=Paper=
{{Paper
|id=Vol-1670/paper-13
|storemode=property
|title=Sampling Methods for Random Subspace Domain Adaptation
|pdfUrl=https://ceur-ws.org/Vol-1670/paper-13.pdf
|volume=Vol-1670
|authors=Christian Pölitz
|dblpUrl=https://dblp.org/rec/conf/lwa/Politz16
}}
==Sampling Methods for Random Subspace Domain Adaptation==
<pdf width="1500px">https://ceur-ws.org/Vol-1670/paper-13.pdf</pdf>
<pre>
     Sampling Methods for Random Subspace Domain
                      Adaptation

                                       Christian Pölitz

                TU Dortmund University, Otto Hahn Str. 12, 44227 Dortmund



       Abstract. Supervised classification tasks like Sentiment Analysis or text clas-
       sification need labelled training data. These labels can be difficult to obtain, es-
       pecially for complicated and ambiguous data like texts. Instead of labelling new
       data, domain adaptation tries to reuse already labelled data from related tasks as
       training data. We propose a greedy selection strategy to identify a small subset
       of data samples that are most suited for domain adaptation. Using these samples
       the adaptation is done on a subspace in a kernel defined feature space. To make
       this kernel approach applicable for large scale data sets, we use random Fourier
       features to approximate kernels by expectations.


Introduction

The usual assumption for most of the Data Mining and Machine Learning tasks is that
the training data used to learn a model has the same distribution as the test data on that
the model is applied. On the other hand, there are many situation where this is not true.
Imagine as Data Mining task Sentiment Analysis on product reviews from Amazon. In
case of a new product or product type, producers might be interested in how their prod-
ucts catches on. Sentiment Analysis now tries to label the reviews of the corresponding
products as being positive or negative. To assign such labels, classification models are
trained on some labelled reviews and than applied on the unlabelled reviews. For new
product types it is reasonable to assume that we have no labelled training data at hand.
Labelling the new reviews can be quite expensive. Especially identifying the sentiment
in texts can be hard - even for experts. Ambiguous words or sarcasm for instance make
this task difficult. Instead of starting to label the new reviews, another possibility is
to reuse already labelled reviews from different products. There might be for instance
already labelled reviews about books and now we get new reviews about DVDs. The
idea is to leverage the reviews about books to train a classifier that is applied on re-
views about DVDs. To accomplish this, we need to find a way to safely transfer the
information from one domain to another.
    We solve this problem by domain adaptation with the following assumptions: We
have two data sets with (possible large) difference in distribution. We have data from a
source domain S that is distributed via ps together with label information y distributed
via ps (y|x). On the other hand, we also have data from a target domain T that is dis-
tributed via pt with no label information. The domain adaptation task now is to use
the source domain together with its label information to find a classifier that labels the
target domain best.
     We expect that many data sets share similarities on latent subspaces. On product
reviews for instance, a book might be described as tedious while a toaster might be
described as malfunctioning. Both words have negative meaning and very likely appear
together with other negative words like bad, poor or poorly. In a latent subspace in
the space spanned by the words, we expect that these words span together a whole
dimension. When we map texts of reviews from books and electronic articles onto such
a subspace the words tedious and malfunctioning can be replaced by their common
meaning. This will make the texts from the different domains more similar. Further,
only terms alone might not be able to find such subspaces. For instance, bi-grams like
little helpful or hardly improving can also span a latent subspace that is helpful for
domain adaptation. Generally, n-grams should also be considered.
     In order to integrate information of multiple combinations of words, kernels like
polynomial kernels can be used. Kernel methods can also integrate structural informa-
tion and even information from probabilistic models. Consequently, we find low dimen-
sional representations of the data from a source and a target domain in a Reproducing
Kernel Hilbert Space. These representations shall keep enough structure from the data
that a classifier trained on the source domain still performs well. On the other hand, the
low dimensional representation shall make the two data sets more similar. This justifies
a safe application of a classifier trained on the source domain, to the target domain.
     To find the subspace for the domain adaptation we propose a greedy selection strat-
egy that finds the most useful data samples in the source domain for the domain adapta-
tion. By this, we reduce the data size and concentrate on those samples that are poten-
tially best suited to transfer knowledge. This idea is based on the assumption that not all
source samples might by equally important for adaptability. This has been investigated
for instance by in [10]. Further, we approximate kernels by random Fourier features as
proposed by [19]. This tackles the quadratically or cubically scaling behaviour of kernel
methods in the number of data samples.


Related Work

We distinguish two main directions in domain adaptation. On the hand, many of the
existing approaches try to find weights for the samples that account for an mismatch
in distribution of a target and a source domain. This is especially useful under the so
call covariate shift assume. Here, we assume that the distribution of the labels given a
sample is the same for both target and source domain. Via the weights, a sample selec-
tion bias shall be corrected. This means, we assume that the source domain is sampled
from the target distribution applied a certain weighting mechanism. Many previous ap-
proaches learn such weights such that the weighted source distributions is most similar
to the target distribution.
    For instance [8] propose density estimators that incorporate sample selection bias to
adapt two distribution, [13] do this by matching the distributions in an RKHS, [14] find
the optimal weights by solving least squares problem and [23] minimize the Kullback-
Leibler divergence of the target distributions and the weighted source distribution, to
name only a few. A theoretical analysis of this adaptation can be found in [6] and [5].
    In contrast to these approaches, several other works try to extract a subspace or
feature representations in the data space that covers invariant parts across the target and
the source distribution. Within such a subspace or feature representations, transferring
knowledge between the source and target domain is expected to be more effective than
in the whole ambient space.
    In [18], Transfer Component Analysis is introduced to find low dimensional repre-
sentations in a kernel defined Hilbert space. In this representation the target and source
domain are more similar than before. The authors in [22] learn a linear subspace that
is suitable for transfer learning by minimizing Bregman divergence of the target and
source distribution in this subspace, [21] transform the target points such that they are a
linear combination of a basis in the source domain, [25] propose to transfer knowledge
in a Hilbert space by aligning a kernel with the target domain, [17] learn domain invari-
ant data transformation to minimize differences in source and target domain distribu-
tions while preserving functional relations of the data with possible label information.
Further, in [9] the authors propose to create subspaces that aligns to the eigenspaces of
the target and source domain.



Background

In this section, we introduce the background on kernel methods, subspaces in Hilbert
Spaces and distances of distributions. The presented information are crucial for our
proposed strategy in the next sections.


Kernel Methods and RKHS

Kernel methods accomplish to apply linear methods on non-linear representations of
data. Any kernel method uses a map X → φ(X) from a compact input space X, for
instance <n , into a so called Reproducing Kernel Hilbert Space (RKHS). In this space,
linear methods are applied to the mapped elements like Linear Regressions or Support
Vector Machines. The RKHS is a space of functions f (y) = φ(x)(y) ∀x ∈ X that
allows point evaluations by an inner product, hence f (y) = φ(x)(y) = hφ(x), φ(y)i.
φ(x) is a function and φ(x)(y) means the function value at y.


Subspace Methods A subspace in an RKHS H is a closed subset H 0 ⊂ H. We identify
this subspace by a projection P that maps all elements of H into H 0 . In this work, we
concentrate only on subspaces that are spanned by the given data points in the RKHS.
This means each element in the subspace    can be written as linear combination of all
data points in the RKHS, hence v = x∈H αi · φ(xi ) for all v ∈ H 0 . This is important
                                     P
since we only need to consider kernel evaluations and not infinite dimensional elements
of the RKHS. Kernel PCA for instance can be used to find an appropriated projection
matrix onto such a subspace. See [20] for further details.
Distance Measures
As proposed by [12] the maximum mean discrepancy (MMD) can be used to estimate
the difference of two distributions ps and pt . For the unit ball H in an RKHS induced
by a universal kernel k, the MMD and its empirical estimate are defined as:

                        M M D(H, ps , pt )2 = kµ[ps ] − µ[pt ]k2H
    respectively

                                1        X
      M M D(H, S, T )2 =                          k(xi , xj )                                             (1)
                               |S|2
                                      xi ,xj ∈S
                                 1          X                            1       X
                           −                           k(xi , xj ) +                       k(xi , xj ).
                               |S||T |                                 |T |2
                                         xi ∈S,xj ∈T                           xi ,xj ∈T


Random Features
To avoid large computational and storage complexity of kernel methods, approxima-
tions of the kernel can be used. Random features for instance approximate the feature
maps in Hilbert spaces by low dimensional random projections. The expectation of
the inner products of these random features evaluate to corresponding kernel values.
Any shift-invariant kernel (as for example the Gaussian kernel) can be represented as
expectation of random features cos(ωx + b) for an appropriate distribution p(ω) and
b uniformly drawn from [0, 2π], see [19]. For Gaussian kernels, ω is drawn from the
                                         2
distribution: p(ω) = (2π)−k/2   √
                                    e−kωk /2 . An unbiased estimate of the expectation is
zω (xi )0 zω (xj ) for zω (x) = k2 [cos(ω1 x), · · · cos(ωk x), sin(ω1 x), · · · sin(ωk x)].
     The deviation of the inner product of the random features of dimension k to the
        √ value is bounded by a tail bound using Hoeffding’s inequality. Since zω ∈
true√kernel
[− 2, 2], we have zω (xi )0 zω (xj ) ∈ [−2, 2]. This and Eω [zω (xi )0 zω (xj )] = k(xi , xj )
justifies the following bound:

                                                                                 2
                    P (|zω (xi )0 zω (xj ) − k(xi , xj )| ≥ ) ≤ 2e−k /8

Domain Adaptation
In a domain adaptation task, we try to use information about a data set S for a clas-
sification task on data from set T . For instance, in online reviews about products we
might have reviews and information about the sentiment of the reviews about lots of
electronic products. Now, the people also start reviewing books. A company might for
instance broaden their offers. Now, the new reviews of books shall also be classified
by their sentiment. Instead of starting from scratch and labelling all book reviews, we
want to leverage the information from all the reviews about electronics that have already
been classified by their sentiment. Using this information, a classifier can be learned on
a transformed representation of the electronic reviews and be applied to transformed
book reviews.
Domain Adaptation via Subspaces
We assume that both data sets lie in the same Hilbert space H by using the same kernel
and that their distributions have the same support. Further, we have for each element a
probability distribution over a label l that is the same for both data sets. This is the so
called Covariate Shift assumption. This means, given an element from H the probability
of label l depends not on the set the elements is in, but only on the element.
     To transfer knowledge, we project all data onto a low dimensional subspace that
captures the structure of the source data and the target data. This is important since
otherwise we might not be able to train a good classifier or even project all data points
onto a single point. In this case the distributions are the same but we can not train a
good classifier.
     The simplest way to find a projection onto a subspaces that captures most of the
structure is using kernel PCA. We have two data sets that should not loose too much of
its structure after projection. The structure of the source domain must be kept to train
a good classifier, but the target domain is the actual data we are interested in. Further,
we expect that not all information from the source is useful. The idea is now to keep
the structure of the target data completely, but for the source data only those parts such
that the source and target distributions are close on the subspace that covers only this
structure.
     Having found a suitable subspace for domain adaptation we project all data or-
thogonally onto this space. An orthogonal projection onto a low dimensional subspace
retracts all data points and makes the distributions of the two data sets more similar.
This is true since kP · µt − P · µs k ≤ kP k · kµt − µs k and kP k = 1 for an orthogonal
projection P and the mean functional µt of the target distribution and µs of the source
distribution.
     Further, the expected distance between classification models on source and target
domain decreases. Since we concentrated on linear classifiers in an RKHS,     P we writes any
classifier from Pthe source, respectively the target domain as: hs (.) =         αi · hφ(xi ), .i
                                t
and ht (.)   =      β j · hφ(x  j ), .i. Hence, we identify  the  classifier by weight  vectors
            αi · φ(xsi ) respectively wt =         βj · φ(xtj ). After projecting all elements
         P                                      P
ws =
                          P , the corresponding weight vectors are wsP =         αi · P · φ(xsi )
                                                                              P
onto the subspace viaP
respectively wtP =          βj · P · φ(xtj ) The distance of any of these classifiers can be
bounded in the following way:

                      Z
                          |wsP (x) − wtP (x)|pt (x)dx                                        (2)
                      Z       X                        X
                  =       |       αi · P · φ(xsi ) −       βj · P · φ(xtj )|pt (x)dx
                         Z X                   X
                  ≤kP k · |     αi · φ(xsi ) −     βj · φ(xtj )|pt (x)dx
                   Z X                   X
                  = |     αi · φ(xti ) −     βj · φ(xtj )|pt (x)dx
                   Z
                  = |ws (x) − wt (x)|pt (x)dx
Fig. 1. Illustration of the samplings. Left: Source (electronic reviews in red) and target (DVD
reviews in blue) data plotted in the space of the first two components of both of them together.
Right: MMD of the selected samples from the source data by Herding based sampling.


    Here, we use the fact that the norm of the orthogonal projection is 1, hence kP k = 1.
The bound shows that the expected distance of the linear classifiers in the subspace is
less than in the original Hilbert space. The inequality cannot become an equality since
we project always on lower dimensional subspace. This shows that these projections
decrease the expected error on the target domain of any classifier trained on source
domain with a different distribution, see (CF-[2]).


Greedy Selection

To find the most promising data points from the source domain for the domain adapta-
tion, we propose a greedy strategy to efficiently select them. The sampled data points
shall be close to the target domain to prevent too much influence of the source domain.
On the other hand, the samples must keep enough structure of the source domain such
that a good classifier can be trained on the source domain data. The proposed strategy is
based on the distance of the the source domain distribution to the target domain distri-
bution. The picture in Figure 1 illustrates our idea on electronic (red) and DVD (blue)
reviews. We assume the reviews of electronics as target domain and the reviews about
DVDs as source domain. The reviews seem to be more similar on one direct than on the
other. The idea now is to prefer points from the source domain that are more prominent
in this direction for the domain adaptation.


Distribution Based Sampling We propose a sampling strategy that is based on the data
distribution. In the Hilbert space we iteratively select mapped samples from the source
domain that are most similar to the target distribution. For µP
                                                              pt the expectation functional
for the target domain in an RKHS, the difference kµpt − n1 x∈S 0 ⊂S φ(x)k2H estimates
the difference of the target distribution and a subset of samples from the source distri-
bution. Similar approaches are proposed by [4], The authors showed that the sampling
strategy introduced by [24] can be used to match empirical and true distributions in an
RKHS. Equation 3 shows the selection strategy based on matching distributions in an
RKHS.


                      xt+1 = argmaxx∈S−{x1 ,··· ,xt } hwt , φ(x)i                             (3)
                              wt+1 = wt + Ept [φ(x)] − φ(xt+1 )

    For deciding when to stop the sampling, we monitor maxx∈S−{x1 ,··· ,xt } hwt , φ(x)i.
As soon as we have only data points from the source data set left that make the distance
in distribution no longer decreasing, we stop. By this, we sample only those points such
that the empirical distributions of samples and the target data are minimal. The picture
on the right of Figure 1 shows an example of the course of the MMD of the samples
from the source domain (electronic reviews) and the target domain (DVD reviews).
We sample as long as the MMD decreases to find all points that make the distribution
similar. This bewares us to sample points that make the two distribution dissimilar.


Analysis of Distribution Based Sampling For µpt = n1t xi ∈T φ(xi ), our sampling
                                                       P
strategy minimizes:
                                     1 X
                          E = kµpt −         φ(xj )k2H .
                                     T     0xj ∈S

   To see this we rewrite
                             2 X                    1          X
        E = hµpt , µpt i −         hµpt , φ(xj )i + 2                     hφ(xi ), φ(xj )i.
                             T   0
                                                   T
                               xj ∈S                        xi ,xj ∈S 0


Since hµpt , µpt i is constant, minimizing E is the same as maximizing

                  2 X                    1       X
                        hµpt , φ(xj )i − 2                  hφ(xi ), φ(xj )i.
                  T   0
                                        T
                    xj ∈S                     xi ,xj ∈S 0


Multiplying the last expression by T results in the greedy sampling as defined above
when we set w0 = µpt . This means the strategy matches the empirical distribution of
the target samples with the empirical distribution of the subset of the samples from the
source distribution.


Random Feature Sampling Our proposed sampling strategy can still result in a large
number of points from the source distribution. We further propose to combine the selec-
tion strategy and the domain adaptation on a subspace by random features of dimension
k. This enables us to perform the domain adaptation task in the linear space spanned by
the random Fourier bases of the random features as defined above.
    We define M M Dω similar as M M D in Equation 2 except that the kernel eval-
uations are replaces by the inner products of the random features. Since M M Dω ∈
[−8, 8], we can apply Hoeffding’s inequality to bound the difference to the true M M D
by:
                                                                  2
                       P (|M M Dω2 − M M D2 | ≤ ) ≤ 2e−k /128 .

    Due to linearity of the expectation we have: Eω M M Dω 2 = M M D2 and from the
definition of the random features we have: k(xi , xj ) = Eω [zω (xi )0 zω (xj )]. All together
results in the bound.
    Further, we need to estimate how much the components for the random features
deviate from the true components the source samples in the RKHS. For this it suffices
to investigate the expected difference of the true kernel matrix K for n data points and
the matrix of the inner products of the random features Kω . An appropriate bound is
proposed by [16]:

                                         r                 r
                                             2n2 log n         2n log n
                       E[kKω − Kk] ≤                   +                .
                                                 k                k

Experiments

We test our proposed method to find projections onto subspaces for domain adaptation
on three standard benchmark data sets that have been used in previous domain adapta-
tion experiments.
    As first data set, we use the Amazon reviews [3] about products from the categories
books (B), DVDs (D), electronics (E) and kitchen (K). The classification task is to
predict a given document as being written in a positive or negative context. We use stop
word removal and keep only the words that appear less than 95% and more often than
5% of the time on all documents. The reviews of a certain product will be used as target
domain and all the others as source domain.
    The second data set is the Reuters-21578 [15] data set. It contains texts about cate-
gories like organizations, people and places. For each two of these categories a classifi-
cation task is set up to distinguish texts by category. Each category is further split into
subcategories and different subcategories are used as source and target domains. The
exact configuration of the tasks is given by [7].
    The third data set is the 20 Newsgroup data set1 . We use the four top-categories
(comp,rec,sci and talk) in the same configuration and splits as in [1]. For each two of
these top-categories a classification task is set up to distinguish texts by category. Each
category is further split into subcategories and different subcategories are used as source
and target domains.
    For the subspace for domain adaptation, we simply extract the first 100 principle
components from the kernel matrix K for all samples from the sampled source do-
main data and the target domain. This means, for each xi , xj ∈ {T ∪ S 0 } we have
K = (k(xi , xj ))i,j . We project all data samples (all source and training data) onto
the subspace spanned by the extracted components and train a classifier on the source
domain in this subspace. Next, we apply this classifier on the target domain in the
 1
     http://qwone.com/ jason/20Newsgroups/
   Method     org vs. places places people comp comp comp rec vs. rec vs. sci vs.
              places vs. org vs.      vs.     vs. rec vs. sci vs. talk sci      talk   talk
                              people places
    KMM       60.1    56.8    58.5    56.2    96.9    84.4     98.5     91.2    98.5   95.4
    TCA       85.4    80.5    76.5    76.5    94.5    87.8     96.2     90.2    94.1   88.9
    GFK       72.9    66.1    68.7    66.4    84.1    74.7     91.9     72.5    86.6   79.02
  Sampling 90         82      83.5    79.2    99.1    92       99.2     98.3    99     96.2
Sampling+RF 84.7      82.9    85.5    77.3    98      88.4     98.7     91.7    98     93.7
Table 1. Accuracies on the Reuters and 20 news groups data sets. We compare our proposed
greedy sampling methods (without and with random features) and projection with Kernel Mean
Matching (KMM) and Transfer Component Analysis (TCA), Gradient Flow Kernel (GFK).


    Method {D ∪ B ∪ K} →E {E ∪ B ∪ K} →D {E ∪ D ∪ K} →B {E ∪ D ∪ B} →K
    KMM              81.0               75.2                72.5                 83.9
     TCA             81.4               77.8                74.7                 84.9
     GFK             68.7               66.3                62.2                 70.7
  Sampling           82.4              79.15                77.25               85.25
 Sample+RF           81.3               79.7                77.65               84.85
Table 2. Accuracies on Amazon reviews using one product as target domains and all the other do-
mains as source domain. We compare our proposed greedy sampling methods (without and with
random features) and projection with Kernel Mean Matching (KMM) and Transfer Component
Analysis (TCA), Gradient Flow Kernel (GFK) and the Landmark method (LM) with projection
for domain adaptation.




subspace. We compare the sampling strategies without and with random features (Sam-
pling, Sampling+RF) with Transfer Component Analyses (TCA) [18], Kernel Mean
Matching (KMM) [13] and Gradient Flow Kernel (GFK) [11]. For TCA we also use
100 components. We use Gaussian kernels with optimized width parameter σ. For the
classification we train an SVM with optimized error weight C. For the random features,
the results are mean values over 10 runs with random features of dimension 10.000.
    The method by [10] has the same objective as our sampling methods. They find
those source domain points that minimizes the MMD to the target domain. Compared
to our method, the points are extracted by solving a quadratic optimization problem with
constraints. This is computationally challenging when we have large source domains.
Further, they do not directly select the points, they propose to learn weights of the points
and remove those points that have weights below a threshold. This threshold has to be
chosen by hand. In the experiments we use the same threshold as they have done in
their experiments.
    The results of the first experiment are shown in Tables 1 and 2. The projections
onto the components result in the best performances for all the domains. The subspace
obviously covers the important invariant parts of the data very well. Using random
features to approximate the kernel values results in the second best accuracies compared
to the other methods.
    We now explore how many source domain points have been chosen from which
domain.
               Fig. 2. Histograms of the selected points from Amazon reviews.



    Figure 2 shows histograms of the selected data points from the source domain for
the different methods. The sampling strategy without and with random features and
the GFK method uses a similar amount of samples from the source domains. The his-
tograms show that for each target domain the methods have always one domain in the
mixture of source domain where most of the samples are drawn from. For sampling
there is always on clear domain from which the method samples most from.
    To investigate this further we calculate the Maximum Mean Discrepancy as defined
in Equation 2 to estimate the difference of the distributions of the target and source do-
mains. Table 3 shows the MMD values using reviews from the domains. For the elec-
tronics reviews (E), the reviews about kitchens (K) are closest in distributions. Com-
paring this result with the accuracies from above, on the target domain with reviews
about electronics, source domain kitchen performs best for domain adaptation. Similar
results can be seen for the other domains. Comparing the MMD of the domains with
the sampled points from the last experiments, we see that the sampling method chooses
the source domain points that results in low MMD best.



                            MMD E        D      B    K
                             E    0 0.0177 0.0207 0.0067
                             D 0.0177 0 0.0174 0.0173
                             B 0.0207 0.0174 0 0.0200
                             K 0.0067 0.0173 0.0200 0

Table 3. Maximum Mean Discrepancy (MMD) measure on the different domains from the cate-
gories from the Amazon reviews.




   Finally, we investigate the influence of the random features on the quality of the
domain adaptation. We perform several runs using different feature sizes.
    The plots in Figure 3 show a fast convergence already after some thousand random
features. Experiments with random features of dimension less than one thousand has let
to poor performance. This might be due to the slower convergence of the kernel matrix
to the matrix of the inner products of the random features in the norm. In the future we
will investigate this further.
Fig. 3. The classification accuracies using different numbers of random features. Left: the Reuters
data set; middle: 20 news groups data set; right: Amazon reviews


Conclusion and Future Work

We proposed a selection strategy on samples from a source domain that are best suited
for domain adaptation to a target domain with a different data distribution. The sam-
ples are selected to keep the structure of the target domain points while adding some
structure from the source domain points. Projecting onto the subspace of the selected
samples and the target samples results in a subspace that is well suited for domain
adaptation from the source to the target domain. To apply this approach also on large
scale data sets, we use random features to approximate kernel values. On benchmark
data sets, we showed that our methods perform well on domain adaptation tasks. In
the future we want to investigate domain adaptation across different feature spaces. In
this context, we want to look at the connections to MKL and domain adaptation using
multiple sources.


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