Learning from Samples: Inverse Problems over measures via Sharpened Fenchel-Young Losses

Estimating parameters from samples of an optimal probability distribution is essential in applications ranging from socio-economic modeling to biological system analysis. In these settings, the probability distribution arises as the solution to an optimization problem that captures either static interactions among agents or the dynamic evolution of a system over time. Our approach relies on minimizing a new class of loss functions, called sharpened Fenchel-Young losses, which measure the sub-optimality gap of the optimization problem over the space of measures. We study the stability of this estimation method when only a finite number of sample is available. The parameters to be estimated typically correspond to a cost function in static problems and to a potential function in dynamic problems. To analyze stability, we introduce a general methodology that leverages the strong convexity of the loss function together with the sample complexity of the forward optimization problem. Our analysis emphasizes two specific settings in the context of optimal transport, where our method provides explicit stability guarantees: The first is inverse unbalanced optimal transport (iUOT) with entropic regularization, where the parameters to estimate are cost functions that govern transport computations; this method has applications such as link prediction in machine learning. The second is inverse gradient flow (iJKO), where the objective is to recover a potential function that drives the evolution of a probability distribution via the Jordan-Kinderlehrer-Otto (JKO) time-discretization scheme; this is particularly relevant for understanding cell population dynamics in single-cell genomics. Finally, we validate our approach through numerical experiments on Gaussian distributions, where closed-form solutions are available, to demonstrate the practical performance of our methods

A Control Architecture with Online Predictive Planning for Position and Torque Controlled Walking of Humanoid Robots

A common approach to the generation of walking patterns for humanoid robots consists in adopting a layered control architecture. This paper proposes an architecture composed of three nested control loops. The outer loop exploits a robot kinematic model to plan the footstep positions. In the mid layer, a predictive controller generates a Center of Mass trajectory according to the well-known table-cart model. Through a whole-body inverse kinematics algorithm, we can define joint references for position controlled walking. The outcomes of these two loops are then interpreted as inputs of a stack-of-task QP-based torque controller, which represents the inner loop of the presented control architecture. This resulting architecture allows the robot to walk also in torque control, guaranteeing higher level of compliance. Real world experiments have been carried on the humanoid robot iCub.