What kind of learning algorithm would you use to build a model of how long it takes a human to solve a given Sudoku situation?

Question

I don't have much experience in machine learning, pattern recognition, data mining, etc. and in their underlying theory and systems.

I would like to develop an artificial model of the time it takes a human to make a move in a given Sudoku puzzle.

So what I'm looking for as an output from the machine learning process is a model that can give predictions on how long does it take for a target human to make a move in a given Sudoku situation.

Same input doesn't always map to same outcome. It takes different times for the human to make a move with the same situation, but my hypothesis is that there's a tendency in the resulting probability distribution. (My educated guess is that it is ~normal.)

I have ideas about the factors that influence the distribution (like #empty slots) but would preferably leave it to the system to figure these patterns out. Please notice, that I'm not interested in the patterns, just the model.

I can generate sample and test data easily by running sudoku puzzles and measuring the times it takes to make the moves.

What kind of learning algorithm would you suggest to use for this?

I was thinking NNs, but I'm not sure if they can have the desired property of giving weighted random outcomes for the same input.

Answer 1

If I understand this correctly you have an input vector of length 81, which contains 1 if the square is filled in and 0 otherwise. You want to learn a function which returns a probability distribution which models the response time of a human to that board position.

My first response would be that this is a regression problem and you should try straightforward linear regression . This will not provide you with a distribution of response times, but a single 'best-guess' response time.

I'm not clear on why you want to model a distribution of response times. However, if you really want to do want to output a distribution then it sounds like you want to look at Bayesian methods . I'm not really an expert on Bayesian inference, so I can't help you much further here.

However, I don't really think your approach is going to work because I agree with your intuition about features such as the number of empty slots being important. There are also other obvious features, such as the number of empty slots per row/column that are likely to be important. Explicitly putting these features in your representation will probably be much more successful than expecting that the learning algorithm will infer something similar on its own.

Answer 2

The monte carlo method seems like it would work well here but would require a stack of solutions the size of the moon to really do it. And it wouldn't give you the time per person, just the time on average.

My understanding of it, tenuous as it is, is that you have a database with a board position and the time it took a human to make the next move. At the very least you have a starting point for most moves. Even if it's not in the database you could start to calculate how long it would take to make a move based on some algorithm. Though I know you had specified you wanted machine learning to do this it might be worth segmenting the problem into something a little smaller then building on it.

Answer 3

If you have some guesstimate as to what influences the function (# of empty cell, etc), try to train a classifier on a vector of features , and not on the 81 cells vector (0/1 or 0..9, doesn't really matter for my argument).

I think that your claim:

we wouldn't have to necessary know the underlying patterns, the "trained patterns" in a learning system automatically encodes these sometimes quite delicate and subtle patterns inside them -- that's one of their great power

is wrong. you do have to give the network the right domain. for example, when trying to detect object in an image, working in the pixel domain is pointless. you'll only get results if you first run some feature detection to detect edges, corners, etc. Theoretically, with enough non-linearity (in NN - enough layers in the network) it can detect such things, but in practice, I have never seen that work, without giving the classifier the right features to work with.

I was thinking NNs, but I'm not sure if they can have the desired property of giving weighted random outcomes for the same input.

You're just trying to learn a function from 2^81 or 10^81 (or a much smaller feature space as I suggest) to R (response time between 0 and Inf) or some discretization of that. So NN and other classifiers can do that.