The BellKor solution to the Netflix Prize

Our final solution (RMSE=0.8712) consists of blending 107 individual results. Since many of these results are close variants, we first describe the main approaches behind them. Then, we will move to describing each individual result. The core components of the solution are published in our ICDM'2007 paper [1] (or, KDD-Cup'2007 paper [2]), and also in the earlier KDD'2007 paper [3]. We assume that the reader is familiar with these works and our terminology there. A movie-oriented k-NN approach was thoroughly described in our KDD-Cup'2007 paper [kNN]. We apply it as a post-processor for most other models. Interestingly, it was most effective when applied on residuals of RBMs [5], thereby driving the Quiz RMSE from 0.9093 to 0.8888. An earlier k-NN approach was described in the KDD'2007 paper ([3], Sec. 3) [Slow-kNN]. It appears that this earlier approach can achieve slightly more accurate results than the newer one, at the expense of a significant increase in running time. Consequently, we dropped the older approach, though some results involving it survive within the final blend. We also tried more naïve k-NN models, where interpolation weights are based on pairwise similarities between movies (see [2], Sec. 2.2). Specifically, we based weights on corr 2 /(1-corr 2) [Corr-kNN], or on mse-10 [MSE-kNN]. Here, corr is the Pearson correlation coefficient between the two respective movies, and mse is the mean squared distance between two movies (see definition of s ij in Sec. 4.1 of [2]). We also tried taking the interpolation weights as the "support-based similarities", which will be defined shortly [Supp-kNN]. Other variants that we tried for computing the interpolation coefficients are: (1) using our KDD-Cup'2007 [2] method on a binary user-movie matrix, which replaces every rating with " 1 " , and sets non-rated user-movie pairs to " 0 " [Bin-kNN]. (2) Taking results of factorization, and regressing the factors associated with the target movie on the factors associated with its neighbors. Then, the resulting regression coefficients are used as interpolation weights [Fctr-kNN]. As explained in our papers, we also tried user-oriented k-NN approaches. Either in a profound way (see: [1], Sec. 4.3; [3], Sec. 5) [User-kNN], or by just taking weights as pairwise similarities among users [User-MSE-kNN], which is the user-oriented parallel of the aforementioned [MSE-kNN]. Prior to computing interpolation weights, one has to choose the set of neighbors. We find the most similar neighbors based on an appropriate similarity measure. In …