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2020


Label Efficient Visual Abstractions for Autonomous Driving
Label Efficient Visual Abstractions for Autonomous Driving

Behl, A., Chitta, K., Prakash, A., Ohn-Bar, E., Geiger, A.

IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), IEEE, October 2020 (conference)

Abstract
It is well known that semantic segmentation can be used as an effective intermediate representation for learning driving policies. However, the task of street scene semantic segmentation requires expensive annotations. Furthermore, segmentation algorithms are often trained irrespective of the actual driving task, using auxiliary image-space loss functions which are not guaranteed to maximize driving metrics such as safety or distance traveled per intervention. In this work, we seek to quantify the impact of reducing segmentation annotation costs on learned behavior cloning agents. We analyze several segmentation-based intermediate representations. We use these visual abstractions to systematically study the trade-off between annotation efficiency and driving performance, ie, the types of classes labeled, the number of image samples used to learn the visual abstraction model, and their granularity (eg, object masks vs. 2D bounding boxes). Our analysis uncovers several practical insights into how segmentation-based visual abstractions can be exploited in a more label efficient manner. Surprisingly, we find that state-of-the-art driving performance can be achieved with orders of magnitude reduction in annotation cost. Beyond label efficiency, we find several additional training benefits when leveraging visual abstractions, such as a significant reduction in the variance of the learned policy when compared to state-of-the-art end-to-end driving models.

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pdf slides video Project Page [BibTex]

2020


pdf slides video Project Page [BibTex]


Convolutional Occupancy Networks
Convolutional Occupancy Networks

Peng, S., Niemeyer, M., Mescheder, L., Pollefeys, M., Geiger, A.

In European Conference on Computer Vision (ECCV), Springer International Publishing, Cham, August 2020 (inproceedings)

Abstract
Recently, implicit neural representations have gained popularity for learning-based 3D reconstruction. While demonstrating promising results, most implicit approaches are limited to comparably simple geometry of single objects and do not scale to more complicated or large-scale scenes. The key limiting factor of implicit methods is their simple fully-connected network architecture which does not allow for integrating local information in the observations or incorporating inductive biases such as translational equivariance. In this paper, we propose Convolutional Occupancy Networks, a more flexible implicit representation for detailed reconstruction of objects and 3D scenes. By combining convolutional encoders with implicit occupancy decoders, our model incorporates inductive biases, enabling structured reasoning in 3D space. We investigate the effectiveness of the proposed representation by reconstructing complex geometry from noisy point clouds and low-resolution voxel representations. We empirically find that our method enables the fine-grained implicit 3D reconstruction of single objects, scales to large indoor scenes, and generalizes well from synthetic to real data.

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pdf suppmat video Project Page [BibTex]

pdf suppmat video Project Page [BibTex]


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Model-Agnostic Counterfactual Explanations for Consequential Decisions

Karimi, A., Barthe, G., Balle, B., Valera, I.

Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), 108, pages: 895-905, Proceedings of Machine Learning Research, (Editors: Silvia Chiappa and Roberto Calandra), PMLR, August 2020 (conference)

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arXiv link (url) [BibTex]

arXiv link (url) [BibTex]


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Deep Graph Matching via Blackbox Differentiation of Combinatorial Solvers

Rolinek, M., Swoboda, P., Zietlow, D., Paulus, A., Musil, V., Martius, G.

In Computer Vision – ECCV 2020, Springer International Publishing, Cham, August 2020 (inproceedings)

Abstract
Building on recent progress at the intersection of combinatorial optimization and deep learning, we propose an end-to-end trainable architecture for deep graph matching that contains unmodified combinatorial solvers. Using the presence of heavily optimized combinatorial solvers together with some improvements in architecture design, we advance state-of-the-art on deep graph matching benchmarks for keypoint correspondence. In addition, we highlight the conceptual advantages of incorporating solvers into deep learning architectures, such as the possibility of post-processing with a strong multi-graph matching solver or the indifference to changes in the training setting. Finally, we propose two new challenging experimental setups.

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Code Arxiv [BibTex]

Code Arxiv [BibTex]


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Fair Decisions Despite Imperfect Predictions

Kilbertus, N., Gomez Rodriguez, M., Schölkopf, B., Muandet, K., Valera, I.

Proceedings of the 23rd International Conference on Artificial Intelligence and Statistics (AISTATS), 108, pages: 277-287, Proceedings of Machine Learning Research, (Editors: Silvia Chiappa and Roberto Calandra), PMLR, August 2020 (conference)

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link (url) [BibTex]

link (url) [BibTex]


Category Level Object Pose Estimation via Neural Analysis-by-Synthesis
Category Level Object Pose Estimation via Neural Analysis-by-Synthesis

Chen, X., Dong, Z., Song, J., Geiger, A., Hilliges, O.

In European Conference on Computer Vision (ECCV), Springer International Publishing, Cham, August 2020 (inproceedings)

Abstract
Many object pose estimation algorithms rely on the analysis-by-synthesis framework which requires explicit representations of individual object instances. In this paper we combine a gradient-based fitting procedure with a parametric neural image synthesis module that is capable of implicitly representing the appearance, shape and pose of entire object categories, thus rendering the need for explicit CAD models per object instance unnecessary. The image synthesis network is designed to efficiently span the pose configuration space so that model capacity can be used to capture the shape and local appearance (i.e., texture) variations jointly. At inference time the synthesized images are compared to the target via an appearance based loss and the error signal is backpropagated through the network to the input parameters. Keeping the network parameters fixed, this allows for iterative optimization of the object pose, shape and appearance in a joint manner and we experimentally show that the method can recover orientation of objects with high accuracy from 2D images alone. When provided with depth measurements, to overcome scale ambiguities, the method can accurately recover the full 6DOF pose successfully.

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Project Page pdf suppmat [BibTex]

Project Page pdf suppmat [BibTex]


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Algorithmic Recourse: from Counterfactual Explanations to Interventions

Karimi, A., Schölkopf, B., Valera, I.

37th International Conference on Machine Learning (ICML), July 2020 (conference) Submitted

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[BibTex]

[BibTex]


Learning Unsupervised Hierarchical Part Decomposition of 3D Objects from a Single RGB Image
Learning Unsupervised Hierarchical Part Decomposition of 3D Objects from a Single RGB Image

Paschalidou, D., Gool, L., Geiger, A.

In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2020, 2020 (inproceedings)

Abstract
Humans perceive the 3D world as a set of distinct objects that are characterized by various low-level (geometry, reflectance) and high-level (connectivity, adjacency, symmetry) properties. Recent methods based on convolutional neural networks (CNNs) demonstrated impressive progress in 3D reconstruction, even when using a single 2D image as input. However, the majority of these methods focuses on recovering the local 3D geometry of an object without considering its part-based decomposition or relations between parts. We address this challenging problem by proposing a novel formulation that allows to jointly recover the geometry of a 3D object as a set of primitives as well as their latent hierarchical structure without part-level supervision. Our model recovers the higher level structural decomposition of various objects in the form of a binary tree of primitives, where simple parts are represented with fewer primitives and more complex parts are modeled with more components. Our experiments on the ShapeNet and D-FAUST datasets demonstrate that considering the organization of parts indeed facilitates reasoning about 3D geometry.

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pdf suppmat Video 2 Project Page Slides Poster Video 1 [BibTex]

pdf suppmat Video 2 Project Page Slides Poster Video 1 [BibTex]


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A Real-Robot Dataset for Assessing Transferability of Learned Dynamics Models

Agudelo-España, D., Zadaianchuk, A., Wenk, P., Garg, A., Akpo, J., Grimminger, F., Viereck, J., Naveau, M., Righetti, L., Martius, G., Krause, A., Schölkopf, B., Bauer, S., Wüthrich, M.

IEEE International Conference on Robotics and Automation (ICRA), 2020 (conference) Accepted

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Project Page PDF [BibTex]

Project Page PDF [BibTex]


Towards Unsupervised Learning of Generative Models for 3D Controllable Image Synthesis
Towards Unsupervised Learning of Generative Models for 3D Controllable Image Synthesis

Liao, Y., Schwarz, K., Mescheder, L., Geiger, A.

In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2020, 2020 (inproceedings)

Abstract
In recent years, Generative Adversarial Networks have achieved impressive results in photorealistic image synthesis. This progress nurtures hopes that one day the classical rendering pipeline can be replaced by efficient models that are learned directly from images. However, current image synthesis models operate in the 2D domain where disentangling 3D properties such as camera viewpoint or object pose is challenging. Furthermore, they lack an interpretable and controllable representation. Our key hypothesis is that the image generation process should be modeled in 3D space as the physical world surrounding us is intrinsically three-dimensional. We define the new task of 3D controllable image synthesis and propose an approach for solving it by reasoning both in 3D space and in the 2D image domain. We demonstrate that our model is able to disentangle latent 3D factors of simple multi-object scenes in an unsupervised fashion from raw images. Compared to pure 2D baselines, it allows for synthesizing scenes that are consistent wrt. changes in viewpoint or object pose. We further evaluate various 3D representations in terms of their usefulness for this challenging task.

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pdf suppmat Video 2 Project Page Video 1 Slides Poster [BibTex]

pdf suppmat Video 2 Project Page Video 1 Slides Poster [BibTex]


Exploring Data Aggregation in Policy Learning for Vision-based Urban Autonomous Driving
Exploring Data Aggregation in Policy Learning for Vision-based Urban Autonomous Driving

Prakash, A., Behl, A., Ohn-Bar, E., Chitta, K., Geiger, A.

In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2020, 2020 (inproceedings)

Abstract
Data aggregation techniques can significantly improve vision-based policy learning within a training environment, e.g., learning to drive in a specific simulation condition. However, as on-policy data is sequentially sampled and added in an iterative manner, the policy can specialize and overfit to the training conditions. For real-world applications, it is useful for the learned policy to generalize to novel scenarios that differ from the training conditions. To improve policy learning while maintaining robustness when training end-to-end driving policies, we perform an extensive analysis of data aggregation techniques in the CARLA environment. We demonstrate how the majority of them have poor generalization performance, and develop a novel approach with empirically better generalization performance compared to existing techniques. Our two key ideas are (1) to sample critical states from the collected on-policy data based on the utility they provide to the learned policy in terms of driving behavior, and (2) to incorporate a replay buffer which progressively focuses on the high uncertainty regions of the policy's state distribution. We evaluate the proposed approach on the CARLA NoCrash benchmark, focusing on the most challenging driving scenarios with dense pedestrian and vehicle traffic. Our approach improves driving success rate by 16% over state-of-the-art, achieving 87% of the expert performance while also reducing the collision rate by an order of magnitude without the use of any additional modality, auxiliary tasks, architectural modifications or reward from the environment.

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pdf suppmat Video 2 Project Page Slides Video 1 [BibTex]

pdf suppmat Video 2 Project Page Slides Video 1 [BibTex]


Learning Situational Driving
Learning Situational Driving

Ohn-Bar, E., Prakash, A., Behl, A., Chitta, K., Geiger, A.

In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2020, 2020 (inproceedings)

Abstract
Human drivers have a remarkable ability to drive in diverse visual conditions and situations, e.g., from maneuvering in rainy, limited visibility conditions with no lane markings to turning in a busy intersection while yielding to pedestrians. In contrast, we find that state-of-the-art sensorimotor driving models struggle when encountering diverse settings with varying relationships between observation and action. To generalize when making decisions across diverse conditions, humans leverage multiple types of situation-specific reasoning and learning strategies. Motivated by this observation, we develop a framework for learning a situational driving policy that effectively captures reasoning under varying types of scenarios. Our key idea is to learn a mixture model with a set of policies that can capture multiple driving modes. We first optimize the mixture model through behavior cloning, and show it to result in significant gains in terms of driving performance in diverse conditions. We then refine the model by directly optimizing for the driving task itself, i.e., supervised with the navigation task reward. Our method is more scalable than methods assuming access to privileged information, e.g., perception labels, as it only assumes demonstration and reward-based supervision. We achieve over 98% success rate on the CARLA driving benchmark as well as state-of-the-art performance on a newly introduced generalization benchmark.

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pdf suppmat Video 2 Project Page Video 1 Slides [BibTex]

pdf suppmat Video 2 Project Page Video 1 Slides [BibTex]


On Joint Estimation of Pose, Geometry and svBRDF from a Handheld Scanner
On Joint Estimation of Pose, Geometry and svBRDF from a Handheld Scanner

Schmitt, C., Donne, S., Riegler, G., Koltun, V., Geiger, A.

In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2020, 2020 (inproceedings)

Abstract
We propose a novel formulation for joint recovery of camera pose, object geometry and spatially-varying BRDF. The input to our approach is a sequence of RGB-D images captured by a mobile, hand-held scanner that actively illuminates the scene with point light sources. Compared to previous works that jointly estimate geometry and materials from a hand-held scanner, we formulate this problem using a single objective function that can be minimized using off-the-shelf gradient-based solvers. By integrating material clustering as a differentiable operation into the optimization process, we avoid pre-processing heuristics and demonstrate that our model is able to determine the correct number of specular materials independently. We provide a study on the importance of each component in our formulation and on the requirements of the initial geometry. We show that optimizing over the poses is crucial for accurately recovering fine details and that our approach naturally results in a semantically meaningful material segmentation.

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pdf Project Page Slides Video Poster [BibTex]

pdf Project Page Slides Video Poster [BibTex]


Optimizing Rank-based Metrics with Blackbox Differentiation
Optimizing Rank-based Metrics with Blackbox Differentiation

Rolinek, M., Musil, V., Paulus, A., Vlastelica, M., Michaelis, C., Martius, G.

In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), pages: 7620-7630, IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2020, 2020, Best paper nomination (inproceedings)

Abstract
Rank-based metrics are some of the most widely used criteria for performance evaluation of computer vision models. Despite years of effort, direct optimization for these metrics remains a challenge due to their non-differentiable and non-decomposable nature. We present an efficient, theoretically sound, and general method for differentiating rank-based metrics with mini-batch gradient descent. In addition, we address optimization instability and sparsity of the supervision signal that both arise from using rank-based metrics as optimization targets. Resulting losses based on recall and Average Precision are applied to image retrieval and object detection tasks. We obtain performance that is competitive with state-of-the-art on standard image retrieval datasets and consistently improve performance of near state-of-the-art object detectors.

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Code Long Oral Short Oral Arxiv Project Page [BibTex]

Code Long Oral Short Oral Arxiv Project Page [BibTex]


Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision
Differentiable Volumetric Rendering: Learning Implicit 3D Representations without 3D Supervision

Niemeyer, M., Mescheder, L., Oechsle, M., Geiger, A.

In Proceedings IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR) 2020, 2020 (inproceedings)

Abstract
Learning-based 3D reconstruction methods have shown impressive results. However, most methods require 3D supervision which is often hard to obtain for real-world datasets. Recently, several works have proposed differentiable rendering techniques to train reconstruction models from RGB images. Unfortunately, these approaches are currently restricted to voxel- and mesh-based representations, suffering from discretization or low resolution. In this work, we propose a differentiable rendering formulation for implicit shape and texture representations. Implicit representations have recently gained popularity as they represent shape and texture continuously. Our key insight is that depth gradients can be derived analytically using the concept of implicit differentiation. This allows us to learn implicit shape and texture representations directly from RGB images. We experimentally show that our single-view reconstructions rival those learned with full 3D supervision. Moreover, we find that our method can be used for multi-view 3D reconstruction, directly resulting in watertight meshes.

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pdf suppmat Video 2 Project Page Video 1 Video 3 Slides Poster [BibTex]

pdf suppmat Video 2 Project Page Video 1 Video 3 Slides Poster [BibTex]


Computer Vision for Autonomous Vehicles: Problems, Datasets and State-of-the-Art
Computer Vision for Autonomous Vehicles: Problems, Datasets and State-of-the-Art

Janai, J., Güney, F., Behl, A., Geiger, A.

Arxiv, Foundations and Trends in Computer Graphics and Vision, 2020 (book)

Abstract
Recent years have witnessed enormous progress in AI-related fields such as computer vision, machine learning, and autonomous vehicles. As with any rapidly growing field, it becomes increasingly difficult to stay up-to-date or enter the field as a beginner. While several survey papers on particular sub-problems have appeared, no comprehensive survey on problems, datasets, and methods in computer vision for autonomous vehicles has been published. This monograph attempts to narrow this gap by providing a survey on the state-of-the-art datasets and techniques. Our survey includes both the historically most relevant literature as well as the current state of the art on several specific topics, including recognition, reconstruction, motion estimation, tracking, scene understanding, and end-to-end learning for autonomous driving. Towards this goal, we analyze the performance of the state of the art on several challenging benchmarking datasets, including KITTI, MOT, and Cityscapes. Besides, we discuss open problems and current research challenges. To ease accessibility and accommodate missing references, we also provide a website that allows navigating topics as well as methods and provides additional information.

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pdf Project Page link Project Page [BibTex]

2016


Patches, Planes and Probabilities: A Non-local Prior for Volumetric {3D} Reconstruction
Patches, Planes and Probabilities: A Non-local Prior for Volumetric 3D Reconstruction

Ulusoy, A. O., Black, M. J., Geiger, A.

In IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), June 2016 (inproceedings)

Abstract
In this paper, we propose a non-local structured prior for volumetric multi-view 3D reconstruction. Towards this goal, we present a novel Markov random field model based on ray potentials in which assumptions about large 3D surface patches such as planarity or Manhattan world constraints can be efficiently encoded as probabilistic priors. We further derive an inference algorithm that reasons jointly about voxels, pixels and image segments, and estimates marginal distributions of appearance, occupancy, depth, normals and planarity. Key to tractable inference is a novel hybrid representation that spans both voxel and pixel space and that integrates non-local information from 2D image segmentations in a principled way. We compare our non-local prior to commonly employed local smoothness assumptions and a variety of state-of-the-art volumetric reconstruction baselines on challenging outdoor scenes with textureless and reflective surfaces. Our experiments indicate that regularizing over larger distances has the potential to resolve ambiguities where local regularizers fail.

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YouTube pdf poster suppmat Project Page [BibTex]

2016


YouTube pdf poster suppmat Project Page [BibTex]


Semantic Instance Annotation of Street Scenes by 3D to 2D Label Transfer
Semantic Instance Annotation of Street Scenes by 3D to 2D Label Transfer

Xie, J., Kiefel, M., Sun, M., Geiger, A.

In IEEE Conf. on Computer Vision and Pattern Recognition (CVPR), IEEE International Conference on Computer Vision and Pattern Recognition (CVPR), June 2016 (inproceedings)

Abstract
Semantic annotations are vital for training models for object recognition, semantic segmentation or scene understanding. Unfortunately, pixelwise annotation of images at very large scale is labor-intensive and only little labeled data is available, particularly at instance level and for street scenes. In this paper, we propose to tackle this problem by lifting the semantic instance labeling task from 2D into 3D. Given reconstructions from stereo or laser data, we annotate static 3D scene elements with rough bounding primitives and develop a probabilistic model which transfers this information into the image domain. We leverage our method to obtain 2D labels for a novel suburban video dataset which we have collected, resulting in 400k semantic and instance image annotations. A comparison of our method to state-of-the-art label transfer baselines reveals that 3D information enables more efficient annotation while at the same time resulting in improved accuracy and time-coherent labels.

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pdf suppmat Project Page Project Page [BibTex]

pdf suppmat Project Page Project Page [BibTex]


Active Uncertainty Calibration in Bayesian ODE Solvers
Active Uncertainty Calibration in Bayesian ODE Solvers

Kersting, H., Hennig, P.

Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence (UAI), pages: 309-318, (Editors: Ihler, A. and Janzing, D.), AUAI Press, June 2016 (conference)

Abstract
There is resurging interest, in statistics and machine learning, in solvers for ordinary differential equations (ODEs) that return probability measures instead of point estimates. Recently, Conrad et al.~introduced a sampling-based class of methods that are `well-calibrated' in a specific sense. But the computational cost of these methods is significantly above that of classic methods. On the other hand, Schober et al.~pointed out a precise connection between classic Runge-Kutta ODE solvers and Gaussian filters, which gives only a rough probabilistic calibration, but at negligible cost overhead. By formulating the solution of ODEs as approximate inference in linear Gaussian SDEs, we investigate a range of probabilistic ODE solvers, that bridge the trade-off between computational cost and probabilistic calibration, and identify the inaccurate gradient measurement as the crucial source of uncertainty. We propose the novel filtering-based method Bayesian Quadrature filtering (BQF) which uses Bayesian quadrature to actively learn the imprecision in the gradient measurement by collecting multiple gradient evaluations.

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link (url) Project Page Project Page [BibTex]

link (url) Project Page Project Page [BibTex]


Automatic LQR Tuning Based on Gaussian Process Global Optimization
Automatic LQR Tuning Based on Gaussian Process Global Optimization

Marco, A., Hennig, P., Bohg, J., Schaal, S., Trimpe, S.

In Proceedings of the IEEE International Conference on Robotics and Automation (ICRA), pages: 270-277, IEEE, IEEE International Conference on Robotics and Automation, May 2016 (inproceedings)

Abstract
This paper proposes an automatic controller tuning framework based on linear optimal control combined with Bayesian optimization. With this framework, an initial set of controller gains is automatically improved according to a pre-defined performance objective evaluated from experimental data. The underlying Bayesian optimization algorithm is Entropy Search, which represents the latent objective as a Gaussian process and constructs an explicit belief over the location of the objective minimum. This is used to maximize the information gain from each experimental evaluation. Thus, this framework shall yield improved controllers with fewer evaluations compared to alternative approaches. A seven-degree- of-freedom robot arm balancing an inverted pole is used as the experimental demonstrator. Results of a two- and four- dimensional tuning problems highlight the method’s potential for automatic controller tuning on robotic platforms.

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Video - Automatic LQR Tuning Based on Gaussian Process Global Optimization - ICRA 2016 Video - Automatic Controller Tuning on a Two-legged Robot PDF DOI Project Page [BibTex]

Video - Automatic LQR Tuning Based on Gaussian Process Global Optimization - ICRA 2016 Video - Automatic Controller Tuning on a Two-legged Robot PDF DOI Project Page [BibTex]


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Batch Bayesian Optimization via Local Penalization

González, J., Dai, Z., Hennig, P., Lawrence, N.

Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 51, pages: 648-657, JMLR Workshop and Conference Proceedings, (Editors: Gretton, A. and Robert, C. C.), May 2016 (conference)

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link (url) Project Page [BibTex]

link (url) Project Page [BibTex]


Probabilistic Approximate Least-Squares
Probabilistic Approximate Least-Squares

Bartels, S., Hennig, P.

Proceedings of the 19th International Conference on Artificial Intelligence and Statistics (AISTATS), 51, pages: 676-684, JMLR Workshop and Conference Proceedings, (Editors: Gretton, A. and Robert, C. C. ), May 2016 (conference)

Abstract
Least-squares and kernel-ridge / Gaussian process regression are among the foundational algorithms of statistics and machine learning. Famously, the worst-case cost of exact nonparametric regression grows cubically with the data-set size; but a growing number of approximations have been developed that estimate good solutions at lower cost. These algorithms typically return point estimators, without measures of uncertainty. Leveraging recent results casting elementary linear algebra operations as probabilistic inference, we propose a new approximate method for nonparametric least-squares that affords a probabilistic uncertainty estimate over the error between the approximate and exact least-squares solution (this is not the same as the posterior variance of the associated Gaussian process regressor). This allows estimating the error of the least-squares solution on a subset of the data relative to the full-data solution. The uncertainty can be used to control the computational effort invested in the approximation. Our algorithm has linear cost in the data-set size, and a simple formal form, so that it can be implemented with a few lines of code in programming languages with linear algebra functionality.

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link (url) Project Page Project Page [BibTex]

link (url) Project Page Project Page [BibTex]


Deep Discrete Flow
Deep Discrete Flow

Güney, F., Geiger, A.

Asian Conference on Computer Vision (ACCV), 2016 (conference) Accepted

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pdf suppmat Project Page [BibTex]

pdf suppmat Project Page [BibTex]


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Extrapolation and learning equations

Martius, G., Lampert, C. H.

2016, arXiv preprint \url{https://arxiv.org/abs/1610.02995} (misc)

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Project Page [BibTex]

Project Page [BibTex]