=Paper=
{{Paper
|id=Vol-2655/paper13
|storemode=property
|title=Robust Multi-Output Learning with Highly Incomplete Data via Restricted Boltzmann Machines
|pdfUrl=https://ceur-ws.org/Vol-2655/paper13.pdf
|volume=Vol-2655
|authors=Giancarlo Fissore, Yufei Han, Aurélien Decelle, Cyril Furtlehner
|dblpUrl=https://dblp.org/rec/conf/ecai/FissoreHDF20
}}
==Robust Multi-Output Learning with Highly Incomplete Data via Restricted Boltzmann Machines==
https://ceur-ws.org/Vol-2655/paper13.pdf
four methods above can handle all the challenges co-currently raised in our work. First, all of these assume
the testing instances to have fully observed feature profiles. They don’t consider coping with incomplete testing
instances by design. Second, all of them are designed specifically for multi-label learning and adapting them to
multi-class classification is not straightforward.
More recently, methods based on Deep Latent Variable Models (DLVM) have been proposed to deal with
missing data. In [MF19], the Variational Autoencoder [KW14] has been adapted to be trained with missing data
and a sampling algorithm for data imputation is proposed. Other approaches based on Generative Adversarial
Networks (GAN) by [GPAM+ 14] are proposed in [YJvdS18] and [LJM19]. Impressive results on image datasets
are displayed for these models, at the price of a rather high model complexity and the need for a large training
set. In addition these works are focused on features reconstruction, and additional specifications and fine-tuning
are required to be able to take partially observed labels into account. The models specifications are quite involved
and any new specificity of the dataset may increase both the cost and the difficulty in training (especially for
the approaches based on GANs).
In this paper we choose to address this problem in a more economical and robust manner. We consider the old
and simple architecture of the Restricted Boltzmann Machine and adapt it to the multi-output learning context
(RBM-MO) with missing data. The RBM-MO method serves as a generative model which collaboratively
learns the marginal distribution of features and label assignments of input data instances, despite the incomplete
observations. Building on the ideas expressed in [NK94, GJ94] we adapt the approach to the more effective
contrastive divergence training procedure [Hin02] and provide results on various real-world datasets. The ad-
vantage of the RBM-MO model is that of providing a robust and flexible method to deal with missing data,
with little additional complexity with respect to the classic RBM. Indeed, the trained model can be naturally
applied to both transductive and inductive scenarios, achieving superior multi-output classification performance
then state-of-the-art baselines. Moreover, it works seamlessly with multi-class and multi-label tasks, providing
a unified framework for multi-output learning.
2 Overview of Restricted Boltzmann Machines
An RBM is a Markov random field with pairwise interactions defined on a bipartite graph formed by two layers
of non-interacting variables: the visible nodes represent instances of the input data while the hidden nodes
provide a latent representation of the data instances. V and H will denote respectively the sets of visible and
hidden variables. In our setting, the visible variables will further split into two subsets Vf and V` corresponding
respectively to features and labels, such that V = Vf +V` . The visible variables form an explicit representation of
the data and are noted v = {vi , i ∈ V}. The hidden nodes h = {hj , j ∈ H} serve to approximate the underlying
dependencies among the visible units.
In this paper, we will work with binary hidden nodes hj ∈ {0, 1}. The variables corresponding to the visible
features will be either real with a Gaussian prior or binary, depending on the data to model, and labels variables
will always be binary (vi ∈ {0, 1}). The joint probability distribution over the nodes is defined through an energy
function
e−E(v,h) X X X
P (v, h) = pprior (v), E(v, h) = − vi wij hj − ai vi − bj hj (1)
Z
i∈V,j∈H i∈V j∈H
where ai and bj are biases acting respectively on the visible and hidden units and wij is the weight matrix that
couples visible and hidden nodes. pprior is in product form and encodes the nature of each
P visible variable, either
with a Gaussian prior pprior = N (0, σv2 ) or a binary prior pprior (v) = δ(s2 − s). Z = v,h pprior (v)e−E(v,h) is
the partition function.P The classical training method consists in maximizing the marginal likelihood over the
visible nodes P (v) = h P (v, h) by tuning the RBM parameters θ = {wij , ai , bj } via gradient ascent of the log
likelihood L(v; θ).
The tractability of the method relies heavily on the fact that the conditional probabilities P (v|h) and P (h|v)
are given in closed forms. In our case these read:
P
Y e j∈H vi wij hj +ai vi pprior (vi ) Y X
P (v|h) = P σ vi wij hj + ai vi , (2)
j∈H vi wij hj +ai vi
P
i∈Vf vi e pprior (vi ) i∈V` j∈H
Y X
P (h|v) = σ vi wij + bj , (3)
j∈H i∈V
where σ(x) = 1/(1 + e−x ) is the logistic function. The gradient of the likelihood w.r.t. the weights (and similarly
w.r.t. the fields ai and bj ) is given by
∂L(v; θ)
= hvi hj p(hj |v)idata − hvi hj iRBM (4)
∂wij
where the brackets hidata and hiRBM respectively indicate the average over the data and over the distribution
(1). The positive term is directly linked to the data and can be estimated exactly with (3), while the negative
term is intractable. Many strategies are used to compute this last term: the contrastive divergence (CD)
approach [Hin02] consists in estimating the term over a finite number of Gibbs sampling steps, starting from
a data point and making alternate use of (2) and (3); in its persistent version (PCD) the chain is maintained
over subsequent mini-batches; using mean-field approximation [MTK15] the term is computed by means of a
low-couplings expansion.
3 Learning RBM with incomplete data
The RBM is a generative model able to learn the joint distribution of some empirical data given as input. As
such, it is intrinsically able to encode the relevant statistical properties found in the training data instances
that relate features and labels, and this makes the RBM particularly suitable to be used in the multi-output
setting in the presence of incomplete observations. In this sense, the most natural way to deal with incomplete
observations is to marginalize over the missing variables; in this section we show how the contrastive divergence
algorithm can be adapted to compute such marginals.
Given a partially-observed instance v, we have a new partition of the visible space V = O + M, where O
is a subset of observed values of v that can correspond both to features and labels. vo = {vi , i ∈ O} and
vm = {vi , i ∈ M} denote respectively the observed and missing values of v. The probability over the observed
variables vo is given by (θ representing the parameters of the model)
Z Y !
ZO [θ] P
a v
Y X
P (vo ) = , ZO [θ] = pprior (vi )dvi × e k∈V k k 1 + exp wkj vk + bj
Z∅ [θ]
i∈M j∈H k∈V
Taking the log-likelihood and then computing the gradient with respect to the weight matrix element wij
(also similarly for the fields ai and bj ), we obtain two different expressions for i ∈ O and i ∈ M.
Z
∂ log ZO [θ] X ∂ log ZO [θ] X
= vi hj p(hj |vo ) i ∈ O, = dvi vi hj p(vi , hj |vo ) i ∈ M (5)
∂wij ∂wij
hj hj
The gradient of the LL over the weights (4) now reads
Z
∂L(v; θ) D X E D X E
= Io (i)vi hj p(hj |vo ) + 1 − Io (i) dvi vi hj p(hj |vo ) − hvi hj iRBM (6)
∂wij data data
hj hj
where Io is the indicator function of the samples dependent set O. The observed variables vi , i ∈ O are pinned
to the values given by the training samples. In terms of our model, the pinned variables play the role of an
additional bias over the hidden variables of a RBM where the ensemble of visible variables is reduced to the
missing ones.
With respect to the non-lossy case where p(hj |v) is given in closed form, here we need to sum over the
missing variables in order to estimate p(hj |vo ). This means that also the positive term of the gradient (6) is now
intractable and we need to approximate it. For CD training, we can simply perform Gibbs sampling over the
missing variables (keeping fixed the observed variables). Details are reported in Alg. 1.
We note that the extra computational burden of Lossy-CD with respect to standard CD is due only to the
extra Gibbs sampling steps in the positive term. Given that the observed variables strongly bias the sampling
procedure speeding up convergence, only few sampling steps are needed to compute this term. Indeed, in our
experiments we observed that a single sampling step (Lossy-CD1) is enough, making the additional complexity
minimal. Finally, we note that the same method can be applied to PCD and mean-field training procedures. In
the first case, it is sufficient to keep track of an additional persistent chain, which requires little extra memory
and no extra computational complexity. In the second case, we only need to substitute Gibbs sampling with
iterative mean-field equations.
Algorithm 1: Lossy-CDk (RBM training with Incomplete data)
1: Data: a training set of N data vectors
2: Randomly initialize the weight matrix W
3: for t = 0 to T (# of epochs) do
4: Divide the training set in m minibatches
5: for all minibatches m do
Positive term:
6: pin variables vi , i ∈ O to their correct value
7: initialize vi , i ∈ M randomly
8: sample h, vm using p(vm | h) and p(h | v) for k steps
9: compute the positive terms in (5)
Negative term:
10: initialize v randomly
11: iterate eq. (2), (3) (k steps) to compute hvi hj imodel
Full update:
12: update W with equation (6)
13: end for
14: end for
4 Mean-field based imputation with RBM
As a generative model, the trained RBM can be used to sample new data. For imputation of missing features
and labels we just need to use the observed portions of our data to bias the sampling procedure in the same
way as for the computation of the positive term in Alg. 1. Namely, we estimate p(vm |vo ) by pinning the
observed variables and iterating CD/PCD or mean-field to approximate the equilibrium values of the missing
variables. In case of a high percentage of missing observations, however, we might expect the observed variables
to be correlated to many different equilibrium configurations, such that the sampling could be biased towards
the wrong sample. To overcome this problem, we simply average over multiple mean-field imputations for each
incomplete data instance.
More in details, let {pi , i ∈ V` } and {qj , j ∈ H} be the marginal probabilities respectively of visible labels
and hidden variables to be activated and {mi , i ∈ Vf } the marginal expectation of the visible features variables.
Mean-field equations at lowest order (O(1/N ), N being the size of the system) express self-consistent relations
among these quantities
X X
mi = wij qj + ai σv2 ∀i ∈ Vf \O pi = σ wij qj + ai ∀i ∈ V` \O (7)
j∈H j∈H
X X
qj = σ wij mi + wij pi + bj (8)
i∈Vf i∈V`
Higher order terms corresponding to TAP equations are discarded [M1́7]. These equations can be efficiently solved
by iteration starting from random configurations until a fixed point is reached. Observed variables are simply
introduced by pinning their corresponding probabilities (0 or 1 for label variables) or their marginal expectation
(for feature variables) to the observed values. In practice we run these fixed-point equations Nf ∼ 10 times and
the imputations are obtained by simple average
Nf Nf
1 X (n) 1 X (n)
m̂i = m pi = p .
Nf n=1 i Nf n=1 i
In the multi-label setting, the predictor is the indicator function p̂i = (pi > t) (t is learned, it is chosen to
maximize the accuracy for known labels), while for class labels we have p̂i = 1 if i = argmaxk (pk )
Model RMSE Averaged AUC Accuracy
qmc % 30% 50% 80% 30% 50% 80% 30% 50% 80%
RBM-MO(qf ea % =50%) 0.183 0.182 0.185 0.969 0.971 0.929 0.950 0.912 0.822
CLE(qf ea % =50%) 0.195 0.195 0.195 0.686 0.718 0.742 0.256 0.232 0.282
NoisyIMC(qf ea % =50%) 0.209 0.210 0.210 0.621 0.578 0.552 0.225 0.232 0.192
MC-1(qf ea % =50%) 0.334 0.335 0.337 0.495 0.493 0.500 0.110 0.111 0.112
RBM-MO(qf ea % =80%) 0.209 0.213 0.211 0.938 0.932 0.906 0.920 0.852 0.733
CLE(qf ea % =80%) 0.206 0.208 0.206 0.673 0.678 0.625 0.230 0.215 0.220
NoisyIMC(qf ea % =80%) 0.212 0.211 0.213 0.652 0.577 0.537 0.230 0.217 0.210
MC-1(qf ea % =80%) 0.334 0.334 0.335 0.500 0.501 0.500 0.112 0.110 0.110
Table 1: Transductive test on MNIST multi-class data set (our method in bold, best result in red)
Model RMSE Micro-AUC Hamming-Accuracy
qml % 30% 50% 80% 30% 50% 80% 30% 50% 80%
RBM-MO(qf ea % =50%) 0.131 0.137 0.123 0.943 0.934 0.888 0.919 0.907 0.873
CLE(qf ea % =50%) 0.130 0.130 0.131 0.905 0.893 0.898 0.885 0.871 0.878
NoisyIMC(qf ea % =50%) 0.132 0.133 0.133 0.865 0.863 0.858 0.845 0.841 0.848
MC-1(qf ea % =50%) 0.258 0.255 0.267 0.522 0.528 0.527 0.826 0.817 0.824
RBM-MO(qf ea % =80%) 0.160 0.158 0.158 0.875 0.867 0.826 0.856 0.858 0.832
CLE(qf ea % =80%) 0.129 0.129 0.128 0.913 0.897 0.899 0.889 0.875 0.876
NoisyIMC(qf ea % =80%) 0.133 0.134 0.134 0.853 0.857 0.849 0.839 0.835 0.826
Table 2: Transductive test on Scene multi-label data set (our method in bold, best result in red)
5 Experimental Study
5.1 Experimental configuration
To evaluate the efficiency of RBM-MO we compare its performance against CLE, NoisyIMC and MC-1, which
provide state-of-the-art baselines.
For the transductive experiments we randomly hide features and labels of the whole dataset to generate
incomplete data for training, and we compute appropriate scores for the reconstruction of missing features and
labels. In the inductive test, instead, we split the whole dataset into non-overlapping training and testing sets.
Concerning the training set the same protocol is used as in the transductive test. For the test set the difference
is that now all labels are hidden. Once the classifier is trained, it is applied on the test set to predict the labels.
We still randomly hide the entries of test features vectors, so as to form an incomplete testing set. Finally,
in the splitting we use 70% of the data instances for training and the remaining 30% for testing.
We denote by qf ea , qml and qmc the percentage of masked features, labels and classes labels respectively. Note
that a masked class label means that all binary variables attached to the classes of a given label are masked
together. These rates of masking are kept identical in the learning and test sets.
In the transductive test, we compute the Root Mean Squared Error (RMSE) to measure the reconstruc-
tion accuracy with respect to the missing feature values. Furthermore, for the reconstructed labels we calculate
Micro-AUC scores and Hamming-accuracy [GKG12] in the multi-label scenario, and Averaged AUC plus
Accuracy [LY15] in the multi-class case. In the tables, we define Hamming-accuracy as 1-Hamming loss
to keep a consistent variation tendency with the AUC scores. In the inductive test we only compute the scores
on the reconstructed labels, since reconstructing missing features is not the goal of inductive classification.
We run the test as described 10 times with different realizations of the missing features and labels. Average and
Model Averaged AUC Accuracy
qmc % 30% 50% 80% 30% 50% 80%
RBM-MO(qf ea % =50%) 0.887 0.914 0.910 0.533 0.673 0.660
CLE(qf ea % =50%) 0.785 0.791 0.791 0.297 0.256 0.268
NoisyIMC(qf ea % =50%) 0.780 0.771 0.781 0.302 0.272 0.265
RBM-MO(qf ea % =80%) 0.891 0.909 0.889 0.562 0.682 0.647
CLE(qf ea % =80%) 0.768 0.664 0.622 0.271 0.200 0.176
NoisyIMC(qf ea % =80%) 0.748 0.687 0.615 0.264 0.220 0.178
Table 3: Inductive test on Pendigits multi-class dataset (our method in bold, best result in red)
Model Micro-AUC Hamming-Accuracy
qml % 30% 50% 80% 30% 50% 80%
RBM-MO(qf ea % =50%) 0.839 0.826 0.793 0.970 0.970 0.965
CLE(qf ea % =50%) 0.705 0.707 0.706 0.700 0.724 0.719
NoisyIMC(qf ea % =50%) 0.704 0.702 0.700 0.710 0.717 0.718
RBM-MO(qf ea % =80%) 0.759 0.791 0.766 0.964 0.964 0.967
CLE(qf ea % =80%) 0.693 0.688 0.694 0.718 0.706 0.718
NoisyIMC(qf ea % =80%) 0.689 0.688 0.685 0.705 0.704 0.704
Table 4: Inductive test on EventCat multi-label data set (our method in bold, best result in red)
Dataset No. of Instances No. of Features No.of Labels No. of Classes
Scene 2,407 294 6 -
Pendigits 10992 16 - 10
MNIST 70,000 784 - 10
EventCat 5,93 72 6 -
Table 5: Summary of 4 public multi-label and multi-class data sets.
standard deviation of the computed scores are recorded to compare the overall performances. In the tables, we
use red fonts to denote the best reconstruction and classification performances among all the algorithms involved
in the empirical study. The bold black font is used to highlight the performance of the proposed RBM-MO
method.
For the baselines, we used grid search to choose the optimal parameter combination following the suggested
ranges of parameters as in [HSSZ18].
The RBM-MO is trained following the guidelines in [Hin10]. We always use binary variables for the hidden
layer, while in the visible layer we use binary variables for MNIST and Gaussian variables for the other datasets.
In all the simulations, we fix the number of hidden nodes to 100. The learning rate η is fixed to 0.001 and the
size of the mini-batches to 10. During training the number of Gibbs steps is set to k = 1 while for imputation
we iterate the mean-field equations 10 times. As a stopping condition, we considered the degradation of the
transductive AUC scores with a look-ahead of 500 epochs
5.2 Summary of datasets
We consider 3 publicly available datasets related to image processing. These datasets cover both multi-label and
multi-class learning tasks, and they are popularly used as benchmark datasets in multi-output learning research.
In addition, we consider the challenging scenario of abnormality detection on IoT devices. The relevant
dataset, that we call EventCat, consists in security telemetry data collected from various network appliances
(e.g. smart watches, smartphones, driving assistance systems...), each reporting a features vector whose entries
indicate the occurring frequency of a specific type of alert (e.g. downloading suspicious files, login failures,
unfixed vulnerabilities...). Multiple labels are assigned to each device in the collected dataset, corresponding to
a variety of categories of security threats.
Some details about the datasets are reported in Table.5.
5.3 Qualitative results on MNIST
A qualitative evaluation of the performance of the RBM-MO model is given by looking at features reconstruction
for the MNIST dataset, as reported in Fig. 1. The model at hand has been trained over a dataset in which
50% of the features were missing. To assess the robustness of the method, we computed the reconstructions in
the highly challenging case in which 80% of the features were missing. Apart from some smoothing due to the
employment of mean-field imputations, the reconstructed samples look reasonably realistic. In general, from the
qualitative point of you the results are comparable to those obtained with more complex and expensive DLVMs
like MIWAE and MisGAN [MF19, LJM19].
5.4 Empirical results
The transductive results for MNIST (multi-class) and Scene (multi-label) datasets are reported in tables 1 and
2. Going into the details, we first observe that RBM-MO is by a large margin more efficient than all of the
Figure 1: Features reconstruction by RBM-MO trained over an incomplete dataset with 50% missing-at-random
features, whose classification accuracy has been measured to be around 91%. The first block shows some complete
testing instances. The second and third block show the same testing instances after hiding respectively 50% and
80% of the pixels. The last two columns show the results of the mean-field imputations over the incomplete
testing instances.
baselines for the inference of class labels (table 1), probably because it is able to encode more complex statistical
properties.
On the multi-label problems, the situation is still in favour of RBM-MO but with less margin (table 2), in
particular at a larger percentage of missing features.
Now if we look at the reconstruction error on these datasets we observe that RBM-MO generally achieves a
higher reconstruction accuracy than the other opponents, especially on the MNIST dataset. The results verify
empirically the basic motivation of using a generative model such as the RBM: incomplete features and
labels can provide complementary information to each other, so as to better recover the missing
elements. The variance of the results is omitted in the tables by lack of space. For RBM-MO the standard
deviation of the derived RMSE, AUC and accuracy scores is not larger than 0.01 over the different datasets.
Although the RMSE scores reported by the baseline methods look comparable to the RBM-MO ones, and in
certain cases they are better, they also come with a slightly higher variance, such that the RBM-MO seems to
be more efficient and robust for features reconstruction.
Except MC-1, all the baseline methods are used for inductive learning. As in the transductive test, we
show only the mean of the derived metrics in the tables. Nevertheless, we have similar variance ranges for the
computed scores as reported in the transductive test. Clearly RBM-MO is much better adapted to this setting
than the baseline methods both for multi-class (table 3) and multi-label learning. The baseline inductive meth-
ods CLE and NoisyIMC are specifically designed for multi-label learning and their performance deteriorates
significatively in the multi-class scenario. By comparison, RBM-MO can be adapted seamlessly to multi-class
and multi-label learning, producing consistently good performances.
For the EventCat dataset, inductive results are reported in table 4. Even with highly incomplete training
data, RBM-MO produces the best predictions over partially observed testing data instances.
6 Conclusion
Machine learning is witnessing a race to high complexity models eager for large data and computational power.
In the context of multi-output classification in a challenging scenario - (i) learning with highly incomplete
features and partially observed labels; ii) applying the learnt classifier with incomplete testing instances) -
we advocate instead for simple probabilistic and interpretable models. After refining the learning of the RBM
model, we give empirical evidences that it can be efficiently adapted to this context on a great variety of datasets.
Experiments are conducted on both public databases and a real-world IoT security dataset, showing various sizes
of training sets as well as features and labels vectors. Our approach consistently outperforms the state-of-the-art
robust multi-class and multi-label learning approaches with imperfect training data, indicating good usability
for practical applications.
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