## Logistic Matrix Factorization for Implicit Feedback Data

Christopher Johnson, Spotify, 2014

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A new matrix factorisation model for behavioural recommendation in the case of implicit feedback is presented.

User-item interactions are encoded in a non-negative interaction matrix. The question as to whether a user-item interaction occurred is then treated as a problem of binary classification. User-item pairs for which an interaction has occurred are regarded as positive outcomes with confidence in constant proportion to the value in the interaction matrix, while the absence of an interaction is regarded as a negative outcome. This binary classification is task is then leveraged to train user- and item-vectors. These vectors reside in dually paired spaces. The dot-product of the vectors, combined with user- and item- bias terms, is then fed into the sigmoid function. What we are really doing is looking for a low-rank approximation to a bilinear form via the sigmoid function.

Confidence values (proportional to the values in the interaction matrix) are used as powers of their corresponding factor in the maximum likelihood function. The constant proportion that defines the confidence values from the interaction matrix is a tuning parameter, but is typically chosen so that the positive outcomes balance the negative outcomes in total confidence. Thus the likelihood function is weighed according to the entries of the interaction matrix. The weighted likelihood function is then maximised using alternating gradient ascent. This optimisation is batch. Negative sampling can be used to speed up convergence, and the confidence parameter is decreased proportionally.

Evalutation
They use a fractional rank type metric to evaluate performance at each iteration. For each user, the interaction probability is computed for each item, and the rank of the target item in this list is determined. This is then averaged over a set-aside collection of user-item pairs. Given that batch gradient descent is used, this is not prohibitively expensive.

The author reports that this logistic matrix factorisation model performs better in low rank than the implicit MF model of Koren et al 2008, though both give a similar fractional rank in high ranks.

Implementation
A very basic implementation in Python is available. The implementation uses AdaGrad to dynamically chose a step size at each iteration, just as is described in the paper. The paper mentions a Spark or Hadoop based implementation, but I couldn’t find this published anywhere.