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SCIENCE CHINA Information Sciences, Volume 62, Issue 11: 212105(2019) https://doi.org/10.1007/s11432-018-9915-8

Recommendation over time: a probabilistic model of time-aware recommender systems

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  • ReceivedDec 31, 2018
  • AcceptedMay 17, 2019
  • PublishedOct 9, 2019

Abstract

In time-aware recommender systems, we have to consider the dynamic aspect of recommendation that is fond of new coming data. Usually, the recent data is more closely related to current recommendation tasks and the early data are useful to indicate overall measurements of the preferences. We propose a probabilistic model that uses the early data to generate the prior distribution and the recent data to capture the change of the states of both users and items in collaborative filtering systems. Our model is dynamic in the sense that it updates every time receiving new data. The time cost of every updating has a constant limit, which is suitable to deal with large scale data for online recommendation. Experiments on real datasets show the improvement performance of our model over the existing time-aware recommender systems.


Acknowledgment

This work was supported by National Natural Science Foundation of China (Grant Nos. 61672049, 61732001).


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  • Figure 1

    Online learning and cold-start performance. (a) MLK; (b) Ep.

  • Figure 2

    Time convergence. (a) MLM; (b) EpEx.

  •   

    Algorithm 1 Update

    function Update($({\rm~user,~item},~T,~l)$);

    ${\rm~RecRating(user).push((user,~item},~T,~l))$;

    ${\rm~RecRating(item).push((user,~item},~T,~l))$;

    ConvertRating(user);

    UpdateParameter(user);

    ConvertRating(item);

    UpdateParameter(item);

    function ConvertRating(${\rm~user}$);

    while ${\rm~RecRating(user).size}()>M$ do

    $({\rm~user,item}_1,~t_1,~l_1)~\leftarrow~{\rm~RecRating(user).front}()$;

    Update $\phi_{{\rm~user},~j}$ by (17);

    ${\rm~RecRating(user).pop}()$;

    $({\rm~user,~item}_2,~t_2,~l_2)~\leftarrow~{\rm~RecRating(user).front}()$;

    if $\tau_{\rm~user}<t_2$ then

    $\tau_{\rm~user}~\leftarrow~t_2$;

    for $j=1,\ldots,~J$

    $f_{{\rm~user},~t,~j}~\leftarrow~x_{{\rm~user},~t,~j}$;

    end for

    end if

    end while

    function UpdateParameter(${\rm~user}$);

    for $t~\in~{\rm~TM(RR(user))}$

    Prepare $E_{{\rm~user},~t,~j}$ by (12);

    end for

    Initial $\alpha_{\tau_{\rm~user},~j}$ as (eqn_alpha_1);

    for $t~\in~{\rm~TM(RR(user))}$, forward

    Calculate $\alpha_{t,~j}$ by (eqn_alpha_2);

    end for

    Initial $\beta_{T,~j}$ as (eqn_beta_1);

    for $t~\in~{\rm~TM(RR(user))}$, backward

    Calculate $\beta_{t,~i}$ by (eqn_beta_2);

    end for

    for $t~\in~{\rm~TM(RR(user))}$

    Calculate $\gamma_{t,~i}$ and $\xi_{t,~i,~j}$ by (eqn_gamma) and (eqn_xi);

    end for

    Update $\pi_i$, $A_{i,~j}$, and $p_{j,~k}$ by (13)–(16);

  • Table 1   Hyperparameters, random variables and parameters
    Symbol Meaning
    $J$ Number of user types
    $K$ Number of item types
    $M$ Max number of recent ratings
    $\lambda_1$ Regularization for $\pi$ and $\omega$
    $\lambda_2$ Regularization for $A$ and $B$
    $X_{{\rm~user},t}$ The type ${\rm~user}$ belongs to at time $t$
    $Y_{{\rm~item},t}$ The type item belongs to at time $t$
    $R_{{\rm~user,item},t}$ The rating that ${\rm~user}$ give to item at time $t$
    $p_{j,k}$ Probability that type $j$ user likes type $k$ item
    $A_{i,j}$ Jump rate matrix for users
    $B_{k,m}$ Jump rate matrix for items
    $\pi_j$ Global prior distribution for users
    $\omega_k$ Global prior distribution for items
    $f_{{\rm~user},t,j}$ Approximation of user variable distribution
    $g_{{\rm~item},t,k}$ Approximation of item variable distribution
  • Table 2   Additional persistent variables
    Variable name Data type Meaning
    $\tau_{\rm~user}$ Number Time of the first recent rating of ${\rm~user}$
    $\tau_{\rm~item}$ Number Time of the first recent rating of item
    $\phi_{{\rm~user},j}$ Number Local prior for users
    $\psi_{{\rm~item},k}$ Number Local prior for items
    $x_{{\rm~user},t,j}$ Number $P(X_{{\rm~user},t}=j|{\rm~Rating})$
    $y_{{\rm~item},t,k}$ Number $P(Y_{{\rm~item},t}=k|{\rm~Rating})$
    ${\rm~RecRating(user)}$ Queue Recent rating for ${\rm~user}$
    ${\rm~RecRating(item)}$ Queue Recent rating for item
  •   

    Algorithm 2 Prediction

    function Predict($({\rm~user,item},t)$);

    $({\rm~user,item}_1,T_1,l_1)~\leftarrow~{\rm~RecRating(user).back}()$;

    for $j=1,\dots,~J$

    $\hat{x}_{{\rm~user},t,j}~\leftarrow~\sum_{i=1}^J~x_{{\rm~user},T_1,i}\exp(A(t-T_1))_{i,j}$;

    end for

    $({\rm~user}_2,{\rm~item},T_2,l_2)~\leftarrow~{\rm~RecRating(item).back}()$;

    for $k=1,\dots,~K$

    $\hat{y}_{{\rm~item},t,k}~\leftarrow~\sum_{m=1}^K~y_{{\rm~item},T_2,l}\exp(B(t-T_2))_{m,k}$;

    end for

    for $n=1,\dots,~N$

    $\hat{r}_n~\leftarrow~\sum\nolimits_{j~=~1}^J~{\sum\nolimits_{k~=~1}^K~{\hat{x}_{{\rm~user},t,j}~\hat{y}_{{\rm~item},t,k}~\Pr(n-1;N-1,p_{j,k})~}~}~$;

    end for

    return $\hat{r}$;

  • Table 3   The experiment datasets with different sizes
    Dataset User Item Rating Density(%)
    MLK (MovieLens100k) [56]944 1683 100000 6.29
    MLM (MovieLens1M) 6040 3706 1000209 4.47
    Ep (Epinions) [57,58]2874 2624 122361 1.62
    EpEx (Epinions extended) [59]11201 109520 5449415 0.44
  • Table 4   The scores of precision
    Setting ClassicalTime-ordered
    Dataset MLK MLM Ep EpEx MLK MLM Ep EpEx
    SRec 0.698 0.728 0.796 0.953 0.662 0.703 0.788 0.943
    TCARS 0.717 0.750 0.783 0.951 0.702 0.724 0.746 0.940
    GRU4Rec 0.652 0.654 0.715 0.936 0.613 0.622 0.705 0.939
    RRN 0.665 0.730 0.740 0.938 0.684 0.702 0.734 0.936
    RT 0.717 0.745 0.831 0.954 0.730 0.717 0.792 0.945
  • Table 5   Normalized discounted cumulative gain
    Setting ClassicalTime-ordered
    Dataset MLK MLM Ep EpEx MLK MLM Ep EpEx
    SRec 0.924 0.933 0.945 0.984 0.938 0.933 0.954 0.979
    TCARS 0.931 0.941 0.936 0.979 0.939 0.940 0.938 0.978
    GRU4Rec 0.903 0.904 0.905 0.974 0.916 0.908 0.923 0.976
    RRN 0.914 0.932 0.915 0.971 0.933 0.933 0.927 0.976
    RT 0.930 0.938 0.956 0.984 0.946 0.939 0.954 0.980
  • Table 6   Mean reciprocal rank
    Setting ClassicalTime-ordered
    Dataset MLK MLM Ep EpEx MLK MLM Ep EpEx
    SRec 0.855 0.897 0.919 0.980 0.850 0.872 0.897 0.971
    TCARS 0.854 0.910 0.880 0.955 0.794 0.876 0.831 0.964
    GRU4Rec 0.797 0.839 0.835 0.969 0.765 0.837 0.815 0.966
    RRN 0.828 0.892 0.843 0.953 0.833 0.875 0.843 0.964
    RT 0.867 0.903 0.939 0.978 0.853 0.884 0.898 0.970

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