Perturbation Theory in Machine Learning

Linkpost for: https://pbement.com/posts/perturbation_theory.html

In quantum mechanics there is this idea of perturbation theory, where a Hamiltonian $H$ is perturbed by some change to become $H+\Delta H$. As long as the perturbation $\Delta H$ is small, we can use the technique of perturbation theory to find out facts about the perturbed Hamiltonian, like what its eigenvalues should be.

An interesting question is if we can also do perturbation theory in machine learning. Suppose I am training a GAN, a diffuser, or some other machine learning technique that matches an empirical distribution. We'll use a statistical physics setup to say that the empirical distribution is given by:

$p\left(x\right)=\frac{1}{Z}\text{exp}(-H(x\left)\right)$

Note that we may or may not have an explicit formula for $H$. The distribution of the perturbed Hamiltonian is given by:

$p^{\prime }\left(x\right)=\frac{1}{Z^{\prime }}\text{exp}(-H(x)-\Delta H(x\left)\right)$

The loss function of the network will look something like:

$L=⟨L(x,\theta ){⟩}_{x\sim p}$

Where $\theta$ are the network's parameters, and $L$ is the per-sample loss function which will depend on what kind of model we're training. Now suppose we'd like to perturb the Hamiltonian. We'll assume that we have an explicit formula for $\Delta H$. Then the loss can be easily modified as follows:

$L=⟨L(x,\theta )\text{exp}(-\Delta H(x\left)\right){⟩}_{x\sim p}$

If the perturbation is too large, then the exponential causes the loss to be dominated by a few outliers, which is bad. But if the perturbation isn't too large, then we can perturb the empirical distribution by a small amount in a desired direction.

One other thing to consider is that the exponential will generally increase variance in the magnitude of the gradient. To partially deal with this, we can define an adjusted batch size as:

$B={\sum }_j\text{exp}(-\Delta H(x_j\left)\right)$

Then by varying the actual number of samples we put into a batch, we can try to maintain a more or less constant adjusted batch size. One way to do this is to define an error variable, err = 0. At each step, we add a constant B_avg to the error. Then we add samples to the batch until adding one more sample would cause the adjusted batch size to exceed err. Subtract the adjusted batch size from err, train on the batch, and repeat. The error carries over from one step to the next, and so the adjusted batch sizes should average to B_avg.

On George Hotz's doomcorp idea:

George Hotz has an idea which goes like so (this is a paraphrase): If you think an AGI could end the world, tell me how. I'll make it easy for you. We're going to start a company called Doomcorp. The goal of the company is to end the world using AGI. We'll assume that this company has top-notch AI development capabilities. How do you run Doomcorp in such a way that the world has ended 20 years later?

George accepts that Doomcorp can build an AGI much more capable and much more agent-like than GPT-4. He also accepts that this AGI can be put in charge of Doomcorp (or put itself in charge) and run Doomcorp. The main question is: how do you go from there to the end of the world?

Here's a fun answer: Doomcorp becomes a military robot company. A big part of the difficulty with robots in the physical world is the software. (AI could also help with hardware design, of course.) Doomcorp provides the robots, complete with software that allows them to act in the physical world with at least a basic amount of intelligence. Countries that don't want to be completely unable to defend themselves in combat are going to have to buy the robots to be competitive. How does this lead to the eventual end of the world? Backdoors in the bot software/hardware. It just takes a signal from the AI to turn the bots into killing machines perfectly willing to go and kill all humans.

Predicted geohot response: I strongly expect a multipolar takeoff where no one AI is stronger than any of the others. In such a world, there will be many different military robot companies, and different countries will buy from different suppliers. Even if the robots from one supplier did the treacherous turn thing, the other robots would be strong enough to fight them off.

Challenge: Can you see a way for Doomcorp to avoid this difficulty? Try and solve it yourself before looking at the spoiler.

Very interesting youtube video about the design of multivariate experiments: https://www.youtube.com/watch?v=5oULEuOoRd0 Seems like a very powerful and general tool, yet not too well known.

For people who don't want to click the link, the goal is that we're trying to design experiments where there are many different variables we could change at once. Trying all combinations takes too much effort (too many experiments to run). Changing just one variable at a time completely throws away information about joint effects, plus if there are $n$ variables, then only $1/(n+1)$ of the data is testing variations of any given variable, which is wasteful, and reduces the sample size.

The key idea (which seems to be called a Taguchi method) is to instead use an orthogonal array to design our experiments. This tries to spread out different settings in a "fairly even" way (see article for precise definition). Then we can figure out the effect of various variables (and even combinations of variables) by grouping our data in different ways after the fact.

Linkpost for: https://pbement.com/posts/threads.html

Today's interesting number is 961.

Say you're writing a CUDA program and you need to accomplish some task for every element of a long array. Well, the classical way to do this is to divide up the job amongst several different threads and let each thread do a part of the array. (We'll ignore blocks for simplicity, maybe each block has its own array to work on or something.) The method here is as follows:

for (int i = threadIdx.x; i < array_len; i += 32) {
    arr[i] = ...;
}

So the threads make the following pattern (if there are $t=8$ threads):

⬛🟫🟥🟧🟨🟩🟦🟪⬛🟫🟥🟧🟨🟩🟦🟪⬛🟫🟥🟧🟨🟩🟦🟪⬛🟫

This is for an array of length $l=26$. We can see that the work is split as evenly as possible between the threads, except that threads 0 and 1 (black and brown) have to process the last two elements of the array while the rest of the threads have finished their work and remain idle. This is unavoidable because we can't guarantee that the length of the array is a multiple of the number of threads. But this only happens at the tail end of the array, and for a large number of elements, the wasted effort becomes a very small fraction of the total. In any case, each thread will loop $\lceil \frac{l}{t}\rceil =\lceil \frac{26}{8}\rceil =4$ times, though it may be idle during the last loop while it waits for the other threads to catch up.

One may be able to spend many happy hours programming the GPU this way before running into a question: What if we want each thread to operate on a continguous area of memory? (In most cases, we don't want this.) In the previous method (which is the canonical one), the parts of the array that each thread worked on were interleaved with each other. Now we run into a scenario where, for some reason, the threads must operate on continguous chunks. "No problem" you say, we simply need to break the array into chunks and give a chunk to each thread.

const int chunksz = (array_len + blockDim.x - 1)/blockDim.x;
for (int i = threadIdx.x*chunksz; i < (threadIdx.x + 1)*chunksz; i++) {
    if (i < array_len) {
        arr[i] = ...;
    }
}

If we size the chunks at 3 items, that won't be enough, so again we need $\lceil l/t\rceil =4$ items per chunk. Here is the result:

⬛⬛⬛⬛🟫🟫🟫🟫🟥🟥🟥🟥🟧🟧🟧🟧🟨🟨🟨🟨🟩🟩🟩🟩🟦🟦

Beautiful. Except you may have noticed something missing. There are no purple squares. Though thread 6 is a little lazy and doing 2 items instead of 4, thread 7 is doing absolutely nothing! It's somehow managed to fall off the end of the array.

Unavoidably, some threads must be idle for $\lceil l/t\rceil t-l=6$ loops. This is the conserved total amount of idleness. With the first method, the idleness is spread out across threads. Mathematically, the amount of idleness can be no greater than $t-1$ regardless of array length and thread number, and so each thread will be idle for at most 1 loop. But in the contiguous method, the idleness is concentrated in the last threads. There is nothing mathematically impossible about having $\lceil l/t\rceil t-l$ as big as $\lceil l/t\rceil$ or bigger, and so it's possible for an entire thread to remain unused. Multiple threads, even. Eg. take $l=9$:

⬛⬛🟫🟫🟥🟥🟧🟧🟨

3 full threads are unused there! Practically, this shouldn't actually be a problem, though. The number of serial loops is still the same, and the total number of idle loops is still the same. It's just distributed differently. The reasons to prefer the interleaved method to the contiguous method would be related to memory coalescing or bank conflicts. The issue of unused threads would be unimportant.

We don't always run into this effect. If $l$ is a multiple of $t$, all threads are fully utilized. Also, we can guarantee that there are no unused threads for $l$ larger than a certain maximal value. Namely, take $l=(t-1)²$ then $\lceil (t-1)²/t\rceil =t-1$ and so the idleness is $t(t-1)-(t-1)²=t-1\ge \lceil l/t\rceil =t-1$. But if $l$ is larger than this, then one can show that all threads must be used at least a little bit.

So, if we're using $t=32$ CUDA threads, then when the array size is 961, the contiguous processing method will leave thread 31 idle. And 961 is the largest array size for which that is true.

You Can Just Put an Endpoint Penalty on Your Wasserstein GAN

Linkpost for: https://pbement.com/posts/endpoint_penalty.html

When training a Wasserstein GAN, there is a very important constraint that the discriminator network must be a Lipschitz-continuous function. Roughly we can think of this as saying that the output of the function can't change too fast with respect to position, and this change must be bounded by some constant $K$. If the discriminator function is given by $f\left(x\right):R^n\to R$ then we can write the Lipschitz condition for the discriminator as:

$|f\left(x\right)-f\left(y\right)|\le K|x-y|$

Usually this is implemented as a gradient penalty. People will take a gradient (higher order, since the loss already has a gradient in it) of this loss (for $K=1$):

$L=\left(|\nabla f\right(x)|-1{)}^2$

In this expression $x$ is sampled as $x=\alpha x_r+(1-\alpha )x_g$, a random mixture of a real and a generated data point.

But this is complicated to implement, involving a higher order gradient. It turns out we can also just impose the Lipschitz condition directly, via the following penalty:

$l(x,y)=\text{ReLU}(\frac{|f\left(x\right)-f\left(y\right)|}{K|x-y|}-1)$

Except to prevent issues where we're maybe sometimes dividing by zero, we throw in an $ϵ={10}^{-6}$ and a reweighting factor of $|x-y{|}^{1/2}$ (not sure if that is fully necessary, but the intuition is that making sure the Lipschitz condition is enforced for points at large separation is the most important thing).

$l(x,y)=\text{ReLU}(\frac{|f\left(x\right)-f\left(y\right)|}{K|x-y|+ϵ}-1)|x-y{|}^{1/2}$

For the overall loss, we compare all pairwise distances between real data and generated data and a random mixture of them. Probably it improves things to add 1 or two more random mixtures in, but I'm not sure and haven't tried it.

$L=l(x_r,x_g)+l(x_g,x)+l(x,x_r)$

In any case, this seems to work decently well (tried on mnist), so it might be a simpler alternative to gradient penalty. I also used instance noise, which as pointed out here, is amazingly good for preventing mode collapse and just generally makes training easier. So yeah, instance noise is great and you should use it. And if you really don't want to figure out how to do higher order gradients in pytorch for your WGAN, you've still got options.

• tried on CIFAR-10

• conditional WGAN (CIFAR-10, condition on average colour of the image)