# What is a "logit probability"?

DeepMind's paper "Mastering the game of Go without human knowledge" states in its "Methods" section on its "Neural network architecture" that the output layer of AlphaGo Zero's policy head is "A fully connected linear layer that outputs a vector of size 19^2+1=362, corresponding to the logit probabilities for all intersections and the pass move" (emphasis mine). I am self-trained regarding neural networks, and I have never heard of a "logit probability" before this paper. I have not been able by searching and reading to figure out what it means. In fact, the Wikipedia page on logit seems to make the term a contradiction. A logit can be converted into a probability using the equation $$p=\frac{e^l}{e^l+1}$$, and a probability can be converted into a logit using the equation $$l=\ln{\frac{p}{1-p}}$$, so the two cannot be the same. The neural network configuration for Leela Zero, which is supposed to have a nearly identical architecture to that described in the paper, seems to indicate that the fully connected layer described in the above quote needs to be followed with a softmax layer to generate probabilities (though I am absolutely new to Caffe and might not be interpreting the definitions of "p_ip1" and "loss_move" correctly). The AlphaGo Zero cheat sheet, which is otherwise very helpful, simply echoes the phrase "logit probability" as though this is a well-known concept. I have seen several websites that refer to "logits" on their own (such as this one), but this is not enough to satisfy me that "logit probability" must mean "a probability generated by passing a logit vector through the softmax function".

What is a logit probability? What sources can I read to help me understand this concept better?

Indeed I haven't seen the term "logit probability" used in many places other than that specific paper. So, I cannot really comment on why they're using that term / where it comes from / if anyone else uses it, but I can confirm that what they mean by "logit probability" is basically the same thing that is more commonly referred to simply as "logits": they are the raw, unbounded scores of which we generally push a vector through a softmax function to generate a discrete probability distribution that nicely adds up to $$1$$.

This definition fits the one you linked from wikipedia (although that link only covers the binary case, and AlphaGo Zero would have multinomial logits since it has more than two outputs for the policy head).

In the AlphaGo Zero paper, the described architecture has a "linear output layer" (i.e. no activation function for the outputs, or the identity function as activation function for the outputs, or however you like to describe it) for the policy head. This means that these outputs are essentially unbounded, they could be any real number. We know for sure that these outputs cannot directly be interpreted as probabilities, even if this isn't stated quite explicitly in the paper.

By calling them logits (or logit probabilities for reasons unknown to me), they are essentially implying that these outputs will still be post-processed by a softmax to convert them into a vector that can be interpreted as a discrete probability distribution over the actions, even if they do not explicitly describe a softmax layer as being a part of the network.

It is indeed possible that in Leela Zero they decided to make the softmax operation explicitly a part of the Neural Network architecture. Mathematically they end up doing the same thing... the AlphaGo Zero paper implies (by using the word "logit") that a softmax is used as a "post-processing" step, and in Leela Zero they explicitly make it a part of the Neural Network.

Here are a couple more sources for the reasoning that usage of the word "logit" basically implies usage of a softmax, though indeed they do not cover the term "logit probability":

• Thanks for your answer. That Wikipedia link to multinomial logistic regression especially helped since it ultimately and explicitly uses softmax to generate the probability distribution. I will likely accept your answer, but I will wait to see if anyone else might have more information about the logit probability phrase in particular. Jan 23, 2019 at 17:20
• @sadakatsu Of course :) I'd love to see if anyone else knows more about where that phrase comes from myself too, I was also wondering about that when I first tried implementing something similar and was looking into all the tiny details. Note that the original AlphaGo paper does explicitly mention using a softmax, and in AlphaGo Zero they also must be doing something to convert their linear outputs into something that can be interpreted as a probability distribution... so it's a fairly safe bet that they're still doing softmax. Jan 23, 2019 at 17:47