Neural reproducing kernel Banach spaces and representer theorems for deep networks

Studying the function spaces defined by neural networks helps to understand the corresponding learning models and their inductive bias. While in some limits neural networks correspond to function spaces that are reproducing kernel Hilbert spaces, these regimes do not capture the properties of the networks used in practice. In contrast, in this paper we show that deep neural networks define suitable reproducing kernel Banach spaces. These spaces are equipped with norms that enforce a form of sparsity, enabling them to adapt to potential latent structures within the input data and their representations. In particular, leveraging the theory of reproducing kernel Banach spaces, combined with variational results, we derive representer theorems that justify the finite architectures commonly employed in applications. Our study extends analogous results for shallow networks and can be seen as a step towards considering more practically plausible neural architectures.

Linear quadratic control of nonlinear systems with Koopman operator learning and the Nyström method

In this paper, we study how the Koopman operator framework can be combined with kernel methods to effectively control nonlinear dynamical systems. While kernel methods have typically large computational requirements, we show how random subspaces (Nystr\"om approximation) can be used to achieve huge computational savings while preserving accuracy. Our main technical contribution is deriving theoretical guarantees on the effect of the Nystr\"om approximation. More precisely, we study the linear quadratic regulator problem, showing that both the approximated Riccati operator and the regulator objective, for the associated solution of the optimal control problem, converge at the rate $m^{-1/2}$, where $m$ is the random subspace size. Theoretical findings are complemented by numerical experiments corroborating our results.

Key Design Choices in Source-Free Unsupervised Domain Adaptation: An In-depth Empirical Analysis

This study provides a comprehensive benchmark framework for Source-Free Unsupervised Domain Adaptation (SF-UDA) in image classification, aiming to achieve a rigorous empirical understanding of the complex relationships between multiple key design factors in SF-UDA methods. The study empirically examines a diverse set of SF-UDA techniques, assessing their consistency across datasets, sensitivity to specific hyperparameters, and applicability across different families of backbone architectures. Moreover, it exhaustively evaluates pre-training datasets and strategies, particularly focusing on both supervised and self-supervised methods, as well as the impact of fine-tuning on the source domain. Our analysis also highlights gaps in existing benchmark practices, guiding SF-UDA research towards more effective and general approaches. It emphasizes the importance of backbone architecture and pre-training dataset selection on SF-UDA performance, serving as an essential reference and providing key insights. Lastly, we release the source code of our experimental framework. This facilitates the construction, training, and testing of SF-UDA methods, enabling systematic large-scale experimental analysis and supporting further research efforts in this field.

RESPRECT: Speeding-up Multi-fingered Grasping with Residual Reinforcement Learning

Deep Reinforcement Learning (DRL) has proven effective in learning control policies using robotic grippers, but much less practical for solving the problem of grasping with dexterous hands -- especially on real robotic platforms -- due to the high dimensionality of the problem. In this work, we focus on the multi-fingered grasping task with the anthropomorphic hand of the iCub humanoid. We propose the RESidual learning with PREtrained CriTics (RESPRECT) method that, starting from a policy pre-trained on a large set of objects, can learn a residual policy to grasp a novel object in a fraction ($\sim 5 \times$ faster) of the timesteps required to train a policy from scratch, without requiring any task demonstration. To our knowledge, this is the first Residual Reinforcement Learning (RRL) approach that learns a residual policy on top of another policy pre-trained with DRL. We exploit some components of the pre-trained policy during residual learning that further speed-up the training. We benchmark our results in the iCub simulated environment, and we show that RESPRECT can be effectively used to learn a multi-fingered grasping policy on the real iCub robot. The code to reproduce the experiments is released together with the paper with an open source license.

Learning Rhythmic Trajectories with Geometric Constraints for Laser-Based Skincare Procedures

The increasing deployment of robots has significantly enhanced the automation levels across a wide and diverse range of industries. This paper investigates the automation challenges of laser-based dermatology procedures in the beauty industry; This group of related manipulation tasks involves delivering energy from a cosmetic laser onto the skin with repetitive patterns. To automate this procedure, we propose to use a robotic manipulator and endow it with the dexterity of a skilled dermatology practitioner through a learning-from-demonstration framework. To ensure that the cosmetic laser can properly deliver the energy onto the skin surface of an individual, we develop a novel structured prediction-based imitation learning algorithm with the merit of handling geometric constraints. Notably, our proposed algorithm effectively tackles the imitation challenges associated with quasi-periodic motions, a common feature of many laser-based cosmetic tasks. The conducted real-world experiments illustrate the performance of our robotic beautician in mimicking realistic dermatological procedures; Our new method is shown to not only replicate the rhythmic movements from the provided demonstrations but also to adapt the acquired skills to previously unseen scenarios and subjects.

Efficient Numerical Integration in Reproducing Kernel Hilbert Spaces via Leverage Scores Sampling

In this work we consider the problem of numerical integration, i.e., approximating integrals with respect to a target probability measure using only pointwise evaluations of the integrand. We focus on the setting in which the target distribution is only accessible through a set of $n$ i.i.d. observations, and the integrand belongs to a reproducing kernel Hilbert space. We propose an efficient procedure which exploits a small i.i.d. random subset of $m<n$ samples drawn either uniformly or using approximate leverage scores from the initial observations. Our main result is an upper bound on the approximation error of this procedure for both sampling strategies. It yields sufficient conditions on the subsample size to recover the standard (optimal) $n^{-1/2}$ rate while reducing drastically the number of functions evaluations, and thus the overall computational cost. Moreover, we obtain rates with respect to the number $m$ of evaluations of the integrand which adapt to its smoothness, and match known optimal rates for instance for Sobolev spaces. We illustrate our theoretical findings with numerical experiments on real datasets, which highlight the attractive efficiency-accuracy tradeoff of our method compared to existing randomized and greedy quadrature methods. We note that, the problem of numerical integration in RKHS amounts to designing a discrete approximation of the kernel mean embedding of the target distribution. As a consequence, direct applications of our results also include the efficient computation of maximum mean discrepancies between distributions and the design of efficient kernel-based tests.

Sim2Real Bilevel Adaptation for Object Surface Classification using Vision-Based Tactile Sensors

In this paper, we address the Sim2Real gap in the field of vision-based tactile sensors for classifying object surfaces. We train a Diffusion Model to bridge this gap using a relatively small dataset of real-world images randomly collected from unlabeled everyday objects via the DIGIT sensor. Subsequently, we employ a simulator to generate images by uniformly sampling the surface of objects from the YCB Model Set. These simulated images are then translated into the real domain using the Diffusion Model and automatically labeled to train a classifier. During this training, we further align features of the two domains using an adversarial procedure. Our evaluation is conducted on a dataset of tactile images obtained from a set of ten 3D printed YCB objects. The results reveal a total accuracy of 81.9%, a significant improvement compared to the 34.7% achieved by the classifier trained solely on simulated images. This demonstrates the effectiveness of our approach. We further validate our approach using the classifier on a 6D object pose estimation task from tactile data.

Shortcuts for causal discovery of nonlinear models by score matching

The use of simulated data in the field of causal discovery is ubiquitous due to the scarcity of annotated real data. Recently, Reisach et al., 2021 highlighted the emergence of patterns in simulated linear data, which displays increasing marginal variance in the casual direction. As an ablation in their experiments, Montagna et al., 2023 found that similar patterns may emerge in nonlinear models for the variance of the score vector $\nabla \log p_{\mathbf{X}}$, and introduced the ScoreSort algorithm. In this work, we formally define and characterize this score-sortability pattern of nonlinear additive noise models. We find that it defines a class of identifiable (bivariate) causal models overlapping with nonlinear additive noise models. We theoretically demonstrate the advantages of ScoreSort in terms of statistical efficiency compared to prior state-of-the-art score matching-based methods and empirically show the score-sortability of the most common synthetic benchmarks in the literature. Our findings remark (1) the lack of diversity in the data as an important limitation in the evaluation of nonlinear causal discovery approaches, (2) the importance of thoroughly testing different settings within a problem class, and (3) the importance of analyzing statistical properties in causal discovery, where research is often limited to defining identifiability conditions of the model.

Assumption violations in causal discovery and the robustness of score matching

When domain knowledge is limited and experimentation is restricted by ethical, financial, or time constraints, practitioners turn to observational causal discovery methods to recover the causal structure, exploiting the statistical properties of their data. Because causal discovery without further assumptions is an ill-posed problem, each algorithm comes with its own set of usually untestable assumptions, some of which are hard to meet in real datasets. Motivated by these considerations, this paper extensively benchmarks the empirical performance of recent causal discovery methods on observational i.i.d. data generated under different background conditions, allowing for violations of the critical assumptions required by each selected approach. Our experimental findings show that score matching-based methods demonstrate surprising performance in the false positive and false negative rate of the inferred graph in these challenging scenarios, and we provide theoretical insights into their performance. This work is also the first effort to benchmark the stability of causal discovery algorithms with respect to the values of their hyperparameters. Finally, we hope this paper will set a new standard for the evaluation of causal discovery methods and can serve as an accessible entry point for practitioners interested in the field, highlighting the empirical implications of different algorithm choices.

A Structured Prediction Approach for Robot Imitation Learning

We propose a structured prediction approach for robot imitation learning from demonstrations. Among various tools for robot imitation learning, supervised learning has been observed to have a prominent role. Structured prediction is a form of supervised learning that enables learning models to operate on output spaces with complex structures. Through the lens of structured prediction, we show how robots can learn to imitate trajectories belonging to not only Euclidean spaces but also Riemannian manifolds. Exploiting ideas from information theory, we propose a class of loss functions based on the f-divergence to measure the information loss between the demonstrated and reproduced probabilistic trajectories. Different types of f-divergence will result in different policies, which we call imitation modes. Furthermore, our approach enables the incorporation of spatial and temporal trajectory modulation, which is necessary for robots to be adaptive to the change in working conditions. We benchmark our algorithm against state-of-the-art methods in terms of trajectory reproduction and adaptation. The quantitative evaluation shows that our approach outperforms other algorithms regarding both accuracy and efficiency. We also report real-world experimental results on learning manifold trajectories in a polishing task with a KUKA LWR robot arm, illustrating the effectiveness of our algorithmic framework.