CS 179: Introduction to Graphical Models: Spring 2022

Homework 6

The submission for this homework should be a single PDF file containing all of the relevant code, figures, and any text explaining your results. When coding your answers, try to write functions to encapsulate and reuse code, instead of copying and pasting the same code multiple times. This will not only reduce your programming efforts, but also make it easier for us to understand and give credit for your work. Show and explain the reasoning behind your work!

In this homework, we will Pyro’s stochastic variational inference procedure to explore a simple model for collaborative filtering, specifically, to predict movie ratings on a (very small) subset of the MovieLens dataset.

For more examples of variational inference in Pyro, please see the course demos on Pyro (VI) and Bayesian linear regression.

Part 1: Loading the data   (20 points)

First, load the training data from                                   .  The first line of the file is text, listing the names of

M = 10 movies; the remaining lines are comma-separated integers between zero and nine, or         (not-a-number),

indicating the value was not observed. The training data file contains a subset of the ratings of N = 200 users

(one line per user). Store the training data in a 200x 10 numpy array called        . You may find           ’s

function helpful.

Part 2: Model Set-up   (30 points)

We will use a Bayesian variant of the SVD low-rank decomposition, or latent space reprsentation, for collaborative filtering described in CS178. If you haven’t seen this and want to read more about it, you can see these data evaluated using non-probabilistic unsupervised learning in the CS178 course notes:

https://canvas.eee.uci.edu/courses/31018/files/folder/Notes

parts 10 (clustering) and 11 (latent space models).

Our model associates a K-dimensional “position” with each user n and each movie m, denoted Un  and Vm , respectively. Then, the degree to which user n likes movie m is a function of their dot product, Un．Vm(T)  =    k Unk Vmk . This will try to place similarly rated movies in a similar direction from the origin, with users that like those movies placed in the same direction, so that the dot product will be large.

A small modification helps the model in practice. Since some users are more positive than others, and some movies are more widely acknowledged as good or bad, we may want to instead predict “relative” preference. For this homework, we will do this very simply, by estimating and subtracting the mean over the movies:

1    movie_avg  =  np .nanmean(Xtr ,  axis=0,  keepdims=True)      #  ignore  NaN  when  averaging                                                   2    Xtr  -=  movie_avg                                                                                                                                                                                             

and then do the same thing for the average over the users (estimate the mean over axis 1, and subtract from X.) (For more sophistication, we could add these as variables in the model and reason over them as well.)

We place simple Gaussian .(0, 1) priors on all the unknown values: the entries in both matrices U, V . Then, we express the probability of our observations as Gaussian with fixed standard deviation:

rnm ～.╱R ; Un．Vm(T)  , 0.1 、.

To do so, define your                      function to generate a collection of M two-dimensional Gaussian random

variables (one for each movie). Your model will be most efficient if you use Pyro’s             for indexing, which informs Pyro that the indexed elements are conditionally independent from one another; see the generation of the theta parameters in

http://sli.ics.uci.edu/extras/cs179/Demo%20-%20Bayesian%20Linear%20Regression%20(Pyro).html

CS