=Paper=
{{Paper
|id=Vol-2038/paper1
|storemode=property
|title=A Half-Life Decaying Model for Recommender Systems with Matrix Factorization
|pdfUrl=https://ceur-ws.org/Vol-2038/paper1.pdf
|volume=Vol-2038
|authors=Panagiotis Ardagelou,Avi Arampatzis
|dblpUrl=https://dblp.org/rec/conf/ercimdl/ArdagelouA17
}}
==A Half-Life Decaying Model for Recommender Systems with Matrix Factorization==
<pdf width="1500px">https://ceur-ws.org/Vol-2038/paper1.pdf</pdf>
<pre>
A Half-Life Decaying Model for Recommender Systems
              with Matrix Factorization

                        Panagiotis Ardagelou and Avi Arampatzis

                      Database & Information Retrieval research unit,
                     Department of Electrical & Computer Engineering,
                   Democritus University of Thrace, Xanthi 67100, Greece.
                    panosard93@gmail.com, avi@ee.duth.gr



       Abstract. We propose the use of an exponential decay function for modeling
       drifts in user interests in collaborative filtering systems. For that purpose, we
       introduce the notion of half-life of ratings and incorporate it as a bias into a ma-
       trix factorization model. Experimental results on movie ratings spanning a period
       of approximately 7 months show that employing a half-life of around 150 days
       yields large improvements in prediction accuracy, confirming that significant user
       interest shifts exist over time and that the proposed model offers a viable strategy.


1   Introduction
The evolution of recommender or recommendation systems in the past decade has in-
fluenced highly our lives, even though many times we are not aware of their existence.
Much of the online information which reaches us derives from various recommendation
techniques. From the news we read and the products we buy, to the movies we watch
and the music we listen to, most of our on-line input is significantly not random. In this
way, both from the user perspective and from the service side as well, everyone can
take advantage of high quality recommendations because, at first, they increase cus-
tomer satisfaction, while they also allow much better quality of services (QoS) from
companies’ side, leading to much higher earnings. For these reasons, well-performing
recommendations techniques are vital for achieving and conserving high degrees of
interaction between companies (services) and customers [4].
    In order to assess users’ preferences, recommender systems need to have some kind
of feedback from users, either implicit or explicit. As explicit feedback we refer to rat-
ings given by users for specific items, based on the rating system each service uses. For
example, as explicit feedback can be characterized a ‘like’, or a 5-stars rating system.
Implicit information is based on user’s actions which can reveal an interest of a user
for a specific item. Such actions include browsing a certain product page, watching a
video, etc., and can many times be proven very useful in order to increase the accuracy
of recommendations [16].
    While there is a variety of ways to produce recommendations, two are the main
techniques. The first is content-based filtering which is crafting a user’s profile based
on known attributes of the items she has liked or bought. Content-based filtering is
capable to provide straight-forward recommendations which are generally easier to un-
derstand and more self-explanatory than those created by other techniques, because it
        Panagiotis Ardagelou, Avi Arampatzis

generates user profiles in isolation using only the user’s history. The second technique
is called collaborative, or memory-based, filtering. It differentiates from content-based
techniques in that it does not build a user’s profile in isolation but it also uses other
users’ likings; furthermore, it does not use any item attributes.
     Collaborative filtering methods can generally be classified into two sub-categories,
called neighborhood models and latent factor models. In this paper we deal only with
latent factor models. In such models, the utility matrix is usually factorized using matrix
factorization techniques, which will be described more thoroughly later in this paper.
We chose to work with matrix factorization mainly for its ease in embedding exter-
nal information in the modeling process. In addition, it generally provides very good
prediction accuracy and relatively fast running times.
     A milestone in the evolution of recommender systems is set by the well-known Net-
flix Prize Challenge which was held between 2007 and 2009 and has revolutionized our
knowledge on recommender systems [3, 5]. Ever since, significant research has taken
place and many more new versatile recommendation approaches have been brought to
life. One of the main methods the winners of the Netflix Prize Challenge employed in
their recommendation techniques was the embedding of temporal information, taking
advantage of some temporal modeling of users’ preferences [9].
     In this paper, we are concerned with temporal modelling schemes. We consider
explicit user feedback and time-dependant recommendation techniques using a matrix
factorization model which takes full advantage of dimensionality reduction methods
for very large and sparse matrices. We propose a novel half-life decaying modeling
embedded into a matrix factorization process which takes advantage of the temporal in-
formation available in the dataset we use for our experimental purposes. In this respect,
we generate a blend of recommendations based on the more recent likings of each spe-
cific user while not completely ignoring her past likings which still carry significant
information about her preferences.


2   Related Work

Several past studies have shown that incorporating new information into modeling can
lead to a boost in prediction accuracy. For example, some studies use external informa-
tion from social networks in order to create social regularization models which capture
the social relationships between users and provide recommendations based on each
user’s social neighborhood [13, 18]. While the social aspect is one important class of
information which can boost the prediction accuracy of a recommender system, another
one which has received increasing attention in the past years is time.
     Taking the most advantage out of the temporal information available in the data
has been proven to be of vital importance in the field of so-called time-aware recom-
mender systems (TARS) as it improves the recommendation performance. One of the
first temporal approaches which leads to increased recommendation accuracy sees the
collaborative filtering task as a time-series problem, binning the data into different tem-
poral bins [19]. Another one, based on implicit purchase data, takes into consideration
the launch time of an item and the time it has been purchased to create recommendation
based on the more recent purchases which reflect better the current preferences of a user
                       A Half-Life Decaying Model for Collab. Filt. with Matrix Fact.

[11, 12]. Other studies, such as the one of the winning team of Netflix’s competition [9],
use external information in order to capture temporal effects within the data and create a
more precise way of modeling users’ preferences in time and items’ popularity as well.
    In this paper, within the general framework of collaborative filtering systems with
temporal information, we introduce a new half-life decaying model by directly employ-
ing a weighting function. The idea comes from [1] where the authors introduced such
a function into training user profiles with Rocchio in a content-based filtering context.
In experiments with the OHSUMED corpus, a collection of medical abstracts spanning
a period of five years, they found that effectiveness peaks when using a half-life of
around 4 years on average in training. Nevertheless, further analysis revealed that there
is a great distribution of optimal half-life values across topics. In our work, we adapt
this idea to a collaborative filtering setup.

3   Matrix Factorization
A widely-used technique in many data mining and machine learning problems is dimen-
sionality reduction. Broadly speaking, when too many features are available to describe
a system’s parameters, these features may form subsets and thus eliminate the need of
referring to each specific one separately. By doing so, the computational cost decreases
significantly and the whole process becomes more efficient as we end up from a high-
dimensional space to a space of fewer dimensions.
     In order to perform dimensionality reduction, we have to factorize the utility matrix.
There are several ways to do so, with singular value decomposition and UV decompo-
sition being two of the most common. We perform UV decomposition which means
that we try to create two new thinner matrices of dimensions m × k and n × k, so that
the product of the first and the second transposed matrix gives the initial utility matrix.
In our case, we do not want the product to reproduce exactly the initial utility matrix
otherwise there will be no variation available in order to fill the initially blank entries,
and thus, generate predictions. An exact reproduction would recreate the blank entries
of the initial utility matrix and so there would be no predictions.
     Let us denote an entry in the initial utility matrix, i.e. a known rating, as ru,i , and
the predicted value for this rating as r̂u,i . Furthermore, each user u is described by a
vector pu ∈ <k and each item i by a vector qi ∈ <k , where k is the dimensionality
of the latent factors space which is always an integer and usually takes values between
5 and 200 [10]. The best number of latent factors is something vital for a model and
varies between different implementations. The only way to discover the best value for
the latent factors is by trial-and-error and there is no reason to assume the same values
will fit in different models.
     Vector p denotes how much interested or not is a specific user for each one of the
total k latent factors. In a similar way, the item vector q shows the correlation of an item
with each latent factor. In addition, the dot product of these two vectors captures the
correlation between item i and user u, namely, the interest user u shows in items with
similar characteristics to the item i and this gives us an approximation of the rating that
the specific user would give on item i:
                                      r̂u,i = pu · qiT .                                 (1)
        Panagiotis Ardagelou, Avi Arampatzis

     The main challenge in such models is how to decompose successfully the initial
utility matrix. Once the decomposition is completed and the two derivative matrices are
filled with the best possible values, we have reached a minimum in our system and we
are ready to compute our final predictions. As said before, the utility matrix in almost
all circumstances is very sparse, hence, only very few of the total entries are known.
     For years in the field of recommender systems, it was common to pre-fill all the
unknown entries with values using techniques such as zero-padding in a pre-processing
step before decomposition. The problem with such approaches is that adding so much
data in a large utility matrix can prove computationally very expensive. Furthermore,
if the imputation is not done correctly, it can add significant amount of noise affecting
tremendously the final predictions, lowering the accuracy.
     Newer approaches suggest that instead of trying to fill the blank entries, we can
model directly the observed ratings, however, we should be really careful with over-
fitting. Thus, the function that we have to minimize is the regularized squared error
function, or also known as loss function [14, 15]:
                        X 
                             (ru,i − pu · qiT )2 + λ(||pu ||2 + ||qi ||2 ) ,
                                                                          
                  min
                  p,q
                        u,i∈κ

where κ is the training set consisting of a percentage of the total known ratings. The
term in the parentheses multiplied with a regularization parameter λ is called regular-
ization term and is responsible for avoiding overfitting.
    In order to perform the minimization of the regularized squared error function, it is
common in matrix factorization models to use either the stochastic gradient descent al-
gorithm or alternating least squares. Both options have their own pros and cons but here
we choose to implement the stochastic gradient descent with regularization algorithm
(SGD-WR) because it provides fast convergence for sparse matrices, achieves high pre-
diction accuracy, and makes it easier to embed external information in our modeling.
    The stochastic gradient descent algorithm works iteratively on each training in-
stance creating a predicted rating r̂u,i in each iteration. Then, the algorithm computes
the prediction error and then applies the update rules shown below in order to perform
the learning:
                         erru,i = ru,i − r̂u,i = ru,i − pu · qiT ,                   (2)
         qi ← qi + γ1 (erru,i pu − λ1 qi ) ,   pu ← pu + γ1 (erru,i qi − λ1 pu ) .
Once the best fit has been reached, the algorithm has converged to a minimum. The
minimum it converges depends on many attributes of the model, such as the learning
rate as well as the initialization of matrices p and q before the optimization begins, thus
fine-tuning is vital for good performance. The algorithm may possibly converge to the
global minimum but this is the ideal case, as it usually converges to a local minimum
since everything depends on the starting point.


3.1   Modeling Biases

Biases, from the user perspective, can be explained as users who tend to rate higher or
lower than others in a systematic way and thus deviate from the mean rating. From the
                      A Half-Life Decaying Model for Collab. Filt. with Matrix Fact.

item perspective, many items tend to go in and out of trend in time and thus receiving
higher or lower ratings accordingly, whereas others may be considered “classics” and
get substantial higher ratings all the time compared to the mean rating of the dataset.
For example, suppose that the overall average ratings in a movies dataset is 3.3 and the
average rating of a certain movie is 4.0. Then, there is a tendency for this movie to be
rated 0.7 above the average rating and so we can say that the bias of this specific movie
is +0.7. Doing exactly the same thing for a user can show us that a specific user, for
example, tends to rate 0.4 below the average so this user has a bias of −0.4.
    Because biases tend to cover a significant percentage of the total information, it is
clear that the better they are modeled, the more accurate the predictions will be. Thus, it
is better to not let the whole rating value be determined just by the interaction between
user and item vectors. As a first and basic approach, we assume that biases are stable in
time, so we can have an approximation of the bias term bu,i for the rating ru,i as:

                                  bu,i = µ + bi + bu .                                  (3)

As µ we denote the overall average rating of the dataset, as bi the item bias, and as
bu the user bias. The equation presented above is usually called baseline predictor; we
follow this specific naming throughout this paper. The reason it is called that way has
to do with the fact that on its own, Equation 3 as written above, can provide a first-
order approximation of the ratings we need to predict, derived purely in a statistical
way without the personalization Equation 1 provides.
    After defining the bias modeling, we can rewrite the prediction function of our
model, including now instead of just the interaction between the user–item vectors,
the biases part as well. A nice way to visualize all the above is to imagine Equation 3
as responsible for the DC part of the information signal and Equation 1 as responsible
for the AC part:
                            r̂u,i = µ + bi + bu + pu · qiT .                         (4)
Now that we have redefined the prediction function, we can also rewrite the squared
error function which we have to minimize as:
        X 
              (ru,i − µ − bi − bu − pu · qiT )2 + λ(||pu ||2 + ||qi ||2 + b2u + b2i ) . (5)
                                                                                     
   min
   p,q
         u,i∈κ

    Previous studies [9, 10] have shown that for the efficient modeling of biases, we
insert them into the learning process of the stochastic gradient descent algorithm and
thus we discover them simultaneously with the vectors p and q. Similarly to the learning
process of the user-latent and item-latent feature vectors, in order to learn the biases the
optimization algorithm has to change their values via applying some update rules in
each iteration, until it converges to a minimum. The update rules for the biases are:

                           bu ← (1 − λ2 γ2 )bu − γ2 erru,i ,
                            bi ← (1 − λ2 γ2 )bi − γ2 erru,i ,
                              µ ← (1 − λ2 γ2 )µ − γ2 erru,i .

We can use different learning rates and regularization parameters from the ones we have
set for the learning of the vectors p and q, but there is no optimal rule for finding the
        Panagiotis Ardagelou, Avi Arampatzis

best values of each parameter other than trial-and-error and one could spend much time
just optimizing by hand all the parameters to achieve fine-tuning.


4   A Half-Life Decaying Model
It is important for recommender systems to adapt accordingly to the user preferences as
they change in time, and there are plenty of different ways to do so [8, 11, 17]. In some
cases it is important to always ‘remember’ all past ratings of a user, while sometimes
it may be more useful to provide ‘fresher’ recommendations based more on her most
recent likings.
     In [2], adaptive filtering systems are categorized as locally or asymptotically adap-
tive. Asymptotically adaptive systems spread the emphasis over the whole time-line,
weighing all events of the past equally. Locally-adaptive systems rely more heavily on
data collected in the recent past. The assumption behind using asymptotic adaptivity is
that user interests or item characteristics are stable over time, while local adaptivity is
suitable when there are drifts in either interests or item characteristics. In this study, we
explore local adaptivity assuming drifts in user interests.
     While locally-adaptive systems typically employ sliding windows over the most re-
cent past, in our model we alter the significance of each rating in time using a half-life
decaying function. Based on the temporal information available about each observed
rating available in the training dataset (i.e. timestamps), we weight each rating in accor-
dance to its recency. The resulting recommendations are based on all of the user’s past
history, as in the ‘classical’ collaborative filtering processes, but are mostly correlated
to the more recent likings of the user and less to the older ones.
     For example, given a user who has recently enjoyed a certain item, she will continue
to receive recommendations based on that specific item for some time until that specific
rating becomes old enough and other newer likings of the user have more significant
impact on the recommendations that are being produced for that user. Nevertheless, that
specific item will continue to have some but lesser impact, depending on the parameter
of the model, on the new recommendations being produced.
     The half-life decaying function is visualized in Figure 1. This exponential decay
function requires the setting of a half-life parameter denoted with the letter h which
shows the number of days that need to be elapsed for a specific rating to acquire half of
its initial significance in the collaborative filtering process. As tn we denote the current
time of the experiment whereas tn − 1 is the day before the current day and so on until
we reach tn − h which is the day after the half-life time has elapsed. At that time, the
rating will only have half of its initial significance to the collaborative filtering process
something which is achieved via under-weighting the values. Thus, the weight of a
rating will be:                                                
                                            ln 0.5
                              wu,i = exp            (tn − tu,i ) .                         (6)
                                               h
    It is very important to clarify that we cannot apply the weighting directly to the
ratings before the learning process begins. Doing so, we would misrepresent the existing
information and thus create false recommendations. For example, before the beginning
of the learning process, a 4-star rating should not be downgraded to 2-star just because
                      A Half-Life Decaying Model for Collab. Filt. with Matrix Fact.




                            Fig. 1. A half-life decaying function



its timestamp is half-life days old, because it has a totally different meaning. A rating of
4 stars may mean that the user enjoyed enough the specific movie whereas a rating of 2
stars may mean that the user was quite disappointed by the movie. So it is clear that if
we apply the time-decaying model, in a static way, as a pre-processing step would alter
the initial information significantly and thus produce wrong recommendations.
     For the reason above, we follow a different route for integrating half-life decay in
our modeling. Instead of direct weighting in a pre-processing step as described above,
we insert the decay in the learning process inside the stochastic gradient descent algo-
rithm. More thoroughly, we now consider the user bias term as well as the users-latent
features matrix as a temporal process which follows the half-life decaying function as
described above. The reason why we pick to insert the decaying characteristics only
into the user parts of the total rating value is based on the assumption that only users’
interests change over time whereas the items’ characteristics stay the same, thus items
can be described in an almost constant way in time with respect to the latent features.
Consequently, Equation 4 is modified as:

                        r̂u,i = µ + bi + wu,i bu + wu,i pu · qiT .                      (7)

    Previous works have shown that models which use temporal information into their
modeling, can achieve an even better accuracy when slightly decreasing the total influ-
ence of temporal effects in the prediction process. In line with this, we also consider
a second technique for the half-life decaying modeling which uses a dampening factor
for this specific effect. The technique is based on the assumption that users’ preferences
do not change entirely over time while not staying stable either. Thus, an approximation
of such situation is to consider both a stable DC part of the user bias modeling, and an
non-stable AC part which is half-life modeled as described previously. Therefore, the
new prediction equation has a stable user bias part and one that is being weighted:

                     r̂u,i = µ + bi + bu + wu,i bu + wu,i pu · qiT .                    (8)
        Panagiotis Ardagelou, Avi Arampatzis

The idea behind this damping factor is not novel as it comes from the previous work
of Koren [9] who used this methodology in a different way in order to achieve a more
accurate modeling of the drift of users’ preferences.


5     Experimental Evaluation
In this section, we provide a thorough description of the different models we imple-
mented as part of our experiments while highlighting the gradual increase in recom-
mendation accuracy as we dive deeper into capturing different effects in the data. First,
we provide some characteristics of the benchmark dataset we used in order to test our
hypotheses and describe the evaluation measure.

5.1   Dataset & Evaluation Measure
The dataset we used in our experiments, Movielens 100k, is standard in the field of
collaborative filtering recommender systems [6, 7]. It has been used widely in many
experiments in academia, thus it is a suitable dataset conducting experiments on and
testing new algorithms because new results can easily be compared to other baselines
and state-of-the-art methods.
    Movielens 100k consists of 100,000 unique ratings, from 943 users for 1,682 movies,
spanning approximately 7 months. In addition, temporal information for each rating is
available, i.e. the exact date and time the rating was given. In short, each row of the
dataset, consists of a unique user-id, a unique movie-id, the given rating in a scale from
1 to 5 stars, and finally the timestamp. The sparsity of the dataset is 6.3%, meaning that
only that amount of the total possible ratings is observed. Therefore, we have to deal
with a sparse utility matrix, as it is usually the case in such tasks.
    To perform our experiments, we use a 80%–20% split, performing the learning on
the 80% of the earliest ratings and then test the model to the rest 20% latest ratings. In
order to compute the prediction accuracy of our model, we use the most common eval-
uation measure in the field of recommender systems, namely, the Root Mean Squared
Error (RMSE), which is computed with the following formula:
                                        s
                                           1X
                             RMSE =              (r̂u,i − ru,i )2 .
                                           n u,i

The set (u, i) denotes all the ratings from a user u to an item i which belong to the
test set. Also, n is equal to the amount of entries in the test set, r̂u,i is the predicted
rating for the rating of user u to item i, and finally, ru,i is the actual rating which our
prediction is compared to.

5.2   Experiments & Results
As part of our experiments, we implement several different models, ranging from the
very basic baseline predictor to the more accurate matrix factorization model with half-
life decay.
                      A Half-Life Decaying Model for Collab. Filt. with Matrix Fact.

    Specifically, the first model we implement is a first rank approximation for the rat-
ings as we described previously given just by the baseline predictor, generating predic-
tions purely in a statistical way using Equation 3 (baseline model). Next, we create a
model with only the collaborative filtering part, described by Equation 1. This specific
model captures only the interaction between the user and the items vector while per-
forming the learning of the values via the stochastic gradient descent algorithm (SGD
model). Our third model is derived from the combination of the two previous equations.
In this way, the model becomes immune to bias effects that exist in the data as described
in a previous section (Biased SGD model). The last two models are our proposed ap-
proaches with the half-life modeling. The first one uses the Equation 7 to compute the
appropriate weights for the half-life decaying and then inserts them in the prediction
equation as shown in Equation 5 (halflife model 1). The second half-life model we in-
vestigate, uses a damping factor to reduce the effect of the half-life decaying taken into
consideration in the prediction process and achieve a smoother usage of the specific
effect, as described in a previous section, using Equation 8 (halflife model 2).
    The iterative process of minimizing the squared error function through the Stochas-
tic Gradient Descent algorithm is performed while revisiting the elements of the utility
matrix which belong in the training set in Round–Robin fashion and applying on them
the update rules of each model as described.
    One of the most important variables of our model is the learning rate γ which can
possibly have different values throughout the learning process of the biases and the
user/item vectors. Its physical meaning is the magnitude of the step is being taken each
time in order to reach a minimum. Although different values of learning rates are possi-
ble, in preliminary experiments (not shown in this paper) the best overall performance
was achieved by a learning rate equal to 0.01 for both the biases and the user/item vec-
tors. This value of learning rate was set through via a grid search for various values of
γ between 0.001 and 0.02 with the best results coming from the γ1 = γ2 = 0.01 value.
    Another important variable for our modeling is the number k of latent features. The
value of k has been set through cross-validation to 20, when using the Biased SGD
model for testing several values, as shown in Table 1. As we can see from the table,
this model achieves better prediction accuracy for 20 latent features. In preliminary
experiments, we found that k = 20 is also a good value for all other implemented
models, thus we keep this variable stable throughout our experiments.

           Table 1. RMSE as a function of k latent features in Biased SGD model

                        k     10     20     30     40     50
                       RMSE 0.9401 0.9344 0.9426 0.9429 0.9460


    Last, we set the regularization parameter λ1 = λ2 = 0.1 after testing several differ-
ent values between 0.001 and 0.2. One can possibly use different regularization values
for each update rule but all our models seem to perform better when these regularization
values are equal.
    As far as the temporal models are concerned, we use a history function in order to
model the time-decay of the ratings. The basic idea is that the older a rating becomes,
        Panagiotis Ardagelou, Avi Arampatzis

the less impact should have to the future likings of a user and thus should be down-
weighted throughout the collaborative filtering process. Due to the fact that there is
always some drift in user interests, we can assume that the older a rating is, the less
correlation it has with the current preferences of a user. On the contrary, the newer the
rating is, the more has to do with the current tastes of the user. As we have said before,
we use Equation 6 to model this decay in the significance of ratings.




                      Fig. 2. Model performance for various values of h


    After testing several half-life values h, we found out that the value of h which yields
the best predictions is about 5 months (or 150 days). This means that the significance of
a rating starts dropping as the rating becomes older and reaches 50% of its initial signif-
icance after 150 days approximately, in the specific dataset we use for our experiments.
The best value for the half-life parameter h was set via a grid search in our first half-life
model while the results seem to confirm on the second model, with the damping factor,
too. The results of the grid search are visualized in Figure 2.


               Table 2. Effectiveness and approximate running time per model

                          Baseline SGD Biased SGD Halflife 1 Halflife 2
                 RMSE 0.9858 0.9428 0.9344         0.9151 0.9122
                Exec. time 1.2 min 2 min  2.4 min   3 min 3.3 min



    As we can see in the figure, small values of the half-life decaying parameter tend to
minimize the impact of the collaborative filtering process resulting to accuracy similar
to our baseline model, which has no collaborative filtering involved in it. While increas-
                        A Half-Life Decaying Model for Collab. Filt. with Matrix Fact.

ing the half-life parameter, we observe some fluctuations in the overall performance of
our model but we can clearly identify that the accuracy maximization is achieved by
h = 150 days. As we can see from Table 2, the incorporation of half-life decay method-
ology works in favor of the recommendations accuracy. Furthermore, the two half-life
models give similar results with the second model with the damping factor having a
slightly better accuracy.
    Comparing our results to the closely related, state-of-the-art, svd++ algorithm [9]
for the Movielens 100k dataset, we see that they are very close to those of svd++ which
scores between 0.91096 and 0.9076, depending on the tuning of model’s parameters, as
reported in the MyMediaLite library’s benchmark results.
    In general, as it can be concluded by the results above, each time we add to our
prediction methodology a specific effect which lets us capture user–item interactions
in greater depth, the model achieves a better accuracy. The first significant increase in
the prediction accuracy happens when we embed the modeling of biases, lowering the
RMSE from the simple collaborative filtering approach from 0.9428 to 0.9344. Bias
modeling has been a well-known technique for improving recommendations accuracy
and many have used it before. When we insert the half-life decaying methodology to our
prediction process, recommendation accuracy increases significantly for both our mod-
els against the simpler models with no temporal information. The improvements in pre-
diction accuracy indicate that there is indeed vital information in temporally-dependent
effects in the data which we can use in order to generate better recommendations.

6   Conclusions and Future Work
We have introduced a method for dealing with drifts in user interests in collaborative
filtering by exploring the notion of half-life of ratings. The method incorporates an ex-
ponential decay function into a matrix factorization model, and comes in two ‘flavors’:
pure decay, or decay with a dampening factor. Experimental results have shown that
both variations yield large improvements in prediction accuracy, confirming that signif-
icant user interest drifts exist. On the Movielens 100k dataset, spanning approximately
7 months of ratings, accuracy was maximized when using a half-life value of approxi-
mately 150 days.
     Continuing this work, we plan to examine temporal effects in data with much longer
timespans. Furthermore, we intend to experiment with similar methods which will learn
via a learning algorithm, such as stochastic gradient descent, the half-life parameter for
each user individually. What motivates us to do so is the fact that not all users are char-
acterized by the same half-life parameter, as some of them tend to change preferences
with a difference pace than others.

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