Understanding and producing sentences like ‘There are more cars than parking lots’, ‘Most of the supporters wear blue t-shirts’, ‘Twenty percent of the trees have been planted last year’, or ‘Seven students passed the exam’, is a fundamental competence which allows speakers to communicate information about quantities. Crucially, the type of information conveyed by these expressions, as well as their underlying cognitive mechanisms, are not equivalent, as suggested by evidence from linguistics, language acquisition, and cognition.
First, comparatives (‘more’, ‘less’), quantifiers (‘some’, ‘most’, ‘all’), and proportions (‘%’, ‘two thirds’) express a comparison or relation between sets (e.g., between the set of cars and the set of parking lots). Such relational information is rather coarse when expressed by comparatives and vague quantifiers, more precise when denoted by proportions. In contrast, numbers (‘one’, ‘six’, ‘twenty-two’) denote the exact, absolute cardinality of the items belonging to one set (e.g., the set of students who passed the exam).
Second, during language acquisition, these expressions are neither learned at the same time nor governed by the same rules. Recent evidence showed that children can understand comparatives at around . years Odic et al. (2013); Bryant (2017), with quantifiers being learned a few months later, at around .-. years Hurewitz et al. (2006); Minai (2006); Halberda et al. (2008). Crucially, knowing the meaning of numbers, an ability that starts not before the age of . years Le Corre and Carey (2007), is not required to understand and use these expressions. As for proportions, they are acquired significantly later, being fully mastered only at the age of or Hartnett and Gelman (1998); Moss and Case (1999); Sophian (2000).
Third, converging evidence from cognition and neuroscience supports the hypothesis that some important components of these expressions of quantity are grounded on a preverbal, non-symbolic system representing magnitudes Piazza (2010). This system, often referred to as Approximate Number System (ANS), is invariant to the sensory modality and almost universal in the animal domain, and consists in the ability of holistically extracting and comparing approximate numerosities Piazza and Eger (2016). In humans, it is present since the youngest age, with -month-old infants being able to automatically compare sets and combine them by means of proto-arithmetical operations Xu and Spelke (2000); McCrink and Wynn (2004). Since it obeys Weber’s law, according to which highly differing sets (e.g. :) are easier to discriminate than highly similar sets (e.g. :), ANS has been recently claimed to be a ratio-based mechanism Sidney et al. (2017); Matthews et al. (2016). In support of this, behavioral findings indicate that, in non-symbolic contexts (e.g. visual scenes), proportional values are extracted holistically, i.e. without relying on the pre-computed cardinalities of the sets Fabbri et al. (2012); Yang et al. (2015). Indeed, people are fairly accurate in providing the proportion of targets in a scene, even in high-speed settings Healey et al. (1996); Treisman (2006). Similarly, in briefly-presented scenes, the interpretation of quantifiers is shown to be best described by proportional information Pezzelle et al. (under review).
Altogether, this suggests that performing () set comparison, () vague quantification, and () proportional estimation, which all rely on information regarding relations among sets, underlies increasingly-complex steps of the same mechanism. Notably, such complexity would range from ‘more/less’ judgements to proportional estimation, as suggested by the increasing precision of ANS through years Halberda and Feigenson (2008), the reported boundary role of ‘half’ in early proportional reasoning Spinillo and Bryant (1991), and the different age of acquisition of the corresponding linguistic expressions. Finally, the ratio-based operation underlying these task would be different from (and possibly conflicting with) that of estimating the absolute numerosity of one set. Indeed, absolute numbers are found to interfere with the access to proportions Fabbri et al. (2012).
Inspired by this converging evidence, the present work proposes a computational framework to explore various quantification tasks in the visual domain (see Figure 1
). In particular, we investigate whether ratio-based quantification tasks can be modeled by a single, multi-task learning neural network. Given a synthetic scene depicting animals (in our setting, the ‘target’ objects) and artifacts (‘non-target’), our model is designed to jointly perform all the tasks by means of an architecture that reflects their increasing complexity.111The dataset and the code can be downloaded from github.com/sandropezzelle/multitask-quant To perform proportional estimation (the most complex), the model builds on the representations learned to perform vague quantification and, in turn, set comparison (the least complex). We show that the multi-task model achieves both higher accuracy and higher generalization power compared to the one-task models. In contrast, we prove that introducing the absolute number task in the loop is not beneficial and indeed hurts the performance.
Our main contribution lies in the novel application and evaluation of a multi-task learning architecture on the task of jointly modeling 3 different quantification operations. On the one hand, our results confirm the interdependency of the mechanisms underlying the tasks of set comparison, vague quantification, and proportional estimation. On the other, we provide further evidence on the effectiveness of these computational architectures.
2 Related Work
2.1 Quantities in Language & Vision
In recent years, the task of extracting quantity information from visual scenes has been tackled via Visual Question Answering (VQA). Given a real image and a natural language question, a VQA computational model is asked to understand the image, the linguistic query, and their interaction to provide the correct answer. So-called count questions, i.e. ‘How many Xs have the property Y?’, are very frequent and have been shown to be particularly challenging for any model Antol et al. (2015); Malinowski et al. (2015); Ren et al. (2015); Fukui et al. (2016). The difficulty of the task has been further confirmed by the similarly poor performance achieved even on the ‘diagnostic’ datasets, which include synthetic visual scenes depicting geometric shapes Johnson et al. (2017); Suhr et al. (2017).
More akin to our work is Stoianov and Zorzi (2012), who showed that hierarchical generative models learn ANS as a statistical property of (synthetic) images. Their networks were tested on the task of set comparison (‘more/less’) and obtained 93% accuracy. A few studies specifically focused on the learning of quantifiers. Sorodoc et al. (2016) proposed a model to assign the correct quantifier to synthetic scenes of colored dots, whereas Sorodoc et al. (2018) operationalized the same task in a VQA fashion, using real images and object-property queries (e.g. ‘How many dogs are black?’). Overall, the results of these studies showed that vague quantification can be learned by neural networks, though the performance is much lower when using real images and complex queries. Finally, Pezzelle et al. (2017) investigated the difference between the learning of cardinals and quantifiers from visual scenes, showing that they require two distinct computational operations. To our knowledge, this is the first attempt to jointly investigate the whole range of quantification mechanisms. Moreover, we are the first to exploit a multi-task learning paradigm for exploring the interactions between set comparison, vague quantification, and proportions.
2.2 Multi-Task Learning
Multi-Task Learning (MTL) has been shown to be very effective for a wide range of applications in machine learning (for an overview, seeRuder (2017)). The core idea is that different and yet related tasks can be jointly learned by a multi-purpose model rather than by separate and highly fine-tuned models. Since they share representations between related (or ‘auxiliary’) tasks, multi-task models are more robust and generalize better than single-task models. Successful applications of MTL have been proposed in CV to improve object classification Girshick (2015)
, face detection and rotationZhang et al. (2014); Yim et al. (2015), and to jointly perform a number of tasks as object detection, semantic segmentation, etc. Misra et al. (2016); Li and Hoiem (2016). Though, recently, a few studies applied MTL techniques to either count or estimate the number of objects in a scene Sun et al. (2017); Sindagi and Patel (2017), to our knowledge none of them were devoted to the learning of various quantification mechanisms.
In the field of natural language processing (NLP), MTL turned out to be beneficial for machine translationLuong et al. (2016) and for a range of tasks such as chunking, tagging, semantic role labelling, etc. Collobert et al. (2011); Søgaard and Goldberg (2016); Bingel and Søgaard (2017). In particular, Søgaard and Goldberg (2016) showed the benefits of keeping low-level tasks at the lower layers of the network, a setting which enables higher-level tasks to make a better use of the shared representations. Since this finding was also in line with previous evidence suggesting a natural order among different tasks Shen and Sarkar (2005), further work proposed MTL models in which several increasingly-complex tasks are hierarchically ordered Hashimoto et al. (2017). The intuition behind this architecture, referred to as ‘joint many-task model’ in the source paper Hashimoto et al. (2017), as well as its technical implementation, constitute the building blocks of the model proposed in the present study.
3 Tasks and Dataset
Given a visual scene depicting a number of animals (targets) and artifacts (non-targets), we explore the following tasks, represented in Figure 1:
set comparison (hence, setComp), i.e. judging whether the targets are ‘more’, ‘same’, ‘less’ than non-targets;
vague quantification (hence, vagueQ
), i.e. predicting the probability to use each of thequantifiers (‘none’, ‘almost none’, ‘few’, ‘the smaller part’, ‘some’, ‘many’, ‘most’, ‘almost all’, ‘all’) to refer to the target set;
proportional estimation (hence, propTarg), i.e. predicting the proportion of targets choosing among ratios, ranging from to %.
Tasks (a) and (c) are operationalized as classification problems and evaluated through accuracy. That is, only one answer out of and , respectively, is considered as correct. Given the vague status of quantifiers, whose meanings are ‘fuzzy’ and overlapping, task (b) is evaluated by means of Pearson’s correlation (r
) between the predicted and the ground-truth probability vector (cf.§ 3.2), for each datapoint.222We also experimented with Mean Average Error and dot product and found the same patterns of results (not reported). The overall r is obtained by averaging these scores. It is worth mentioning that we could either evaluate (b) in terms of a classification task or operationalize (a) and (c) in terms of a correlation with human responses. The former evaluation is straightforward and can be easily carried out by picking the quantifier with the highest probability. The latter, in contrast, implies relying on behavioral data assessing the degree of overlap between ground-truth classes and speakers’ choice. Though interesting, such evaluation is less crucial given the discrete, non-overlapping nature of the classes in tasks (a) and (c).
The tasks are explored by means of a MTL network that jointly performs the three quantification operations (see § 4.2). The intuition is that solving the lower-level tasks would be beneficial for tackling the higher-level ones. In particular, providing a proportional estimation (‘%’) after performing vagueQ (‘most’) and setComp (‘more’) should lead to a higher accuracy in the highest-level task, which represents a further step in complexity compared to the previous ones. Moreover, lower-level tasks might be boosted in accuracy by the higher-level ones, since the latter include all the operations that are needed to carry out the former. In addition to the MTL model, we test a number of ‘one-task’ networks specifically designed to solve one task at a time (see § 4.1).
We built a large dataset of synthetic visual scenes depicting a variable number of animals and artifacts on the top of a neutral, grey background (see Figure 2). In doing so, we employed the same methodology and materials used in Pezzelle et al. (under review), where the use of quantifiers in grounded contexts was explored by asking participants to select the most suitable quantifier for a given scene. Since the category of animals was always treated as the ‘target’, and that of artifacts as the ‘non-target’, we will henceforth use this terminology throughout the paper. The scenes were automatically generated by an in-house script using the following pipeline: (a) Two natural images, one depicting a target object (e.g. a butterfly) and one depicting a non-target (e.g. a mug) were randomly picked up from a sample of the dataset by Kiani et al. (2007). The sample was obtained by Pezzelle et al. (under review), who manually selected pictures depicting whole items (not just parts) and whose color, orientation and shape were not deceptive. In total, 100 unique instances of animals and 145 unique instances of artifacts were included; (b) The proportion of targets in the scene (e.g. %) was chosen by selecting one among pre-defined ratios between targets:non-targets (e.g. :, ‘four non-targets to one target’). Out of ratios, were positive (targets %), negative (targets %), and equal (targets = %); (c) The absolute number of targets/non-targets was chosen to equally represent the various combinations available for a given ratio (e.g., for ratio :: -, -, -, -), with the constraint of having a number of total objects in the scene (targets+non-targets) ranging from to . In total, combinations were represented in the dataset, with an average of combinations/ratio (min , max ); (d) To inject some variability, the instances of target/non-target objects were randomly resized according to one of three possible sizes (i.e. medium, big, and small) and flipped on the vertical axis before being randomly inserted onto a *-cell virtual grid. As reported in Table 1, K scenes balanced per ratio (K scenes/ratio) were generated and further split into train (%), validation (%), and test (%).
Ground-truth classes for the tasks of setComp and propTarg were automatically assigned to each scene while generating the data. For vagueQ, we took the probability distributions obtained on a dataset ofscenes by Pezzelle et al. (under review) and we applied them to our datapoints, which were built in the exact same way. These probability distributions had been collected by asking participants to select, from a list of quantifiers (reported in § 3.1), the most suitable one to describe the target objects in a visual scene presented for second. In particular, they were computed against the proportion of targets in the scene, which in that study was shown to be the overall best predictor for quantifiers. To illustrate, given a scene containing of targets (cf. leftmost panel in Figure 2), the probability of choosing ‘few’ (ranging from to ) is , ‘almost none’ , ‘the smaller part’ , etc. It is worth mentioning that, for scenes containing either % or % targets the probability of choosing ‘all’ and ‘none’, respectively, is around . In all other cases, the distribution of probabilities is fuzzier and reflects the largely overlapping use of quantifiers, as in the example above. On average, the probability of the most-chosen quantifier across ratios is . Though this number cannot be seen as a genuine inter-annotator agreement score, it suggests that, on average, there is one quantifier which is preferred over the others.
In this section, we describe the various models implemented to perform the tasks. For each model, several settings and parameters were evaluated by means of a thorough ablation analysis. Based on a number of factors like performance, speed, and stability of the networks, we opted for using ReLU nonlinear activation at all hidden layers and the simple and effective Stochastic Gradient Descent (SGD) as optimizer (lr =). We run each model for epochs and saved weights and parameters of the epoch with the lowest validation loss. The best model was then used to obtain the predictions in the test set. All models were implemented using Keras.333https://keras.io/
4.1 One-Task Models
We implemented separate models to tackle one task at a time. For each task, in particular, both a network using ‘frozen’ (i.e. pretrained) visual features and one computing the visual features in an ‘end-to-end’ fashion were tested.
These models are simple,
-layer (ReLU) Multi-Layer Perceptron (MLP) networks that take as input a-d frozen representation of the scene and output a vector containing softmax probability values. The frozen representation of the scene had been previously extracted using the state-of-art Inception v3 CNN Szegedy et al. (2016)
pretrained on ImageNetDeng et al. (2009). In particular, the network is fed with the average of the features computed by the last Convolutional layer, which has size *.
These models are MLP networks that take as input the *-pixel image and compute the visual features by means of the embedded Inception v3 module, which outputs *-d vectors (the grey and colored box in Figure 1). Subsequently, the feature vectors are reduced twice via ReLU hidden layers, then concatenated, reduced (ReLU), and fed into a softmax layer to obtain the probability values.
4.2 Multi-Task Model
The multi-task-prop model performs tasks at the same time with an architecture that reproduces in its order the conjectured complexity (see Figure 3 and its caption for technical details). The model has a core structure, represented by layers - in the figure, which is shared across tasks and trained with multiple outputs. In particular, (a) layers , , and are trained using information regarding the output of all
tasks. That is, these layers are updated three times by as many backpropagation passes: One on the top of setComp output, the second on the top of vagueQ output, the third on the top of propTarg output; (b) layersand are affected by information regarding the output of vagueQ and propTarg, and thus updated twice; (c) layers and are updated once, on the top of the output of propTarg. Importantly, the three lower layers in Figure 3 (concatenation, ReLU, softmax) are not shared between the tasks, but specialized to output each a specific prediction. As can be noted, the order of the tasks reflects their complexity, since the last task in the pipeline has more layers than the preceding one and more than the first one.
Table 2 reports the performance of each model in the various tasks (note that the lowest row and the rightmost column report results described in § 6.1). In setComp, all the models are neatly above chance/majority level (). In particular, the one-task-end2end model achieves a remarkable acc., which is more than % better compared to the simple one-task-frozen model (). The same pattern of results can be observed for vagueQ, where the Pearson’s correlation (r) between the ground-truth and the predicted probability vector is around , that is more than % over the simpler model (). This gap increases even more in propTarg, where the accuracy of the frozen model is more than points below the one achieved by the one-task-end2end model ( against ). These results firmly indicate that, on the one hand, the frozen representation of the visual scene encodes little information about the proportion of targets (likely due to the the different task for which they were pretrained, i.e. object classification). On the other hand, computing the visual features in an end-to-end fashion leads to a significant improvement, suggesting that the network learns to pay attention to features that are helpful for specific tasks.
The most interesting results, however, are those achieved by the multi-task model, which turns out to be the best in all the tasks. As reported in Table 2, sharing the weights between the various tasks is especially beneficial for propTarg, where the accuracy reaches , that is, more than points over the end-to-end, one-task model. An almost perfect performance of the model in this task can be observed in Figure 4
, which reports the confusion matrix with the errors made by the model. As can be seen, the few errors are between ‘touching’ classes, e.g. between ratio: (% of targets) and ratio : (%). Since these classes differ by a very small percentage, we gain indirect evidence that the model is learning some kind of proportional information rather than trivial associations between scenes and orthogonal classes.
To further explore this point, one way is to inspect the last layer of the proportional task (i.e. the 32-d turquoise vector in Figure 3). If the vectors contain information regarding the proportion of targets, we should expect scenes depicting the same proportion to have a similar representation. Also, scenes with similar proportions (e.g. % and %) would be closer to each other than are scenes with different proportions (e.g. % and %). Figure 5 depicts the results of a two-dimensional PCA analysis performed on the vectors of the last layer of the proportional task (the -d vectors).444We used https://projector.tensorflow.org/ As can be noted, scenes depicting the same proportion clearly cluster together, thus indicating that using these representations in a retrieval task would lead to a very high precision. Crucially, the clusters are perfectly ordered with respect to proportion. Starting from the purple cluster on the left side (%) and proceeding clockwise, we find % (green), % (turquoise), % (brown), and so on, until reaching % (light blue). Proportions % (blue) and % (yellow) are neatly separated from the other clusters, being at the extremes of the ‘clock’.
An improvement in the results can be also observed for setComp and vaqueQ, where the model achieves acc. and r, respectively. Figure 6 reports, for each quantifier, the probability values predicted by the model against the ground-truth ones. As can be seen, the red lines (model) approximate very closely the green ones (humans). In the following section, we perform further experiments to provide a deeper evaluation of the results.
6 In-Depth Evaluation
6.1 Absolute Numbers in the Loop
As discussed in § 1, the cognitive operation underlying setComp, vagueQ, and propTarg is different compared to that of estimating the absolute number of objects included in one set. To investigate whether such dissociation emerges at the computational level, we tested a modified version of our proposed multi-task model where propTarg task has been replaced with nTarg, namely the task of predicting the absolute number of targets. One-task models were also tested to evaluate the difficulty of the task when performed in isolation. Since the number of targets in the scenes ranges from to , nTarg is evaluated as a -class classification task (majority class ).
As reported in Table 2, the accuracy achieved by the one-task-end2end model is extremely high, i.e. around . This suggests that, when learned in isolation, the task is fairly easy, but only if the features are computed within the model. In fact, using frozen features results in a quite low accuracy, namely . This pattern of results is even more interesting if compared against the results of the multi-task-number model. When included in the multi-task pipeline, in fact, nTarg has a huge, -point accuracy drop (). Moreover, both setComp and vagueQ turn out to be significantly hurt by the highest-level task, and experience a drop of around and points compared to the one-task-end2end model, respectively. These findings seem to corroborate the incompatibility of the operations needed for solving the tasks.
6.2 Reversing the Architecture
Previous work exploring MTL suggested that defining a hierarchy of increasingly-complex tasks is beneficial for jointly learning related tasks (see § 2.2). In the present work, the order of the tasks was inspired by cognitive and linguistic abilities (see § 1). Though cognitively implausible, it might still be the case that the model is able to learn even when reversing the order of the tasks, i.e. from the conjectured highest-level to the lowest-level one. To shed light on this issue, we tested the multi-task-prop model after reversing its architecture. That is, propTarg is now the first task, followed by vagueQ, and setComp.
In contrast with the pattern of results obtained by the original pipeline, no benefits are observed for this version of MTL model compared to one-task networks. In particular, both vagueQ ( r) and propTarg ( acc.) performance are around chance level, with setComp reaching just acc., i.e. point lower than the one-task-end2end model. The pipeline of increasing complexity motivated theoretically is thus confirmed at the computational level.
6.3 Does MTL Generalize?
As discussed in § 2.2, MTL is usually claimed to allow a higher generalization power. To investigate whether our proposed multi-task-prop model genuinely learns to quantify from visual scenes, and not just associations between patterns and classes, we tested it with unseen combinations of targets/non-targets. The motivation is that, even in the most challenging propTarg task, the model might learn to match a given combination, e.g. :, to a given proportion, i.e. %. If this is the case, the model would solve the task by learning “just” to assign a class to each of the possible combinations included in the dataset. If it learns a more abstract representation of the proportion of targets depicted in the scene, in contrast, it should be able to generalize to unseen combinations.
We built an additional dataset using the exact same pipeline described in § 3.2. This time, however, we randomly selected one combination per ratio ( combinations in total) to be used only for validation and testing. The remaining combinations were used for training. A balanced number of datapoints for each combination were generated in val/test, whereas datapoints in training set were balanced with respect to ratios, by randomly selecting scenes among the remaining combinations. The unseen dataset included around K datapoints (% train, % val, % test). Table 3 reports the results of the models on the unseen dataset. Starting from setComp, we note a similar and fairly high accuracy achieved by the two one-task models ( and , respectively). In vagueQ, in contrast, the one-task-end2end model neatly outperforms the simpler model ( vs. r). Finally, in propTarg both models are at chance level, with an accuracy that is lower than . Overall, this pattern of results suggests that propTarg is an extremely hard task for the separate models, which are not able to generalize to unseen combinations. The multi-task-prop model, in contrast, shows a fairly high generalization power. In particular, it achieves acc. in propTarg, that is, almost times chance level. The overall good performance in predicting the correct proportion can be appreciated in Figure 7, where the errors are represented by means of a heatmap. The error analysis reveals that end-of-the-scale proportions (% and %) are the easiest, followed by proportions % (:), % (:), % (:), and % (:). More in general, negative ratios (targets %) are mispredicted to a much greater extent than are positive ones. Moreover, the model shows a bias toward some proportions, that the model seems ‘to see everywhere’. However, the fact that the errors are found among the adjacent ratios (similar proportions) seems to be a convincing evidence that the model learns representations encoding genuine proportional information. Finally, it is worth mentioning that in setComp and vagueQ the model achieves very high results, acc. and r, respectively.
In the present study, we investigated whether ratio-based quantification mechanisms, expressed in language by comparatives, quantifiers, and proportions, can be computationally modeled in vision exploiting MTL. We proved that sharing a common core turned out to boost the performance in all the tasks, supporting evidence from linguistics, language acquisition, and cognition. Moreover, we showed (a) the increasing complexity of the tasks, (b) the interference of absolute number, and (c) the high generalization power of MTL. These results lead to many additional questions. For instance, can these methods be successfully applied to datasets of real scenes? We firmly believe this to be the case, though the results might be affected by the natural biases contained in those images. Also, is this pipeline of increasing complexity specific to vision (non-symbolic level), or is it shared across modalities, in primis language? Since linguistic expressions of quantity are grounded on a non-symbolic system, we might expect that a model trained on one modality can be applied to another, at least to some extent. Even further, jointly learning representations from both modalities might represent an even more natural, human-like way to learn and refer to quantities. Further work is needed to explore all these issues.
We kindly acknowledge Gemma Boleda and the AMORE team (UPF), Raquel Fernández and the Dialogue Modelling Group (UvA) for the feedback, advice and support. We are also grateful to Aurélie Herbelot, Stephan Lee, Manuela Piazza, Sebastian Ruder, and the anonymous reviewers for their valuable comments. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 715154). We gratefully acknowledge the support of NVIDIA Corporation with the donation of GPUs used for this research. This paper reflects the authors’ view only, and the EU is not responsible for any use that may be made of the information it contains.
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