There are no dedekind cuts or cauchy sequences on digital computers so the fact that the analytical equations map to algorithms at all is very non-obvious.
Discretizing e.g. time or space is perhaps a bigger issue, but the issues are usually well understood and mitigated by e.g. advanced numerical integration schemes, discrete-continuous formulations or just cranking up the discretization resolution.
Analytical tools for discrete formulations are usually a lot less developed and don't as easily admit closed-form solutions.
For instance we know that algorithms like the leapfrog integrator not only approximate a physical system quite well but even conserve the energy, or rather a quantity that approximates the true energy.
There are plenty of theorems about the accuracy and other properties of numerical algorithms.
So I guess one might want to do a similar exercise to deriving numerical dispersion for example in order to see just how discretizing the diffusion process affects it and the relation to optimal control theory.
[1]: https://en.wikipedia.org/wiki/Numerical_dispersion
[2]: https://en.wikipedia.org/wiki/Courant%E2%80%93Friedrichs%E2%...
I'm not saying that this is the case here, but there generally needs to be some justification to say that a certain result that is proven for a continuous function also holds for some discrete version of it.
For a somewhat famous real-world example, it's not currently known how to produce a version of QM/QFT that works with discrete spacetime coordinates, the attempted discretizations fail to maintain the properties of the continuous equations.
https://en.wikipedia.org/wiki/Finite_difference
I'm not sure about applications of real numbers outside of calculus, and how to replace them there.
If your definition of "algorithm" is "list of instructions", then there is nothing surprising. It's very obvious. The "algorithm" isn't perfect, but a mapping with an error exists.
If your definition of "algorithm" is "error free equivalent of the equations", then the analytical equations do not map to "algorithms". "Algorithms" do not exist.
I mean, your objection is kind of like questioning how a construction material could hold up a building when it is inevitably bound to decay and therefore result in structural collapse. Is it actually holding the entire time or is it slowly collapsing the entire time?
I am unsure of the next course of action or if software will survive another 5 years and how my career will look like in the future. Seems like I am engaged in the ice trade and they are about to invent the refrigerator.
More often than not this is duplicated work (mathematically speaking) and there is a lot to be gained by sharing advances in either field by running it through a "translation". This has happened many times historically - a lot of the "we met at a cafe and worked it out on a napkin" inventions are exactly that.
Math proficiency helps a lot at that. The level of abstraction you deal with is naturally high.
Recently, the problem of actually knowing every field enough, just cursorily, to make connections is easier with AI. Modern LLMs do approximate retrieval and still need a planner + verifier, the mathematician can be that.
This is somewhat adjacent to what terry tao spoke about, and the setup is sort of what alpha evolve does.
You get that impression because such advances are high impact and rare (because they are difficult). Most advances come as a sequence of field-specific assumption, field-specific empirical observation, field-specific theorem, and so on. We only see the advances that are actually made, leading to an observation bias.
(One of) The value(s) that a math grad brings is debugging and fixing these ML models when training fails. Many would not have an idea about how to even begin debugging why the trained model is not working so well, let alone how to explore fixes.
Instead of people "hacking" university education to make them purely fotm job training centers. The real hack would be something that really drills down at the fundamentals. CS, Math, Physics, and Philosophy to get an all around education in approaching problems from fundamentals I think would be the optimal school experience.
So in my specific case I stopped thinking about mathematics as: how to interpret a sequence of symbols
But instead I decided to start thinking about it as “the symbols tell me about the multidimensional topological coordinate space that I need to inhabit
So now when I look at a equation (or whatever) my first step is “OK how do I turn this into a topology so that I can explore the toplogical space the way that a number would”
Kind of like if you were to extend Nagle’s “what it’s like to be a bat” but instead of being a bat you’re a number
The way I like to look at it is that I'm engaged in the ice trade and they are about to invent everything else that will end mine and every other current trade. Which leaves me with two practical options: a) deep despair. b) to become a Jacks of all trades, master of none, but oftentimes better than a master of one. The Jacks can, for now, capitalize in the thing that the Machines currently lack, which is agency.
Once the fundamental concepts are understood, what problem is being solved and where the key difficulties are, only then the equations will start to make sense. If you start out with the math, you're making your life unnecessarily hard.
Also, not universally true but directionally true as a rule of thumb, the more equations a text contains the less likely it is that the author itself has truly grasped the subject. People who really grasp a subject can usually explain it well in plain language.
That's very much a matter of style. An equation is often the plainest way of expressing something
I'm most interested in how the equation can be implemented step by step in an ML library - worked examples would be very helpful.
Thank you!
It includes both mathematical formulas and PyTorch code.
I found it a bit more practical than the Sutton & Barto book, which is a classic but doesn't cover some of the more modern methods used in deep reinforcement learning.
It's also nice that Sutton & Barto belabors a lot of old stuff that is no longer obsessed over, and this skims through that and gets to the stuff that is much more relevant today.
Read here: http://incompleteideas.net/book/the-book-2nd.html
This is because they work assuming you know a model of the data. Most real world RL is model-free RL. Or, like in LLMs, "model is known but too big to practically use" RL.
Apart from the resources you use (good ones in other comments already), try to get the initial mental model of the whole field right, that is important since everything you read can then fit in the right place of that mental model. I will try to give one below.
- the absolute core raison d'etre of RL as a separate field: the quality of data you train on only improves as your algorithm improves. As opposed to other ML where you have all your data beforehand.
- first basic bellman equation solving (this is code wise just solving a system of linear equations)
- an algo you will come across called policy iteration (code wise, a bunch of for loops..)
- here you will be able to see how different parts of the algo become impossible in different setups, and what approximations can be done for each of them (and this is where the first neural network - called "function approximator" in RL literature - comes into play). Here you can recognise approximate versions of the bellman equation.
- here you learn DDPG, SAC algos. Crucial. Called "actor critic" in parlance.
- you also notice problems of this approach that arise because a) you don't have much high quality data and b) learning recursivelt with neural networks is very unstable, this motivates stuff like PPO.
- then you can take a step back, look at deep RL, and re-cast everything in normal ML terms. For example, techniques like TD learning (the term you would have used so far) can be re-cast as simply "data augmentation", which you do in ML all the time.
- at this point you should get in the weeds of actually engineering at scale real RL algos. Stuff like atari benchmarks. You will find that in reality, the algos as learnt are more or less a template and you need lots of problems specific detailing to actually make it work. And you will also learn engineering tricks that are crucial. This is mostly computer science stuff (increasing throughout on gpu etc - but correctly! without changing the model assumptions)
- learn goal conditioned RL, imitation learning, some model based RL like alphazero/dreamer after all of the above. You will be able to easily understand it in the overall context at this point. First two are used in robotics quite a bit. You can run a few small robotics benchmarks at this point.
- learn stuff like HRL, offline RL as extras since they are not that practically relevant yet.
Q-learning (the usual application of the Bellman equation) is generally model-free. It is also commonly found in reinforcement learning libraries.
https://www.andrew.cmu.edu/course/10-703/textbook/BartoSutto...
This will have a gentler learning curve. After this you can move on to more advanced material.
The other resource I will recommend is everything by Bertsekas. In this context, his books on dynamic programming and neurodyanamic programming.
Happy reading.
