Exploiting recommendation confidence in decision-aware recommender systems

Abstract

The main goal of a Recommender System is to suggest relevant items to users, although other utility dimensions – such as diversity, novelty, confidence, possibility of providing explanations – are often considered. In this work, we investigate about confidence but from the perspective of the system: what is the confidence a system has on its own recommendations; more specifically, we focus on different methods to embed awareness into the recommendation algorithms about deciding whether an item should be suggested. Sometimes it is better not to recommend than fail because failure can decrease user confidence in the system. In this way, we hypothesise the system should only show the more reliable suggestions, hence, increasing the performance of such recommendations, at the expense of, presumably, reducing the number of potential recommendations. Different from other works in the literature, our approaches do not exploit or analyse the input data but intrinsic aspects of the recommendation algorithms or of the components used during prediction are considered. We propose a taxonomy of techniques that can be applied to some families of recommender systems allowing to include mechanisms to decide if a recommendation should be generated. In particular, we exploit the uncertainty in the prediction score for a probabilistic matrix factorisation algorithm and the family of nearest-neighbour algorithms, the support of the prediction score for nearest-neighbour algorithms, and a method independent of the algorithm. We study how the performance of a recommendation algorithm evolves when it decides not to recommend in some situations. If the decision of avoiding a recommendation is sensible – i.e., not random but related to the information available to the system about the target user or item –, the performance is expected to improve at the expense of other quality dimensions such as coverage, novelty, or diversity. This balance is critical, since it is possible to achieve a very high precision recommending only one item to a unique user, which would not be a very useful recommender. Because of this, on the one hand, we explore some techniques to combine precision and coverage metrics, an open problem in the area. On the other hand, a family of metrics (correctness) based on the assumption that it is better to avoid a recommendation rather than providing a bad recommendation is proposed herein. In summary, the contributions of this paper are twofold: a taxonomy of techniques that can be applied to some families of recommender systems allowing to include mechanisms to decide if a recommendation should be generated, and a first exploration to the combination of evaluation metrics, mostly focused on measures for precision and coverage. Empiric results show that large precision improvements are obtained when using these approaches at the expense of user and item coverage and with varying levels of novelty and diversity.

Appendix: Complete derivation of variational Bayesian

The goal of this algorithm is to minimise the objective function in (6) using a probabilistic model where R are the observations, U and I are the parameters, and some regularisation is introduced through priors on these parameters. As before, user u is represented by the u-th row of matrix U and item i is represented by the i-th column of matrix I; nonetheless, from now on, we shall consider these column vectors as row vectors.

The first hypothesis of the model is that the rating probability, considering matrices U and I, follows a Normal distribution with mean u^⊤i and variance τ²:

$$ P(r_{ui}\mid U,I) = N(u^{\top} \cdot i, \tau^{2}) $$

(32)

Moreover, the entries of these matrices U and I follow independent distributions; in particular, $u_{l} \sim N(0,{\sigma _{l}^{2}})$ and $i_{l} \sim N(0,{\rho _{l}^{2}})$. As a consequence, the density functions of U and I are formulated as in (33):

$$ P(U) = \prod\limits_{u = 1}^{V}\prod\limits_{l = 1}^{D}N(u_{l}\mid 0,{\sigma_{l}^{2}}) \qquad P(I) = \prod\limits_{i = 1}^{J}{\prod}_{l = 1}^{D}N(i_{l}\mid 0,{\rho_{l}^{2}}) $$

(33)

The goal now is to compute or approximate the posterior probability P(U, I∣R), so, in this way, we can compute a predictive distribution on ratings given the observation matrix, as in (34):

$$ P(\hat{r}_{ui}\mid R) = \int P(\hat{r}_{ui}\mid u,i,\tau^{2})P(U,I\mid R) $$

(34)

Since computing this posterior probability P(U, I∣R) is unfeasible, the proposed algorithm aims at approximating such distribution by using Bayesian inference, more specifically, through the method known as Mean-Field Variational Inference.

1.1 Bayesian inference

Bayesian inference is a method that approximates a posterior distribution of a set of unobserved variables Z = {Z₁, ⋯ , Z_D} knowing a subset of information R, P(Z∣R), through a variational distribution Q(Z) (Box and Tiao 2011). For this, a function Q(Z) is sought that minimises the dissimilarity function d(Q, P). In the specific case of the Mean-Field Variational Inference method, such a dissimilarity function is the Kullback-Leibler divergence between P and Q (Bishop 2006):

$$ D_{KL}(Q \mid \mid P) = \sum\limits_{Z}Q(Z) \log{\frac{Q(Z)}{P(Z \mid R)}} = \sum\limits_{Z} Q(Z) \log \frac{Q(Z)}{P(Z,R)} + \log P(R) $$

(35)

where we use that P(Z∣R) = P(Z, R)/P(R).

Furthermore, by solving (35) for log P(R) we obtain the following (36):

$$ \log P(R) = D_{KL}(Q \mid \mid P) - \sum\limits_{Z}Q(Z) \log \frac{Q(Z)}{P(Z,R)} = D_{KL}(Q \mid \mid P) + \mathscr{F}(Q) $$

(36)

If we set log P(R) as a constant in (36), it is enough with maximising $\mathscr{F}(Q)$ to minimise D_KL(Q∣∣P).

Indeed, $\mathscr{F}(Q)$ is denoted as the (negative) variational free energy, since it can be written as a sum of Q’s entropy and an energy (37):

$$\begin{array}{@{}rcl@{}} \mathscr{F}(Q) &=& -\sum\limits_{Z}Q(Z) \log Q(Z) + \sum\limits_{Z}{Q(Z) \log{P(Z,R)}}\\ &=& H(Q) + \mathbb{E}[\log P(Z,R)] = \mathbb{E}_{Q(z)}[\log P(Z,R) - \log Q(Z)] \end{array} $$

(37)

1.2 Bayesian inference in VB

Once we apply the Mean-Field Variational Inference method presented above where Z = {U, I} are the unobserved variables, and the rating matrix R is the known information, the goal is to estimate a variational distribution function Q(U, I) that approximates the distribution P(U, I∣R).

For this, we aim at maximising the so-called (negative) variational free energy defined as in (38):

$$\begin{array}{@{}rcl@{}} \mathscr{F}(Q(U,I)) &=& \mathbb{E}_{Q(U,I)}[\log P(U,I,R) - \log Q(U,I)]\\ &=& -\sum Q(U,I)(\log Q(U,I) - \log P(U,I,R)) \end{array} $$

(38)

In practice, it is intractable to maximise F(U, I) and, because of this, this algorithm applies another simplification by considering that Q(U, I) = Q(U)Q(I). We now arrive to the definitive formulation for $\mathscr{F}(Q(U)Q(I))$, (39), as it is defined in Lim and Teh (2007):

$$\begin{array}{@{}rcl@{}} \mathscr{F}(Q(U)Q(I)) &=& \mathbb{E}_{Q(U)Q(I)}[\log P(U,I,R) - \log Q(U,I)]\\ &=& \mathbb{E}_{Q(U)Q(I)}\left[\log \frac{P(R \mid U,I)}{P(U,I)}\right] + H\left( Q(U,I)\right)\\ &=& \mathbb{E}_{Q(U)Q(I)}\left[\log P(R \mid U,I) - \log P(U) - \log P(I)\right] + H(Q(U,I))\\ &=& -\frac{K}{2} \log (2\pi \tau^{2}) -\frac{1}{2}\sum\limits_{(u,i)}\frac{\mathbb{E}_{Q(U)Q(I)}[(r_{ui}-u^{\top} i)^{2}]}{\tau^{2}}\\ &&- \frac{V}{2} \sum\limits_{l = 1}^{D}\log (2\pi {\sigma_{l}^{2}}) -\frac{1}{2}\sum\limits_{l = 1}^{D}\frac{\sum\limits_{u = 1}^{V}\mathbb{E}_{Q(U)}[{u_{l}^{2}}]}{{\sigma_{l}^{2}}}\\ &&-\frac{J}{2}\sum\limits_{l = 1}^{D}\log (2\pi {\rho_{l}^{2}}) -\frac{1}{2}\sum\limits_{l = 1}^{D} \frac{\sum\limits_{i = 1}^{J}\mathbb{E}_{Q(I)}[{i_{l}^{2}}]}{{\rho_{l}^{2}}}+ H(Q(U,I)) \end{array} $$

(39)

where K is the number of observed entries in R, or, in other terms, the number of ratings or interactions known by the system at training time.

Maximising $\mathscr{F}(Q(U)Q(I))$ with respect to Q(U) keeping Q(I) fixed, and viceversa, we obtain the distributions for items and users. Equations (40) and (41) describe the covariance matrix ϕ_u and user’s u mean $\overline {u}$ in Q(U), respectively.

$$ \phi_{u} = \left( \left( \begin{array}{cccc} \frac{1}{{\sigma_{1}^{2}}} & & & 0 \\ & & {\ddots} & \\ 0 & & & \frac{1}{{\sigma_{D}^{2}}} \end{array}\right) + \sum\limits_{i \in O(u)}\frac{\psi_{i} + \overline{i}^{\top} \overline{i}}{\tau^{2}} \right)^{-1} $$

(40)

$$ \overline{u} = \phi_{u} \left( \sum\limits_{i \in O(u)}\frac{r_{ui}\overline{i}}{\tau^{2}} \right) $$

(41)

where O(u) are the items observed by user u, which corresponds to the identifiers i such that r_ui was observed in the rating matrix R.

Similarly, (42) and (43) show the formulation for the covariance matrix ψ_i and item’s i mean $\overline {i}$ in Q(I), respectively.

$$ \psi_{i} = \left( \left( \begin{array}{cccc} \frac{1}{{\rho_{1}^{2}}} & & & 0 \\ & & {\ddots} & \\ 0 & & & \frac{1}{{\rho_{D}^{2}}} \end{array}\right) + \sum\limits_{u \in O(i)}\frac{\phi_{u} + \overline{u}^{\top} \overline{u}}{\tau^{2}} \right)^{-1} $$

(42)

$$ \overline{i} = \psi_{i} \left( \sum\limits_{u \in O(i)}\frac{r_{ui}\overline{u}}{\tau^{2}} \right) $$

(43)

where O(i) denotes the set of users that have rated item i.

The variances σ_l, ρ_l, and τ can also be estimated and learned by the model. By differentiating (39) with respect to σ_l, ρ_l, and τ, setting the derivatives to zero, and solving for the optimal parameters, we obtain (44), (45), and (46).

$$ {\sigma_{l}^{2}} = \frac{1}{V-1}\sum\limits_{u = 1}^{V}(\phi_{u})_{ll} + \overline{u_{l}}^{2} $$

(44)

$$ {\rho_{l}^{2}} = \frac{1}{J-1}\sum\limits_{i = 1}^{J}(\psi_{i})_{ll} + \overline{v_{l}}^{2} $$

(45)

$$ \tau^{2} = \frac{1}{K-1}\sum\limits_{(u,i)}r_{ui}^{2}-2r_{ui}\overline{u}^{\top} \overline{i} + trace[(\phi_{u} + \overline{u}^{\top} \overline{u})(\psi_{i} + \overline{i}^{\top} \overline{i})] $$

(46)

1.3 Rating prediction in VB

Once the optimal approximation for P(U, I∣R) is calculated, that is, Q(U, I), it can be used to predict future interactions between users and the system. For this, given a matrix R, we find that the distribution of a new rating is the following (47):

$$ P(\hat{r}_{ui}\mid R) \sim \int P(r_{ui}\mid u^{\top} i,\tau) Q(U,I) dUdI = \int N(\hat{r}_{ui}\mid u^{\top} i, \tau^{2})Q(U,I)dUdI $$

(47)

Therefore, we can obtain its mean and, hence, the estimated or predicted rating:

$$ \mathbb{E}(\hat{r}_{ui}\mid R) = \int \int \hat{r}_{ui} N(\hat{r}_{ui}\mid u^{\top} i,\tau^{2}) Q(U,I) dUdId\hat{r}_{ui} $$

(48)

Finally, when we apply Fubini’s theorem, we obtain the following formulation:

$$ \mathbb{E}(\hat{r}_{ui}\mid R) = \int \underbrace{\int \hat{r}_{ui} N(\hat{r}_{ui}\mid u^{\top} i,\tau^{2}) d\hat{r}_{ui}}_{u^{\top} i} Q(U,I) dUdI = \mathbb{E}(u^{\top} i) = \overline{u}^{\top} \overline{i} $$

(49)

1.4 Standard deviation of predicted rating in VB

We use the following explicit formulation for the standard deviation, derived by using mean-field variational inference. First, we use the general form to compute the standard deviation when estimating rating $\hat {r}_{ui}$ for user u and item i:

$$ Var(\hat{r}_{ui}\mid R) = \mathbb{E}(\hat{r}_{ui}^{2}\mid R) - \mathbb{E}(\hat{r}_{ui}\mid R)^{2} $$

(50)

Considering the formulation presented in (49), we know that $\mathbb {E}(\hat {r}_{ui}\mid R)^{2} = (\overline {u}^{\top } \overline {i})^{2} = \overline {i}^{\top } \overline {u} \overline {u}^{\top } \overline {i}$. Hence:

$$\begin{array}{@{}rcl@{}} \mathbb{E}(\hat{r}_{ui}^{2}\mid R) &=& \int \int \hat{r}_{ui}^{2}N(\hat{r}_{ui}\mid u^{\top} i,\tau^{2})Q(U,I)dUdId\hat{r}_{ui}\\ &=& \int \underbrace{\int \hat{r}_{ui}^{2}N(\hat{r}_{ui}\mid u^{\top} i,\tau^{2})d\hat{r}_{ui}}_{\mathbb{E}(r_{ui}^{2}) = \tau^{2}-\mathbb{E}(r_{ui})^{2} = \tau^{2} + i^{\top} uu^{\top} i} Q(U,I)dUdV\\ &=& \mathbb{E}(\tau^{2} + i^{\top} uu^{\top} i)\\ &=& \tau^{2} + \mathbb{E}(i^{\top} uu^{\top} i)\\ &=& \tau^{2} + trace((\phi_{u} - \overline{u} \overline{u}^{\top} )(\psi_{i} - \overline{i} \overline{i}^{\top} )) \end{array} $$

(51)

And therefore:

$$ Var(\hat{r}_{ui}) = Var(\hat{r}_{ui}\mid R) = \tau^{2} + \text{trace}\left( \left( \phi_{u} + \overline{u} \overline{u}^{\top}\right)\left( \psi_{i} + \overline{i} \overline{i}^{\top}\right)\right) - \overline{i}^{\top} \overline{u} \overline{u}^{\top} \overline{i} $$

(52)

which gives us an explicit estimation of the deviation (uncertainty) on the predicted rating $\hat {r}_{ui}$.