Mathematics of Machine Learning
What is this talk about?
The main emphasis is to give you a short and pedestrian introduction to the basic mathematics of machine learning methods. And why this could (or should) be of interest.
A simple perspective on the interface between ML and Physics
AI/ML and some statements you may have heard (and what do they mean?)
- Fei-Fei Li on ImageNet: map out the entire world of objects (The data that transformed AI research)
- Russell and Norvig in their popular textbook: relevant to any intellectual task; it is truly a universal field (Artificial Intelligence, A modern approach)
- Woody Bledsoe puts it more bluntly: in the long run, AI is the only science (quoted in Pamilla McCorduck, Machines who think)
If you wish to have a critical read on AI/ML from a societal point of view, see Kate Crawford's recent text Atlas of AI
Here: with AI/ML we intend a collection of machine learning methods with an emphasis on statistical learning and data analysisScientific Machine Learning
An important and emerging field is what has been dubbed as scientific ML, see the article by Deiana et al Applications and Techniques for Fast Machine Learning in Science, arXiv:2110.13041
The authors discuss applications and techniques for fast machine learning (ML) in science – the concept of integrating power ML methods into the real-time experimental data processing loop to accelerate scientific discovery. The report covers three main areas
- applications for fast ML across a number of scientific domains;
- techniques for training and implementing performant and resource-efficient ML algorithms;
- and computing architectures, platforms, and technologies for deploying these algorithms.
Machine Learning software
Machine learning is an extremely rich field, in spite of its young age. The increases we have seen during the last three decades in computational capabilities have been followed by developments of methods and techniques for analyzing and handling large date sets, relying heavily on statistics, computer science and mathematics. The field is rather new and developing rapidly.
Popular software packages written in Python for ML are
and more. These are all freely available at their respective GitHub sites. They encompass communities of developers in the thousands or more. And the number of code developers and contributors keeps increasing.
Lots of room for creativity
Not all the algorithms and methods can be given a rigorous mathematical justification, opening up thereby for experimenting and trial and error and thereby exciting new developments.
A solid command of linear algebra, multivariate theory, probability theory, statistical data analysis, optimization algorithms, understanding errors and Monte Carlo methods is important in order to understand many of the various algorithms and methods.
Job market, a personal statement: A familiarity with ML is almost becoming a prerequisite for many of the most exciting employment opportunities. And add quantum computing and there you are!
Types of machine learning
The approaches to machine learning are many, but are often split into two main categories. In supervised learning we know the answer to a problem, and let the computer deduce the logic behind it. On the other hand, unsupervised learning is a method for finding patterns and relationship in data sets without any prior knowledge of the system. Some authours also operate with a third category, namely reinforcement learning. This is a paradigm of learning inspired by behavioural psychology, where learning is achieved by trial-and-error, solely from rewards and punishment.
Another way to categorize machine learning tasks is to consider the desired output of a system. Some of the most common tasks are:
- Classification: Outputs are divided into two or more classes. The goal is to produce a model that assigns inputs into one of these classes. An example is to identify digits based on pictures of hand-written ones. Classification is typically supervised learning.
- Regression: Finding a functional relationship between an input data set and a reference data set. The goal is to construct a function that maps input data to continuous output values.
- Clustering: Data are divided into groups with certain common traits, without knowing the different groups beforehand. It is thus a form of unsupervised learning.
Examples
The large amount of degrees of freedom pertain to both theory and experiment in the physical sciences. With increasingly complicated experiments that produce large amounts data, automated classification of events becomes increasingly important. Here, deep learning methods offer a plethora of interesting research avenues.
- Reconstruction of particle trajectories or classification of events are typical examples where ML methods are being used. However, since these data can often be extremely noisy, the precision necessary for discovery in physics requires algorithmic improvements. Research along such directions, interfacing nuclear and particle physics with AI/ML is expected to play a significant role in physics discoveries related to new facilities. The treatment of corrupted data in imaging and image processing is also a relevant topic.
- Design of detectors represents an important area of applications for ML/AI methods in nuclear physics.
And more
- An important application of AI/ML methods is to improve the estimation of bias or uncertainty due to the introduction of or lack of physical constraints in various theoretical models.
- In theory, we expect to use AI/ML algorithms and methods to improve our knowledge about correlations of physical model parameters in data for quantum many-body systems. Deep learning methods show great promise in circumventing the exploding dimensionalities encountered in quantum mechanical many-body studies.
- Merging a frequentist approach (the standard path in ML theory) with a Bayesian approach, has the potential to infer better probabilitity distributions and error estimates. As an example, methods for fast Monte-Carlo- based Bayesian computation of nuclear density functionals show great promise in providing a better understanding
- Machine Learning and Quantum Computing is a very interesting avenue to explore. See for example talk of Sofia Vallecorsa.
Selected references
What are the basic ingredients?
Almost every problem in ML and data science starts with the same ingredients:
- The dataset \( \mathbf{x} \) (could be some observable quantity of the system we are studying)
- A model which is a function of a set of parameters \( \mathbf{\alpha} \) that relates to the dataset, say a likelihood function \( p(\mathbf{x}\vert \mathbf{\alpha}) \) or just a simple model \( f(\mathbf{\alpha}) \)
- A so-called loss/cost/risk function \( \mathcal{C} (\mathbf{x}, f(\mathbf{\alpha})) \) which allows us to decide how well our model represents the dataset.
We seek to minimize the function \( \mathcal{C} (\mathbf{x}, f(\mathbf{\alpha})) \) by finding the parameter values which minimize \( \mathcal{C} \). This leads to various minimization algorithms. It may surprise many, but at the heart of all machine learning algortihms there is an optimization problem.
Quantum Monte Carlo and deep learning
Given a hamiltonian \( H \) and a trial wave function \( \Psi_T \), the variational principle states that the expectation value of \( \langle H \rangle \), defined through
$$
\langle E \rangle =
\frac{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})H(\boldsymbol{R})\Psi_T(\boldsymbol{R})}
{\int d\boldsymbol{R}\Psi^{\ast}_T(\boldsymbol{R})\Psi_T(\boldsymbol{R})},
$$
is an upper bound to the ground state energy \( E_0 \) of the hamiltonian \( H \), that is
$$
E_0 \le \langle E \rangle.
$$
In general, the integrals involved in the calculation of various expectation values are multi-dimensional ones. Traditional integration methods such as the Gauss-Legendre will not be adequate for say the computation of the energy of a many-body system. Basic philosophy: Let a neural network find the optimal wave function
The electron gas in three dimensions with \( N=14 \) electrons
Jane Kim, Bryce Fore, Alessandro Lovato and MHJ, in preparation
Extrapolations and model interpretability
When you hear phrases like predictions and estimations and correlations and causations, what do you think of? May be you think of the difference between classifying new data points and generating new data points. Or perhaps you consider that correlations represent some kind of symmetric statements like if \( A \) is correlated with \( B \), then \( B \) is correlated with \( A \). Causation on the other hand is directional, that is if \( A \) causes \( B \), \( B \) does not necessarily cause \( A \).
Statistical learning and data analysis
The above concepts are in some sense the difference between old-fashioned machine learning and statistics and Bayesian learning. In machine learning and prediction based tasks, we are often interested in developing algorithms that are capable of learning patterns from given data in an automated fashion, and then using these learned patterns to make predictions or assessments of newly given data. In many cases, our primary concern is the quality of the predictions or assessments, and we are less concerned about the underlying patterns that were learned in order to make these predictions.
Statistical learning points however to approaches that give us both predictions and correlations as well as being able to produce error estimates and understand causations. This leads us to the very interesting field of Bayesian statistics and Bayesian machine learning and much more.
Bayes' Theorem
Bayes' theorem
$$
p(X\vert Y)= \frac{p(X,Y)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}=\frac{p(Y\vert X)p(X)}{\sum_{i=0}^{n-1}p(Y\vert X=x_i)p(x_i)}.
$$
The quantity \( p(Y\vert X) \) on the right-hand side of the theorem is evaluated for the observed data \( Y \) and can be viewed as a function of the parameter space represented by \( X \). This function is not necessarily normalized and is normally called the likelihood function.
The function \( p(X) \) on the right hand side is called the prior while the function on the left hand side is the called the posterior probability. The denominator on the right hand side serves as a normalization factor for the posterior distribution.