x) = |x-2| + |0.5x-5| + |x+1|</annotation></semantics></math>f(x)=∣x−2∣+∣0.5x−5∣+∣x+1∣ at <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">x = 2</annotation></semantics></math>x=2?

$$\theta = \frac{1}{n} \sum_{i=1}^n x_i$$

$$\theta = \frac{1}{n} \sum_{i=1}^n x^2_i$$

$$\theta = \frac{1}{n} \sum_{i=1}^n \sqrt{x_i}$$

Given the observed data points $x_1, x_2, ..., x_n$, which is the correct maximum likelihood estimate $\theta^*$ of the following distribution?

$$p(x, \theta) = \frac{\theta^x e^{-\theta}}{x!}$$

Given the observed data points <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub></mrow><annotation encoding="application/x-tex">x_1, x_2, ..., x_n</annotation></semantics></math>x1​,x2​,...,xn​, which is the correct maximum likelihood estimate <math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>θ</mi><mo>∗</mo></msup></mrow><annotation encoding="application/x-tex">\theta^*</annotation></semantics></math>θ∗ of the following distribution?
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>θ</mi><mo stretchy="false">)</mo><mo>=</mo><mfrac><mrow><msup><mi>θ</mi><mi>x</mi></msup><msup><mi>e</mi><mrow><mo>−</mo><mi>θ</mi></mrow></msup></mrow><mrow><mi>x</mi><mo stretchy="false">!</mo></mrow></mfrac></mrow><annotation encoding="application/x-tex">p(x, \theta) = \frac{\theta^x e^{-\theta}}{x!}
</annotation></semantics></math>p(x,θ)=x!θxe−θ​

Evaluate your convex optimization concepts via an MCQ quiz.

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 Convex Optimization