Luca Biggio

Lorenzo Noci, Sotiris Anagnostidis, Luca Biggio et al. · eth-zurich

Transformers have achieved remarkable success in several domains, ranging from natural language processing to computer vision. Nevertheless, it has been recently shown that stacking self-attention layers - the distinctive architectural component of Transformers - can result in rank collapse of the tokens' representations at initialization. The question of if and how rank collapse affects training is still largely unanswered, and its investigation is necessary for a more comprehensive understanding of this architecture. In this work, we shed new light on the causes and the effects of this phenomenon. First, we show that rank collapse of the tokens' representations hinders training by causing the gradients of the queries and keys to vanish at initialization. Furthermore, we provide a thorough description of the origin of rank collapse and discuss how to prevent it via an appropriate depth-dependent scaling of the residual branches. Finally, our analysis unveils that specific architectural hyperparameters affect the gradients of queries and values differently, leading to disproportionate gradient norms. This suggests an explanation for the widespread use of adaptive methods for Transformers' optimization.

Sotiris Anagnostidis, Arne Thomsen, Tomasz Kacprzak et al. · eth-zurich

In recent years, deep learning approaches have achieved state-of-the-art results in the analysis of point cloud data. In cosmology, galaxy redshift surveys resemble such a permutation invariant collection of positions in space. These surveys have so far mostly been analysed with two-point statistics, such as power spectra and correlation functions. The usage of these summary statistics is best justified on large scales, where the density field is linear and Gaussian. However, in light of the increased precision expected from upcoming surveys, the analysis of -- intrinsically non-Gaussian -- small angular separations represents an appealing avenue to better constrain cosmological parameters. In this work, we aim to improve upon two-point statistics by employing a \textit{PointNet}-like neural network to regress the values of the cosmological parameters directly from point cloud data. Our implementation of PointNets can analyse inputs of $\mathcal{O}(10^4) - \mathcal{O}(10^5)$ galaxies at a time, which improves upon earlier work for this application by roughly two orders of magnitude. Additionally, we demonstrate the ability to analyse galaxy redshift survey data on the lightcone, as opposed to previously static simulation boxes at a given fixed redshift.

Tommaso Bendinelli, Luca Biggio, Pierre-Alexandre Kamienny · meta-ai

In symbolic regression, the goal is to find an analytical expression that accurately fits experimental data with the minimal use of mathematical symbols such as operators, variables, and constants. However, the combinatorial space of possible expressions can make it challenging for traditional evolutionary algorithms to find the correct expression in a reasonable amount of time. To address this issue, Neural Symbolic Regression (NSR) algorithms have been developed that can quickly identify patterns in the data and generate analytical expressions. However, these methods, in their current form, lack the capability to incorporate user-defined prior knowledge, which is often required in natural sciences and engineering fields. To overcome this limitation, we propose a novel neural symbolic regression method, named Neural Symbolic Regression with Hypothesis (NSRwH) that enables the explicit incorporation of assumptions about the expected structure of the ground-truth expression into the prediction process. Our experiments demonstrate that the proposed conditioned deep learning model outperforms its unconditioned counterparts in terms of accuracy while also providing control over the predicted expression structure.

Enea Monzio Compagnoni, Luca Biggio, Antonio Orvieto et al.

We study the SAM (Sharpness-Aware Minimization) optimizer which has recently attracted a lot of interest due to its increased performance over more classical variants of stochastic gradient descent. Our main contribution is the derivation of continuous-time models (in the form of SDEs) for SAM and two of its variants, both for the full-batch and mini-batch settings. We demonstrate that these SDEs are rigorous approximations of the real discrete-time algorithms (in a weak sense, scaling linearly with the learning rate). Using these models, we then offer an explanation of why SAM prefers flat minima over sharp ones~--~by showing that it minimizes an implicitly regularized loss with a Hessian-dependent noise structure. Finally, we prove that SAM is attracted to saddle points under some realistic conditions. Our theoretical results are supported by detailed experiments.

Luca Biggio, Tommaso Bendinelli, Chetan Kulkarni et al.

Electrochemical batteries are ubiquitous devices in our society. When they are employed in mission-critical applications, the ability to precisely predict the end of discharge under highly variable environmental and operating conditions is of paramount importance in order to support operational decision-making. While there are accurate predictive models of the processes underlying the charge and discharge phases of batteries, the modelling of ageing and its effect on performance remains poorly understood. Such a lack of understanding often leads to inaccurate models or the need for time-consuming calibration procedures whenever the battery ages or its conditions change significantly. This represents a major obstacle to the real-world deployment of efficient and robust battery management systems. In this paper, we propose for the first time an approach that can predict the voltage discharge curve for batteries of any degradation level without the need for calibration. In particular, we introduce Dynaformer, a novel Transformer-based deep learning architecture which is able to simultaneously infer the ageing state from a limited number of voltage/current samples and predict the full voltage discharge curve for real batteries with high precision. Our experiments show that the trained model is effective for input current profiles of different complexities and is robust to a wide range of degradation levels. In addition to evaluating the performance of the proposed framework on simulated data, we demonstrate that a minimal amount of fine-tuning allows the model to bridge the simulation-to-real gap between simulations and real data collected from a set of batteries. The proposed methodology enables the utilization of battery-powered systems until the end of discharge in a controlled and predictable way, thereby significantly prolonging the operating cycles and reducing costs.

Elior Benarous, Sotiris Anagnostidis, Luca Biggio et al.

Recent advancements in deep learning have been primarily driven by the use of large models trained on increasingly vast datasets. While neural scaling laws have emerged to predict network performance given a specific level of computational resources, the growing demand for expansive datasets raises concerns. To address this, a new research direction has emerged, focusing on the creation of synthetic data as a substitute. In this study, we investigate how neural networks exhibit shape bias during training on synthetic datasets, serving as an indicator of the synthetic data quality. Specifically, our findings indicate three key points: (1) Shape bias varies across network architectures and types of supervision, casting doubt on its reliability as a predictor for generalization and its ability to explain differences in model recognition compared to human capabilities. (2) Relying solely on shape bias to estimate generalization is unreliable, as it is entangled with diversity and naturalism. (3) We propose a novel interpretation of shape bias as a tool for estimating the diversity of samples within a dataset. Our research aims to clarify the implications of using synthetic data and its associated shape bias in deep learning, addressing concerns regarding generalization and dataset quality.

Freya Behrens, Luca Biggio, Lenka Zdeborová

Next to scaling considerations, architectural design choices profoundly shape the solution space of transformers. In this work, we analyze the solutions simple transformer blocks implement when tackling the histogram task: counting items in sequences. Despite its simplicity, this task reveals a complex interplay between predictive performance, vocabulary and embedding sizes, token-mixing mechanisms, and feed-forward layer capacity. We identify two theoretical counting strategies transformers adopt, relation-based and inventory-based counting, each defining distinct learning regimes for the task. These strategies dictate how functionality is distributed between attention and feed-forward layers. We further show that adding softmax and beginning-of-sequence tokens allow for more robustness when embedding dimensions are comparatively small. Empirical introspection of trained models closely confirms both the learning regimes of the various architectures and the formation of these strategies during training. We demonstrate how a basic task that requires only aggregation and selection is significantly impacted by minor design changes.

Jerome Garnier-Brun, Luca Biggio, Davide Beltrame et al.

Generalization in generative modeling is defined as the ability to learn an underlying distribution from a finite dataset and produce novel samples, with evaluation largely driven by held-out performance and perceived sample quality. In practice, training is often stopped at the minimum of the test loss, taken as an operational indicator of generalization. We challenge this viewpoint by identifying a phase of biased generalization during training, in which the model continues to decrease the test loss while favoring samples with anomalously high proximity to training data. By training the same network on two disjoint datasets and comparing the mutual distances of generated samples and their similarity to training data, we introduce a quantitative measure of bias and demonstrate its presence on real images. We then study the mechanism of bias, using a controlled hierarchical data model where access to exact scores and ground-truth statistics allows us to precisely characterize its onset. We attribute this phenomenon to the sequential nature of feature learning in deep networks, where coarse structure is learned early in a data-independent manner, while finer features are resolved later in a way that increasingly depends on individual training samples. Our results show that early stopping at the test loss minimum, while optimal under standard generalization criteria, may be insufficient for privacy-critical applications.

Vittorio Erba, Emanuele Troiani, Luca Biggio et al.

Current progress in artificial intelligence is centered around so-called large language models that consist of neural networks processing long sequences of high-dimensional vectors called tokens. Statistical physics provides powerful tools to study the functioning of learning with neural networks and has played a recognized role in the development of modern machine learning. The statistical physics approach relies on simplified and analytically tractable models of data. However, simple tractable models for long sequences of high-dimensional tokens are largely underexplored. Inspired by the crucial role models such as the single-layer teacher-student perceptron (aka generalized linear regression) played in the theory of fully connected neural networks, in this paper, we introduce and study the bilinear sequence regression (BSR) as one of the most basic models for sequences of tokens. We note that modern architectures naturally subsume the BSR model due to the skip connections. Building on recent methodological progress, we compute the Bayes-optimal generalization error for the model in the limit of long sequences of high-dimensional tokens, and provide a message-passing algorithm that matches this performance. We quantify the improvement that optimal learning brings with respect to vectorizing the sequence of tokens and learning via simple linear regression. We also unveil surprising properties of the gradient descent algorithms in the BSR model.

Riccardo Alberghi, Elizaveta Demyanenko, Luca Biggio et al.

Recent advances in natural language processing highlight two key factors for improving reasoning in large language models (LLMs): (i) allocating more test-time compute tends to help on harder problems but often introduces redundancy in the reasoning trace, and (ii) compute is most effective when reasoning is systematic and incremental, forming structured chains of thought (CoTs) akin to human problem-solving. To study these factors in isolation, we introduce a controlled setting based on shortest-path tasks in layered graphs. We train decoder-only transformers on question-trace-answer triples using a custom tokenizer, comparing models trained on optimal bottom-up dynamic programming traces with those trained on longer, valid traces involving backtracking. Surprisingly, with the same training-token budget, models trained on inefficient traces generalize better to unseen graphs. This benefit is not due to length alone-injecting arbitrary redundancy into reasoning traces fails to help and can even hurt performance. Instead, we find that generalization correlates with the model's confidence in next-token prediction, suggesting that long, coherent, and locally incremental traces make the training signal easier to optimize.

Tommaso Bendinelli, Luca Biggio, Daniel Nyfeler et al.

The value of luxury goods, particularly investment-grade gemstones, is greatly influenced by their origin and authenticity, sometimes resulting in differences worth millions of dollars. Traditionally, human experts have determined the origin and detected treatments on gemstones through visual inspections and a range of analytical methods. However, the interpretation of the data can be subjective and time-consuming, resulting in inconsistencies. In this study, we propose Gemtelligence, a novel approach based on deep learning that enables accurate and consistent origin determination and treatment detection. Gemtelligence comprises convolutional and attention-based neural networks that process heterogeneous data types collected by multiple instruments. Notably, the algorithm demonstrated comparable predictive performance to expensive laser-ablation inductively-coupled-plasma mass-spectrometry (ICP-MS) analysis and visual examination by human experts, despite using input data from relatively inexpensive analytical methods. Our innovative methodology represents a major breakthrough in the field of gemstone analysis by significantly improving the automation and robustness of the entire analytical process pipeline.

Sotiris Anagnostidis, Dario Pavllo, Luca Biggio et al.

Autoregressive Transformers adopted in Large Language Models (LLMs) are hard to scale to long sequences. Despite several works trying to reduce their computational cost, most of LLMs still adopt attention layers between all pairs of tokens in the sequence, thus incurring a quadratic cost. In this study, we present a novel approach that dynamically prunes contextual information while preserving the model's expressiveness, resulting in reduced memory and computational requirements during inference. Our method employs a learnable mechanism that determines which uninformative tokens can be dropped from the context at any point across the generation process. By doing so, our approach not only addresses performance concerns but also enhances interpretability, providing valuable insight into the model's decision-making process. Our technique can be applied to existing pre-trained models through a straightforward fine-tuning process, and the pruning strength can be specified by a sparsity parameter. Notably, our empirical findings demonstrate that we can effectively prune up to 80\% of the context without significant performance degradation on downstream tasks, offering a valuable tool for mitigating inference costs. Our reference implementation achieves up to $2\times$ increase in inference throughput and even greater memory savings.

Venkat Nemani, Luca Biggio, Xun Huan et al.

On top of machine learning models, uncertainty quantification (UQ) functions as an essential layer of safety assurance that could lead to more principled decision making by enabling sound risk assessment and management. The safety and reliability improvement of ML models empowered by UQ has the potential to significantly facilitate the broad adoption of ML solutions in high-stakes decision settings, such as healthcare, manufacturing, and aviation, to name a few. In this tutorial, we aim to provide a holistic lens on emerging UQ methods for ML models with a particular focus on neural networks and the applications of these UQ methods in tackling engineering design as well as prognostics and health management problems. Toward this goal, we start with a comprehensive classification of uncertainty types, sources, and causes pertaining to UQ of ML models. Next, we provide a tutorial-style description of several state-of-the-art UQ methods: Gaussian process regression, Bayesian neural network, neural network ensemble, and deterministic UQ methods focusing on spectral-normalized neural Gaussian process. Established upon the mathematical formulations, we subsequently examine the soundness of these UQ methods quantitatively and qualitatively (by a toy regression example) to examine their strengths and shortcomings from different dimensions. Then, we review quantitative metrics commonly used to assess the quality of predictive uncertainty in classification and regression problems. Afterward, we discuss the increasingly important role of UQ of ML models in solving challenging problems in engineering design and health prognostics. Two case studies with source codes available on GitHub are used to demonstrate these UQ methods and compare their performance in the life prediction of lithium-ion batteries at the early stage and the remaining useful life prediction of turbofan engines.

Dimitri von Rütte, Luca Biggio, Yannic Kilcher et al.

Generating music with deep neural networks has been an area of active research in recent years. While the quality of generated samples has been steadily increasing, most methods are only able to exert minimal control over the generated sequence, if any. We propose the self-supervised description-to-sequence task, which allows for fine-grained controllable generation on a global level. We do so by extracting high-level features about the target sequence and learning the conditional distribution of sequences given the corresponding high-level description in a sequence-to-sequence modelling setup. We train FIGARO (FIne-grained music Generation via Attention-based, RObust control) by applying description-to-sequence modelling to symbolic music. By combining learned high level features with domain knowledge, which acts as a strong inductive bias, the model achieves state-of-the-art results in controllable symbolic music generation and generalizes well beyond the training distribution.

Enea Monzio Compagnoni, Anna Scampicchio, Luca Biggio et al.

Many finance, physics, and engineering phenomena are modeled by continuous-time dynamical systems driven by highly irregular (stochastic) inputs. A powerful tool to perform time series analysis in this context is rooted in rough path theory and leverages the so-called Signature Transform. This algorithm enjoys strong theoretical guarantees but is hard to scale to high-dimensional data. In this paper, we study a recently derived random projection variant called Randomized Signature, obtained using the Johnson-Lindenstrauss Lemma. We provide an in-depth experimental evaluation of the effectiveness of the Randomized Signature approach, in an attempt to showcase the advantages of this reservoir to the community. Specifically, we find that this method is preferable to the truncated Signature approach and alternative deep learning techniques in terms of model complexity, training time, accuracy, robustness, and data hungriness.

Luca Biggio, Tommaso Bendinelli, Alexander Neitz et al.

Symbolic equations are at the core of scientific discovery. The task of discovering the underlying equation from a set of input-output pairs is called symbolic regression. Traditionally, symbolic regression methods use hand-designed strategies that do not improve with experience. In this paper, we introduce the first symbolic regression method that leverages large scale pre-training. We procedurally generate an unbounded set of equations, and simultaneously pre-train a Transformer to predict the symbolic equation from a corresponding set of input-output-pairs. At test time, we query the model on a new set of points and use its output to guide the search for the equation. We show empirically that this approach can re-discover a set of well-known physical equations, and that it improves over time with more data and compute.

Luca Biggio, Alexander Wieland, Manuel Arias Chao et al.

Remaining Useful Life (RUL) estimation is the problem of inferring how long a certain industrial asset can be expected to operate within its defined specifications. Deploying successful RUL prediction methods in real-life applications is a prerequisite for the design of intelligent maintenance strategies with the potential of drastically reducing maintenance costs and machine downtimes. In light of their superior performance in a wide range of engineering fields, Machine Learning (ML) algorithms are natural candidates to tackle the challenges involved in the design of intelligent maintenance systems. In particular, given the potentially catastrophic consequences or substantial costs associated with maintenance decisions that are either too late or too early, it is desirable that ML algorithms provide uncertainty estimates alongside their predictions. However, standard data-driven methods used for uncertainty estimation in RUL problems do not scale well to large datasets or are not sufficiently expressive to model the high-dimensional mapping from raw sensor data to RUL estimates. In this work, we consider Deep Gaussian Processes (DGPs) as possible solutions to the aforementioned limitations. We perform a thorough evaluation and comparison of several variants of DGPs applied to RUL predictions. The performance of the algorithms is evaluated on the N-CMAPSS (New Commercial Modular Aero-Propulsion System Simulation) dataset from NASA for aircraft engines. The results show that the proposed methods are able to provide very accurate RUL predictions along with sensible uncertainty estimates, providing more reliable solutions for (safety-critical) real-life industrial applications.