Author name cluster

Luca Biggio

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8 papers

2 author rows

ICML Conference 2025 Conference Paper

Counting in Small Transformers: The Delicate Interplay between Attention and Feed-Forward Layers

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.

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NeurIPS Conference 2025 Conference Paper

On the Bias of Next-Token Predictors Toward Systematically Inefficient Reasoning: A Shortest-Path Case Study

Riccardo Alberghi
Elizaveta Demyanenko
Luca Biggio
Luca Saglietti

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, under the same training-token budget, the latter models 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.

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ICML Conference 2023 Conference Paper

An SDE for Modeling SAM: Theory and Insights

Enea Monzio Compagnoni
Luca Biggio
Antonio Orvieto
Frank Norbert Proske
Hans Kersting
Aurélien Lucchi

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.

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ICML Conference 2023 Conference Paper

Controllable Neural Symbolic Regression

Tommaso Bendinelli
Luca Biggio
Pierre-Alexandre Kamienny

In symbolic regression, the objective 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.

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NeurIPS Conference 2023 Conference Paper

Dynamic Context Pruning for Efficient and Interpretable Autoregressive Transformers

Sotiris Anagnostidis
Dario Pavllo
Luca Biggio
Lorenzo Noci
Aurelien Lucchi
Thomas Hofmann

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.

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ICLR Conference 2023 Conference Paper

FIGARO: Controllable Music Generation using Learned and Expert Features

Dimitri von Rütte
Luca Biggio
Yannic Kilcher
Thomas Hofmann 0001

Recent symbolic music generative models have achieved significant improvements in the quality of the generated samples. Nevertheless, it remains hard for users to control the output in such a way that it matches their expectation. To address this limitation, high-level, human-interpretable conditioning is essential. In this work, we release FIGARO, a Transformer-based conditional model trained to generate symbolic music based on a sequence of high-level control codes. To this end, we propose description-to-sequence learning, which consists of automatically extracting fine-grained, human-interpretable features (the description) and training a sequence-to-sequence model to reconstruct the original sequence given only the description as input. FIGARO achieves state-of-the-art performance in multi-track symbolic music generation both in terms of style transfer and sample quality. We show that performance can be further improved by combining human-interpretable with learned features. Our extensive experimental evaluation shows that FIGARO is able to generate samples that closely adhere to the content of the input descriptions, even when they deviate significantly from the training distribution.

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NeurIPS Conference 2022 Conference Paper

Signal Propagation in Transformers: Theoretical Perspectives and the Role of Rank Collapse

Lorenzo Noci
Sotiris Anagnostidis
Luca Biggio
Antonio Orvieto
Sidak Pal Singh
Aurelien Lucchi

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, keys and values differently, leading to disproportionate gradient norms. This suggests an explanation for the widespread use of adaptive methods for Transformers' optimization.

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ICML Conference 2021 Conference Paper

Neural Symbolic Regression that scales

Luca Biggio
Tommaso Bendinelli
Alexander Neitz
Aurélien Lucchi
Giambattista Parascandolo

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.

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