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1、1The BellKor Solution to the Netflix Grand PrizeYehuda Koren August 2009I. INTRODUCTIONThis article describes part of our contribution to the “Bell-Kors Pragmatic Chaos” final solution, which won the Netflix Grand Prize. The other portion of the contribution was created while working at AT in our im
2、plementation=0.4. We introduce a single new parameter for each user calleduso that we get our first definition of a time-dependent user-bias:b(1)u(t) = bu+udevu(t)(7)This simple linear model for approximating a drifting behavior requires learning two parameters per user: buandu. The linear function
3、for modeling the user bias meshes well with gradual drifts in the user behavior. However, we also observe sudden drifts emerging as “spikes” associated with a single day or session. For example, we have found that multiple ratings a user gives in a single day, tend to concentrate around a single val
4、ue. Such an effect need not span more thana single day. This may reflect the mood of the user that day, the impact of ratings given in a single day on each other, or changes in the actual rater in multi-person accounts. To address such short lived effects, we assign a single parameter per userand da
5、y, absorbing the day-specific variability. This parameter is denoted by but.In the Netflix data, a user rates on 40 different days on average. Thus, working with butrequires, on average, 40 parameters to describe each user bias. It is expected that but is inadequate as a standalone for capturing the
6、 user bias, since it misses all sorts of signals that span more than a single day. Thus, it serves as an additive component within the previously described schemes. The user bias model (7) becomesb(3)u(t) = bu+udevu(t)+but.(8)The discussion so far leads to the baseline predictorbui=+bu+udevu(tui)+bu
7、,tui+bi+bi,Bin(tui).(9)If used as a standalone predictor, its resulting RMSE would be 0.9605. Another effect within the scope of baseline predictors is related to the changing scale of user ratings. While bi(t) is a user-independent measure for the merit of item i at time t, users tend to respond to
8、 such a measure differently. For example, different users employ different rating scales, and a single user can change his rating scale over time. Accordingly, the raw value of the movie bias is not completely user- independent. To address this, we add a time-dependent scaling feature to the baselin
9、e predictors, denoted by cu(t). Thus, the baseline predictor (9) becomesbui=+bu+udevu(tui)+bu,tui+(bi+bi,Bin(tui)cu(tui). (10)All discussed ways to implement bu(t) would be valid for implementing cu(t) as well. We chose to dedicate a separate parameter per day, resulting in: cu(t) = cu+cut. As usual
10、, cu is the stable part of cu(t), whereas cutrepresents day-specific variability. Adding the multiplicative factor cu(t) to the baseline pre- dictor (as per (10) lowers RMSE to 0.9555. Interestingly, this basic model, which captures just main effects disregarding user-item interactions, can explain
11、almost as much of the datavariability as the commercial Netflix Cinematch recommender system, whose published RMSE on the same Quiz set is 0.9514 4.B. FrequenciesIt was brought to our attention by our colleagues at the Pragmatic Theory team (PT) that the number of ratings a usergave on a specific da
12、y explains a significant portion of the variability of the data during that day. Formally, denote by Fuithe overall number of ratings that user u gave on day tui. The value of Fuiwill be henceforth dubbed a “frequency”, following PTs notation. In practice we work with a rounded logarithm of Fui, den
13、oted by fui= logaFui.1 Interestingly, even though fuiis solely driven by user u,it will influence the item-biases, rather than the user-biases. Accordingly, for each item i we introduce a term bif, capturingthe bias specific for the item i at log-frequency f. Baseline predictor (10) is extended to b
14、ebui=+bu+udevu(tui)+bu,tui+(bi+bi,Bin(tui)cu(tui)+bi,fui. (11) We note that it would be sensible to multiply bi,fuiby cu(tui), but we have not experimented with this. The effect of adding the frequency term to the movie bias is quite dramatic. RMSE drops from 0.9555 to 0.9278. Notably, it shows a ba
15、seline predictor with a prediction accuracysignificantly better than that of the original Netflix Cinematch algorithm. Here, it is important to remind that a baseline predictor, no matter how accurate, cannot yield personalized recommenda- tions on its own, as it misses all interactions between user
16、s and items. In a sense, it is capturing the portion of the data that is less relevant for establishing recommendations and in doing so enables deriving accurate recommendations by subjecting other models to cleaner data. Nonetheless, we included two of the more accurate baseline predictors in our blend. Why frequencies work?: In order to grasp the source of frequencies contribution, we make two empirical observations. First, we could see that frequencies are extremely
