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Aurelien Lucchi

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19 papers

1 author row

NeurIPS Conference 2025 Conference Paper

Enhancing Optimizer Stability: Momentum Adaptation of The NGN Step-size

Rustem Islamov
Niccolò Ajroldi
Antonio Orvieto
Aurelien Lucchi

Modern optimization algorithms that incorporate momentum and adaptive step-size offer improved performance in numerous challenging deep learning tasks. However, their effectiveness is often highly sensitive to the choice of hyperparameters, especially the step-size. Tuning these parameters is often difficult, resource-intensive, and time-consuming. Therefore, recent efforts have been directed toward enhancing the stability of optimizers across a wide range of hyperparameter choices [Schaipp et al. , 2024]. In this paper, we introduce an algorithm that matches the performance of state-of-the-art optimizers while improving stability to the choice of the step-size hyperparameter through a novel adaptation of the ${\sf NGN}$ step-size method [Orvieto and Xiao, 2024]. Specifically, we propose a momentum-based version ${\sf NGN}$-${\sf M}$ that attains the standard convergence rate of $\mathcal{O}(1/\sqrt{K})$ under less restrictive assumptions, without the need for interpolation condition or assumptions of bounded stochastic gradients or iterates, in contrast to previous approaches. Additionally, we empirically demonstrate that the combination of the ${\sf NGN}$ step-size with momentum results in enhanced robustness to the choice of the step-size hyperparameter while delivering performance that is comparable to or surpasses other state-of-the-art optimizers.

TMLR Journal 2025 Journal Article

Optimization Guarantees for Square-Root Natural-Gradient Variational Inference

Navish Kumar
Thomas Möllenhoff
Mohammad Emtiyaz Khan
Aurelien Lucchi

Variational inference with natural-gradient descent often shows fast convergence in practice, but its theoretical convergence guarantees have been challenging to establish. This is true even for the simplest cases that involve concave log-likelihoods and use a Gaussian approximation. We show that the challenge can be circumvented for such cases using a square-root parameterization for the Gaussian covariance. This approach establishes novel convergence guarantees for natural-gradient variational-Gaussian inference and its continuous-time gradient flow. Our experiments demonstrate the effectiveness of natural gradient methods and highlight their advantages over algorithms that use Euclidean or Wasserstein geometries.

NeurIPS Conference 2024 Conference Paper

A Comprehensive Analysis on the Learning Curve in Kernel Ridge Regression

Tin Sum Cheng
Aurelien Lucchi
Anastasis Kratsios
David Belius

This paper conducts a comprehensive study of the learning curves of kernel ridge regression (KRR) under minimal assumptions. Our contributions are three-fold: 1) we analyze the role of key properties of the kernel, such as its spectral eigen-decay, the characteristics of the eigenfunctions, and the smoothness of the kernel; 2) we demonstrate the validity of the Gaussian Equivalent Property (GEP), which states that the generalization performance of KRR remains the same when the whitened features are replaced by standard Gaussian vectors, thereby shedding light on the success of previous analyzes under the Gaussian Design Assumption; 3) we derive novel bounds that improve over existing bounds across a broad range of setting such as (in)dependent feature vectors and various combinations of eigen-decay rates in the over/underparameterized regimes.

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NeurIPS Conference 2024 Conference Paper

Loss Landscape Characterization of Neural Networks without Over-Parametrization

Rustem Islamov
Niccoló Ajroldi
Antonio Orvieto
Aurelien Lucchi

Modern machine learning heavily depends on the effectiveness of optimization techniques. While deep learning models have achieved remarkable empirical results in training, their theoretical underpinnings remain somewhat elusive. Ensuring the convergence of optimization methods requires imposing specific structures on the objective function which often do not hold in practice. One prominent example is the widely recognized Polyak-Lojasiewicz (PL) inequality, which has garnered considerable attention in recent years. However, validating such assumptions for deep neural networks entails substantial and often impractical levels of over-parametrization. In order to address this limitation, we propose a novel class of functions that can characterize the loss landscape of modern deep models without requiring extensive over-parametrization and can also include saddle points. Crucially, we prove that gradient-based optimizers possess theoretical guarantees of convergence under this assumption. Finally, we validate the soundness of our assumption through both theoretical analysis and empirical experimentation across a diverse range of deep learning models.

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NeurIPS Conference 2024 Conference Paper

Theoretical Characterisation of the Gauss Newton Conditioning in Neural Networks

Jim Zhao
Sidak Pal Singh
Aurelien Lucchi

The Gauss-Newton (GN) matrix plays an important role in machine learning, most evident in its use as a preconditioning matrix for a wide family of popular adaptive methods to speed up optimization. Besides, it can also provide key insights into the optimization landscape of neural networks. In the context of deep neural networks, understanding the GN matrix involves studying the interaction between different weight matrices as well as the dependencies introduced by the data, thus rendering its analysis challenging. In this work, we take a first step towards theoretically characterizing the conditioning of the GN matrix in neural networks. We establish tight bounds on the condition number of the GN in deep linear networks of arbitrary depth and width, which we also extend to two-layer ReLU networks. We expand the analysis to further architectural components, such as residual connections and convolutional layers. Finally, we empirically validate the bounds and uncover valuable insights into the influence of the analyzed architectural components.

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NeurIPS Conference 2023 Conference Paper

A Theoretical Analysis of the Test Error of Finite-Rank Kernel Ridge Regression

Tin Sum Cheng
Aurelien Lucchi
Anastasis Kratsios
Ivan Dokmanić
David Belius

Existing statistical learning guarantees for general kernel regressors often yield loose bounds when used with finite-rank kernels. Yet, finite-rank kernels naturally appear in a number of machine learning problems, e. g. when fine-tuning a pre-trained deep neural network's last layer to adapt it to a novel task when performing transfer learning. We address this gap for finite-rank kernel ridge regression (KRR) by deriving sharp non-asymptotic upper and lower bounds for the KRR test error of any finite-rank KRR. Our bounds are tighter than previously derived bounds on finite-rank KRR and, unlike comparable results, they also remain valid for any regularization parameters.

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.

NeurIPS Conference 2022 Conference Paper

On the Theoretical Properties of Noise Correlation in Stochastic Optimization

Aurelien Lucchi
Frank Proske
Antonio Orvieto
Francis Bach
Hans Kersting

Studying the properties of stochastic noise to optimize complex non-convex functions has been an active area of research in the field of machine learning. Prior work~\citep{zhou2019pgd, wei2019noise} has shown that the noise of stochastic gradient descent improves optimization by overcoming undesirable obstacles in the landscape. Moreover, injecting artificial Gaussian noise has become a popular idea to quickly escape saddle points. Indeed, in the absence of reliable gradient information, the noise is used to explore the landscape, but it is unclear what type of noise is optimal in terms of exploration ability. In order to narrow this gap in our knowledge, we study a general type of continuous-time non-Markovian process, based on fractional Brownian motion, that allows for the increments of the process to be correlated. This generalizes processes based on Brownian motion, such as the Ornstein-Uhlenbeck process. We demonstrate how to discretize such processes which gives rise to the new algorithm ``fPGD''. This method is a generalization of the known algorithms PGD and Anti-PGD~\citep{orvieto2022anti}. We study the properties of fPGD both theoretically and empirically, demonstrating that it possesses exploration abilities that, in some cases, are favorable over PGD and Anti-PGD. These results open the field to novel ways to exploit noise for training machine learning models.

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.

NeurIPS Conference 2021 Conference Paper

On the Second-order Convergence Properties of Random Search Methods

Aurelien Lucchi
Antonio Orvieto
Adamos Solomou

We study the theoretical convergence properties of random-search methods when optimizing non-convex objective functions without having access to derivatives. We prove that standard random-search methods that do not rely on second-order information converge to a second-order stationary point. However, they suffer from an exponential complexity in terms of the input dimension of the problem. In order to address this issue, we propose a novel variant of random search that exploits negative curvature by only relying on function evaluations. We prove that this approach converges to a second-order stationary point at a much faster rate than vanilla methods: namely, the complexity in terms of the number of function evaluations is only linear in the problem dimension. We test our algorithm empirically and find good agreements with our theoretical results.

AAAI Conference 2021 Conference Paper

Scalable Graph Networks for Particle Simulations

Karolis Martinkus
Aurelien Lucchi
Nathanaël Perraudin

Learning system dynamics directly from observations is a promising direction in machine learning due to its potential to significantly enhance our ability to understand physical systems. However, the dynamics of many real-world systems are challenging to learn due to the presence of nonlinear potentials and a number of interactions that scales quadratically with the number of particles N, as in the case of the N-body problem. In this work, we introduce an approach that transforms a fully-connected interaction graph into a hierarchical one which reduces the number of edges to O(N). This results in linear time and space complexity while the precomputation of the hierarchical graph requires O(N log(N)) time and O(N) space. Using our approach, we are able to train models on much larger particle counts, even on a single GPU. We evaluate how the phase space position accuracy and energy conservation depend on the number of simulated particles. Our approach retains high accuracy and efficiency even on large-scale gravitational N-body simulations which are impossible to run on a single machine if a fully-connected graph is used. Similar results are also observed when simulating Coulomb interactions. Furthermore, we make several important observations regarding the performance of this new hierarchical model, including: i) its accuracy tends to improve with the number of particles in the simulation and ii) its generalisation to unseen particle counts is also much better than for models that use all O(N2 ) interactions.

NeurIPS Conference 2020 Conference Paper

Batch normalization provably avoids ranks collapse for randomly initialised deep networks

Hadi Daneshmand
Jonas Kohler
Francis Bach
Thomas Hofmann
Aurelien Lucchi

Randomly initialized neural networks are known to become harder to train with increasing depth, unless architectural enhancements like residual connections and batch normalization are used. We here investigate this phenomenon by revisiting the connection between random initialization in deep networks and spectral instabilities in products of random matrices. Given the rich literature on random matrices, it is not surprising to find that the rank of the intermediate representations in unnormalized networks collapses quickly with depth. In this work we highlight the fact that batch normalization is an effective strategy to avoid rank collapse for both linear and ReLU networks. Leveraging tools from Markov chain theory, we derive a meaningful lower rank bound in deep linear networks. Empirically, we also demonstrate that this rank robustness generalizes to ReLU nets. Finally, we conduct an extensive set of experiments on real-world data sets, which confirm that rank stability is indeed a crucial condition for training modern-day deep neural architectures.

NeurIPS Conference 2020 Conference Paper

Convolutional Generation of Textured 3D Meshes

Dario Pavllo
Graham Spinks
Thomas Hofmann
Marie-Francine Moens
Aurelien Lucchi

While recent generative models for 2D images achieve impressive visual results, they clearly lack the ability to perform 3D reasoning. This heavily restricts the degree of control over generated objects as well as the possible applications of such models. In this work, we bridge this gap by leveraging recent advances in differentiable rendering. We design a framework that can generate triangle meshes and associated high-resolution texture maps, using only 2D supervision from single-view natural images. A key contribution of our work is the encoding of the mesh and texture as 2D representations, which are semantically aligned and can be easily modeled by a 2D convolutional GAN. We demonstrate the efficacy of our method on Pascal3D+ Cars and CUB, both in an unconditional setting and in settings where the model is conditioned on class labels, attributes, and text. Finally, we propose an evaluation methodology that assesses the mesh and texture quality separately.

NeurIPS Conference 2019 Conference Paper

A Domain Agnostic Measure for Monitoring and Evaluating GANs

Paulina Grnarova
Kfir Y. Levy
Aurelien Lucchi
Nathanael Perraudin
Ian Goodfellow
Thomas Hofmann
Andreas Krause

Generative Adversarial Networks (GANs) have shown remarkable results in modeling complex distributions, but their evaluation remains an unsettled issue. Evaluations are essential for: (i) relative assessment of different models and (ii) monitoring the progress of a single model throughout training. The latter cannot be determined by simply inspecting the generator and discriminator loss curves as they behave non-intuitively. We leverage the notion of duality gap from game theory to propose a measure that addresses both (i) and (ii) at a low computational cost. Extensive experiments show the effectiveness of this measure to rank different GAN models and capture the typical GAN failure scenarios, including mode collapse and non-convergent behaviours. This evaluation metric also provides meaningful monitoring on the progression of the loss during training. It highly correlates with FID on natural image datasets, and with domain specific scores for text, sound and cosmology data where FID is not directly suitable. In particular, our proposed metric requires no labels or a pretrained classifier, making it domain agnostic.

NeurIPS Conference 2019 Conference Paper

Continuous-time Models for Stochastic Optimization Algorithms

Antonio Orvieto
Aurelien Lucchi

We propose new continuous-time formulations for first-order stochastic optimization algorithms such as mini-batch gradient descent and variance-reduced methods. We exploit these continuous-time models, together with simple Lyapunov analysis as well as tools from stochastic calculus, in order to derive convergence bounds for various types of non-convex functions. Guided by such analysis, we show that the same Lyapunov arguments hold in discrete-time, leading to matching rates. In addition, we use these models and Ito calculus to infer novel insights on the dynamics of SGD, proving that a decreasing learning rate acts as time warping or, equivalently, as landscape stretching.

NeurIPS Conference 2019 Conference Paper

Shadowing Properties of Optimization Algorithms

Antonio Orvieto
Aurelien Lucchi

Ordinary differential equation (ODE) models of gradient-based optimization methods can provide insights into the dynamics of learning and inspire the design of new algorithms. Unfortunately, this thought-provoking perspective is weakened by the fact that, in the worst case, the error between the algorithm steps and its ODE approximation grows exponentially with the number of iterations. In an attempt to encourage the use of continuous-time methods in optimization, we show that, if some additional regularity on the objective is assumed, the ODE representations of Gradient Descent and Heavy-ball do not suffer from the aforementioned problem, once we allow for a small perturbation on the algorithm initial condition. In the dynamical systems literature, this phenomenon is called shadowing. Our analysis relies on the concept of hyperbolicity, as well as on tools from numerical analysis.

NeurIPS Conference 2017 Conference Paper

Stabilizing Training of Generative Adversarial Networks through Regularization

Kevin Roth
Aurelien Lucchi
Sebastian Nowozin
Thomas Hofmann

Deep generative models based on Generative Adversarial Networks (GANs) have demonstrated impressive sample quality but in order to work they require a careful choice of architecture, parameter initialization, and selection of hyper-parameters. This fragility is in part due to a dimensional mismatch or non-overlapping support between the model distribution and the data distribution, causing their density ratio and the associated f -divergence to be undefined. We overcome this fundamental limitation and propose a new regularization approach with low computational cost that yields a stable GAN training procedure. We demonstrate the effectiveness of this regularizer accross several architectures trained on common benchmark image generation tasks. Our regularization turns GAN models into reliable building blocks for deep learning.

NeurIPS Conference 2016 Conference Paper

Adaptive Newton Method for Empirical Risk Minimization to Statistical Accuracy

Aryan Mokhtari
Hadi Daneshmand
Aurelien Lucchi
Thomas Hofmann
Alejandro Ribeiro

We consider empirical risk minimization for large-scale datasets. We introduce Ada Newton as an adaptive algorithm that uses Newton's method with adaptive sample sizes. The main idea of Ada Newton is to increase the size of the training set by a factor larger than one in a way that the minimization variable for the current training set is in the local neighborhood of the optimal argument of the next training set. This allows to exploit the quadratic convergence property of Newton's method and reach the statistical accuracy of each training set with only one iteration of Newton's method. We show theoretically that we can iteratively increase the sample size while applying single Newton iterations without line search and staying within the statistical accuracy of the regularized empirical risk. In particular, we can double the size of the training set in each iteration when the number of samples is sufficiently large. Numerical experiments on various datasets confirm the possibility of increasing the sample size by factor 2 at each iteration which implies that Ada Newton achieves the statistical accuracy of the full training set with about two passes over the dataset.

NeurIPS Conference 2015 Conference Paper

Variance Reduced Stochastic Gradient Descent with Neighbors

Thomas Hofmann
Aurelien Lucchi
Simon Lacoste-Julien
Brian McWilliams

Stochastic Gradient Descent (SGD) is a workhorse in machine learning, yet it is also known to be slow relative to steepest descent. Recently, variance reduction techniques such as SVRG and SAGA have been proposed to overcome this weakness. With asymptotically vanishing variance, a constant step size can be maintained, resulting in geometric convergence rates. However, these methods are either based on occasional computations of full gradients at pivot points (SVRG), or on keeping per data point corrections in memory (SAGA). This has the disadvantage that one cannot employ these methods in a streaming setting and that speed-ups relative to SGD may need a certain number of epochs in order to materialize. This paper investigates a new class of algorithms that can exploit neighborhood structure in the training data to share and re-use information about past stochastic gradients across data points. While not meant to be offering advantages in an asymptotic setting, there are significant benefits in the transient optimization phase, in particular in a streaming or single-epoch setting. We investigate this family of algorithms in a thorough analysis and show supporting experimental results. As a side-product we provide a simple and unified proof technique for a broad class of variance reduction algorithms.