Sticky Post – Read this first. Categories and Links in Mathemafrica

The navigability of Mathemafrica isn’t ideal, so I have created this post which might guide you to what you are looking for. Here are a number of different categories of post which you might like to take a look at:

Always write in a comment if there is anything you would like to see us write about, or you would like to write about.…

By | January 17th, 2018|Uncategorized|0 Comments

On useful study habits

I’ve been teaching MAM1000W for around 9 years now, and I am learning all the time. I learn both about new ways to think about old subjects (and how to try and best explain them), and I learn about the way students study, about what works and what doesn’t, and what are some of the habits of students who succeed. Not all of these ideas will be perfect for everyone, but I hope that they will help.

Passive versus active learning

Trying to teach as clearly as possible is a double-edged sword. Of course I want students to come away feeling like they have understood the subject, but if they come away with too much confidence, then they won’t do the one thing which they have to do to actually understand it…and that is practice, but practice of a very particular kind. There is a balance that we should all be thinking about when trying to improve on something (be it sports, music, languages, or maths), and that is finding the right questions to practice on which are hard enough to make us have to sweat a little, but not so hard so as to make us give up completely.…

By | May 13th, 2022|Courses, First year, MAM1000, Undergraduate|2 Comments

In pursuit of Zeta-3 – The World’s Most Mysterious Unsolved Math Problem, by Paul Nahin – a review

NB. I was sent this book as a review copy.

I have to admit that I felt very skeptical when I started reading this book. In the prologue it is stated that the book is aimed at enthusiastic readers of mathematics with an AP level of high school maths. Then, diving into the book one sees what looks at first sight like a pure maths textbook at graduate level. But Paul Nahin isn’t one to pull a fast one like that, so I read further. In fact, I raced through it, hugely enjoyed it, and in the end agree with Nahin that someone with a US AP level of high school maths, or here in South Africa a confident first year undergraduate could actually understand everything in the book.

The book is not written as a textbook on mathematics, much as it might look like one, but rather it is taking an historical path through the investigations into the mysteries of zeta(3).…

By | April 23rd, 2022|Book reviews, Reviews|1 Comment

When least is best, by Paul Nahin – a review

NB. I was sent this book as a review copy.

For my review of Nahin’s superb book “How to fall slower than gravity”, see here.

While not often taught as a topic with such wide-ranging uses in maths classes, finding the maxima or minima of functions is one of the most important areas in all of applied mathematics. I say this as a practitioner of machine learning, where most of what we do is trying to find the minimum of a loss function, and as a physicist where in quantum field theory, the dynamical equations come from trying to extremise an action. While these areas aren’t discussed in the book (the closest it gets is looking at the classical Euler-Lagrange problem), to get students to think about how useful it is to find the maxima and minima of a function is really a powerful thing.

Nahin takes on this challenge and succeeds in the same way that he succeeded in making the problems in the previous book of his that I reviewed both fascinating and easy to follow.…

By | April 23rd, 2022|Book reviews, Reviews|0 Comments

A course in Complex Analysis, by Saeed Zakeri – a review

NB. I was sent this book as a review copy.

This is a no-nonsense, clearly written graduate level textbook on complex analysis, and while it is written for a graduate audience, I think that the way it is laid out, with clear examples throughout, a keen undergraduate with a background in analysis and topology. As such it is far more approachable than many other books on complex analysis and I would say that it would be perfectly suited for physics students wanting to go into areas like quantum field theory, particularly string theorists where the sections on conformal metrics and the modular group would be very helpful.

One thing to look out for in a book like this is the clarity of the proofs, and the number of intermediate lines which are included, and in this case I think that there is just the right amount to make everything easy to follow, but not overwhelming the material.…

By | April 23rd, 2022|Book reviews, Reviews|0 Comments

Visual Differential Geometry and Forms – a mathematical drama in five acts, by Tristan Needham – a review

NB. I was sent this book as a review copy.

Studying physics, some two decades ago at The University of Bristol, I found the majority of what we covered relatively intuitive. Even the arcane world of quantum mechanics, while impossible to truly visualise, is, paradoxically, often relatively simple to calculate, and the objects that you use are directly from the world of complex numbers, differential equations and linear algebra. What stumped me however were tensors. I found it so hard to really picture what was going on with these objects. Vectors were ok, and the metric tensor I could handle, but as soon as you got onto differential forms, all my intuition went out the window. The world of differential geometry, while I could plug and chug, felt like putting together sentences in a foreign language where all I had were rules for using the syntax and grammar, without a deep understanding of what the objects were

This book would have answered all of my prayers back then.…

By | December 11th, 2021|Book reviews, Reviews|0 Comments

CRL Task 5: Learning Causal Models

We’ve now come to one of the most vital aspects of this theory – how can we learn causal models? Learning models is often an exceptionally computationally intensive process, so getting this right is crucial. We now develop some mathematical results which guarantee bounds on our learning. We’ll start by discussing the current state of this field in relation to causal inference and reinforcement learning.

This Series

  1. Causal Reinforcement Learning
  2. Preliminaries for CRL
  3. CRL Task 1: Generalised Policy Learning
  4. CRL Task 2: Interventions – When and Where?
  5. CRL Task 3: Counterfactual Decision Making
  6. CRL Task 4: Generalisability and Robustness
  7. Task 5: Learning Causal Models
  8. (Coming soon) Task 6: Causal Imitation Learning
  9. (Coming soon) Wrapping Up: Where To From Here?

Learning Causal Models

Perhaps one of the most computationally difficult processes in the field of causal inference is that of learning underlying causal structure by algorithmically identifying cause-effect relationships. In recent years there has been a surge of interest in learning such relationships in the fields of machine learning and artificial intelligence, though it has been relatively prevalent in the social sciences for many years now (e.g.…

By | September 19th, 2021|Uncategorized|0 Comments

CRL Task 3: Counterfactual Decision Making

In the previous blog post we discussed some theory of how to select optimal and possibly optimal interventions in a causal framework. For those interested in the decision science, this blog post may be more inspiring. This next task involves applying counterfactual quantities to boost learning performance. This is clearly very important for an RL agent where its entire learning mechanism is based on interventions in a system. What if intervention isn’t possible? Let’s begin!

This Series

  1. Causal Reinforcement Learning
  2. Preliminaries for CRL
  3. CRL Task 1: Generalised Policy Learning
  4. CRL Task 2: Interventions – When and Where?
  5. CRL Task 3: Counterfactual Decision Making
  6. CRL Task 4: Generalisability and Robustness
  7. Task 5: Learning Causal Models
  8. (Coming soon) Task 6: Causal Imitation Learning
  9. (Coming soon) Wrapping Up: Where To From Here?

Counterfactual Decision Making

A key feature of causal inference is its ability to deal with counterfactual queries. Reinforcement learning, by its nature, deals with interventional quantities in a trial-and-error style of learning.…

By | July 10th, 2021|English, Level: intermediate, Uncategorized|6 Comments

CRL Task 2: Interventions – When and Where?

In the previous blog post we discussed the gorey details of generalised policy learning – the first task of CRL. We went into some very detailed mathematical description of dynamic treatment regimes and generalised modes of learning for data processing agents. The next task is a bit more conceptual and focuses on the question on how to identfy optimal areas of intervention in a system. This is clearly very important for an RL agent where its entire learning mechanism is based on these very interventions in some system with a feedback mechanism. Let’s begin!

This Series

  1. Causal Reinforcement Learning
  2. Preliminaries for CRL
  3. CRL Task 1: Generalised Policy Learning
  4. CRL Task 2: Interventions – When and Where?
  5. CRL Task 3: Counterfactual Decision Making
  6. CRL Task 4: Generalisability and Robustness
  7. Task 5: Learning Causal Models
  8. (Coming soon) Task 6: Causal Imitation Learning
  9. (Coming soon) Wrapping Up: Where To From Here?

Interventions – When and Where?

By | July 6th, 2021|English, Level: intermediate|4 Comments

CRL Task 1: Generalised Policy Learning

In the previous blog post we developed some ideas and theory needed to discuss a causal approach to reinforcement learning. We formalised notions of multi-armed bandits (MABs), Markov Decision Processes (MDPs), and some causal notions. In this blog post we’ll finally get to developing some causal reinforcement learning ideas. The first of which is dubbed Task 1, for CRL can help solve. This is Generalised Policy Learning. Let’s begin.

This Series

  1. Causal Reinforcement Learning
  2. Preliminaries for CRL
  3. CRL Task 1: Generalised Policy Learning
  4. CRL Task 2: Interventions – When and Where?
  5. CRL Task 3: Counterfactual Decision Making
  6. CRL Task 4: Generalisability and Robustness
  7. Task 5: Learning Causal Models
  8. (Coming soon) Task 6: Causal Imitation Learning
  9. (Coming soon) Wrapping Up: Where To From Here?

Generalised Policy Learning

Reinforcement learning typically involves learning and optimising some policy about how to interact in an environment to maximise some reward signal. Typical reinforcement learning agents are trained in isolation, exploiting copious amounts of computing power and energy resources.…

By | July 1st, 2021|Background, English, Level: intermediate|4 Comments

Preliminaries for CRL

In the previous blog post we discussed and motivated the need for a causal approach to reinforcement learning. We argued that reinforcement learning naturally falls on the interventional rung of the ladder of causation. In this blog post we’ll develop some ideas necessary for understanding the material covered in this series. This might get quite technical, but don’t worry. There is still always something to take away. Let’s begin.

This Series

  1. Causal Reinforcement Learning
  2. Preliminaries for CRL
  3. CRL Task 1: Generalised Policy Learning
  4. CRL Task 2: Interventions – When and Where?
  5. CRL Task 3: Counterfactual Decision Making
  6. CRL Task 4: Generalisability and Robustness
  7. Task 5: Learning Causal Models
  8. (Coming soon) Task 6: Causal Imitation Learning
  9. (Coming soon) Wrapping Up: Where To From Here?

Preliminaries

As you probably recall from high school, probability and statistics are almost entirely formulated on the idea of drawing random samples from an experiment. One imagines observing realisations of outcomes from some set of possibilities when drawing from an assortment of independent and identically distributed (i.i.d.) events.…

By | April 6th, 2021|Background, English, Level: Simple|5 Comments