Metadata Embeddings for User and Item Cold-start Recommendations

Maciej Kula (Lyst)
CBRecSys 2015 (arxiv)

Kula presents a model for cold start recommendation, which he calls “LightFM”.  Users and items are considered as sets of binary features. For example:

\text{alice} = \{ \text{domain}:\text{gmail} \}

\text{itemXYZ} = \{\text{description}:\text{pleated}, \text{description}:\text{skirt}, \text{tag}:\text{chanel} \}.

Each of these features (e.g. each tag, each word and each email domain) corresponds to a parameter vector and a bias term.  A user vector (or item vector) is then the sum of the vectors associated to its constituent features.  Similarly, a user (item) bias term is just the sum of the bias terms associated to its features.

The probability \hat{r}(u, i) of an interaction between a user u and an item i is modelled as the sigmoid of the dot product of the user vector and the item vector, along with the bias terms associated with the user and the item:

\hat{r}(u, i) := \sigma (vec(u) \cdot vec(i) + bias(u) + bias(i))

The model is trained on a set S_{+} of user-item pairs observed as having interacted, and on a set S_{-} of user-item pairs that were not observed to have interacted (in the case of implicit feedback recommendation) or to have interacted negatively (in the case of explicit feedback recommendation).  Specifically, these interactions and non-interactions are assumed independent and the likelihood

\displaystyle L = \prod_{(u, i) \in S_{+}} \hat{r}(u,i) \cdot \prod_{(u, i) \in S_{-}} (1 - \hat{r}(u,i))

is then maximised using stochastic gradient descent and with adaptive per-parameter learning rates determined by Adagrad.

Trivial featurisation gives matrix factorisation

Note that users (or items) can be featurised trivially using their ids.   We create one user feature for each user id, so that the user-feature matrix is the identity matrix.  In this case, we have a separate parameter vector for each user.  If we do this for both users and items, then the model is just a (sigmoid-) factorisation of the user-item interaction matrix. This is then the case of Johnson’s logistic matrix factorization.

Evaluation

Performance is evaluated on MovieLens for explicit feedback recommendation and on CrossValidated (one of the StackExchange websites) for implicit feedback recommendation.  In both cases, warm- and cold-start scenarios are tested.  Warm start is tested by holding out interactions in such a way that every item and every user is still represented in the training interaction data.  Cold start is tested by holding out all interactions for some items.  Model accuracy is measured by considering each user in the set of test interactions, considering the binary classification task of labelling each item as having been interacted with or not and then measuring the area under the curve of the associated ROC curve.  The mean is that taken over all users in the test set.

LightFM seems to perform well in both cold and warm start scenarios.

Engineering Notes

Kula included some interesting notes on the production use of LightFM at Lyst.  Training is incremental with model state stored in the database.

Implementation and Examples

Available on GitHub and extensively documented.  Written in Cython.  In addition to the logistic loss used above, Bayesian Personalised Ranking and WARP are supported.

 

Collaborative Filtering for Implicit Feedback Datasets

Hu, Koren and Volinsky (AT&T, Yahoo!), 2008.

A well-written paper.

PDF

The authors give a good description of the distinctions between explicit and implicit feedback datasets, pointing out in particular that:

  1. implicit feedback data is inherently noisy, since a user might decide that they do not like an item after viewing it — interaction does not necessarily indicate interest.
  2. the numerical value in explicit feedback indicates preference whereas in the implicit case indicates confidence.

The authors describe their model as being based on SVD, but this is not accurate, since they weight squared difference summands in the cost function according to a confidence value (which is proportional to the number of interactions for that user-item pair).

The input matrix is the user-item matrix.

Optimisation is via alternating least squares.

Their evaluation metric is percentile rank based.

Their model, which we’ll call “weighted SVD” (they speak of “confidence intervals”) compares favourably with the baseline popularity method and also with an old-school item-based neighbourhood method, in terms of the expected percentile rank (Figure 1). Interestingly, the differences are less marked when the probability that a desired item is in the top (say) 1% is considered (Figure 2).

The unweighted SVD on the user-item matrix is shown to perform terribly, with a significant but insufficient improvement obtained with regularisation.

Logistic Matrix Factorization for Implicit Feedback Data

Christopher Johnson, Spotify, 2014

PDF

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.