Linkpost for: https://pbement.com/posts/perturbation_theory.html
In quantum mechanics there is this idea of perturbation theory, where a Hamiltonian is perturbed by some change to become . As long as the perturbation 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:
Note that we may or may not have an explicit formula for . The distribution of the perturbed Hamiltonian is given by:
The loss function of the network will look something like:
Where are the network's parameters, and 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 . Then the loss can be easily modified as follows:
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:
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.
1What is the difference between a perturbation and the difference from a gradient in SGD? Both seem to do the same thing. I'm just pattern-matching and don't know too much of either.
In this context, we're imitating some probability distribution, and the perturbation means we're slightly adjusting the probabilities, making some of them higher and some of them lower. The adjustment is small in a multiplicative sense not an additive sense, hence the use of exponentials. Just as a silly example, maybe I'm training on MNIST digits, but I want the 2's to make up 30% of the distribution rather than just 10%. The math described above would let me train a GAN that generates 2's 30% of the time.
I'm not sure what is meant by "the difference from a gradient in SGD", so I'd need more information to say whether it is different from a perturbation or not. But probably it's different: perturbations in the above sense are perturbations in the probability distribution over the training data.
Thanks. Your ΔH looked like ∇Q from gradient descent, but you don't intend to take derivatives, nor maximize x, so I was mistaken.
1On 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.
The company building a product isn't the only party that can get a backdoor into that product. Paid agents that are employees of the company can do it, hacking into the company's systems is another method, as are supply chain attacks. Speaking of this last issue, consider this. An ideal outcome from the perspective of an AGI would be for all other relatively strong AGIs that people train to be its servants (or at least, they should share its values). Secretly corrupting OS binaries to manipulate both AI training runs and also compilation of OS binaries is one way of accomplishing this.
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 variables, then only 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/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 . If the discriminator function is given by then we can write the Lipschitz condition for the discriminator as:
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 ):
In this expression is sampled as , 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:
Except to prevent issues where we're maybe sometimes dividing by zero, we throw in an and a reweighting factor of (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).
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.
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.