From N to N+1: Multiclass Transfer Incremental Learning
Ilja Kuzborskij1,2 , Francesco Orabona3 , and Barbara Caputo1
1

2

Idiap Research Institute, Switzerland,École Polytechnique Fédérale de Lausanne (EPFL), Switzerland,3
Toyota Technological Institute at Chicago, USA,ilja.kuzborskij@idiap.ch, francesco@orabona.com, barbara.caputo@idiap.ch

Abstract
Since the seminal work of Thrun [17], the learning tolearn paradigm has been defined as the ability of an agent toimprove its performance at each task with experience, with
the number of tasks. Within the object categorization domain, the visual learning community has actively declined
this paradigm in the transfer learning setting. Almost all
proposed methods focus on category detection problems,
addressing how to learn a new target class from few samples by leveraging over the known source. But if one thinksof learning over multiple tasks, there is a need for multiclasstransfer learning algorithms able to exploit previous source
knowledge when learning a new class, while at the sametime optimizing their overall performance. This is an open
challenge for existing transfer learning algorithms. The
contribution of this paper is a discriminative method that
addresses this issue, based on a Least-Squares Support Vector Machine formulation. Our approach is designed to balance between transferring to the new class and preserving
what has already been learned on the source models. Extensive experiments on subsets of publicly available datasetsprove the effectiveness of our approach.

1. Introduction
Vision-based applications like Google Goggle, assisted
ambient living, home robotics and intelligent car driver assistants all share the need to distinguish between several object categories. They also share the need to update theirknowledge over time, by learning new category models
whenever faced with unknown objects. Consider for instance the case of a service robot, designed for cleaning
up kitchens in public hospitals. Its manufacturers will haveequipped it with visual models of objects expected to befound in a kitchen, but inevitably the robot will encountersomething not anticipated at design time – perhaps an object out of context, such as a personal item forgotten by apatient on her food tray, or a new type of food processor
that entered the market after the robot. To learn such new
object, the robot will generally have to rely on little data andexplanation from its human supervisor. Also, it will have topreserve its current range of competences while adding thenew object to its set of known visual models. This challenge, which holds for any intelligent system equipped witha camera, can be summarized as follows: suppose you havea system that knows N objects (source). Now you need toextend its object knowledge to the N + 1-th (target), using only few new annotated samples, without having thepossibility to re-train everything from scratch. Can you
add effectively the new target N + 1-th class model to the
known N source models by leveraging over them, while atthe same time preserving their classification abilities?
As of today, we are not aware of previous work addressing this issue, nor of existing algorithms able to capture allits nuances. The problem of how to learn a new object
category from few annotated samples by exploiting priorknowledge has been extensively studied [20, 11, 7]. The
majority of previous work focused on object category detection (i.e. binary classification) rather than the multiclasscase [1, 19, 18]. It is natural to ask if such previous methods would work well in the scenario depicted, by just extending them to the multiclass. We argue that to solve theN −→ N + 1 transfer learning problem one needs to address a deeper algorithmic challenge.
In addition, learning from scratch and preserving training sets from all the source tasks might be infeasible due tothe large number of tasks or when acquiring tasks incrementally, especially for large datasets [15]. In object categorization case this might come as training source classifiers fromlarge scale visual datasets, in abundance of data.
Consider the following example: a transfer learning taskof learning a dog detector, given that the system has alreadywhen going from N to N + 1.
As in [18], we learn how much to transfer from eachof the source classifiers, by minimizing the Leave-One-Out(LOO) error, which is an unbiased estimator of the generalization error for a classifier [4]. We call our algorithm
MULticlass Transfer Incremental LEarning (MULTIpLE).
Experiments on various subsets of the Caltech-256 [9]and Animals with Attributes (AwA) datasets [13] show that
our algorithm outperforms the One-Versus-All (OVA) extension of [18], as well as other baselines [11, 20, 1]. Moreover, its performance often is comparable to what it would
be obtained by re-training the whole N + 1 classifier fromall data, without the need to store the source training data.
The paper is organized as follows: after a review of previous work (Section 2), we describe our setting (Section 3)
and our algorithm (Section 4). Experiments are reported in
Section 5, followed by conclusions in Section 6.
Figure 1: Binary (left) versus N −→ N + 1 transfer learning (right). In both cases, transfer learning implies that thetarget class is learned close to where informative sources
models are. This is likely to affect negatively performancein the N −→ N + 1 case, where one aims for optimal accuracy on the sources and target classes simultaneously.

learned other kind of animal detectors. This is achieved, in
one form or another, by constraining the dog model to besomehow “similar” to the horse and cat detectors learned
before [11, 18]. Success in this setting is defined as optimizing the accuracy of the dog detector, with a minimal
number of annotated training samples (Figure 1, left).
But if we consider the multiclass case, the different tasks
now “overlap”. Hence we are faced with two opposite
needs: on one side, we want to learn to recognize dogs fromfew samples, and for that we need to impose that the dog
model is close to the horse and cat models learned before.
On the other side, we want to optimize the overall system
performance, which means that we need to avoid mispredictions between classes at hand (Figure 1, right). These
two seemingly contradictory requirements are true for manyN −→ N + 1 transfer learning scenarios: how to reconcile
them in a principled manner is the contribution of this paper.
We build on the algorithm of Tommasi et al. [18], atransfer learning method based on the multiclass extensionof Least-Squares Support Vector Machine (LSSVM) [16].
Thanks to the linear nature of LSSVM, we cast transfer
learning as a constraint for the classifier of the N + 1 target
class to be close to a subset of the N source classifiers. At
the same time, we impose a stability to the system, biasing
the formulation towards solutions close to the hyperplanesof the N source classes. In practice, given N source models, we require that these models would not change much

2. Related Work
Prior work in transfer learning addresses mostly the binary classification problem (object detection). Some approaches transfer information through samples belonging toboth source and target domains during the training process,as in [14] for reinforcement learning. Feature space approaches consider transferring or sharing feature space representations between source and target domains. Typically,in this setting source and target domain samples are available to the learner. In that context, Blitzer et al. [2] proposeda heuristic for finding corresponding features, that appear
frequently in both domains. Daumé [6] showed a simple
and effective way to replicate feature spaces for performing adaptation for the case of natural language processing.
Yao and Doretto [20] proposed an AdaBoost-based method
using multiple source domains for the object detection task.
Another research line favors model-transfer (orparameter-transfer) methods, where the only knowledge
available to the learner is “condensed” within a modeltrained on the source domain. Thus, samples from source
domain are not preserved. Model-transfer is theoreticallysound as was shown by Kuzborskij and Orabona [12],since relatedness of the source and target tasks enables
quick convergence of the empirical error estimate to thetrue error. Within this context, Yang et al. [19] proposed akernelizable SVM-like classifier with a biased regularization term. There, instead of the standard `2 regularization,
the goal of the algorithm is to keep the target domainclassifier “close” to the one trained on the source domain.
Tommasi et al. [18] proposed a multi-source transfer modelwith a similar regularizer, where each source classifier was
weighted by learned coefficients. The method obtained
strong results on the visual object detection task, using
only a small amount of samples from the target domain.
Aytar and Zisserman [1] proposed a similar model, with alinear formulation for the problem of object localization.
Both methods rely on weighted source classifiers, which is
crucial when attempting to avoid negative transfer. Several
Multiple Kernel Learning (MKL) methods were proposed
for solving transfer learning problems. Jie et al. [11]suggested to use MKL kernel weights as source classifier
weights, proposing one of the few truly multiclass transfer
learning models. An MKL approach was also proposed byDuan et al. [7]. There, kernel weights affect both the sourceclassifiers and the representation of the target domain.

3. Problem Setting and Definitions
In the following we denote with small and capital bold
letters respectively column vectors and matrices, e.g. α =[α1 , α2 , . . . , αN ]T ∈ RN and A ∈ RM ×N with Aji corresponding to the (j, i) element. When only one subscriptedindex is present, it represents the column index: e.g. Ai is
the i-th column of the matrix A.
As in related literature, we define a set of M training
samples consisting of a feature vector xi ∈ Rd and thecorresponding label yi ∈ Y = {1, . . . , N, N + 1} fori ∈ {1, 2, . . . , M }. We will denote by X the samplecolumn matrix, i.e. X = [x1 , · · · , xM ]. We use the formalism of linear classifiers, so that a multiclass classifier is
described as a matrix W = [W 1 , . . . , W N ], where eachcolumn vector W n represents the hyperplane that separates
one of the N classes from the rest. Hence, the label associated to a given sample x is predicted as fW (x) :=argmax W >n x + bn . Note that it is straightforward to lift

n=1,...,Nthe theory to the non-linear domain by using kernels, andfor clarity we describe the algorithm in linear notation.
A common way to find the set of hyperplanes W is bysolving a regularized problem with a convex loss function,that upper bounds the 0/1 loss. For reasons that will become
clear later, we base our method on the OVA variant of the
LSSVM [16], which combines a square loss function with
`2 regularization. Defining the label matrix Y such that Yin
is equal to 1 if yi = n and −1 otherwise, we obtain themulticlass LSSVM objective function
M N
C XX
1
2
(W >
min kW k2F +n xi + bn − Yin ) ,W ,b 22 i=1 n=1

where k · kF is the Frobenius norm.
In our setting of interest, there are two types of information. First, we have a set of models that were obtained
from the source N class problem. These source models are
encoded as a set of N hyperplanes, that we again represent
in matrix form as W 0 = [W 01 , . . . , W 0N ]. Note that weassume no access to the samples used to train the sourceclassifiers. Second, we have a small training set composedfrom samples belonging to all the N + 1 classes, target andsource classes.

4. MULTIpLE
The aim of our approach is to find a new set of hyperplanes W = [W 1 , . . . , W N ], wN +1 , such that i) performance on the target N + 1-th class improves by transferringfrom the source models, and ii) performance on the source
N classes should not deteriorate or even improve comparedto the former. Thanks to the model linearity, we obtain ametric between classifiers, that could be used to find classifiers with similar performance by enforcing the distance
between them to be small. We propose to achieve both aims
above through the use of distance-based regularizers.
The first objective can be recognized as the transfer
learning problem. It has been shown that this can be implemented using the regularizer kwN +1 − W 0 βk2 [18]. This
term enforces the target model wN +1 to be close to a linear combination of the source models, while negative transfer is prevented by weighing the amount of transfer of eachsource model using the coefficient vector β = [β1 . . . βN ]> .
The second objective of avoiding degradation of existingmodels W 0 has been ignored in the transfer learning literature. However, as explained before, adding a target class
may affect the performance of the source models and it istherefore useful to transfer the novel information back to
the N source models. To prevent negative transfer, we enforce the new hyperplanes W to remain close to the source
hyperplanes W 0 using the term kW − W 0 k2F . With both
regularizers in the LSSVM objective function, we obtain
min

W ,wN +1 ,b1
1
kW − W 0 k2F + kwN +1 − W 0 βk2F22
M N +1

+

CXX
2
(W >
n xi + bn − Yin ) .
2 i=1 n=1

The solution to this minimization problem is given by
W n = W 0n +MX

Ain xi , n = 1, · · · , N

(1)

i=1

wN +1 =NX
n=1

βn W 0n +M
X

Ai(N +1) xi ,

(2)

i=1

i
h
>
and b = b0 − b00 b00 β , where A = A0 − [A00 A00 β],




A0
Y
:=
M
(3)
>
0
b0


 > 0 
A00
X W
(4)
:= M
>
0
b00
 >
−1
X X + C1 I 1
M :=
.
(5)
1>
0
The solution of the tranfer learning problem is completely
defined once we set the parameters β. In the next section
we describe how to automatically tune these parameters.

4.1. Self-tuning of Transfer Parameters
We want to set the transfer coefficients β to improve
the performance by exploiting only relevant source models while preventing negative transfer. With this in mind,
we extend the method of [18] to our setting and to our objective function. We optimize the coefficients β automatically using an objective based on the LOO error, which isan almost unbiased estimator of the generalization error of aclassifier [4]. An advantage of LSSVM over other methods
is that it allows the LOO error to be computed efficiently inclosed-form.
Specifically, we cast the optimization of β as the minimization of a convex upper bound of the LOO error. The
LOO prediction for a sample i with respect to hyperplane
W n is given by (derivation is available in supplementarymaterial1 )
Ain
.
(6)
Ỹin (β) := Yin −Mii
In matrix form we haveỸ(β) = Y − (M ◦ I)−1 (A0 − [A00 A00 β]) .

(7)

We stress that (7) is a linear function of β.
We now need a convex multiclass loss to measure the
LOO errors. We could choose the the convex multiclass loss
presented in [5], which keeps samples of different classes at
the unit marginal distance:L(β, i) = max|1 + Yir (β) − Yiyi (β)|+ ,
r6=yi

(8)

where |x|+ := max(x, 0). However, from (1) and (2) itis possible to see that by changing β we only change thescores of the target N + 1-th class. Thus, when using thisloss almost all samples are neglected during optimization
with respect to β. We address this issue by proposing amodified version of (8), Lmod (β, i) as(|1 + Yi(N +1) (β) − Yiyi (β)|+ : yi 6= N + 1
max|1 + Yir (β) − Yiyi (β)|+ : yi = N + 1
r6=yi

The rationale behind this loss is to enforce a margin of 1
between the target N + 1-th class and the correct one, evenwhen the N + 1-th class has not the highest score. This has
the advantage of forcing the use of all the samples in the
optimization of β.
Given the LOO errors and the multiclass loss function,
we can obtain β by solving the convex problem
min
β

M
X

Lmod (β, i)

i=1s.t. kβk2 ≤ 1,βi ≥ 0, i = 1, . . . , N .

1 http://www.idiap.ch/

˜ikuzbor(9)

Constraining β within a unit `2 ball is a form of regularization on β, that prevents the overfitting of the parameters
β. This optimization procedure can be implemented elegantly using projected subgradient descent [3], which is notaffected by the fact that the objective function in (9) is notdifferentiable everywhere. The pseudocode of the optimization algorithm is summarized in Alg. 1.
Algorithm 1 Projected subgradient descent to find β
Input: M, A0 , A00 , T
Output: β
1: β ← 0
2: for t = 1 . . . T do
3:Ỹ ← Y − (M ◦ I)−1 (A0 − [A00 A00 β])
4:
∆←0
5:
for i = 1 . . . M do
6:
if yi 6= N + 1 then7:
if 1 + Yi(N +1) − Yiyi > 0 thenA00i∆ ← ∆ + Mii
end ifelse if maxr6=yi (1 + Yir − Yiyi ) > 0 then
A00
i
11:
∆ ← ∆ − Mii
12:
end if13:
end for14:β ← β − M∆√t15:
βi ← max(βi , 0) ∀i = 1, . . . , N16:
if kβk2 > 1 thenβ17:
β = kβk218:
end if
19: end for

8:
9:
10:

Computational complexity for obtaining A0 , A00 and M
is in O(M 3 +M 2 (N +1)), which comes from matrix operations (3)-(5). Note that this complexity is better than the oneof a classical OVA SVM which in worst case is known to bein O(M 3 N ) [10]. The Alg. 1 is in O(M N (T + 1)), where
we assume that most terms in (7) are precomputed. Each
iteration of the algorithm is very efficient since it depends
linearly on both training set size and number of classes.
To conclude this section, a compact description of the
MULTIpLE algorithm is: i) find the optimal tranfer weights
β with Alg. 1; ii) calculate the final solution using (1)-(5).
The source code of MULTIpLE is available online1 .

5. Experiments
We present here a series of experiments designed to investigate the behavior of our algorithm when (a) the sourceclasses and the target class are related/unrelated, and when(b) the overall number of classes increases. All experiments
were conducted on subsets of two different public datasets,
and the results were benchmarked against several baselines.
In the rest of the section we first describe our experimental

setup (section 5.1), then we describe the chosen baselines(section 5.2). Section 5.3 reports our findings.

5.1. Data setup
We run all experiments on subsets of the Caltech-256
database [9] and of the Animal with Attributes (AwA)
database [13]. From the Caltech-256 database, we selected
a total of 14 classes and for the AwA dataset, 42 classes. We
did not carry out any image pre-selection or pre-processing.
Moreover, for both databases we used pre-computed features available online2 . Specifically, for the Caltech-256
experiments we used the following features: oriented andunoriented PHOG shape descriptors, SIFT appearance descriptors, region covariance and local binary patterns totalling in 14 descriptor types [8]. For the AwA experiments
the chosen features were SIFT, rgSIFT, SURF, PHOG, RGB
color histograms and local self-similarity histograms [13].
For each class considered, we randomly selected 80 image samples. These were then split in three disjoint sets:
30 samples for the source classifier, 20 samples for training
and 30 samples for test. The samples of the source classifier
were used for training the N models W 0 (Section 4).
The performance of each method (see Section 5.2) was
evaluated using progressively {5, 10, 15, 20} training samples for each of the N +1 classes. The experiments were repeated 10 times, using different randomly sampled training
and test sets, which we refer to as data splits. Furthermore,
to get a reliable estimate of the performance of transfer withrespect to different classes, we used a leave-one-class-out
approach, considering in turn each class as the N + 1 target class, and the other N as source classifiers. We reportresults averaged over all data splits and leave-one-class-out
evaluations.

5.2. Algorithmic setup
We compared MULTIpLE against two categories ofbaselines. The first, that we call no transfer baselines, consists of a group of algorithms addressing the N −→ N + 1
problem without leveraging over source models; the second, that we call transfer baselines, consists of a group of
methods attempting to solve the N −→ N + 1 problem byleveraging over source models. The no transfer baselines
are the following:
No transfer corresponds to LSSVM trained only on the
new training data.
Batch corresponds to a LSSVM trained using all available
samples, i.e. assuming to have access to all the data used
to build the source models plus the new training data. The
performance of this method might be seen as an indicatorof the best performance achievable on the problem, thus as
2 Caltech-256: http://www.vision.caltech.edu/Image_
Datasets/Caltech256/
AwA: http://attributes.kyb.tuebingen.mpg.de/

an important reference for assessing the results obtained bytransfer learning methods.
Source is the LSSVM N -class source classifier. In thiscase, classification on the sample belonging to N + 1-thclass is assigned 0 accuracy.
Source+1 corresponds to a binary LSSVM trained to discriminate between the target class vs the source classes
given the training data. It is evaluated on the N + 1 problem
by combining it with Source in a OVA setting. It is arguably
the simplest possible approach to address the N −→ N + 1
problem.
Source+1 (hinge) is the scheme analogous to Source+1, bututilizing the hinge loss `(x, z) = |1 − xz|+ , thus corresponding to a classical SVM formulation.
As transfer baselines, we chose the following methods:
MKTL We compared against Multi Kernel Transfer Learning (MKTL) [11], which is one of the few existing discriminative transfer learning algorithm in multiclass formulation.
MultiKT-OVA We implemented an OVA multiclass extension of the binary transfer learning method by Tommasi etal. [18] as follows: as in the standard OVA formulation,
we train MultiKT instance to discriminate between one of
N + 1 classes and the rest N . At the same time we use
Source as the source classifier. Thus, eventually we obtainN + 1 MultiKT instances.
PMT-SVM-OVA We also implemented an OVA multiclassextension of the binary transfer learning method by Aytar
and Zisserman [1], as done for MultiKT-OVA.
MultisourceTrAdaBoost-OVA As a final transfer learning baseline, we implemented an OVA extension of MultisourceTrAdaBoost [20], where each source corresponds
to a subset of samples designated for the source classifier,while belonging to a specific class. We follow the authors
by using linear SVM as weak learner.
Apart for PMT-SVM-OVA and MultisourceTrAdaBoostOVA, which cannot be kernelized, we used all the features
available for each dataset via kernel averaging [8], computing the average of RBF kernels over all available features from the dataset at hand and RBF hyperparametersγ ∈ {2−5 , 2−6 , . . . , 28 }. The trade-off hyperparameterC ∈ {10−5 , 10−6 , . . . , 108 } was tuned by 5-fold crossvalidation for the no transfer baselines. In case of modeltransfer algorithms, source model’s C value was reused.
Since MultisourceTrAdaBoost-OVA is a non-kernel
baseline, to test its performance over multiple features weconcatenated them. This approach proved computationally
unfeasible for PMT-SVM-OVA (we used the implementation made available by the authors). Thus, to compare fairly
with it, we also did run experiments using, for all methods, alinear kernel and a single feature (SIFT for the Caltech-256
and PHOG for the AwA). We refer to this setting as linear.

0.64

Batch
No transfer
MULTIpLE
Source+1
SourceSource+1 (hinge)

0.55
0.504

6

8

10 12 14 16 18 20
Number of samples

Caltech-256 5 unrelated
linear, transfer baselines

0.75

N+1 accuracy

0.70
0.65

0.60
0.55
0.504

6

MKTL
MultisourceTrAdaBoost-OVA
MultiKT-OVA
MULTIpLE
PMT-SVM-OVA

8 10 12 14 16 18 20 22
Number of samples

N+1 accuracy

Caltech-256 5 unrelated
0.85 kernel, transfer & selected no transfer
0.80
0.75
MKTL
0.70
MultisourceTrAdaBoost-OVA
MultiKT-OVA
Batch
0.65
MULTIpLE
Source+1
0.60
Source+1 (hinge)
0.55
0.50
0.454 6 8 10 12 14 16 18 20 22
Number of samples

0.60
0.58

0.564
0.65
0.64
0.63
0.62
0.61
0.60
0.59
0.58
0.574

6

8

10 12 14 16 18 20
Number of samples

Caltech-256 5 mixed
linear, transfer baselines

Caltech-256 5 related
linear, transfer baselines

0.40
0.38
0.36

N+1 accuracy

0.60

0.62

0.34

6

8

MKTL
MultisourceTrAdaBoost-OVA
MultiKT-OVA
MULTIpLE
PMT-SVM-OVA

10 12 14 16 18 20
Number of samples

Caltech-256 5 mixed
0.74 kernel, transfer & selected no transfer
0.72
0.70
0.68
0.66
0.64 MKTL
MultisourceTrAdaBoost-OVA
0.62 MultiKT-OVA
Batch
0.60 MULTIpLE
Source+1
Source+1 (hinge)
0.584 6 8 10 12 14 16 18 20
Number of samples

N+1 accuracy

0.65

Caltech-256 5 related
linear, no transfer baselines
0.44
0.42
0.40
0.38
0.36
0.34
0.32 Batch
transfer
0.30 No
MULTIpLE
Source+10.28 Source
Source+1 (hinge)
0.264 6 8 10 12 14 16 18 20
Number of samples

N+1 accuracy

0.70

N+1 accuracy

0.66

N+1 accuracy

0.75

Caltech-256 5 mixed
linear, no transfer baselines

Batch
No transfer
MULTIpLE
Source+1
SourceSource+1 (hinge)

N+1 accuracy

0.68

0.32

0.30
0.28

0.264

6

MKTL
MultisourceTrAdaBoost-OVA
MultiKT-OVA
MULTIpLE
PMT-SVM-OVA

8 10 12 14 16 18 20 22
Number of samples

Caltech-256 5 related
0.50 kernel, transfer & selected no transfer
0.45
MKTL
MultisourceTrAdaBoost-OVA
MultiKT-OVA
Batch
MULTIpLE
Source+1
Source+1 (hinge)

N+1 accuracy

Caltech-256 5 unrelated
linear, no transfer baselines

0.80

0.40
0.35

0.30
0.254

6

8 10 12 14 16 18 20 22
Number of samples

Figure 2: Experimental results for N + 1 = 5, Caltech-256. From left to right, columns report results for the unrelated,
mixed and related settings. Top row: no transfer baselines, linear case. Middle row: transfer learning baselines, linear
case. Bottom row: transfer and competitive no transfer baselines, average of RBF kernels over all features. Stars represent
statistical significance of MULTIpLE over MultiKT-OVA, p < 0.05.

5.3. Evaluation results
Mimicking the setting proposed in Tommasi et al. [18],we performed experiments on different groups of related,unrelated and mixed categories for both databases.
For the Caltech-256 database, the related classes werechosen from the “quadruped animals” subset; the unrelatedclasses were chosen randomly from the whole dataset, andthe mixed classes were taken from the “quadruped animals”
and the “ground transportation” subsets, sampled in equal

proportions. For the AwA database, the related classes werechosen from the “quadruped animals” subset; the unrelatedclasses were randomly chosen from the whole dataset, andthe mixed classes were sampled in equal proportions fromthe subsets “quadruped animals” and “aquatic animals”.
This setting allows us to evaluate how MULTIpLE, and thechosen baselines, are able to exploit the source knowledge
in different situations, while considering the overall accuracy. To assess the performance of all methods as the overall number of classes grows, we repeated all experiments in-

0.35

0.35

0.35

0.30

0.30

0.25

0.20
0.154

6

MKTL
MultiKT-OVA
Batch
MULTIpLE
Source+1
Source+1 (hinge)

8 10 12 14 16 18 20 22
Number of samples

0.30

0.25

0.20
0.154

N+1 accuracy

AwA 20 relatedtransfer & selected no transfer
0.40 kernel,
MKTL

N+1 accuracy

AwA 20 mixed
0.40 kernel, transfer & selected no transfer

N+1 accuracy

AwA 20 unrelated
0.40 kernel, transfer & selected no transfer

6

MKTL
MultiKT-OVA
Batch
MULTIpLE
Source+1
Source+1 (hinge)

8 10 12 14 16 18 20 22
Number of samples

MultiKT-OVA
Batch
MULTIpLE
Source+1
Source+1 (hinge)

0.25

0.20
0.154

6

8 10 12 14 16 18 20 22
Number of samples

Figure 3: Results for N + 1 = 20, AwA, transfer and competitive no transfer baselines, average of RBF kernels, all features.
Left to right: unrelated, mixed and related settings. Stars represent statistical significance of MULTIpLE over MultiKT-OVA.
creasing progressively their number, with N +1 = 5, 10, 20
respectively. Because of space constraint and redundancy,
only a subset of all experiments is reported here3 .
Figure 2 shows the results obtained for N + 1 = 5. The
left column shows the results for the unrelated setting; thecenter column shows the results for the mixed setting, andthe right column shows the results for the related setting.
The first row compares the results obtained by MULTIpLE
with those of the no transfer baselines (Section 5.2), using
a single feature and a linear kernel. We see that the performance of MULTIpLE is always better than no transfer, andin two cases out of three is better or on par with Source andSource+1 (hinge) (unrelated and mixed), while it is always
equivalent to Source+1. This is not the case anymore whenusing multiple features through kernel averaging (Figure 2,
bottom row): when using the kernelized version of all algorithms, our approach always performs equal or better thanmost baselines, apart for Batch and in rare cases, Source+1(hinge). Compared to Batch, in two cases out of three (unrelated, related) MULTIpLE performs on par with it. Thisis a remarkable result, as the Batch method constitutes animportant reference for the behavior of transfer learning algorithms in this setting (Section 5.2).
Figure 2, middle row, reports results obtained for
MULTIpLE and all transfer learning baselines, as definedin Section 5.2, for one feature and the linear kernel. We seethat our algorithm obtains a better performance comparedto all the others, especially in the small sample regime. As
our method builds on the MultiKT algorithm, we tested thestatistical significance of our performance with respect toit, using the Wilcoxon signed-rank test (p < 0.05). In twocases out of three (related, unrelated), MULTIpLE is significantly better than its competitor. This is the case also
when using all features via kernel averaging. We mark thesecases with a star on the plots (Figure 2, middle and bottom
3 All experimental results and the source code are available at
http://www.idiap.ch/˜ikuzbor.

row). With respect to the transfer baselines, the related setting seems to be the one more favorable to our approach.
With respect to the no transfer baselines, MULTIpLE seems
to perform better in the unrelated case.
The performance of PMT-SVM-OVA and MultisourceTrAdaBoost-OVA is disappointing, compared with whatachieved by the other two transfer learning baselines, i.e.
MultiKT and MKTL. This is true for all settings (related, unrelated and mixed). Particularly, the performanceof MultisourceTrAdaBoost-OVA does not seem to benefit
from using multiple features (Figure 2, middle and bottomrow). On the basis of these results, we did not consider these
two baseline algorithms in the rest of our experiments.
Figure 3 shows results for N + 1 = 20 classes on the
AwA dataset, for the unrelated (left), mixed (center) and related (right) settings, all features (averaged RBF kernels).
For sake of readability, we report here only the baselines
which were competitive with, or better than, MULTIpLE in
the N + 1 = 5 case, in at least one setting. We see that here
our algorithm consistently outperforms all transfer learning
baselines, especially with a small training set, while obtaining a performance remarkably similar to Batch, in terms ofaccuracy and behavior. The Wilcoxon signed-rank test (p <0.05) indicates that, in all these experiments MULTIpLE isagain significantly better than MultiKT-OVA. These results
suggest that, as the number of sources grows, our method
gets closer to the Batch performance while using only a considerably smaller amount of data – the ultimate goal of any
effective transfer learning method. Results obtained on thewhole AwA dataset support this claim3 .

6. Discussion and Conclusions
All results confirm our claim that the mere extension tomulticlass of existing binary transfer learning algorithms is
not sufficient to address the N −→ N + 1 problem. This iswell illustrated by the gap in performance between MULTIpLE and MultiKT, which is consistent across datasets,

AwA 20 unrelated
performance on target classtransfer & selected no transfer
0.40 kernel,
MKTL
0.35 MultiKT-OVA
Batch
MULTIpLE
0.30 Source+1
Source+1 (hinge)
0.25
0.20
0.15
MKTL
MultiKT-OVA
0.10
Batch
MULTIpLE
0.05
Source+1
Source+1 (hinge)
8 10 12 14 16 18 20 0.004 6 8 10 12 14 16 18 20
Number of samples
Number of samples

AwA 20 unrelated
performance on N source classes
0.40 kernel, transfer & selected no transfer

target accuracy

N source accuracy

0.35

0.30
0.25

0.204

6

Figure 4: Results for N +1 = 20, AwA, unrelated: accuracy
over the N sources (left) and over the +1 target (right).
settings and the number of classes. The main difference
between the two algorithms is the term we added into theobjective, that allows to learn the new class, while preserving the performance on the old classes. The results we haveshown demonstrate the importance of such a term in the behavior of the algorithm. One might argue that the worse performance of the transfer learning baselines depends on how
we implemented the OVA extension for such binary methods. Still, the results obtained by MKTL, the only transfer
learning baseline with a multiclass formulation, clearly indicate that the ability to handle multiple sources by itself
is not the solution. To gain a better understanding on how
MULTIpLE balances the need to preserve performance overthe sources, and the learning of the target class, we show theaccuracy plots for the AWA experiments, N + 1 = 20, unrelated, for the N sources and for the +1 target separately(Figure 4). MULTIpLE and Batch present similar behaviors, as they both preserve the accuracy over the N sources.
Both methods do not aggressively leverage over sources
for learning the target class, as done by MultiKT-OVA andMKTL (to a lesser extent), although MULTIpLE seems tobe able to do so better than Batch. Thus, our choice of optimizing the overall accuracy has resulted in a method able
to reproduce the behavior and the performance achievable
if all training data would be accessible. Note training withall the data might not be possible, nor desirable, in all applications. As opposed to this, the OVA extensions of existingbinary transfer learning algorithms are more biased towards
a strong exploitation of source knowledge when learning
the target class, at the expenses of the overall performance.
How to combine these two aspects, namely how to designprincipled methods able to obtain an overall accuracy comparable to that of the Batch method while at the same time
boosting the learning of the target class, remains the open
challenge of the N −→ N + 1 transfer learning problem.

7. Acknowledgments
This work is partially supported by the Swiss National
Science Foundation Sinergia project Ninapro (I.K. & B.C.).

We are thankful to Arjan Gijsberts and Tatiana Tommasi for
helping with the coding.

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