Demand for high-performance, robust, and safe autonomous systems has grown substantially in recent years. These objectives motivate the desire for efficient safety-theoretic reasoning that can be embedded in core decision-making tasks such as motion planning, particularly in constrained environments. On one hand, Monte-Carlo (MC) and other sampling-based techniques provide accurate collision probability estimates for a wide variety of motion models but are cumbersome in the context of continuous optimization. On the other, "direct" approximations aim to compute (or upper-bound) the failure probability as a smooth function of the decision variables, and thus are convenient for optimization. However, existing direct approaches fundamentally assume discrete-time dynamics and can perform unpredictably when applied to continuous-time systems ubiquitous in the real world, often manifesting as severe conservatism. State-of-the-art attempts to address this within a conventional discrete-time framework require additional Gaussianity approximations that ultimately produce inconsistency of their own. In this paper we take a fundamentally different approach, deriving a risk approximation framework directly in continuous time and producing a lightweight estimate that actually converges as the underlying discretization is refined. Our approximation is shown to significantly outperform state-of-the-art techniques in replicating the MC estimate while maintaining the functional and computational benefits of a direct method. This enables robust, risk-aware, continuous motion-planning for a broad class of nonlinear and/or partially-observable systems.

Estimating a quantity of interest from finite measurements is a central problem in a number of fields including machine learning but also statistics and signal processing. In this context, a key idea is that reliable estimation requires imposing some prior assumptions on the problem at hand. The theory of inverse problems provides a principled framework to formalize this idea [27]. The quantity of interest is typically seen as a function, or a vector, and prior assumptions take the form of suitable functionals, called regularizers. Following this idea, Tikhonov regularization provides a classic approach to estimate solutions [83, 84]. Indeed, the latter are found by minimizing an empirical objective where a data fit term is penalized adding the chosen regularizer. Other regularization approaches are classic in inverse problems, and in particular iterative regularization has become popular in machine learning, see e.g.

Abstract: In recent years there has been growing attention to interpretable machine learning models which can give explanatory insights on their behavior. Thanks to their interpretability, decision trees have been intensively studied for classification tasks, and due to the remarkable advances in mixed-integer programming (MIP), various approaches have been proposed to formulate the problem of training an Optimal Classification Tree (OCT) as a MIP model. We present a novel mixed-integer quadratic formulation for the OCT problem, which exploits the generalization capabilities of Support Vector Machines for binary classification. Our model, denoted as Maximum Margin Optimal Classification Tree (MARGOT), encompasses the use of maximum margin multivariate hyperplanes nested in a binary tree structure. To enhance the interpretability of our approach, we analyse two alternative versions of MARGOT, which include feature selection constraints inducing local sparsity of the hyperplanes.

Decision-making problems are commonly formulated as optimization problems, which are then solved to make optimal decisions. In this work, we consider the inverse problem where we use prior decision data to uncover the underlying decision-making process in the form of a mathematical optimization model. This statistical learning problem is referred to as data-driven inverse optimization. We focus on problems where the underlying decision-making process is modeled as a convex optimization problem whose parameters are unknown. We formulate the inverse optimization problem as a bilevel program and propose an efficient block coordinate descent-based algorithm to solve large problem instances. Numerical experiments on synthetic datasets demonstrate the computational advantage of our method compared to standard commercial solvers. Moreover, the real-world utility of the proposed approach is highlighted through two realistic case studies in which we consider estimating risk preferences and learning local constraint parameters of agents in a multiplayer Nash bargaining game.

Optimally developing an oil and gas field requires predicting future production using a reservoir model, whose key material properties are tuned in a process called history matching. This process of adjusting the key parameters is non-unique and computationally challenging. Typically, the reservoir model is divided into cells that match the geology of the field. The key properties of these cells, such as porosity and permeability, are assigned initially using core sample data, where available. For computational efficiency, the geological model is converted to a reservoir model using upscaling [6, 20, 49] to reduce the number of the cells in the model. Due to the challenges of finding the key properties in each cell, history matching is used to adjust the values of these properties so the model reflects historical production data [19, 28, 9]. History matching is typically done by matching the computed pressure and saturation data (oil, gas and water rates) from the simulation model and comparing it the actual historical data. The difference between the actual data and data generated by the reservoir model is then computed using an objective function that quantifies the mismatch between the two quantities.

We introduce MuJoCo MPC (MJPC), an open-source, interactive application and software framework for real-time predictive control, based on MuJoCo physics. MJPC allows the user to easily author and solve complex robotics tasks, and currently supports three shooting-based planners: derivative-based iLQG and Gradient Descent, and a simple derivative-free method we call Predictive Sampling. Predictive Sampling was designed as an elementary baseline, mostly for its pedagogical value, but turned out to be surprisingly competitive with the more established algorithms. This work does not present algorithmic advances, and instead, prioritises performant algorithms, simple code, and accessibility of model-based methods via intuitive and interactive software. MJPC is available at: github.com/deepmind/mujoco_mpc, a video summary can be viewed at: dpmd.ai/mjpc.

Profile extrusion is a continuous production process for manufacturing plastic profiles from molten polymer. Especially interesting is the design of the die, through which the melt is pressed to attain the desired shape. However, due to an inhomogeneous velocity distribution at the die exit or residual stresses inside the extrudate, the final shape of the manufactured part often deviates from the desired one. To avoid these deviations, the shape of the die can be computationally optimized, which has already been investigated in the literature using classical optimization approaches. A new approach in the field of shape optimization is the utilization of Reinforcement Learning (RL) as a learning-based optimization algorithm. RL is based on trial-and-error interactions of an agent with an environment. For each action, the agent is rewarded and informed about the subsequent state of the environment. While not necessarily superior to classical, e.g., gradient-based or evolutionary, optimization algorithms for one single problem, RL techniques are expected to perform especially well when similar optimization tasks are repeated since the agent learns a more general strategy for generating optimal shapes instead of concentrating on just one single problem. In this work, we investigate this approach by applying it to two 2D test cases. The flow-channel geometry can be modified by the RL agent using so-called Free-Form Deformation, a method where the computational mesh is embedded into a transformation spline, which is then manipulated based on the control-point positions. In particular, we investigate the impact of utilizing different agents on the training progress and the potential of wall time saving by utilizing multiple environments during training.

In this work, we propose ModelPred, a framework that helps to understand the impact of changes in training data on a trained model. This is critical for building trust in various stages of a machine learning pipeline: from cleaning poor-quality samples and tracking important ones to be collected during data preparation, to calibrating uncertainty of model prediction, to interpreting why certain behaviors of a model emerge during deployment. Specifically, ModelPred learns a parameterized function that takes a dataset $S$ as the input and predicts the model obtained by training on $S$. Our work differs from the recent work of Datamodels [1] as we aim for predicting the trained model parameters directly instead of the trained model behaviors. We demonstrate that a neural network-based set function class is capable of learning the complex relationships between the training data and model parameters. We introduce novel global and local regularization techniques to prevent overfitting and we rigorously characterize the expressive power of neural networks (NN) in approximating the end-to-end training process. Through extensive empirical investigations, we show that ModelPred enables a variety of applications that boost the interpretability and accountability of machine learning (ML), such as data valuation, data selection, memorization quantification, and model calibration.

This paper considers a combination of actuation tendons and measurement strings to achieve accurate shape sensing and direct kinematics of continuum robots. Assuming general string routing, a methodical Lie group formulation for the shape sensing of these robots is presented. The shape kinematics is expressed using arc-length-dependent curvature distributions parameterized by modal functions, and the Magnus expansion for Lie group integration is used to express the shape as a product of exponentials. The tendon and string length kinematic constraints are solved for the modal coefficients and the configuration space and body Jacobian are derived. The noise amplification index for the shape reconstruction problem is defined and used for optimizing the string/tendon routing paths, and a planar simulation study shows the minimal number of strings/tendons needed for accurate shape reconstruction. A torsionally stiff continuum segment is used for experimental evaluation, demonstrating mean (maximal) end-effector absolute position error of less than 2% (5%) of total length. Finally, a simulation study of a torsionally compliant segment demonstrates the approach for general deflections and string routings. We believe that the methods of this paper can benefit the design process, sensing and control of continuum and soft robots.

Image classification with small datasets has been an active research area in the recent past. However, as research in this scope is still in its infancy, two key ingredients are missing for ensuring reliable and truthful progress: a systematic and extensive overview of the state of the art, and a common benchmark to allow for objective comparisons between published methods. This article addresses both issues. First, we systematically organize and connect past studies to consolidate a community that is currently fragmented and scattered. Second, we propose a common benchmark that allows for an objective comparison of approaches. It consists of five datasets spanning various domains (e.g., natural images, medical imagery, satellite data) and data types (RGB, grayscale, multispectral). We use this benchmark to re-evaluate the standard cross-entropy baseline and ten existing methods published between 2017 and 2021 at renowned venues. Surprisingly, we find that thorough hyper-parameter tuning on held-out validation data results in a highly competitive baseline and highlights a stunted growth of performance over the years. Indeed, only a single specialized method dating back to 2019 clearly wins our benchmark and outperforms the baseline classifier.