Alberto Testolin

Kuinan Hou, Marco Zorzi, Alberto Testolin

Reliably counting and generating sequences of items remain a significant challenge for neural networks, including Large Language Models (LLMs). Indeed, although this capability is readily handled by rule-based symbolic systems based on serial computation, learning to systematically deploy counting procedures is difficult for neural models, which should acquire these skills through learning. Previous research has demonstrated that recurrent architectures can only approximately track and enumerate sequences of events, and it remains unclear whether modern deep learning systems, including LLMs, can deploy systematic counting procedures over sequences of discrete symbols. This paper aims to fill this gap by investigating the sequential enumeration abilities of five state-of-the-art LLMs, including proprietary, open-source, and reasoning models. We probe LLMs in sequential naming and production tasks involving lists of letters and words, adopting a variety of prompting instructions to explore the role of chain-of-thought in the spontaneous emerging of counting strategies. We also evaluate open-source models with the same architecture but increasing size to see whether the mastering of counting principles follows scaling laws, and we analyze the embedding dynamics during sequential enumeration to investigate the emergent encoding of numerosity. We find that some LLMs are indeed capable of deploying counting procedures when explicitly prompted to do so, but none of them spontaneously engage in counting when simply asked to enumerate the number of items in a sequence. Our results suggest that, despite their impressive emergent abilities, LLMs cannot yet robustly and systematically deploy counting procedures, highlighting a persistent gap between neural and symbolic approaches to compositional generalization.

Matteo Zambra, Alberto Testolin, Marco Zorzi

Deep belief networks (DBNs) are stochastic neural networks that can extract rich internal representations of the environment from the sensory data. DBNs had a catalytic effect in triggering the deep learning revolution, demonstrating for the very first time the feasibility of unsupervised learning in networks with many layers of hidden neurons. These hierarchical architectures incorporate plausible biological and cognitive properties, making them particularly appealing as computational models of human perception and cognition. However, learning in DBNs is usually carried out in a greedy, layer-wise fashion, which does not allow to simulate the holistic maturation of cortical circuits and prevents from modeling cognitive development. Here we present iDBN, an iterative learning algorithm for DBNs that allows to jointly update the connection weights across all layers of the model. We evaluate the proposed iterative algorithm on two different sets of visual stimuli, measuring the generative capabilities of the learned model and its potential to support supervised downstream tasks. We also track network development in terms of graph theoretical properties and investigate the potential extension of iDBN to continual learning scenarios. DBNs trained using our iterative approach achieve a final performance comparable to that of the greedy counterparts, at the same time allowing to accurately analyze the gradual development of internal representations in the deep network and the progressive improvement in task performance. Our work paves the way to the use of iDBN for modeling neurocognitive development.

Burla Nur Korkmaz, Roee Diamant, Gil Danino et al.

Effective conservation of maritime environments and wildlife management of endangered species require the implementation of efficient, accurate and scalable solutions for environmental monitoring. Ecoacoustics offers the advantages of non-invasive, long-duration sampling of environmental sounds and has the potential to become the reference tool for biodiversity surveying. However, the analysis and interpretation of acoustic data is a time-consuming process that often requires a great amount of human supervision. This issue might be tackled by exploiting modern techniques for automatic audio signal analysis, which have recently achieved impressive performance thanks to the advances in deep learning research. In this paper we show that convolutional neural networks can indeed significantly outperform traditional automatic methods in a challenging detection task: identification of dolphin whistles from underwater audio recordings. The proposed system can detect signals even in the presence of ambient noise, at the same time consistently reducing the likelihood of producing false positives and false negatives. Our results further support the adoption of artificial intelligence technology to improve the automatic monitoring of marine ecosystems.

Alberto Testolin

Creating learning models that can exhibit sophisticated reasoning skills is one of the greatest challenges in deep learning research, and mathematics is rapidly becoming one of the target domains for assessing scientific progress in this direction. In the past few years there has been an explosion of neural network architectures, data sets, and benchmarks specifically designed to tackle mathematical problems, reporting notable success in disparate fields such as automated theorem proving, numerical integration, and discovery of new conjectures or matrix multiplication algorithms. However, despite these impressive achievements it is still unclear whether deep learning models possess an elementary understanding of quantities and symbolic numbers. In this survey we critically examine the recent literature, concluding that even state-of-the-art architectures often fall short when probed with relatively simple tasks designed to test basic numerical and arithmetic knowledge.

Flavio Petruzzellis, Alberto Testolin, Alessandro Sperduti

Solving symbolic reasoning problems that require compositionality and systematicity is considered one of the key ingredients of human intelligence. However, symbolic reasoning is still a great challenge for deep learning models, which often cannot generalize the reasoning pattern to out-of-distribution test cases. In this work, we propose a hybrid system capable of solving arithmetic problems that require compositional and systematic reasoning over sequences of symbols. The model acquires such a skill by learning appropriate substitution rules, which are applied iteratively to the input string until the expression is completely resolved. We show that the proposed system can accurately solve nested arithmetical expressions even when trained only on a subset including the simplest cases, significantly outperforming both a sequence-to-sequence model trained end-to-end and a state-of-the-art large language model.

Samuel Cognolato, Alberto Testolin

Mathematical reasoning is one of the most impressive achievements of human intellect but remains a formidable challenge for artificial intelligence systems. In this work we explore whether modern deep learning architectures can learn to solve a symbolic addition task by discovering effective arithmetic procedures. Although the problem might seem trivial at first glance, generalizing arithmetic knowledge to operations involving a higher number of terms, possibly composed by longer sequences of digits, has proven extremely challenging for neural networks. Here we show that universal transformers equipped with local attention and adaptive halting mechanisms can learn to exploit an external, grid-like memory to carry out multi-digit addition. The proposed model achieves remarkable accuracy even when tested with problems requiring extrapolation outside the training distribution; most notably, it does so by discovering human-like calculation strategies such as place value alignment.

Kuinan Hou, Marco Zorzi, Alberto Testolin

Humans share with many animal species the ability to perceive and approximately represent the number of objects in visual scenes. This ability improves throughout childhood, suggesting that learning and development play a key role in shaping our number sense. This hypothesis is further supported by computational investigations based on deep learning, which have shown that numerosity perception can spontaneously emerge in neural networks that learn the statistical structure of images with a varying number of items. However, neural network models are usually trained using synthetic datasets that might not faithfully reflect the statistical structure of natural environments, and there is also growing interest in using more ecological visual stimuli to investigate numerosity perception in humans. In this work, we exploit recent advances in computer vision algorithms to design and implement an original pipeline that can be used to estimate the distribution of numerosity and non-numerical magnitudes in large-scale datasets containing thousands of real images depicting objects in daily life situations. We show that in natural visual scenes the frequency of appearance of different numerosities follows a power law distribution. Moreover, we show that the correlational structure for numerosity and continuous magnitudes is stable across datasets and scene types (homogeneous vs. heterogeneous object sets). We suggest that considering such "ecological" pattern of covariance is important to understand the influence of non-numerical visual cues on numerosity judgements.

Flavio Petruzzellis, Ling Xuan Chen, Alberto Testolin

Acquiring mathematical skills is considered a key challenge for modern Artificial Intelligence systems. Inspired by the way humans discover numerical knowledge, here we introduce a deep reinforcement learning framework that allows to simulate how cognitive agents could gradually learn to solve arithmetic problems by interacting with a virtual abacus. The proposed model successfully learn to perform multi-digit additions and subtractions, achieving an error rate below 1% even when operands are much longer than those observed during training. We also compare the performance of learning agents receiving a different amount of explicit supervision, and we analyze the most common error patterns to better understand the limitations and biases resulting from our design choices.

Kuinan Hou, Jing Mi, Marco Zorzi et al.

Counting the number of items in a visual scene remains a fundamental yet challenging task in computer vision. Traditional approaches to solving this problem rely on domain-specific counting architectures, which are trained using datasets annotated with a predefined set of object categories. However, recent progress in creating large-scale multimodal vision-language models (VLMs) suggests that these domain-general architectures may offer a flexible alternative for open-set object counting. In this study, we therefore systematically compare the performance of state-of-the-art specialized counting architectures against VLMs on two popular counting datasets, as well as on a novel benchmark specifically created to have a finer-grained control over the visual properties of test images. Our findings show that most VLMs can approximately enumerate the number of items in a visual scene, matching or even surpassing the performance of specialized computer vision architectures. Notably, enumeration accuracy significantly improves when VLMs are prompted to generate intermediate representations (i.e., locations and verbal labels) of each object to be counted. Nevertheless, none of the models can reliably count the number of objects in complex visual scenes, showing that further research is still needed to create AI systems that can reliably deploy counting procedures in realistic environments.

Flavio Petruzzellis, Alberto Testolin, Alessandro Sperduti

Large Language Models (LLMs) achieve impressive performance in a wide range of tasks, even if they are often trained with the only objective of chatting fluently with users. Among other skills, LLMs show emergent abilities in mathematical reasoning benchmarks, which can be elicited with appropriate prompting methods. In this work, we systematically investigate the capabilities and limitations of popular open-source LLMs on different symbolic reasoning tasks. We evaluate three models of the Llama 2 family on two datasets that require solving mathematical formulas of varying degrees of difficulty. We test a generalist LLM (Llama 2 Chat) as well as two fine-tuned versions of Llama 2 (MAmmoTH and MetaMath) specifically designed to tackle mathematical problems. We observe that both increasing the scale of the model and fine-tuning it on relevant tasks lead to significant performance gains. Furthermore, using fine-grained evaluation measures, we find that such performance gains are mostly observed with mathematical formulas of low complexity, which nevertheless often remain challenging even for the largest fine-tuned models.

Flavio Petruzzellis, Alberto Testolin, Alessandro Sperduti

Large Language Models (LLMs) have revolutionized the field of Natural Language Processing thanks to their ability to reuse knowledge acquired on massive text corpora on a wide variety of downstream tasks, with minimal (if any) tuning steps. At the same time, it has been repeatedly shown that LLMs lack systematic generalization, which allows to extrapolate the learned statistical regularities outside the training distribution. In this work, we offer a systematic benchmarking of GPT-4, one of the most advanced LLMs available, on three algorithmic tasks characterized by the possibility to control the problem difficulty with two parameters. We compare the performance of GPT-4 with that of its predecessor (GPT-3.5) and with a variant of the Transformer-Encoder architecture recently introduced to solve similar tasks, the Neural Data Router. We find that the deployment of advanced prompting techniques allows GPT-4 to reach superior accuracy on all tasks, demonstrating that state-of-the-art LLMs constitute a very strong baseline also in challenging tasks that require systematic generalization.

Flavio Petruzzellis, Alberto Testolin, Alessandro Sperduti

Modern neural network architectures still struggle to learn algorithmic procedures that require to systematically apply compositional rules to solve out-of-distribution problem instances. In this work, we focus on formula simplification problems, a class of synthetic benchmarks used to study the systematic generalization capabilities of neural architectures. We propose a modular architecture designed to learn a general procedure for solving nested mathematical formulas by only relying on a minimal set of training examples. Inspired by rewriting systems, a classic framework in symbolic artificial intelligence, we include in the architecture three specialized and interacting modules: the Selector, trained to identify solvable sub-expressions; the Solver, mapping sub-expressions to their values; and the Combiner, replacing sub-expressions in the original formula with the solution provided by the Solver. We benchmark our system against the Neural Data Router, a recent model specialized for systematic generalization, and a state-of-the-art large language model (GPT-4) probed with advanced prompting strategies. We demonstrate that our approach achieves a higher degree of out-of-distribution generalization compared to these alternative approaches on three different types of formula simplification problems, and we discuss its limitations by analyzing its failures.

Alberto Testolin, Kuinan Hou, Marco Zorzi

Many animal species can approximately judge the number of objects in a visual scene at a single glance, and humans can further determine the exact cardinality of a set by deploying systematic counting procedures. In contrast, it has been observed that even state-of-the-art AI systems have very limited enumeration skills. In this work, we propose two benchmark tasks inspired by cognitive science that allow to precisely evaluate the visual enumeration capabilities of multimodal foundation models, thereby providing an objective measure of their number sense and counting level. We consider popular visual question answering models (BLIP, LLaVA and ViLT) as well as advanced image-to-text (Gemini, GPT and Qwen) and text-to-image (DALL-E, FLUX and Stable Diffusion) AI systems. Our analyses show that even the most advanced models cannot reliably name the number of objects in simple visual stimuli or generate images containing a target number of items, as indexed by their low accuracy in both types of tasks. Especially for numbers outside the subitizing range, their responses are often far from the target numerosity, and, in stark contrast with human behavior, in many cases the distribution of errors depends on the object category. We also observe some striking mistakes with small numbers. Our findings demonstrate that developing an intuitive visual understanding of number remains challenging for AI models and that merely increasing model size might not be a viable strategy to promote the emergence of systematic counting skills. We release the full code of our benchmark to facilitate the evaluation of enumeration skills in future AI systems.

Flavio Petruzzellis, Alberto Testolin, Alessandro Sperduti

Building neural systems that can learn to execute symbolic algorithms is a challenging open problem in artificial intelligence, especially when aiming for strong generalization and out-of-distribution performance. In this work, we introduce a general framework for learning convergent term rewriting systems using a neuro-symbolic architecture inspired by the rewriting algorithm itself. We present two modular implementations of such architecture: the Neural Rewriting System (NRS) and the Fast Neural Rewriting System (FastNRS). As a result of algorithmic-inspired design and key architectural elements, both models can generalize to out-of-distribution instances, with FastNRS offering significant improvements in terms of memory efficiency, training speed, and inference time. We evaluate both architectures on four tasks involving the simplification of mathematical formulas and further demonstrate their versatility in a multi-domain learning scenario, where a single model is trained to solve multiple types of problems simultaneously. The proposed system significantly outperforms two strong neural baselines: the Neural Data Router, a recent transformer variant specifically designed to solve algorithmic problems, and GPT-4o, one of the most powerful general-purpose large-language models. Moreover, our system matches or outperforms the latest o1-preview model from OpenAI that excels in reasoning benchmarks.

Lorenzo Tausani, Alberto Testolin, Marco Zorzi

Generative neural networks can produce data samples according to the statistical properties of their training distribution. This feature can be used to test modern computational neuroscience hypotheses suggesting that spontaneous brain activity is partially supported by top-down generative processing. A widely studied class of generative models is that of Restricted Boltzmann Machines (RBMs), which can be used as building blocks for unsupervised deep learning architectures. In this work, we systematically explore the generative dynamics of RBMs, characterizing the number of states visited during top-down sampling and investigating whether the heterogeneity of visited attractors could be increased by starting the generation process from biased hidden states. By considering an RBM trained on a classic dataset of handwritten digits, we show that the capacity to produce diverse data prototypes can be increased by initiating top-down sampling from chimera states, which encode high-level visual features of multiple digits. We also found that the model is not capable of transitioning between all possible digit states within a single generation trajectory, suggesting that the top-down dynamics is heavily constrained by the shape of the energy function.

Federico Venturini, Federico Mason, Francesco Pase et al.

Over the past few years, the use of swarms of Unmanned Aerial Vehicles (UAVs) in monitoring and remote area surveillance applications has become widespread thanks to the price reduction and the increased capabilities of drones. The drones in the swarm need to cooperatively explore an unknown area, in order to identify and monitor interesting targets, while minimizing their movements. In this work, we propose a distributed Reinforcement Learning (RL) approach that scales to larger swarms without modifications. The proposed framework relies on the possibility for the UAVs to exchange some information through a communication channel, in order to achieve context-awareness and implicitly coordinate the swarm's actions. Our experiments show that the proposed method can yield effective strategies, which are robust to communication channel impairments, and that can easily deal with non-uniform distributions of targets and obstacles. Moreover, when agents are trained in a specific scenario, they can adapt to a new one with minimal additional training. We also show that our approach achieves better performance compared to a computationally intensive look-ahead heuristic.

Mattia Lecci, Paolo Testolina, Mattia Rebato et al.

With the advent of millimeter wave (mmWave) communications, the combination of a detailed 5G network simulator with an accurate antenna radiation model is required to analyze the realistic performance of complex cellular scenarios. However, due to the complexity of both electromagnetic and network models, the design and optimization of antenna arrays is generally infeasible due to the required computational resources and simulation time. In this paper, we propose a Machine Learning framework that enables a simulation-based optimization of the antenna design. We show how learning methods are able to emulate a complex simulator with a modest dataset obtained from it, enabling a global numerical optimization over a vast multi-dimensional parameter space in a reasonable amount of time. Overall, our results show that the proposed methodology can be successfully applied to the optimization of thinned antenna arrays.

Matteo Zambra, Alberto Testolin, Amos Maritan

Network science can offer fundamental insights into the structural and functional properties of complex systems. For example, it is widely known that neuronal circuits tend to organize into basic functional topological modules, called "network motifs". In this article we show that network science tools can be successfully applied also to the study of artificial neural networks operating according to self-organizing (learning) principles. In particular, we study the emergence of network motifs in multi-layer perceptrons, whose initial connectivity is defined as a stack of fully-connected, bipartite graphs. Our simulations show that the final network topology is primarily shaped by learning dynamics, but can be strongly biased by choosing appropriate weight initialization schemes. Overall, our results suggest that non-trivial initialization strategies can make learning more effective by promoting the development of useful network motifs, which are often surprisingly consistent with those observed in general transduction networks.

Paolo Testolina, Mattia Lecci, Mattia Rebato et al.

Complex phenomena are generally modeled with sophisticated simulators that, depending on their accuracy, can be very demanding in terms of computational resources and simulation time. Their time-consuming nature, together with a typically vast parameter space to be explored, make simulation-based optimization often infeasible. In this work, we present a method that enables the optimization of complex systems through Machine Learning (ML) techniques. We show how well-known learning algorithms are able to reliably emulate a complex simulator with a modest dataset obtained from it. The trained emulator is then able to yield values close to the simulated ones in virtually no time. Therefore, it is possible to perform a global numerical optimization over the vast multi-dimensional parameter space, in a fraction of the time that would be required by a simple brute-force search. As a testbed for the proposed methodology, we used a network simulator for next-generation mmWave cellular systems. After simulating several antenna configurations and collecting the resulting network-level statistics, we feed it into our framework. Results show that, even with few data points, extrapolating a continuous model makes it possible to estimate the global optimum configuration almost instantaneously. The very same tool can then be used to achieve any further optimization goal on the same input parameters in negligible time.

Alberto Cenzato, Alberto Testolin, Marco Zorzi

The ability to accurately predict the surrounding environment is a foundational principle of intelligence in biological and artificial agents. In recent years, a variety of approaches have been proposed for learning to predict the physical dynamics of objects interacting in a visual scene. Here we conduct a systematic empirical evaluation of several state-of-the-art unsupervised deep learning models that are considered capable of learning the spatio-temporal structure of a popular dataset composed by synthetic videos of bouncing objects. We show that most of the models indeed obtain high accuracy on the standard benchmark of predicting the next frame of a sequence, and one of them even achieves state-of-the-art performance. However, all models fall short when probed with the more challenging task of generating multiple successive frames. Our results show that the ability to perform short-term predictions does not imply that the model has captured the underlying structure and dynamics of the visual environment, thereby calling for a careful rethinking of the metrics commonly adopted for evaluating temporal models. We also investigate whether the learning outcome could be affected by the use of curriculum-based teaching.

Alberto Testolin, Serena Dolfi, Mathijs Rochus et al.

Numerosity perception is foundational to mathematical learning, but its computational bases are strongly debated. Some investigators argue that humans are endowed with a specialized system supporting numerical representation; others argue that visual numerosity is estimated using continuous magnitudes, such as density or area, which usually co-vary with number. Here we reconcile these contrasting perspectives by testing deep networks on the same numerosity comparison task that was administered to humans, using a stimulus space that allows to measure the contribution of non-numerical features. Our model accurately simulated the psychophysics of numerosity perception and the associated developmental changes: discrimination was driven by numerosity information, but non-numerical features had a significant impact, especially early during development. Representational similarity analysis further highlighted that both numerosity and continuous magnitudes were spontaneously encoded even when no task had to be carried out, demonstrating that numerosity is a major, salient property of our visual environment.

Alberto Testolin, Michele Piccolini, Samir Suweis

Thanks to the availability of large scale digital datasets and massive amounts of computational power, deep learning algorithms can learn representations of data by exploiting multiple levels of abstraction. These machine learning methods have greatly improved the state-of-the-art in many challenging cognitive tasks, such as visual object recognition, speech processing, natural language understanding and automatic translation. In particular, one class of deep learning models, known as deep belief networks, can discover intricate statistical structure in large data sets in a completely unsupervised fashion, by learning a generative model of the data using Hebbian-like learning mechanisms. Although these self-organizing systems can be conveniently formalized within the framework of statistical mechanics, their internal functioning remains opaque, because their emergent dynamics cannot be solved analytically. In this article we propose to study deep belief networks using techniques commonly employed in the study of complex networks, in order to gain some insights into the structural and functional properties of the computational graph resulting from the learning process.