arXiv subjects

Massimiliano Pontil

Publications and source records attributed to Massimiliano Pontil.

At least 37 records · Page 2Linked to original sources

arXiv2025

DeltaProduct: Improving State-Tracking in Linear RNNs via Householder Products

Linear Recurrent Neural Networks (linear RNNs) have emerged as competitive alternatives to Transformers for sequence modeling, offering efficient training and linear-time inference. However, existing architectures face a fundamental trade-off between expressivity and efficiency, dictated by the structure of their state-transition matrices. Diagonal matrices, used in models such as Mamba, GLA, or mLSTM, yield fast runtime but have limited expressivity. To address this, recent architectures such as DeltaNet and RWKV-7 adopted a diagonal plus rank--1 structure, which allows simultaneous token and channel mixing, improving associative recall and, as recently shown, state-tracking when allowing state-transition matrices to have negative eigenvalues. Building on the interpretation of DeltaNet's recurrence as performing one step of online gradient descent per token on an associative recall loss, we introduce DeltaProduct, which instead takes multiple ($n_h$) steps per token. This naturally leads to diagonal plus rank--$n_h$ state-transition matrices, formed as products of $n_h$ generalized Householder transformations, providing a tunable mechanism to balance expressivity and efficiency. We provide a detailed theoretical characterization of the state-tracking capability of DeltaProduct in finite precision, showing how it improves by increasing $n_h$. Our extensive experiments demonstrate that DeltaProduct outperforms DeltaNet in both state-tracking and language modeling, while also showing significantly improved length extrapolation capabilities.

cs.LG↗

arXiv2025

The seeds of the future are in the present: A blind exploration of metastable states

In this work, we present a novel type of molecular dynamics simulation that aims at discovering, in a blind way, new metastable states. Using only data coming from an initial unbiased simulation, and with the help of an appropriately defined loss function, we compute a bias that favors sampling yet unexplored configurational space regions, encouraging the system to leave the initial basin. In our work, we take advantage of what is normally thought to be a defect, namely the difficulty of neural networks to generalize. Contrary to most other enhanced sampling methods, which need previous knowledge of the reactive process, we are able to discover in a blind way new metastable states, overcoming otherwise insuperable kinetic bottlenecks. We illustrate the workings of the method with a number of instructive examples.

physics.chem-ph↗

arXiv2025

An Empirical Bernstein Inequality for Dependent Data in Hilbert Spaces and Applications

Learning from non-independent and non-identically distributed data poses a persistent challenge in statistical learning. In this study, we introduce data-dependent Bernstein inequalities tailored for vector-valued processes in Hilbert space. Our inequalities apply to both stationary and non-stationary processes and exploit the potential rapid decay of correlations between temporally separated variables to improve estimation. We demonstrate the utility of these bounds by applying them to covariance operator estimation in the Hilbert-Schmidt norm and to operator learning in dynamical systems, achieving novel risk bounds. Finally, we perform numerical experiments to illustrate the practical implications of these bounds in both contexts.

cs.LG↗

arXiv2025

Laplace Transform Based Low-Complexity Learning of Continuous Markov Semigroups

Markov processes serve as a universal model for many real-world random processes. This paper presents a data-driven approach for learning these models through the spectral decomposition of the infinitesimal generator (IG) of the Markov semigroup. The unbounded nature of IGs complicates traditional methods such as vector-valued regression and Hilbert-Schmidt operator analysis. Existing techniques, including physics-informed kernel regression, are computationally expensive and limited in scope, with no recovery guarantees for transfer operator methods when the time-lag is small. We propose a novel method that leverages the IG's resolvent, characterized by the Laplace transform of transfer operators. This approach is robust to time-lag variations, ensuring accurate eigenvalue learning even for small time-lags. Our statistical analysis applies to a broader class of Markov processes than current methods while reducing computational complexity from quadratic to linear in the state dimension. Finally, we illustrate the behaviour of our method in two experiments.

cs.LG↗

arXiv2025

Neural Conditional Probability for Uncertainty Quantification

We introduce Neural Conditional Probability (NCP), an operator-theoretic approach to learning conditional distributions with a focus on statistical inference tasks. NCP can be used to build conditional confidence regions and extract key statistics such as conditional quantiles, mean, and covariance. It offers streamlined learning via a single unconditional training phase, allowing efficient inference without the need for retraining even when conditioning changes. By leveraging the approximation capabilities of neural networks, NCP efficiently handles a wide variety of complex probability distributions. We provide theoretical guarantees that ensure both optimization consistency and statistical accuracy. In experiments, we show that NCP with a 2-hidden-layer network matches or outperforms leading methods. This demonstrates that a a minimalistic architecture with a theoretically grounded loss can achieve competitive results, even in the face of more complex architectures.

cs.LG↗

arXiv2025

Self-Supervised Evolution Operator Learning for High-Dimensional Dynamical Systems

We introduce an encoder-only approach to learn the evolution operators of large-scale non-linear dynamical systems, such as those describing complex natural phenomena. Evolution operators are particularly well-suited for analyzing systems that exhibit complex spatio-temporal patterns and have become a key analytical tool across various scientific communities. As terabyte-scale weather datasets and simulation tools capable of running millions of molecular dynamics steps per day are becoming commodities, our approach provides an effective tool to make sense of them from a data-driven perspective. The core of it lies in a remarkable connection between self-supervised representation learning methods and the recently established learning theory of evolution operators. To show the usefulness of the proposed method, we test it across multiple scientific domains: explaining the folding dynamics of small proteins, the binding process of drug-like molecules in host sites, and autonomously finding patterns in climate data. Code and data to reproduce the experiments are made available open source.

cs.LG↗

arXiv2025

Convergence Properties of Stochastic Hypergradients

Bilevel optimization problems are receiving increasing attention in machine learning as they provide a natural framework for hyperparameter optimization and meta-learning. A key step to tackle these problems is the efficient computation of the gradient of the upper-level objective (hypergradient). In this work, we study stochastic approximation schemes for the hypergradient, which are important when the lower-level problem is empirical risk minimization on a large dataset. The method that we propose is a stochastic variant of the approximate implicit differentiation approach in (Pedregosa, 2016). We provide bounds for the mean square error of the hypergradient approximation, under the assumption that the lower-level problem is accessible only through a stochastic mapping which is a contraction in expectation. In particular, our main bound is agnostic to the choice of the two stochastic solvers employed by the procedure. We provide numerical experiments to support our theoretical analysis and to show the advantage of using stochastic hypergradients in practice.

stat.ML↗

arXiv2025

Fast and Fourier Features for Transfer Learning of Interatomic Potentials

Training machine learning interatomic potentials that are both computationally and data-efficient is a key challenge for enabling their routine use in atomistic simulations. To this effect, we introduce franken, a scalable and lightweight transfer learning framework that extracts atomic descriptors from pretrained graph neural networks and transfer them to new systems using random Fourier features-an efficient and scalable approximation of kernel methods. Franken enables fast and accurate adaptation of general-purpose potentials to new systems or levels of quantum mechanical theory without requiring hyperparameter tuning or architectural modifications. On a benchmark dataset of 27 transition metals, franken outperforms optimized kernel-based methods in both training time and accuracy, reducing model training from tens of hours to minutes on a single GPU. We further demonstrate the framework's strong data-efficiency by training stable and accurate potentials for bulk water and the Pt(111)/water interface using just tens of training structures. Our open-source implementation (https://franken.readthedocs.io) offers a fast and practical solution for training potentials and deploying them for molecular dynamics simulations across diverse systems.

physics.comp-ph↗

arXiv2025

Morphological Symmetries in Robotics

We present a comprehensive framework for studying and leveraging morphological symmetries in robotic systems. These are intrinsic properties of the robot's morphology, frequently observed in animal biology and robotics, which stem from the replication of kinematic structures and the symmetrical distribution of mass. We illustrate how these symmetries extend to the robot's state space and both proprioceptive and exteroceptive sensor measurements, resulting in the equivariance of the robot's equations of motion and optimal control policies. Thus, we recognize morphological symmetries as a relevant and previously unexplored physics-informed geometric prior, with significant implications for both data-driven and analytical methods used in modeling, control, estimation and design in robotics. For data-driven methods, we demonstrate that morphological symmetries can enhance the sample efficiency and generalization of machine learning models through data augmentation, or by applying equivariant/invariant constraints on the model's architecture. In the context of analytical methods, we employ abstract harmonic analysis to decompose the robot's dynamics into a superposition of lower-dimensional, independent dynamics. We substantiate our claims with both synthetic and real-world experiments conducted on bipedal and quadrupedal robots. Lastly, we introduce the repository MorphoSymm to facilitate the practical use of the theory and applications outlined in this work.

cs.RO↗

arXiv2025

Unlocking State-Tracking in Linear RNNs Through Negative Eigenvalues

Linear Recurrent Neural Networks (LRNNs) such as Mamba, RWKV, GLA, mLSTM, and DeltaNet have emerged as efficient alternatives to Transformers for long sequences. However, both Transformers and LRNNs struggle to perform state-tracking, which may impair performance in tasks such as code evaluation. In one forward pass, current architectures are unable to solve even parity, the simplest state-tracking task, which non-linear RNNs can handle effectively. Recently, Sarrof et al. (2024) demonstrated that the failure of LRNNs like Mamba to solve parity stems from restricting the value range of their diagonal state-transition matrices to $[0, 1]$ and that incorporating negative values can resolve this issue. We extend this result to non-diagonal LRNNs such as DeltaNet. We prove that finite precision LRNNs with state-transition matrices having only positive eigenvalues cannot solve parity, while non-triangular matrices are needed to count modulo $3$. Notably, we also prove that LRNNs can learn any regular language when their state-transition matrices are products of identity minus vector outer product matrices, each with eigenvalues in the range $[-1, 1]$. Our experiments confirm that extending the eigenvalue range of Mamba and DeltaNet to include negative values not only enables them to solve parity but consistently improves their performance on state-tracking tasks. We also show that state-tracking enabled LRNNs can be pretrained stably and efficiently at scale (1.3B parameters), achieving competitive performance on language modeling and showing promise on code and math tasks.

cs.LG↗

arXiv2025

Leveraging Symmetry in RL-based Legged Locomotion Control

Model-free reinforcement learning is a promising approach for autonomously solving challenging robotics control problems, but faces exploration difficulty without information of the robot's kinematics and dynamics morphology. The under-exploration of multiple modalities with symmetric states leads to behaviors that are often unnatural and sub-optimal. This issue becomes particularly pronounced in the context of robotic systems with morphological symmetries, such as legged robots for which the resulting asymmetric and aperiodic behaviors compromise performance, robustness, and transferability to real hardware. To mitigate this challenge, we can leverage symmetry to guide and improve the exploration in policy learning via equivariance/invariance constraints. In this paper, we investigate the efficacy of two approaches to incorporate symmetry: modifying the network architectures to be strictly equivariant/invariant, and leveraging data augmentation to approximate equivariant/invariant actor-critics. We implement the methods on challenging loco-manipulation and bipedal locomotion tasks and compare with an unconstrained baseline. We find that the strictly equivariant policy consistently outperforms other methods in sample efficiency and task performance in simulation. In addition, symmetry-incorporated approaches exhibit better gait quality, higher robustness and can be deployed zero-shot in real-world experiments.

cs.RO↗

arXiv2024

From Biased to Unbiased Dynamics: An Infinitesimal Generator Approach

We investigate learning the eigenfunctions of evolution operators for time-reversal invariant stochastic processes, a prime example being the Langevin equation used in molecular dynamics. Many physical or chemical processes described by this equation involve transitions between metastable states separated by high potential barriers that can hardly be crossed during a simulation. To overcome this bottleneck, data are collected via biased simulations that explore the state space more rapidly. We propose a framework for learning from biased simulations rooted in the infinitesimal generator of the process and the associated resolvent operator. We contrast our approach to more common ones based on the transfer operator, showing that it can provably learn the spectral properties of the unbiased system from biased data. In experiments, we highlight the advantages of our method over transfer operator approaches and recent developments based on generator learning, demonstrating its effectiveness in estimating eigenfunctions and eigenvalues. Importantly, we show that even with datasets containing only a few relevant transitions due to sub-optimal biasing, our approach recovers relevant information about the transition mechanism.

cs.LG↗

arXiv2024

Adaptive AI-Driven Material Synthesis: Towards Autonomous 2D Materials Growth

Two-dimensional (2D) materials are poised to revolutionize current solid-state technology with their extraordinary properties. Yet, the primary challenge remains their scalable production. While there have been significant advancements, much of the scientific progress has depended on the exfoliation of materials, a method that poses severe challenges for large-scale applications. With the advent of artificial intelligence (AI) in materials science, innovative synthesis methodologies are now on the horizon. This study explores the forefront of autonomous materials synthesis using an artificial neural network (ANN) trained by evolutionary methods, focusing on the efficient production of graphene. Our approach demonstrates that a neural network can iteratively and autonomously learn a time-dependent protocol for the efficient growth of graphene, without requiring pretraining on what constitutes an effective recipe. Evaluation criteria are based on the proximity of the Raman signature to that of monolayer graphene: higher scores are granted to outcomes whose spectrum more closely resembles that of an ideal continuous monolayer structure. This feedback mechanism allows for iterative refinement of the ANN's time-dependent synthesis protocols, progressively improving sample quality. Through the advancement and application of AI methodologies, this work makes a substantial contribution to the field of materials engineering, fostering a new era of innovation and efficiency in the synthesis process.

cond-mat.mes-hall↗

arXiv2024

Operator World Models for Reinforcement Learning

Policy Mirror Descent (PMD) is a powerful and theoretically sound methodology for sequential decision-making. However, it is not directly applicable to Reinforcement Learning (RL) due to the inaccessibility of explicit action-value functions. We address this challenge by introducing a novel approach based on learning a world model of the environment using conditional mean embeddings. Leveraging tools from operator theory we derive a closed-form expression of the action-value function in terms of the world model via simple matrix operations. Combining these estimators with PMD leads to POWR, a new RL algorithm for which we prove convergence rates to the global optimum. Preliminary experiments in finite and infinite state settings support the effectiveness of our method

cs.LG↗

arXiv2024

Dynamics Harmonic Analysis of Robotic Systems: Application in Data-Driven Koopman Modelling

We introduce the use of harmonic analysis to decompose the state space of symmetric robotic systems into orthogonal isotypic subspaces. These are lower-dimensional spaces that capture distinct, symmetric, and synergistic motions. For linear dynamics, we characterize how this decomposition leads to a subdivision of the dynamics into independent linear systems on each subspace, a property we term dynamics harmonic analysis (DHA). To exploit this property, we use Koopman operator theory to propose an equivariant deep-learning architecture that leverages the properties of DHA to learn a global linear model of the system dynamics. Our architecture, validated on synthetic systems and the dynamics of locomotion of a quadrupedal robot, exhibits enhanced generalization, sample efficiency, and interpretability, with fewer trainable parameters and computational costs.

cs.RO↗

arXiv2024

Nonsmooth Implicit Differentiation: Deterministic and Stochastic Convergence Rates

We study the problem of efficiently computing the derivative of the fixed-point of a parametric nondifferentiable contraction map. This problem has wide applications in machine learning, including hyperparameter optimization, meta-learning and data poisoning attacks. We analyze two popular approaches: iterative differentiation (ITD) and approximate implicit differentiation (AID). A key challenge behind the nonsmooth setting is that the chain rule does not hold anymore. We build upon the work by Bolte et al. (2022), who prove linear convergence of nonsmooth ITD under a piecewise Lipschitz smooth assumption. In the deterministic case, we provide a linear rate for AID and an improved linear rate for ITD which closely match the ones for the smooth setting. We further introduce NSID, a new stochastic method to compute the implicit derivative when the contraction map is defined as the composition of an outer map and an inner map which is accessible only through a stochastic unbiased estimator. We establish rates for the convergence of NSID, encompassing the best available rates in the smooth setting. We also present illustrative experiments confirming our analysis.

stat.ML↗

arXiv2024

Learning the Infinitesimal Generator of Stochastic Diffusion Processes

We address data-driven learning of the infinitesimal generator of stochastic diffusion processes, essential for understanding numerical simulations of natural and physical systems. The unbounded nature of the generator poses significant challenges, rendering conventional analysis techniques for Hilbert-Schmidt operators ineffective. To overcome this, we introduce a novel framework based on the energy functional for these stochastic processes. Our approach integrates physical priors through an energy-based risk metric in both full and partial knowledge settings. We evaluate the statistical performance of a reduced-rank estimator in reproducing kernel Hilbert spaces (RKHS) in the partial knowledge setting. Notably, our approach provides learning bounds independent of the state space dimension and ensures non-spurious spectral estimation. Additionally, we elucidate how the distortion between the intrinsic energy-induced metric of the stochastic diffusion and the RKHS metric used for generator estimation impacts the spectral learning bounds.

stat.ML↗

arXiv2024

High Probability Bounds for Stochastic Subgradient Schemes with Heavy Tailed Noise

In this work we study high probability bounds for stochastic subgradient methods under heavy tailed noise. In this setting the noise is only assumed to have finite variance as opposed to a sub-Gaussian distribution for which it is known that standard subgradient methods enjoys high probability bounds. We analyzed a clipped version of the projected stochastic subgradient method, where subgradient estimates are truncated whenever they have large norms. We show that this clipping strategy leads both to near optimal any-time and finite horizon bounds for many classical averaging schemes. Preliminary experiments are shown to support the validity of the method.

math.OC↗