### 2021　Theory of Statistical Mathematics

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Graduate major in Mathematical and Computing Science
Instructor(s)
Kanamori Takafumi
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Tue7-8()  Fri7-8()
Group
-
Course number
MCS.T507
Credits
2
2021
Offered quarter
1Q
Syllabus updated
2021/3/19
Lecture notes updated
-
Language used
English
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### Course description and aims

Some advanced topics and theories related to statistics and machine learning are taught. More specifically, a nonparametric method called kernel method, statistical properties of training and prediction errors, prediction error bound using Rademacher complexity, are taught.

### Student learning outcomes

[Objectives] Statistical science and machine learning are disciplines in which useful information is extracted from data to aid
prediction and decision making. Students will learn methodology not simply as knowledge but also learning the background theory including the validity of those methods to promote understanding the essence. Students will broadly apply all kinds of techniques to a variety of problems, learning to construct new techniques on one's own.
[Topics] Students in this course will learn several of statistical science's more advanced techniques, based on their connection to various application fields. We will focus in particular on the connection with machine learning, introducing central topics from both statistical science and machine learning.

### Keywords

machine learning, statistics, kernel methods, prediction error, Rademacher complexity, statistical consistency

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills ✔ Critical thinking skills Practical and/or problem-solving skills

### Class flow

Online Lectures using slides.

### Course schedule/Required learning

Course schedule Required learning
Class 1 Introduction Introduction of Statistical Mathematics. Understand the problem setup of machine learning through some practical examples.
Class 2 Regression Analysis Understand the statistical modeling with kernels, regularization, and kernel ridge regressions in regression analysis.
Class 3 Kernel methods I: positive definite kernels Review positive definite matrix. Understand the definition of positive definite kernels, and learn some properties of kernel functions and examples.
Class 4 Kernel methods II: reproducing property, representer theorem, etc. Ｌｅａｒ　ｔｈｅ　ｓｔａｔｉｓｔｉｃａｌ　ｍｏｄｅｌｉｎｇ　ｗｉｔｈ　ｒｅｐｒｏｄｕｃｉｎｇ　ｋｅｒｎｅｌ　Ｈｉｌｂｅｒｔ　ｓｐａｃｅ．
Class 5 Spline smoothing and kernel methods Learn the relationship between spline smoothing methods and kernel methods.
Class 6 Spline smoothing and kernel methods II Learn B-spline and multi-dimensional spline regression
Class 7 Classification analysis and kernel methods Learn kernel-based support vector machine for classification problems
Class 8 Review of Probability theory Review the probability theory used in Machine Leaning
Class 9 Inequalities in Probability Theory. Understand some probabilistic inequalities used in machine learning.
Class 10 Problem setup of statistical learning theory Understand the problem setup of statistical learning theory. Learn the concepts of hypothesis class, training errors, prediction errors, Bayes errors and Bayes rules.
Class 11 Prediction error and Model Selection Learn the prediction error of statistical learning and model selection methods.
Class 12 Rademacher Complexity Learn Rademacher complexity to measures the statistical models
Class 13 Uniform law of Large Numbers and Statistical Consistency Learning Algorithms Learn uniform law of large numbers (ULLN) that is an extension of the law of large numbers. Understand the proof of statistical consistency of learning algorithms.
Class 14 Summary Summarize Stasistical Learning

### Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Unspecified.

### Reference books, course materials, etc.

Course materials are provided during class.
Reference book: Shai Shalev-Shwartz and Shai Ben-David, Understanding Machine Learning: From Theory to Algorithms, Cambridge University Press, 2014.

### Assessment criteria and methods

Evaluated by report submission.

### Related courses

• MCS.T223 ： Mathematical Statistics
• MCS.T402 ： Mathematical Optimization: Theory and Algorithms
• MCS.T403 ： Statistical Learning Theory

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

It is preferred that students know the basics of statistics and probability theory.