Math

The last time, we ran our first self-play training loop on a simple MLP model and observed catastrophic policy collapse. Let’s first understand some of the math behind what happened, and then how to combat it. What is entropy? Given a probability distribution \(p=(p_1,\ldots,p_C)\) over a number of categories \(i=1,\ldots,C\), such as the distribution over the columns our Connect 4 model outputs for a given board state, entropy measures the “amount of randomness” and is defined as1 ...

With the REINFORCE algorithm under our belt, we can finally attempt to start training some models for Connect 4. However, as we’ll see, there are still some hurdles in our way before we get anywhere. It’s good to set your expectations accordingly because rarely if ever do things go smoothly the first time in RL. A simple MLP model As a fruitfly of Connect 4-playing models, let’s start with a simple multilayer perceptron (MLP) model that follows the model protocol we outlined earlier: that means that it has an input layer taking a 6x7 int8 board state tensor, a few simple hidden layers consisting of just a linear layer and a ReLU activation function each, and an output layer of 7 neurons without any activation function—that’s exactly what we meant earlier when we said that the model should output raw logits. ...

Let’s say we have a Connect 4-playing model and we let it play a couple of games. (We haven’t really talked about model architecture until now, so for now just imagine a simple multilayer perceptron with a few hidden layers which outputs 7 raw logits, as discussed in the previous post.) As it goes in life, our model wins some and loses some. How do we make it actually learn from its experiences? How does the magic happen? ...

With the triangle self-intersection algorithm ready to go, we can start gathering the training data for our machine learning setup. But first we have to think about how exactly we want to represent it. Canonical triangles The curved triangles we work with are specified by six 3D vectors, so that would mean 18 floating point numbers as our input data. But an important insight is that whether a triangle intersects itself doesn’t change when we rotate it, translate it, or uniformly scale it—it’s well known that affine transformations of spline control points result in affine transformations of the surface itself. ...

In the last post, we found pairs of subtriangles of our curved triangle which intersect. The subtriangles were linear approximations, which means that the intersection points we found are also only approximate. This might be good enough for our purposes, but in the interest of getting training data that’s as accurate as possible, we will refine these intersections by projecting them onto the exact curved triangle. To be precise, we are looking for two distinct parameter pairs \((u_1, v_1)\) and \((u_2, v_2)\) within the triangle’s domain such that their mappings coincide, ...

Before we can apply ML to the triangle problem, we need to be able to compute self-intersections of a curved triangle in an accurate and efficient way so that we can generate enough training data. The basic approach is: Subdivide the curved triangle into smaller subtriangles Find potentially intersecting pairs of subtriangles Check for actual intersections among these candidate pairs Subdividing the triangle We split the original triangle into a list of sufficiently flat subtriangles by a simple recursive procedure, starting with the full triangle {(0,0), (1,0), (0,1)}: ...

As the starting point for a little machine learning project, I chose the following geometric problem. We are given a curved triangle in 3D space. It’s specified via its three vertices plus three additional vector-valued coefficients associated to its three edges. These coefficients are interpreted as control points of a quadratic triangular Bezier surface. Such representations are commonly used in CAD systems to represent curved surfaces. Mathematically speaking, we map parameters \((u,v)\) which lie in the parameter-space triangle \( 0 \le u, v;\ u+v \le 1\) to ...