Deep neural networks (DNNs) have met with success multiple tasks, and testified a constantly increasing popularity, being able to deal with the vast heterogeneity of data and to provide state-of-the-art results across many fields and domains [1, 2]
. Convolutional Neural Networks (CNNs)[3, 4] are one of the protagonists of this success. Starting from AlexNet , until the most recent convolutional-based architectures [6, 7, 8]
CNNs have proved to be especially useful in the field of computer vision, improving the classification accuracy in many datasets[9, 10].
However, a common caveat of large CNNs is that they require a lot of training data in order to work well. In the presence of classification tasks on small datasets, typically those networks are pre-trained
in a very large dataset like ImageNet, and then finetuned on the dataset the problem is set on. The idea is that the pre-trained network has stored a decent amount of information regarding features which are common to the majority of images, and in many cases this knowledge can be transferred to different datasets or to solve different problems (image segmentation, localization, detection, etc.). This technique is referred as transfer learning  and has been an important ingredient in the success and popularization of CNNs. Another important technique – very often paired with the previous one – is data augmentation, through which small transformations are directly applied on the images. A nice characteristic of data augmentation is its agnosticism toward algorithms and datasets.  used this technique to achieve state-of-the-art results in MNIST dataset , while  used the method almost without any changes to improve the accuracy of their CNN in the ImageNet dataset . Since then, data augmentation has been used in virtually every implementation of CNNs in the field of computer vision.
Despite the practicality of the above-mentioned techniques, when the number of images per class is extremely small, the performances of CNNs rapidly degrade and leave much to be desired. The high availability of unlabeled data only solves half of the problem, since the manual labeling process is usually costly, tedious and prone to human error. Under these assumptions, we propose a new method to perform an automatic labeling, called transductive label augmentation. Starting from a very small labeled dataset, we set an automatic label propagation procedure, that relies on graph transduction techniques, to label a large unlabeled set of data. This method takes advantage of second-order similarity information among the data objects, a source of information which is not directly exploited by traditional techniques. To assess our statements, we perform a series of experiments with different CNN architectures and datasets, comparing the results with a first-order “label propagator”.
In summary, our contributions in this article are as follows: a) by using graph transductive approaches, we propose and develop the aforementioned label augmentation method and use it to improve the accuracy of state-of-the-art CNNs in datasets where the number of labels is limited; b) by gradually increasing the number of labeled objects, we give detailed results in three standard computer vision datasets and compare the results with the results of CNNs; c) we replace our transductive algorithm with linear support vector machines (SVM) to perform label augmentation and compare the results; d) we give directions for future work and how the method can be used on other domains.
I-a Related Work
Semi-supervised label propagation has a long history of usage in the field of machine learning
. Starting from an initial large dataset, with a small portion of labeled observations the traditional way of using semi-supervised learning is to train a classifier only in the labeled part, and then use the classifier to predict labels for the unlabeled part. The labels predicted in this way are calledpseudo-labels. The classifier is then trained in the entire dataset, considering the pseudo-labels as if they were real labels.
Different methods with the same intent have been previously proposed. In deep learning in particular, there have been devised algorithms to use data with a small number of labeled observations. 
trained the network jointly in both the labeled and unlabeled points. The final loss function is a weighted loss of both labeled and unlabeled points, where in the case of the unlabeled points, the pseudo-label is determined by the highest score proposed by the model. optimized a CNN on such a way as to produce embeddings that have high similarities for the observations that belong to the same class.  used a totally different approach, developing a generative model that allows for effective generalization from small labeled datasets to large unlabeled ones.
In all the mentioned methods, the way how the unlabeled data has been used can be considered as an intrinsic property of their engineered neural networks. Our choice of CNNs as the algorithm used for the experiments was motivated because CNNs are state-of-the-art models in computer vision, but the approach is more general than that. The method presented in this article does not even require a neural network and in principle, non-feature based observations (i.e graphs) can be considered, as long as a similarity measure can be derived for them. At the same time, the method shows good results in relatively complex image datasets, improving over the results of state-of-the-art CNNs.
Ii Graph Transduction Game
Graph Transduction (GT) is a subfamily of semi-supervised learning that aims to classify unlabeled objects starting from a small set of labeled ones. In particular, in GT the data is modeled as a graph whose vertices are the objects in a dataset. The provided label information is then propagated all over the unlabeled objects through the edges, weighted according to the consistency of object pairs. The reader is encouraged to refer to  for a detailed description of algorithms and applications on graph transduction.
More formally, let be a graph. is the vertex set of the objects and can be partitioned in two sets: contains the labeled objects, where is a real-valued vector describing the object (features), and is its associated label, while is the set of unlabeled objects. is the set of edges connecting the vertices and is a weight function that assigns a non-negative similarity measure to each edge in , and can be summarized in a weight matrix .
In , GT takes in input along with initial probability distributions for every objects – one-hot labels for , soft labels for – and iteratively applies a function where is the standard simplex. At each iteration, if the distributions of labeled objects have changed, they are reset. Once the algorithm reaches the convergence, the resulting final probabilities give a labeling over the entire set of objects.
In this article, we follow the approach proposed in , where the authors interpret the graph transduction task as a non-cooperative multiplayer game. The same methodology has been successfully applied in different context, e.g. bioinformatics  and matrix factorization .
In graph transduction game (GTG), objects of a dataset are represented as players and their labels as strategies. In synthesis, a non-cooperative multiplayer game is played among the objects, until an equilibrium condition is reached, the Nash Equilibria 
. Here, we provide some basic knowledge on game theory in order to be self-contained. Given a set of playersand a set of possible pure strategies :
mixed strategy: a mixed strategy is a probability distribution over the possible strategies for player . Then , where
is the standard -dimensional simplex and is the probability of player choosing the pure strategy .
mixed strategy space: it corresponds to the set of all mixed strategies of the players
utility function: it represents the gain obtained by a player when it chooses a certain mixed strategy, in particular .
Here, it is assumed that the payoffs associated to each player are additively separable, thus the algorithm is a member of polymatrix games. In GTG, the aforementioned definitions turns into the following:
The strategy space is the starting point of the game and contains all the mixed strategies. The space can be initialized in different ways based on the fact that some prior knowledge exists or not. Here, we distinguish the initialization based on the type of object, labeled or unlabeled. For the labeled object, since their class is known, a one-hot vector is assigned:
. For the unlabeled objects all the labels have the same probability of being associated to an object, thus:
The utility function reflects the likelihood of choosing a particular label and considers the similarity between labeled and unlabeled players. Similar players influence each other more in picking one of the possible strategies (labels). Once the game reaches an equilibrium, every player play their best strategies which correspond to a consistent labeling  not only for the player itself but also for the others. Under equilibrium conditions the label of player is given by the strategy played with the highest probability. Formally, given a player and a strategy :
where is the utility received by player when it plays the mixed strategy and is the partial payoff matrix between players and . As in , where is the similarity between player and and
is the identity matrix of size. The similarity function between players (objects) can be given or computed starting from the features. Given two objects and their features , , their similarity is computed following the method proposed by :
where corresponds to the distance between and its -nearest- neighbors. Similarity values are stored in matrix .
Finding Nash Equilibria
The last component of our method is an algorithm for finding equilibrium conditions in this game. In  a result from Evolutionary Game Theory , named Replicator Dynamics (RD)  is used. The RD are a class of dynamical systems that perform a natural selection process on a multi-population of strategies. The idea is to lead the fittest strategies to survive while the others to go extinct. More specifically the RD are defined as follow:
where is the probability of strategy at time for player .
The RD are iterated until convergence, this means either the distance between two successive steps is zero (formally ) or a certain amount of iterations is reached (See  for a detailed analysis). In practical applications one could set the to a small number but typically 10-20 iterations are sufficient.
Iii Label Generation
The previously explained framework can be applied to a dataset with many unlabeled objects to perform an automatic labeling and thus increase the availability of training objects. In this article we deal with datasets for image classification, but our approach can be applied in other domains too.
Preliminary step: both the labeled and unlabeled sets can be refined to obtain more informative feature vectors. In this article, we used fc7 features of CNNs trained on ImageNet, but in principle, any type of features can be considered. Our particular choice was motivated because fc7 features work significantly better than traditional computer vision features (SIFT  and its variations). While this might seem counter-intuitive (using pre-trained CNNs on ImageNet, while we are solving the problem of limited labeled data), we need to consider that our datasets are different from ImageNet (they come from different distributions), and by using some other dataset to pre-train our networks, we are not going against the spirit of the idea of the paper.
: the objects are assigned to initial probability distributions, needed to start the GTG. The labeled ones use their respective one-hot label representations, while the unlabeled ones can be set to a uniform distribution among all the labels. In presence of previous possessed information, some labels can be directly excluded in order to start from a multi-peaked distribution, which if chosen wisely, can improve the final results.
Step 2: the extracted features are used to compute the similarity matrix . The literature  presents multiple methods to obtain a matrix and extra care should be taken when performing this step, since an incorrect choice in its computation can determine a failure in the transductive labeling.
Step 3: once is computed, graph transduction game can be played (up to convergence) among the objects to obtain the final probabilities which determine the label for the unlabeled objects.
The resulting labeled dataset can then be used to train a classification model. This is very convenient for several reasons: 1) CNNs are fully parametric models, so we do not need to store the training set in memory like in the case of graph transduction. In some aspect, the CNN is approximating in a parametric way the GTG algorithm; 2) the inference stage on CNNs is extremely fast (real-time); 3) CNN features can be used for other problems, like image segmentation, detection and classification, something that we cannot do with graph-transduction or with classical machine learning methods (like SVM). In the next section we will report the results obtained from state-of-the-art CNNs, and compare those results with the same CNNs trained only on the labeled part of the dataset.
|accuracy 2% labeled||caltech||indoors||scenenet|
|GTG + CNN||0.532||0.620||0.486||0.538||0.430||0.495|
|SVM + CNN||0.473||0.539||0.434||0.468||0.370||0.417|
|F score 2% labeled||caltech||indoors||scenenet|
|GTG + CNN||0.468||0.559||0.357||0.396||0.399||0.457|
|SVM + CNN||0.388||0.455||0.319||0.327||0.352||0.377|
|accuracy 5% labeled||caltech||indoors||scenenet|
|GTG + CNN||0.625||0.698||0.568||0.613||0.563||0.621|
|SVM + CNN||0.605||0.675||0.516||0.580||0.511||0.601|
|F score 5% labeled||caltech||indoors||scenenet|
|GTG + CNN||0.571||0.653||0.454||0.508||0.536||0.608|
|SVM + CNN||0.542||0.626||0.426||0.505||0.501||0.590|
|accuracy 10% labeled||caltech||indoors||scenenet|
|GTG + CNN||0.667||0.727||0.598||0.645||0.624||0.686|
|SVM + CNN||0.658||0.724||0.576||0.635||0.622||0.660|
|F score 10% labeled||caltech||indoors||scenenet|
|GTG + CNN||0.622||0.694||0.509||0.574||0.609||0.700|
|SVM + CNN||0.612||0.686||0.515||0.579||0.612||0.650|
In order to assess the quality of the algorithm, we used it to automatically label three known realistic datasets, namely Caltech-256 , Indoor Scene Recognition  and SceneNet-100 . Caltech-256 contains images belonging to different categories and it is used for object recognition tasks. Indoor Scene Recognition is a dataset containing images of different common places (restaurants, bedrooms, etc.), divided in
categories and, as the name says, it is used for scene recognition.SceneNet-100 database is a publicly available online ontology for scene understanding that organizes scene categories according to their perceptual relationships. The dataset contains real-world images, separated into different classes.
Each dataset was split in a training (70%) and a testing (30%) set. In addition, we further randomly split the training set in a small labeled part and a large unlabeled one, according to three different percentages for labeled objects (2%, 5%, 10%). For feature representation, we used two models belonging to state-of-the-art CNN families of architectures, ResNet and DenseNet. In particular we used the smallest models offered in PyTorch library, the choice motivated by the fact that our datasets are relatively small, and so models with smaller number of parameters are expected to work better. The features were combined to generate the similarity matrix, as described in Eq. 5. The matrix for GTG model was initialized as described in the previous section. We ran the GTG algorithm up to convergence, with the pseudo-labels being computed by doing an over the final probability vectors.
We then trained ResNet18 (RN18) and DenseNet121 (DN121) in the entire dataset, by not having a distinction between labels and pseudo-labels, using Adam optimizer  with learning rate. We think that the results reported in this section are conservative, and can be improved with a more careful training of the networks, and by doing an exhaustive search over the space of hyper-parameters.
For comparison, we performed an alternative approach, by replacing GTG with a first-order information algorithm, namely linear SVM. While we experimented also with kernel SVM, we saw that its results are significantly worse than those of linear SVM, most likely because the features were generated from a CNN and so they are already quite good, having transformed the feature space in order to solve the classification problem linearly. No other transductive methods have been taken into consideration, since GTG has already been compared with them in [20, 21], showing that it performs better.
On Table I we give the results of the accuracy and F score on the testing set, in all three datasets, while the number of labels is only 2% for each of the datasets ( observations for Caltech-256, observations for Indoor, and observations for Scenenet). In all three datasets, and both CNNs, our results are significantly better than those of CNNs trained only in the labeled data, or the results of the alternative approach when a linear SVM is used instead of GTG. Table II and Table III give the results of the accuracy and F score while the number of labeled images is 5%, respectively 10%. It can be seen that with the number of labeled points increasing, the performance boost of our model becomes smaller, but our performance still gives better (or equal) results to the alternative approach in all bar three cases, and it gives significantly better results than CNN in all cases.
Figure 3 shows the results of our approach compared with the other approach and with the results of CNN. We plotted the relative improvement of our model and the alternative approach over CNN. When the number of labels is very small (2%), in all three datasets we have significantly better improvements compared with the alternative approach. Increasing the number of labels to 5% and 10%, this trend persists. In all cases, our method gives significant improvements compared to CNN trained on only the labeled part of the dataset, with the most interesting case (only 2% of labeled observations), our model gives 36.24% relative improvement over CNN for ResNet18 and 50.29% relative improvement for DenseNet121.
V Conclusions and Future Work
In this paper, we proposed and developed a game-theoretic model which can be used as a semi-supervised learning algorithm in order to label the unlabeled observations and so augment datasets. Different types of algorithms (including state-of-the-art CNNs) can then be trained on the extended dataset, where the “pseudo-labels” can be treated as normal labels.
Our method is not the only semi-supervised learning model used to train deep learning methods, and at this stage, we do not claim that our method is the best one. However, to the best of our knowledge, the other methods are directed towards deep learning and incorporated within the learning algorithm itself. On the contrary, we offer a different perspective, developing a model which is algorithm-agnostic, and which doesn’t even need the data to be on feature-based format.
Part of the future work will consist on tailoring our model specifically towards convolutional neural networks and to make comparisons with other semi-supervised learning algorithms. In addition to this, we believe that the true potential of the model can be unleashed when the data is in some non-traditional format. In particular, we plan to use our model in the fields of bio-informatics and natural language processing, where non-conventional learning algorithms need to be developed. A direct extension of this work is to embed into the model the similarity between classes which has been proven to significantly boost the performances of learning algorithms.
This work was supported by Samsung Global Research Outreach Program. We thank the anonymous reviewers for their suggestions to improve the paper.
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