Pre-release Prediction of Crowd Opinion on Movies by Label Distribution Learning

1 Introduction

The movie industry is a worldwide business worth tens of billions of dollars. Thousands of new movies are produced and shown in movie theatres each year, among which some are successful, many are not. For movie producers, the increasing cost and competition boosts the investment risk. For movie audience, the prevalent immodest advertisement and promotion makes it hard to choose a movie worth watching. Therefore, both sides demand a reliable prediction of what people will think about a particular movie before it is actually released or even during its planing phase. However, to our best knowledge, there is little work, up to the present, on pre-release prediction of the crowd opinion about movies.

Aside from the unstructured reviews and discussions about the movie [Diao et al., 2014], an explicit and well-structured reflection of the crowd opinion might be the distribution of ratings given by the audience who have watched the movie, just as many movie review web sites collect from their users, such as IMDb and Netflix. Note that the average rating is not a good indicator of the crowd opinion because the averaging process mingles those who like the movie and those who dislike it. From the marketing point of view, a movie with controversial crowd opinions (i.e., the rating distribution concentrates at both low and high ratings) is generally much more successful than another movie with a consistent medium rating. But they might have similar average ratings. Fig. 1 gives a typical example of such case. According to the data from IMDb, the movies Twilight and I, Frankenstein both have the same average rating 5.2/10. But the top two popular ratings in the rating distribution of Twilight are the lowest rating 1 (15.7%) and the highest rating 10 (15.3%), respectively, while those of I, Frankenstein concentrate at the medium ratings 6 (21.4%) and 5 (20.1%). As a result, the budget/gross ratio of Twilight is $37m/$191m and that of I, Frankenstein is $65m/$19m. Obviously, the former movie is more worthy to invest and watch. Note that the usage of the rating distribution is not limited to gross prediction, but includes marketing strategy, advertising design, movie recommendation, etc.

∗This research was supported by NSFC (61273300, 61232007), JiangsuSF (BK20140022), and the Key Lab of Computer Network and Information Integration of Ministry of Education of China.

It is worth emphasizing, as will be further discussed in Section 2, that predicting the crowd opinion (overall rating distribution) is quite different from predicting the individual opinion (person-specific rating). The latter problem has been extensively studied in the area of recommender systems [Adomavicius and Tuzhilin, 2005]. While personalized rating is valuable when recommending a movie to a particular user, it does not mean much when analysing the crowd opinion toward a movie. Moreover, recommendation according to the individual rating prediction is worthwhile even after many users have already watched the movie, as long as the target user has not. But crowd opinion prediction is generally only useful before any user ratings are available. As a result, the prediction should only be based on the metadata available before the movie is released.

Instead of putting the pre-release crowd opinion prediction in the recommender system scenario, this paper regards it as a Label Distribution Learning (LDL) [Geng et al., 2013; 2010] problem since the rating distribution can be naturally viewed as a label distribution for each movie. According to the characteristics of the movie rating distribution, we propose a novel Label Distribution Support Vector Regressor (LDSVR), which can give multivariate and probabilistic output. The key idea of LDSVR is to fit a sigmoid function to each component of the distribution simultaneously by a support vector machine.

The rest of this paper is organized as follows. Section 2 introduces some existing work related to pre-release crowd opinion prediction. After that, the proposed method LDSVR is described in Section 3. Then, experimental results are reported in Section 4. Finally, conclusions and discussions are given in Section 5.

2 RelatedWork

Movie rating prediction (abbreviated as RP) is a widely investigated topic in the context of recommender systems [Adomavicius and Tuzhilin, 2005], where the problem of movie recommendation is reduced to predicting movie ratings for a particular user and then recommend the movie with the highest rating to the user. The RP methods in such case are usually classified into three categories [Marovic et al., 2011], i.e., content-based methods [Soares and Viana, 2014; Pil´aszy and Tikk, 2009], collaborative methods [Bresler et al., 2014; Diao et al., 2014], and hybrid methods [Amolochitis et al., 2014; Jin et al., 2005]. However, the pre-release crowd opinion prediction (abbreviated as PCOP) problem raised in this paper is fundamentally different from the aforementioned RP problem mainly in the following three aspects.

1. RP usually aims to predict the rating for a particular user while PCOP aims to predict the overall rating distribution generated from many users' ratings, i.e., the crowd opinion rather than the individual opinion.
2. Existing RP methods usually require previous ratings on the already watched movies from the target user (content-based methods) or other users with similar preferences (collaborative methods). On the other hand, PCOP methods do not require any previous rating data once the model is trained because they do not need to learn the preference of any particular user.
3. Most RP methods, especially the prevailing collaborative methods, cannot predict the rating of a movie until it is officially released. Many of them further rely on a sufficient number of users who have already watched and rated that movie. However, PCOP methods can make predictions before the movie is actually released or even as early as during its planning phase.

Another more related work is Label Distribution Learning (LDL) [Geng and Ji, 2013] recently proposed to deal with a new machine learning paradigm where each instance is annotated by a label distribution rather than a single label (singlelabel learning) or multiple labels (multi-label learning). The label distribution covers a certain number of labels, representing the degree to which each label describes the instance. Let X = Rq denote the input space, Y = {y1, y2, · · · , yc} denote the complete set of labels, and dyx denote the description degree of the label y ∈ Y to the instance x ∈ X. Given a training set S = {(x1, d1), (x2, d2), · · · , (xn, dn)}, where di = [dy1 xi , dy2 xi , · · · , dyc xi ]T is the label distribution associated with the instance xi, the goal of LDL is to learn a mapping from x ∈ Rq to d ∈ Rc based on S. Geng et al. [2013] construct the mapping via a conditional mass function p(y|x) formulated as a maximum entropy model. Then they use Improved Iterative Scaling (IIS) [Pietra et al., 1997] or BFGS [Nocedal and Wright, 2006] to minimize the KullbackLeibler divergence between the predicted distribution and the ground truth distribution, resulting in two LDL algorithms named IIS-LLD and BFGS-LLD, respectively. They also propose a neural-network-based approach named CPNN when the labels can be ordered [Geng et al., 2013]. As a learning framework more general than single-label and multilabel learning, LDL has been successfully applied to various problems, such as facial age estimation [Geng et al., 2013; 2010], head pose estimation [Geng and Xia, 2014], and multilabel ranking for natural scene images [Geng and Luo, 2014]. If we regard the movie pre-release feature vector as the instance x, the rating as the label y and the rating distribution as the label distribution d, then the PCOP problem can be naturally viewed as an LDL problem.

3 Label Distribution Support Vector Regression

In this section, we propose a Label Distribution Support Vector Regressor (LDSVR) for pre-release crowd opinion prediction. Compared with standard SVR [Drucker et al., 1996], LDSVR must address two additional issues: (1) How to output a distribution composed by multiple components? (2) How to constrain each component of the distribution within the range of a probability, i.e., [0, 1]? The first issue might be tackled by building a single-output SVR for each component respectively. But as pointed out in [P´erez-Cruz et al., 2002], this will cause the problem that some examples beyond the insensitive zone might be penalized more than once. It is also prohibitive to view the distribution as a structure and solve the problem via structured prediction [Tsochantaridis et al., 2004] because there is no direct way to define the auxiliary discriminant function that measures the compatibility between a movie and its rating distribution. A more rational solution to this issue might root in the Multivariate Support Vector Regression (M-SVR) [Fern´andez et al., 2004], which can output multiple variables simultaneously. For the second issue, we are inspired by the common practice in classification tasks to fit a sigmoid function after the SVM when a probabilistic output for each class is expected [Platt, 1999]. For regression tasks, the sigmoid function could directly act as the target model of the regression instead of a post-process. Thus, the basic idea of LDSVR is, in short, to fit a sigmoid function to each component of the label distribution simultaneously by a support vector machine.

Suppose the label distribution d of the instance x is modeled by an element-wise sigmoid vector

where ϕ(x) is a nonlinear transformation of x to a higherdimensional feature space RH, W ∈ Rc×H and b ∈ Rc are the model parameters, z = Wϕ(x) + b, and g(z) is a short representation for the vector obtained by applying the sigmoid function g(·) to each element of z. Then, we can generalize the single-output SVR by minimizing the sum of the target functions on all dimensions

where wj is the transpose of the j-th row of W and L(ui) is the loss function for the i-th example. In standard SVR [Drucker et al., 1996], the unidimensional loss is defined as a hinge loss function

where dji and f j (xi) are the j-th elements in the corresponding vectors. This will create an insensitive zone determined by ε around the estimate, i.e., the loss less than ε will be ignored. If we directly sum the loss functions on all dimensions, i.e., L(ui) = ∑j Lh(uji ), then, as pointed out in [P´erezCruz et al., 2002], some examples beyond the insensitive zone might be penalized more than once. Fig. 2 illustrates

0 < uj- = g(zj ) - g(zj - 4ε) ≤ ε

Figure 2: The insensitive zones around the estimate f(x) = g(z) in the bivariate (d1 and d2 ) output space. The black square represents the hyper-cubic insensitive zone for the two single-output SVRs. The blue circle represents the hyperspherical insensitive zone for M-SVR. The shaded area represents the insensitive zone for LDSVR.

this problem via a bivariate regression case, where the black square represents the hyper-cubic insensitive zone for the two single-output SVRs. As can be seen, all the examples falling into the area ρ1 will be penalized once while those falling into the area ρ2 will be penalized twice. Also, the L1-norm uji in Eq. (3) is calculated dimension by dimension, which makes the solution complexity grow linearly with the increase of dimensionality [Fern´andez et al., 2004]. As suggested in [Fern´andez et al., 2004], we can instead use the L2-norm to define L(ui) so that all dimensions can join the same constraint and yield the same support vector, i.e.,

This will generate a hyper-spherical insensitive zone with the radius ε, which is represented by the blue circle in Fig. 2. Unfortunately, substituting Eq. (1), (5)-(7) into Eq. (2) does not lead to a convex quadratic form and the optimization process will not just depend on inner product. Therefore, it is hard to find the optimum as well as to apply the kernel trick.

To solve this problem, we propose an alternative loss function which can reform the minimization of Eq. (2) to a convex quadratic programming process depending only on inner product. Note that Eq. (6) calculates the Euclidean distance from the estimate f(xi) = g(zi) to the ground truth di. We can instead measure the loss by calculating how far away from zi another point z′i ∈ Rc should move to get the same output with the ground truth, i.e., g(z′i ) = di. Solving this

z

Figure 3: The relationship between ui and u′i . The boundaries of the insensitive zone defined on u′i have equal distance to the sigmoid curve horizontally, but not vertically.

equation yields z′i = - log(1/di - 1). Thus, the distance u′i from z′i to zi can be calculated by

The relationship between ui and u′i is illustrated in Fig.3 and quantified in the following lemma.

Lemma 1. u′i ≥ 4ui for any xi, di, W, and b.

The proof of Lemma 1 is given in the Appendix.

Replacing ui with u′i/4 in the loss function Eq. (5) will generate an insensitive zone around the sigmoid function as represented by the shaded area in Fig. 3. Note that the vertical distance (along the d axis) from the two boundaries of the insensitive zone to the sigmoid curve might be different. This will result in an insensitive zone in the bivariate output space as represented by the shaded area in Fig. 2. It can be derived from Lemma 1 that, in Fig. 2, 0 < uj+ ≤ ε and 0 < uj- ≤ ε for j = 1, 2. Thus, although not strictly isotropic, the shaded area is a reasonable approximation to the ideal hyper-spherical insensitive zone (the blue circle) so long as ε is small.

Replacing ui with u′i/4 in Eq. (2) yields a new target function Γ′ (W, b). It is trivial to prove the following theorem with Lemma 1.

Theorem 1. Γ′ (W, b) is an upper bound for Γ(W, b).

Therefore, minimizing Γ′ (W, b) is equivalent to minimizing an upper bound of Γ(W, b).

It is still hard to minimize Γ′ (W, b) as standard SVR does via solving its dual problem. Instead, we directly solve the primal problem with an iterative quasi-Newton method called Iterative Re-Weighted Least Square (IRWLS) [P´erez-Cruz et al., 2000]. Firstly, Γ′ (W, b) is approximated by its first order Taylor expansion at the solution of the current k-th iteration, denoted byW (k) and b(k) :

where e′(k) i and u′(k) i are calculated from W (k) and b(k) . Then, a quadratic approximation is further constructed from Eq. (10):

where

and τ is a constant term that does not depend on W and b. Eq. (11) is a weighted least square problem whose optimum can be effectively found by letting the gradient equal zero and then solving a system of linear equations for j = 1, . . . , c:

where Φ = [ϕ(x1), ..., ϕ(xn)]T, a = [a1, ..., an]T, (Da)ij = aiδij (δij is the Kronecker's delta function), dj = [dj1 , ..., djn ]T, 1 is a vector of all ones. Then, the direction of the optimal solution of Eq. (11) is used as the descending direction for the optimization of Γ′ (W, b), and the solution for the next iteration (W (k+1) and b(k+1) ) is obtained via a line search algorithm [Nocedal and Wright, 2006] along this direction.

According to the Representer Theorem [Schlk¨opf and Smola, 2001], wj may be represented as a linear combination of the training examples in the feature space, i.e., wj = ΦTβj . Substituting this expression into Eq. (13) yields

where Kij = k(xi, xj ) = ϕT(xi)ϕ(xj ) is the kernel matrix. Accordingly, the later line search for Γ′ (W, b) can be performed in terms of βj . Finally, after the optimal solution ˆ βj is obtained, ˆwj = ΦT ˆ βj and ˆbj are substituted into Eq. (1) and the label distribution can be calculated indirectly in the original input space X, rather than the high-dimensional feature space RH, via the kernel matrix K.

4 Experiments

4.1 Methodology

The data set used in the experiments includes 7, 755 movies and 54, 242, 292 ratings from 478, 656 different users. The ratings come from Netflix, which are on a scale from 1 to 5 integral stars. Each movie has, on average, 6, 994 ratings. The rating distribution is calculated for each movie as an indicator for the crowd opinion on that movie. The pre-release metadata are crawled from IMDb according to the unique movie IDs. Table 1 lists all the metadata included in the data set. Note that all the attributes in Table 1 can be retrieved before the movie is officially released or even during its planning phase. No post-release attributes are included in the data set, although some of them might be closely related to the crowd opinion, such as the box office gross. There are both numeric (N) and categorical (C) attributes in this data set. Some categorical attributes, typically human names, might have too many different values. In such case, we set a threshold θ and re-assign a new value 'other' to those who appear less times in the data set than the threshold. This will filter out most categories with limited influence on the crowd opinion, e.g., unfamous actors. The categorical attributes are then transformed into numeric ones by replacing the k-valued categorical attribute by k binary attributes, one for each value indicating whether the attribute has that value or not. Finally, all the attributes are normalized to the same scale through the min-max normalization.

As mentioned in Section 3, LDSVR deals with two challenges simultaneously: (1) multivariate output and (2) probability output. In order to show the advantages of solving these two problems simultaneously, LDSVR is compared with two baseline variants. The first is to fit a sigmoid function to each dimension separately, i.e., replace uji in Eq. (3) by the absolute value of the j-th element of e′i calculated by Eq. (9), and then use Lh(|e′j i |) as the loss function. This variant is named as S-SVR standing for Sigmoid SVR, which solves the challenge (2) but not (1). The second variant is to firstly run a standard M-SVR [Fern´andez et al., 2004], and then perform a post-process where the outputs are subtracted by a common bias determined by the minimum regression output over the training set and then divided by the sum of all elements. This variant is named as M-SVRp standing for M-SVR plus postprocess, which solves the challenge (1) but not (2). Note that the output of all regression methods should be finally normalized by dividing with the sum of all elements to make sure that ∑j dj = 1. This is reasonable in most cases where only the relative relationship matters in the distribution. LDSVR is also compared with existing typical LDL algorithms including BFGS-LLD [Geng and Ji, 2013], IIS-LLD [Geng et al., 2010], AA-kNN [Geng and Ji, 2013], and CPNN [Geng et al., 2013].

The performance of the algorithms is evaluated by those commonly used measures in LDL, i.e., the average distance or similarity between the predicted and ground truth label distributions. As suggested in [Geng and Ji, 2013], six measures are used in the experiments, which include four distance measures (the smaller the better), i.e., Euclidean, Sørensen, Squared χ2 , and Kullback-Leibler (K-L), and two similarity measures (the larger the better), i.e., Intersection and Fidelity.

The algorithm parameters used in the experiments are empirically determined. The parameter selection process is nested into the 10-fold cross validation. In detail, the whole data set is first randomly split into 10 chunks. Each time, one chunk is used as the test set, another is used as the validation set, and the rest 8 chunks are used as the training set. Then, the model is trained with different parameter settings on the training set and tested on the validation set. This procedure is repeated 10 folds, and the parameter setting with the best average performance is selected. After that, the original validation set is merged into the training set and the test set remains unchanged. The model is trained with the selected parameter setting on the updated training set and tested on the test set. This procedure is repeated 10 folds and the mean value and standard deviation of each evaluation measure is reported. The final parameter settings for the compared algorithms are as follows. All kernel based methods (LDSVR, S-SVR and M-SVRp) use the RBF kernel with the scaling factor σ equal to the average distance among the training examples. The penalty parameter C in Eq. (2) is set to 1. The insensitivity parameter ε is set to 0.1. All iterative algorithms terminate their iteration when the difference between adjacent steps is smaller than 10-10 . The number of neighbors k in AA-kNN is set to 10, and the number of hidden neurons in CPNN is set to 80.

4.2 Results

The experimental results of the seven algorithms on the movie data set are tabulated in Table 2. For the four distance measures (Euclidean, Sørensen, Squared χ2 , and K-L), '↓' indicates 'the smaller the better'. For the two similarity measures (Intersection and Fidelity), '↑' indicates 'the larger the better'. The best performance on each measure is highlighted by boldface. On each measure, the algorithms are ranked in decreasing order of their performance. The ranks are given in the brackets right after the measure values.

As can be seen from Table 2, LDSVR performs best on all of the six measures. The two variants of LDSVR, S-SVR and M-SVRp, both perform significantly worse than LDSVR. This proves the advantage of LDSVR gained by means of solving the multivariate output problem and the probability output problem simultaneously, rather than one by one. The performance of LDSVR is also significantly better than that of the compared LDL algorithms (BFGS-LLD, IIS-LLD, AA-kNN, and CPNN). The reason might be two-fold. First, while most existing LDL algorithms seek to directly minimize the K-L divergence between the predicted and ground truth distributions, LDSVR takes advantage of the large margin regression by a support vector machine. Second, application of the kernel trick makes it possible for LDSVR to solve the problem in a higher-dimensional thus more discriminative feature space without loss of computational feasibility.

5 Conclusion and Discussion

This paper investigates possibilities to predict the crowd opinion about a movie before it is released. The crowd opinion is represented by the distribution of ratings given by a sufficient number of people who have watched the movie. The pre-release prediction of the crowd opinion could be a crucial indicator of whether the movie will be successful or not, and thus has vast potential applications in movie production and marketing.

This paper regards the pre-release crowd opinion prediction as a Label Distribution Learning (LDL) problem, and proposes a Label Distribution Support Vector Regressor (LDSVR) for it. The nature of user rating distribution requires the output of LDSVR to be both multivariate and probabilistic. Thus, the basic idea of LDSVR is to fit a sigmoid function to each component of the rating distribution simultaneously by a multi-output support vector machine. Experiments on a data set including 7, 755 movies reveal that LDSVR can perform significantly better than not only its two variants, but also four typical LDL algorithms.

One of the key ideas of LDSVR is to measure the loss by the distance needed to move in the input space to get the same output as the ground truth. This works fine for most cases, but sometimes could be risky when the ground truth outputs of some training examples are very close to 0 or 1. In such case, the loss of those examples will tend to be so large that they will dominate the training process. Although this is not a big problem for the movie rating distributions so long as there are a sufficient number of ratings for each movie, it might be troublesome for some other data. Therefore, a mechanism of compensating the loss of those examples with outputs close to 0 or 1 might be an important future work to make LDSVR applicable to more general cases.

Appendix: Proof of Lemma 1

Proof. Firstly, consider the situation in the (zj , dj )-space (j = 1, . . . , c), where zj and dj represent the j-th dimension of z and d, respectively. The projections of ui and u′i in such space are denoted by uji and u′j i , respectively. Since ui = √∑j (uji )2 and u′i = √∑j (u′j i )2 , if we can prove u′j i ≥ 4uji for j = 1, . . . , c, then we have u′i ≥ 4ui.

In the (zj , dj )-space, the sigmoid function dj = g(zj ) is symmetric about the point (0, 0.5). So we only need to consider the case zj ≥ 0, where g(zj ) is concave as shown in Fig. 4. When the point is above the curve like P ′ , we can always find its counterpart P below the curve with same vertical and horizontal distance to the curve. Thus, we only need to consider the case when the point is below the curve.

Since g(zj ) is concave, the line segment AB (blue dash line) is always below the curve. Suppose its slope is θ, and the slope of the g(zj ) curve's tangent line (red dash line) at the point A is θ′ , then θ ≤ θ′ . The derivative of g(zj ) is

Letting the second derivative of g(zj ) equal zero yields zj = 0. Thus, the maximum of θ′ is g′ (0) = 1/4. Therefore,