{"title": "Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control", "book": "Advances in Neural Information Processing Systems", "page_first": 659, "page_last": 669, "abstract": "Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on these models. However, existing error bounds require restrictive assumptions. In this paper, we employ the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions. Furthermore, we demonstrate how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of our bound. Finally, we derive safety conditions for the control of unknown dynamical systems based on Gaussian process models and evaluate them in simulations of a robotic manipulator.", "full_text": "Uniform Error Bounds for Gaussian Process\nRegression with Application to Safe Control\n\nArmin Lederer\n\nTechnical University of Munich\n\narmin.lederer@tum.de\n\nJonas Umlauft\n\nTechnical University of Munich\n\njonas.umlauft@tum.de\n\nSandra Hirche\n\nTechnical University of Munich\n\nhirche@tum.de\n\nAbstract\n\nData-driven models are subject to model errors due to limited and noisy training\ndata. Key to the application of such models in safety-critical domains is the\nquanti\ufb01cation of their model error. Gaussian processes provide such a measure\nand uniform error bounds have been derived, which allow safe control based on\nthese models. However, existing error bounds require restrictive assumptions. In\nthis paper, we employ the Gaussian process distribution and continuity arguments\nto derive a novel uniform error bound under weaker assumptions. Furthermore,\nwe demonstrate how this distribution can be used to derive probabilistic Lipschitz\nconstants and analyze the asymptotic behavior of our bound. Finally, we derive\nsafety conditions for the control of unknown dynamical systems based on Gaussian\nprocess models and evaluate them in simulations of a robotic manipulator.\n\n1\n\nIntroduction\n\nThe application of machine learning techniques in control tasks bears signi\ufb01cant promises. The\nidenti\ufb01cation of highly nonlinear systems through supervised learning techniques [1] and the auto-\nmated policy search in reinforcement learning [2] enables the control of complex unknown systems.\nNevertheless, the application in safety-critical domains, like autonomous driving, robotics or aviation\nis rare. Even though the data-ef\ufb01ciency and performance of self-learning controllers is impressive,\nengineers still hesitate to rely on learning approaches if the physical integrity of systems is at risk,\nin particular, if humans are involved. Empirical evaluations, e.g. for autonomous driving [3], are\navailable, however, this might not be suf\ufb01cient to reach the desired level of reliability and autonomy.\nLimited and noisy training data lead to imperfections in data-driven models [4]. This makes the\nquanti\ufb01cation of the uncertainty in the model and the knowledge about a model\u2019s ignorance key for the\nutilization of learning approaches in safety-critical applications. Gaussian process models provide this\nmeasure for their own imprecision and therefore gained attention in the control community [5, 6, 7].\nThese approaches heavily rely on error bounds of Gaussian process regression and are therefore\nlimited by the strict assumptions made in previous works on GP uniform error bounds [8, 9, 10, 11].\nThe main contribution of this paper is therefore the derivation of a novel GP uniform error bound,\nwhich requires less prior knowledge and assumptions than previous approaches and is therefore\napplicable to a wider range of problems. Furthermore, we derive a Lipschitz constant for the samples\nof GPs and investigate the asymptotic behavior in order to demonstrate that arbitrarily small error\nbounds can be guaranteed with suf\ufb01cient computational resources and data. The proposed GP bounds\nare employed to derive safety guarantees for unknown dynamical systems which are controlled based\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\fon a GP model. By employing Lyapunov theory [12], we prove that the closed-loop system - here\nwe take a robotic manipulator as example - converges to a small fraction of the state space and can\ntherefore be considered as safe.\nThe remainder of this paper is structured as follows: We brie\ufb02y introduce Gaussian process regression\nand discuss related error bounds in Section 2. The novel proposed GP uniform error bound, the\nprobabilistic Lipschitz constant and the asymptotic analysis are presented in Section 3. In Section 4\nwe show safety of a GP model based controller and evaluate it on a robotic manipulator in Section 5.\n\n2 Background\n\n2.1 Gaussian Process Regression and Uniform Error Bounds\n\nGaussian process regression is a Bayesian machine learning method based on the assumption that\nany \ufb01nite collection of random variables1 yi \u2208 R follows a joint Gaussian distribution with prior\nmean 0 and covariance kernel k : Rd \u00d7 Rd \u2192 R+ [13]. Therefore, the variables yi are observations\nof a sample function f : X \u2282 Rd \u2192 R of the GP distribution perturbed by zero mean Gaussian\nn \u2208 R+,0. By concatenating N input data points xi in a matrix XN the\nnoise with variance \u03c32\nelements of the GP kernel matrix K(XN , XN ) are de\ufb01ned as Kij = k(xi, xj), i, j = 1, . . . , N\nand k(XN , x) denotes the kernel vector, which is de\ufb01ned accordingly. The probability distribution\nof the GP at a point x conditioned on the training data concatenated in XN and yN is then given\nas a normal distribution with mean \u03bdN (x) = k(x, XN )(K(XN , XN ) + \u03c32\nnIN )\u22121yN and variance\nN (x, x(cid:48)) = k(x, x(cid:48)) \u2212 k(x, XN )(K(XN , XN ) + \u03c32\nnIN )\u22121k(XN , x(cid:48)).\n\u03c32\nA major reason for the popularity of GPs and related approaches in safety critical applications is the\nexistence of uniform error bounds for the regression error, which is de\ufb01ned as follows.\nDe\ufb01nition 2.1. Gaussian process regression exhibits a uniformly bounded error on a compact\nset X \u2282 Rd if there exists a function \u03b7(x) such that\n\n(1)\nIf this bound holds with probability of at least 1 \u2212 \u03b4 for some \u03b4 \u2208 (0, 1), it is called a probabilistic\nuniform error bound.\n\n|\u03bdN (x) \u2212 f (x)| \u2264 \u03b7(x) \u2200x \u2208 X.\n\n2.2 Related Work\n\nFor many methods closely related to Gaussian process regression, uniform error bounds are very\ncommon. When dealing with noise-free data, i.e. in interpolation of multivariate functions, results\nfrom the \ufb01eld of scattered data approximation with radial basis functions can be applied [14]. In fact,\nmany of the results from interpolation with radial basis functions can be directly applied to noise-free\nGP regression with stationary kernels. The classical result in [15] employs Fourier transform methods\nto derive an error bound for functions in the reproducing kernel Hilbert space (RKHS) attached to\nthe interpolation kernel. By additionally exploiting properties of the RKHS a uniform error bound\nwith increased convergence rate is derived in [16]. Typically, this form of bound crucially depends\non the so called power function, which corresponds to the posterior standard deviation of Gaussian\nprocess regression under certain conditions [17]. In [18], a Lp error bound for data distributed on\na sphere is developed, while the bound in [19] extends existing approaches to functions from Sobolev\nspaces. Bounds for anisotropic kernels and the derivatives of the interpolant are developed in [20].\nA Sobolev type error bound for interpolation with Mat\u00e9rn kernels is derived in [21]. Moreover, it is\nshown that convergence of the interpolation error implies convergence of the GP posterior variance.\nRegularized kernel regression is a method which extends many ideas from scattered data interpolation\nto noisy observations and it is highly related to Gaussian process regression as pointed out in [17].\nIn fact, the GP posterior mean function is identical to kernel ridge regression with squared cost\n\n1Notation: Lower/upper case bold symbols denote vectors/matrices and R+/R+,0 all real positive/non-\nnegative numbers. N denotes all natural numbers, In the n \u00d7 n identity matrix, the dot in \u02d9x the derivative of x\nwith respect to time and (cid:107) \u00b7 (cid:107) the Euclidean norm. A function f (x) is said to admit a modulus of continuity\n\u03c9 : R+ \u2192 R+ if and only if |f (x) \u2212 f (x\n(cid:107)). The \u03c4-covering number M (\u03c4 , X) of a set X\n(with respect to the Euclidean metric) is de\ufb01ned as the minimum number of spherical balls with radius \u03c4 which\nis required to completely cover X. Big O notation is used to describe the asymptotic behavior of functions.\n\n)| \u2264 \u03c9((cid:107)x \u2212 x\n\n(cid:48)\n\n(cid:48)\n\n2\n\n\ffunction [13]. Many error bounds such as [22] depend on the empirical L2 covering number and the\nnorm of the unknown function in the RKHS attached to the regression kernel. In [23], the effective\ndimension of the feature space, in which regression is performed, is employed to derive a probabilistic\nuniform error bound. The effect of approximations of the kernel, e.g. with the Nystr\u00f6m method, on\nthe regression error is analyzed in [24]. Tight error bounds using empirical L2 covering numbers are\nderived under mild assumptions in [25]. Finally, error bounds for general regularization are developed\nin [26], which depend on regularization and the RKHS norm of the function.\nUsing similar RKHS-based methods for Gaussian process regression, probabilistic uniform error\nbounds depending on the maximal information gain and the RKHS norm have been developed in [8].\nThese constants pose a high hurdle which has prevented the rigorous application of this work in\ncontrol and typically heuristic constants without theoretical foundations are applied, see e.g. [27].\nWhile regularized kernel regression allows a wide range of observation noise distributions, the bound\nin [8] only holds for bounded sub-Gaussian noise. Based on this work an improved bound is derived\nin [9] in order to analyze the regret of an upper con\ufb01dence bound algorithm in multi-armed bandit\nproblems. Although these bounds are frequently used in safe reinforcement learning and control,\nthey suffer from several issues. On the one hand, they depend on constants which are very dif\ufb01cult\nto calculate. While this is no problem for theoretical analysis, it prohibits the integration of these\nbounds into algorithms and often estimates of the constants must be used. On the other hand, they\nsuffer from the general problem of RKHS approaches: The space of functions, for which the bounds\nhold, becomes smaller the smoother the kernel is [19]. In fact, the RKHS attached to a covariance\nkernel is usually small compared to the support of the prior distribution of a Gaussian process [28].\nThe latter issue has been addressed by considering the support of the prior distribution of the Gaussian\nprocess as belief space. Based on bounds for the suprema of GPs [29] and existing error bounds for\ninterpolation with radial basis functions, a probabilistic uniform error bound for Kriging (alternative\nterm for GP regression for noise-free training data) is derived in [30]. However, the uniform error of\nGaussian process regression with noisy observations has not been analyzed with the help of the prior\nGP distribution to the best of our knowledge.\n\n3 Probabilistic Uniform Error Bound\n\nWhile probabilistic uniform error bounds for the cases of noise-free observations and the restriction\nto subspaces of a RKHS are widely used, they often rely on constants which are hard to determine\nand are typically limited to unnecessarily small function spaces. The inherent probability distribution\nof GPs, which is the largest possible function space for regression with a certain GP, has not been\nexploited to derive uniform error bounds for Gaussian process regression with noisy observations.\nUnder the weak assumption of Lipschitz continuity of the covariance kernel and the unknown function,\na directly computable probabilistic uniform error bound is derived in Section 3.1. We demonstrate\nhow Lipschitz constants for unknown functions directly follow from the assumed distribution over the\nfunction space in Section 3.2. Finally, we show that an arbitrarily small error bound can be reached\nwith suf\ufb01ciently many and well-distributed training data in Section 3.3.\n\n3.1 Exploiting Lipschitz Continuity of the Unknown Function\n\nIn contrast to the RKHS based approaches in [8, 9], we make use of the inherent probability\ndistribution over the function space de\ufb01ned by Gaussian processes. We achieve this through the\nfollowing assumption.\nAssumption 3.1. The unknown function f (\u00b7) is a sample from a Gaussian process GP(0, k(x, x(cid:48)))\nand observations y = f (x) + \u0001 are perturbed by zero mean i.i.d. Gaussian noise \u0001 with variance \u03c32\nn.\nThis assumption includes abundant information about the regression problem. The space of sample\nfunctions F is limited through the choice of the kernel k(\u00b7,\u00b7) of the Gaussian process. Using\nMercer\u2019s decomposition [31] \u03c6i(x), i = 1, . . . ,\u221e of the kernel k(\u00b7,\u00b7), this space is de\ufb01ned through\n\nF =\n\nf (x) : \u2203\u03bbi, i = 1, . . . ,\u221e such that f (x) =\n\n(2)\nwhich contains all functions that can be represented in terms of the kernel k(\u00b7,\u00b7). By choosing\na suitable class of covariance functions k(\u00b7,\u00b7), this space can be designed in order to incorporate\n\n\u03bbi\u03c6i(x)\n\n,\n\n(cid:40)\n\n(cid:41)\n\n\u221e(cid:88)\n\ni=1\n\n3\n\n\f(cid:13)(cid:13)(cid:13)(cid:13)(cid:104) \u2202k(x,x(cid:48))\n\n\u2202x1\n\n(cid:105)T(cid:13)(cid:13)(cid:13)(cid:13) .\n\nprior knowledge of the unknown function f (\u00b7). For example, for covariance kernels k(\u00b7,\u00b7) which\nare universal in the sense of [32], continuous functions can be learned with arbitrary precision.\nMoreover, for the squared exponential kernel, the space of sample functions corresponds to the space\nof continuous functions on X, while its RKHS is limited to analytic functions [28]. Furthermore,\nAssumption 3.1 de\ufb01nes a prior GP distribution over the sample space F which is the basis for\nthe calculation of the posterior probability. The prior distribution is typically shaped by the\nhyperparameters of the covariance kernel k(\u00b7,\u00b7), e.g. slowly varying functions can be assigned a\nhigher probability than functions with high derivatives. Finally, Assumption 3.1 allows Gaussian\nobservation noise which is in contrast to the bounded noise required e.g. in [8, 9].\nIn addition to Assumption 3.1, we need Lipschitz continuity of the kernel k(\u00b7,\u00b7) and the unknown\nfunction f (\u00b7). We de\ufb01ne the Lipschitz constant of a differentiable covariance kernel k(\u00b7,\u00b7) as\n\n\u2202k(x,x(cid:48))\n\n\u2202xd\n\n. . .\n\nLk := max\nx,x(cid:48)\u2208X\n\n(3)\nSince most of the practically used covariance kernels k(\u00b7,\u00b7), such as squared exponential and Mat\u00e9rn\nkernels, are Lipschitz continuous [13], this is a weak restriction on covariance kernels. However,\nit allows us to prove continuity of the posterior mean function \u03bdN (\u00b7) and the posterior standard\ndeviation \u03c3N (\u00b7), which is exploited to derive a probabilistic uniform error bound in the following\ntheorem. The proofs for all following theorems can be found in the supplementary material.\nTheorem 3.1. Consider a zero mean Gaussian process de\ufb01ned through the continuous covariance\nkernel k(\u00b7,\u00b7) with Lipschitz constant Lk on the compact set X. Furthermore, consider a continuous\nunknown function f : X \u2192 R with Lipschitz constant Lf and N \u2208 N observations yi satisfying\nAssumption 3.1. Then, the posterior mean function \u03bdN (\u00b7) and standard deviation \u03c3N (\u00b7) of a Gaussian\nprocess conditioned on the training data {(xi, yi)}N\ni=1 are continuous with Lipschitz constant L\u03bdN\nand modulus of continuity \u03c9\u03c3N (\u00b7) on X such that\n\n\u221a\n\n(cid:115)\nL\u03bdN \u2264 Lk\n\n\u03c9\u03c3N (\u03c4 ) \u2264\n\n2\u03c4 Lk\n\nN(cid:13)(cid:13)(K(XN , XN ) + \u03c32\n(cid:18)\n\nnIN )\u22121yN\n1 + N(cid:107)(K(XN , XN ) + \u03c32\n\n(cid:13)(cid:13)\n\nnIN )\u22121(cid:107) max\nx,x(cid:48)\u2208X\n\nk(x, x(cid:48))\n\nMoreover, pick \u03b4 \u2208 (0, 1), \u03c4 \u2208 R+ and set\n\n(cid:18) M (\u03c4 , X)\n\n(cid:19)\n\n\u03b4\n\n\u03b2(\u03c4 ) = 2 log\n\n\u03b3(\u03c4 ) = (L\u03bdN + Lf ) \u03c4 +(cid:112)\u03b2(\u03c4 )\u03c9\u03c3N (\u03c4 ).\n(cid:16)|f (x) \u2212 \u03bdN (x)| \u2264(cid:112)\u03b2(\u03c4 )\u03c3N (x) + \u03b3(\u03c4 ), \u2200x \u2208 X\n\n(cid:17) \u2265 1 \u2212 \u03b4.\n\nThen, it holds that\n\nP\n\n(cid:19)\n\n.\n\n(4)\n\n(5)\n\n(6)\n\n(7)\n\n(8)\n\nThe parameter \u03c4 is in fact the grid constant of a grid used in the derivation of the theorem. The error\non the grid can be bounded by exploiting properties of the Gaussian distribution [8] resulting in a\ndependency on the number of grid points. Eventually, this leads to the constant \u03b2(\u03c4 ) de\ufb01ned in (6)\nsince the covering number M (\u03c4 , X) is the minimum number of points in a grid over X with grid\nconstant \u03c4. By employing the Lipschitz constant L\u03bdN and the modulus of continuity \u03c9\u03c3N (\u00b7), which\nare trivially obtained due Lipschitz continuity of the covariance kernel k(\u00b7,\u00b7), as well as the Lipschitz\nconstant Lf , the error bound is extended to the complete set X, which results in (8).\nNote, that most of the equations in Theorem 3.1 can be directly evaluated. Although our expression\nfor \u03b2(\u03c4 ) depends on the covering number of X, which is in general dif\ufb01cult to calculate, upper bounds\ncan be computed trivially. For example, for a hypercubic set X \u2282 Rd the covering number can be\nbounded by\n\nM (\u03c4 , X) \u2264(cid:16)\n\n(cid:17)d\n\n1 +\n\nr\n\u03c4\n\n,\n\n(9)\n\nwhere r is the edge length of the hypercube. Furthermore, (4) and (5) depend only on the training\ndata and kernel expressions, which can be calculated analytically in general. Therefore, (8) can\n\n4\n\n\fbe computed for \ufb01xed \u03c4 and \u03b4 if an upper bound for the Lipschitz constant Lf of the unknown\nfunction f (\u00b7) is known. Prior bounds on the Lipschitz constant Lf are often available for control\nsystems, e.g. based on simpli\ufb01ed \ufb01rst order physical models. However, we demonstrate a method to\nobtain probabilistic Lipschitz constants from Assumption 3.1 in Section 3.2. Therefore, it is trivial\nto compute all expressions in Theorem 3.1 or upper bounds thereof, which emphasizes the high\napplicability of Theorem 3.1 in safe control of unknown systems.\nMoreover, it should be noted that \u03c4 can be chosen arbitrarily small such that the effect of the\n\nconstant \u03b3(\u03c4 ) can always be reduced to an amount which is negligible compared to(cid:112)\u03b2(\u03c4 )\u03c3N (x).\n\nEven conservative approximations of the Lipschitz constants L\u03bdN and Lf and a loose modulus of\ncontinuity \u03c9\u03c3N (\u03c4 ) do not affect the error bound (8) much since (6) grows merely logarithmically with\ndiminishing \u03c4. In fact, even the bounds (4) and (5), which grow in the order of O(N ) and O(N 1\n2 ),\nrespectively, as shown in the proof of Theorem 3.3 and thus are unbounded, can be compensated\nsuch that a vanishing uniform error bound can be proven under weak assumptions in Section 3.3.\n\n3.2 Probabilistic Lipschitz Constants for Gaussian Processes\nIf little prior knowledge of the unknown function f (\u00b7) is given, it might not be possible to directly\nderive a Lipschitz constant Lf on X. However, we indirectly assume a certain distribution of the\nderivatives of f (\u00b7) with Assumption 3.1. Therefore, it is possible to derive a probabilistic Lipschitz\nconstant Lf from this assumption, which is described in the following theorem.\nTheorem 3.2. Consider a zero mean Gaussian process de\ufb01ned through the covariance kernel k(\u00b7,\u00b7)\nwith continuous partial derivatives up to the fourth order and partial derivative kernels\n\nLf =\n\n6d max\n\n6d max\n\nmax\nx\u2208X\n\nmax\nx\u2208X\n\n\u22022\n\n\u2202xi\u2202x(cid:48)i\n\nk(x, x(cid:48)) \u2200i = 1, . . . , d.\n\nk\u2202i(x, x(cid:48)) =\n\n(10)\nk denote the Lipschitz constants of the partial derivative kernels k\u2202i(\u00b7,\u00b7) on the set X with\nLet L\u2202i\nmaximal extension r = maxx,x(cid:48)\u2208X (cid:107)x \u2212 x(cid:48)(cid:107). Then, a sample function f (\u00b7) of the Gaussian process\n(cid:113)\nis almost surely continuous on X and with probability of at least 1 \u2212 \u03b4L, it holds that\n(cid:113)\n\n(cid:112)k\u22021(x, x) + 12\n(cid:112)k\u2202d(x, x) + 12\n\n(cid:112)k\u22021(x, x),\n(cid:112)k\u2202d(x, x),\n\n(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) (11)\n\uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb\n\n(cid:16) 2d\n(cid:16) 2d\n\n\u03b4L\n\n\u03b4L\n\n(cid:114)\n(cid:114)\n\n(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)\n\uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0\n\n(cid:27)\n(cid:27)\n\n(cid:26)\n(cid:26)\n\n\u221a\n\n...\n\u221a\n\nrL\u22021\nk\n\nrL\u2202d\nk\n\n2 log\n\n2 log\n\n(cid:17)\n(cid:17)\n\nmax\nx\u2208X\n\nmax\nx\u2208X\n\nis a Lipschitz constant of f (\u00b7) on X.\nNote that a higher differentiability of the covariance kernel k(\u00b7,\u00b7) is required compared to Theorem 3.1.\nThe reason for this is that the proof of Theorem 3.2 exploits the fact that the partial derivative\nk\u2202i(\u00b7,\u00b7) of a differentiable kernel is again a covariance function, which de\ufb01nes a derivative Gaussian\nprocess [33]. In order to obtain continuity of the samples of these derivative processes, the derivative\nkernels k\u2202i(\u00b7,\u00b7) must be continuously differentiable [34]. Using the metric entropy criterion [34]\nand the Borell-TIS inequality [35], we exploit the continuity of sample functions and bound their\nmaximum value, which directly translates into the probabilistic Lipschitz constant (11).\nNote that all the values required in (11) can be directly computed. The maximum of the derivative ker-\nnels k\u2202i(\u00b7,\u00b7) as well as their Lipschitz constants L\u2202i\nk can be calculated analytically for many kernels.\nTherefore, the Lipschitz constant obtained with Theorem 3.2 can be directly used in Theorem 3.1\nthrough application of the union bound. Since the Lipschitz constant Lf has only a logarithmic depen-\ndence on the probability \u03b4L, small error probabilities for the Lipschitz constant can easily be achieved.\nRemark 3.1. The work in [36] derives also estimates for the Lipschitz constants. However, they only\ntake the Lipschitz constant of the posterior mean function, which neglects the probabilistic nature of\nthe GP and thereby underestimates the Lipschitz constants of samples of the GP.\n\n3.3 Analysis of Asymptotic Behavior\n\nIn safe reinforcement learning and control of unknown systems an important question regards\nthe existence of lower bounds for the learning error because they limit the achievable control\n\n5\n\n\fperformance. It is clear that the available data and constraints on the computational resources pose\nsuch lower bounds in practice. However, it is not clear under which conditions, e.g. requirements of\ncomputational power, an arbitrarily low uniform error can be guaranteed. The asymptotic analysis of\nthe error bound, i.e. investigation of the bound (8) in the limit N \u2192 \u221e can clarify this question. The\nfollowing theorem is the result of this analysis.\nTheorem 3.3. Consider a zero mean Gaussian process de\ufb01ned through the continuous covariance\nkernel k(\u00b7,\u00b7) with Lipschitz constant Lk on the set X. Furthermore, consider an in\ufb01nite data stream\nof observations (xi, yi) of an unknown function f : X \u2192 R with Lipschitz constant Lf and maximum\nabsolute value \u00aff \u2208 R+ on X which satis\ufb01es Assumption 3.1. Let \u03bdN (\u00b7) and \u03c3N (\u00b7) denote the mean\nand standard deviation of the Gaussian process conditioned on the \ufb01rst N observations. If there\n, \u2200x \u2208 X, then it\n\nexists a \u0001 > 0 such that the standard deviation satis\ufb01es \u03c3N (x) \u2208 O(cid:16)\n(cid:19)\n\nholds for every \u03b4 \u2208 (0, 1) that\n\n2\u2212\u0001(cid:17)\n\nlog(N )\u2212 1\n\n(cid:18)\n\n(cid:107)\u03bdN (x) \u2212 f (x)(cid:107) \u2208 O(log(N )\u2212\u0001)\n\n\u2265 1 \u2212 \u03b4.\n\nP\n\nsup\nx\u2208X\n\n(12)\n\nIn addition to the conditions of Theorem 3.1 the absolute value of the unknown function is required\nto be bounded by a value \u00aff. This is necessary to bound the Lipschitz constant L\u03bdN of the posterior\nmean function \u03bdN (\u00b7) in the limit of in\ufb01nite training data. Even if no such constant is known, it\ncan be derived from properties of the GP under weak conditions similarly to Theorem 3.2. Based\non this restriction, it can be shown that the bound of the Lipschitz constant L\u03bdN grows at most\nwith rate O(N ) using the triangle inequality and the fact that the squared norm of the observation\nnoise (cid:107)\u0001(cid:107)2 follows a \u03c72\nN distribution with probabilistically bounded maximum value [37]. Therefore,\nwe pick \u03c4 (N ) \u2208 O(N\u22122) such that \u03b3(\u03c4 (N )) \u2208 O(N\u22121) and \u03b2(\u03c4 (N )) \u2208 O(log(N )) which\nimplies (12).\nThe condition on the convergence rate of the posterior standard deviation \u03c3N (\u00b7) in Theorem 3.3\ncan be seen as a condition for the distribution of the training data, which depends on the structure\nof the covariance kernel. In [38, Corollary 3.2], the condition is formulated as follows: Let B\u03c1(x)\n(cid:12)(cid:12)B\u03c1(N )(x)(cid:12)(cid:12) = \u221e holds. This is achieved, e.g. if a constant fraction of all samples\ndenote a set of training points around x with radius \u03c1 > 0, then the posterior variance converges\nto zero if there exists a function \u03c1(N ) for which \u03c1(N ) \u2264 k(x, x)/Lk \u2200N, limN\u2192\u221e \u03c1(N ) = 0\nand limN\u2192\u221e\nlies on the point x.\nIn fact, it is straightforward to derive a similar condition for the uniform\nerror bounds in [8, 9]. However, due to their dependence on the maximal information gain, the\nsatisfy \u03c3N (\u00b7) \u2208 O(cid:16)\nrequired decrease rates depend on the covariance kernel k(\u00b7,\u00b7) and are typically higher. For example,\nthe posterior standard deviation of a Gaussian process with a squared exponential kernel must\n\nfor [8] and \u03c3N (\u00b7) \u2208 O(cid:16)\n\n2 \u22122(cid:17)\n\nlog(N )\u2212 d+1\n\nlog(N )\u2212 d\n\nfor [9].\n\n(cid:17)\n\n2\n\n4 Safety Guarantees for Control of Unknown Dynamical Systems\n\nSafety guarantees for dynamical systems, in terms of upper bounds for the tracking error, are\nbecoming more and more relevant as learning controllers are applied in safety-critical applications\nlike autonomous driving or robots working in close proximity to humans [39, 40, 4]. We therefore\nshow how the results in Theorem 3.1 can be applied to control safely unknown dynamical systems. In\nSection 4.1 we propose a tracking control law for systems which are learned with GPs. The stability\nof the resulting controller is analyzed in Section 4.2.\n\n4.1 Tracking Control Design\n\nConsider the nonlinear control af\ufb01ne dynamical system\n\n\u02d9x1 = x2,\n\n(13)\n(cid:124) \u2208 X \u2282 R2 and control input u \u2208 U \u2286 R. While the structure of the\nwith state x = [x1 x2]\ndynamics (13) is known, the function f (\u00b7) is not. However, we assume that it is a sample from a GP\nwith kernel k(\u00b7,\u00b7). Systems of the form (13) cover a large range of applications including Lagrangian\ndynamics and many physical systems.\n\n\u02d9x2 = f (x) + u,\n\n6\n\n\fThe task is to de\ufb01ne a policy \u03c0 : X \u2192 U for which the output x1 tracks the desired trajectory xd(t)\n(cid:124) vanishes over time,\n(cid:124)\nsuch that the tracking error e = [e1 e2]\n(cid:107)e(cid:107) = 0. For notational simplicity, we introduce the \ufb01ltered state r = \u03bbe1 + e2, \u03bb \u2208 R+.\ni.e. limt\u2192\u221e\nA well-known method for tracking of control af\ufb01ne systems is feedback linearization [12], which\naims for a model-based compensation of the non-linearity f (\u00b7) using an estimate \u02c6f (\u00b7) and then applies\nlinear control principles for the tracking. The feedback linearizing policy reads as\n\n= x \u2212 xd with xd = [xd \u02d9xd]\n\nu = \u03c0(x) = \u2212 \u02c6f (x) + \u03bd,\n\n\u03bd = \u00a8xd \u2212 kcr \u2212 \u03bbe2,\n\n\u02d9r = f (x) \u2212 \u02c6f (x) \u2212 kcr.\n\n(14)\n\n(15)\n\n(16)\n\nwhere the linear control law \u03bd is the PD-controller\n\nwith control gain kc \u2208 R+. This results in the dynamics of the \ufb01ltered state\n\nAssuming training data of the real system yi = f (xi) + \u0001, i = 1, . . . , N, \u0001 \u223c N (0, \u03c32\nn) are\navailable, we utilize the posterior mean function \u03bdN (\u00b7) for the model estimate \u02c6f (\u00b7). This implies,\nthat observations of \u02d9x2 are corrupted by noise, while x is measured free of noise. This is of course\ndebatable, but in practice measuring the time derivative is usually realized with \ufb01nite difference\napproximations, which injects signi\ufb01cantly more noise than a direct measurement.\n\n4.2 Stability Analysis\n\nDue to safety constraints, e.g. for robots interacting with humans, it is usually necessary to verify that\nthe model \u02c6f (\u00b7) is suf\ufb01ciently precise and the parameters of the controller kc, \u03bb are chosen properly.\nThese safety certi\ufb01cates can be achieved if there exists an upper bound for the tracking error as\nde\ufb01ned in the following.\nDe\ufb01nition 4.1 (Ultimate Boundedness). The trajectory x(t) of a dynamical system \u02d9x = f (x, u) is\nglobally ultimately bounded, if there exist a positive constants b \u2208 R+ such that for every a \u2208 R+,\nthere is a T = T (a, b) \u2208 R+ such that\n\n(cid:107)x(t0)(cid:107) \u2264 a \u21d2 (cid:107)x(t)(cid:107) \u2264 b, \u2200t \u2265 t0 + T .\n\nSince the solutions x(t) cannot be computed analytically, a stability analysis is necessary, which\nallows conclusions regarding the closed-loop behavior without running the policy on the real\nsystem [12].\nTheorem 4.1. Consider a control af\ufb01ne system (13), where f (\u00b7) admits a Lipschitz constant Lf\non X \u2282 Rd. Assume that f (\u00b7) and the observations yi, i = 1, . . . , N, satisfy the conditions of\nAssumption 3.1. Then, the feedback linearizing controller (14) with \u02c6f (\u00b7) = \u03bdN (\u00b7) guarantees with\nprobability 1 \u2212 \u03b4 that the tracking error e converges to\n\n(cid:40)\nx \u2208 X\n\n(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:107)e(cid:107) \u2264\n\nB =\n\n(cid:112)\u03b2(\u03c4 )\u03c3N (x) + \u03b3(\u03c4 )\n\n(cid:41)\n\n\u221a\n\nkc\n\n\u03bb2 + 1\n\n,\n\n(17)\n\nwith \u03b2(\u03c4 ) and \u03b3(\u03c4 ) de\ufb01ned in Theorem 3.1.\n\nBased on Lyapunov theory, it can be shown that the tracking error converges if the feedback\nterm |kcr| dominates the model error |f (\u00b7) \u2212 \u02c6f (\u00b7)|. As Theorem 3.1 bounds the model error, the\n\nset for which holds |kcr| >(cid:112)\u03b2(\u03c4 )\u03c3N (x) + \u03b3(\u03c4 ) can be computed. It can directly be seen, that\n\nthe ultimate bound can be made arbitrarily small, by increasing the gains \u03bb, kc or with more training\npoints to decrease \u03c3N (\u00b7). Computing the set B allows to check whether the controller (14) adheres\nto the safety requirements.\n\n5 Numerical Evaluation\n\nWe evaluate our theoretical results in two simulations.2 In Section 5.1, we investigate the effect of\napplying Theorem 3.2 to determine a probabilistic Lipschitz constant for an unknown synthetic\n\n2Matlab code is online available: https://gitlab.lrz.de/ga68car/GPerrorbounds4safecontrol\n\n7\n\n\fFigure 1: Snapshots of the state trajectory (blue) as it approaches the desired trajectory (green). In low\nuncertainty areas (yellow background), the set B (red) is signi\ufb01cantly smaller then in high uncertainty\nareas (blue background).\n\nFigure 2: Higher uncertainty in the model leads to a larger ultimate bound (red). Similarly, the\ntracking error (blue) increases in areas with a less precise model.\n\nsystem. Furthermore, we analyze the effect of unevenly distributed training samples on the tracking\nerror bound from Theorem 4.1. In Section 5.2, we apply the feedback linearizing controller (14) to a\ntracking problem with a robotic manipulator.\n\n5.1 Synthetic System with Unknown Lipschitz Constant Lf\nAs an example for a system of form (13), we consider f (x) = 1\u2212 sin(x1) +\n1+exp(\u2212x2). Based on a\nuniform grid over [0 3]\u00d7 [\u22123 3] the training set is formed of 81 points with \u03c32\nn = 0.01. The reference\ntrajectory is a circle xd(t) = 2 sin(t) and the controller gains are kc = 2 and \u03bb = 1. We choose a\nprobability of failure \u03b4 = 0.01, \u03b4L = 0.01 and set \u03c4 = 10\u22128. The state space is the rectangle X =\n[\u22126 4] \u00d7 [\u22124 4]. A squared exponential kernel with automatic relevance determination is utilized,\nfor which Lk and maxx,x(cid:48)\u2208X k(x, x(cid:48)) is derived analytically for the optimized hyperparameters. We\nmake use of Theorem 3.2 to estimate the Lipschitz constant Lf , and it turns out to be a conservative\nbound (factor 10 \u223c 100). However, this is not crucial, because \u03c4 can be chosen arbitrarily small\n\nand \u03b3(\u03c4 ) is dominated by(cid:112)\u03b2(\u03c4 )\u03c9\u03c3N (\u03c4 ). As Theorems 3.1 and 3.2 are subsequently utilized in this\n\n1\n\nexample, a union bound approximation can be applied to combine \u03b4 and \u03b4L.\nThe results are shown in Figs. 1 and 2. Both plots show, that the safety bound here is rather\nconservative, which also results from the fact that the violation probability was set to 1%.\n\n5.2 Robotic Manipulator with 2 Degrees of Freedom\n\nWe consider a planar robotic manipulator in the z1-z2-plane with 2 degrees of freedom (DoFs), with\nunit length and unit masses / inertia for all links. For this example, we consider Lf to be known and\nextend Theorem 3.1 to the multidimensional case using the union bound. The state space is here\nfour dimensional [q1 \u02d9q1 q2 \u02d9q2] and we consider X = [\u2212\u03c0 \u03c0]4. The 81 training points are distributed\n\n8\n\n\u22126\u22124\u22122024\u22124\u22122024x1x2\u22126\u22124\u22122024\u22124\u22122024x1\u03c32(x)BXNx(t)xd(t)010203010\u2212410\u22122100tkek100101volumeBkekvolumeB\fFigure 3: The task space of the robot (left) shows the robot is guaranteed to remain in B (red) after a\ntransient phase. Hence, the remaining state space X \\ B (green) can be considered as safe. The joint\nangles and velocities (right) converge to the desired trajectories (dashed lines) over time.\n\nin [\u22121 1]4 and the control gain is kc = 7, while other constants remain the same as in Section 5.1.\nThe desired trajectories for both joints are again sinusoidal as shown in Fig. 3 on the right side. The\nrobot dynamics are derived according to [41, Chapter 4].\nTheorem 3.1 allows to derive an error bound in the joint space of the robot according to Theorem 4.1,\nwhich can be transformed into the task space as shown in Fig. 3 on the left. Thus, based on the\nlearned (initially unknown) dynamics, it can be guaranteed, that the robot will not leave the depicted\narea and can thereby be considered as safe.\nPrevious error bounds for GPs are not applicable to this practical setting, because they i) do not allow\nthe observation noise on the training data to be Gaussian [8], which is a common assumption in\nrobotics, ii) utilize constants which cannot be computed ef\ufb01ciently (e.g. maximal information gain\nin [42]) or iii) make assumptions dif\ufb01cult to verify in practice (e.g. the RKHS norm of the unknown\ndynamical system [6]).\n\n6 Conclusion\n\nThis paper presents a novel uniform error bound for Gaussian process regression. By exploiting\nthe inherent probability distribution of Gaussian processes instead of the reproducing kernel Hilbert\nspace attached to the covariance kernel, a wider class of functions can be considered. Furthermore,\nwe demonstrate how probabilistic Lipschitz constants can be estimated from the GP distribution and\nderive suf\ufb01cient conditions to reach arbitrarily small uniform error bounds. We employ the derived\nresults to show safety bounds for a tracking control algorithm and evaluate them in simulation for a\nrobotic manipulator.\n\nAcknowledgments\n\nArmin Lederer gratefully acknowledges \ufb01nancial support from the German Academic Scholarship\nFoundation.\n\nReferences\n[1] P. M. N\u00f8rg\u00e5rd, O. Ravn, N. K. Poulsen, and L. K. Hansen, Neural Networks for Modelling and\n\nControl of Dynamical Systems - A Practicioner\u2019s Handbook. London: Springer, 2000.\n\n[2] M. P. Deisenroth, \u201cA Survey on Policy Search for Robotics,\u201d Foundations and Trends in\n\nRobotics, vol. 2, no. 1-2, pp. 1\u2013142, 2013.\n\n[3] B. Huval, T. Wang, S. Tandon, J. Kiske, W. Song, J. Pazhayampallil, M. Andriluka,\nP. Rajpurkar, T. Migimatsu, R. Cheng-Yue, F. Mujica, A. Coates, and A. Y. Ng, \u201cAn Empirical\n\n9\n\n\u22122\u22121012\u22122\u22121012z1z2X\\BBXN0246\u2212202tq,\u02d9qq1\u02d9q1q2\u02d9q2\fEvaluation of Deep Learning on Highway Driving,\u201d pp. 1\u20137, 2015. [Online]. Available:\nhttp://arxiv.org/abs/1504.01716\n\n[4] J. Umlauft, Y. Fanger, and S. Hirche, \u201cBayesian Uncertainty Modeling for Programming by\nDemonstration,\u201d in Proceedings of the IEEE Conference on Robotics and Automation, 2017, pp.\n6428\u20136434.\n\n[5] T. Beckers, D. Kuli\u00b4c, and S. Hirche, \u201cStable Gaussian Process based Tracking Control of\n\nEuler\u2013Lagrange Systems,\u201d Automatica, vol. 103, no. 23, pp. 390\u2013397, 2019.\n\n[6] F. Berkenkamp, R. Moriconi, A. P. Schoellig, and A. Krause, \u201cSafe Learning of Regions of\nAttraction for Uncertain, Nonlinear Systems with Gaussian Processes,\u201d in Proceedings of the\nIEEE Conference on Decision and Control, 2016, pp. 4661\u20134666.\n\n[7] Y. Fanger, J. Umlauft, and S. Hirche, \u201cGaussian Processes for Dynamic Movement Primitives\nwith Application in Knowledge-based Cooperation,\u201d in Proceedings of the IEEE Conference on\nIntelligent Robots and Systems, 2016, pp. 3913\u20133919.\n\n[8] N. Srinivas, A. Krause, S. M. Kakade, and M. W. Seeger, \u201cInformation-Theoretic Regret Bounds\nfor Gaussian Process Optimization in the Bandit Setting,\u201d IEEE Transactions on Information\nTheory, vol. 58, no. 5, pp. 3250\u20133265, 2012.\n\n[9] S. R. Chowdhury and A. Gopalan, \u201cOn Kernelized Multi-armed Bandits,\u201d in Proceedings of the\n\nInternational Conference on Machine Learning, 2017, pp. 844\u2013853.\n\n[10] J. Umlauft, L. P\u00f6hler, and S. Hirche, \u201cAn Uncertainty-Based Control Lyapunov Approach for\nControl-Af\ufb01ne Systems Modeled by Gaussian Process,\u201d IEEE Control Systems Letters, vol. 2,\nno. 3, pp. 483\u2013488, 2018.\n\n[11] J. Umlauft, T. Beckers, and S. Hirche, \u201cScenario-based Optimal Control for Gaussian Process\n\nState Space Models,\u201d in Proceedings of the European Control Conference, 2018.\n\n[12] H. K. Khalil, Nonlinear Systems; 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 2002.\n\n[13] C. E. Rasmussen and C. K. I. Williams, Gaussian Processes for Machine Learning. Cambridge,\n\nMA: The MIT Press, 2006.\n\n[14] H. Wendland, Scattered Data Approximation. Cambridge University Press, 2004.\n\n[15] Z. M. Wu and R. Schaback, \u201cLocal Error Estimates for Radial Basis Function Interpolation of\n\nScattered Data,\u201d IMA Journal of Numerical Analysis, vol. 13, no. 1, pp. 13\u201327, 1993.\n\n[16] R. Schaback, \u201cImproved Error Bounds for Scattered Data Interpolation by Radial Basis Func-\n\ntions,\u201d Mathematics of Computation, vol. 68, no. 225, pp. 201\u2013217, 2002.\n\n[17] M. Kanagawa, P. Hennig, D. Sejdinovic, and B. K. Sriperumbudur, \u201cGaussian Processes and\nKernel Methods: A Review on Connections and Equivalences,\u201d pp. 1\u201364, 2018. [Online].\nAvailable: http://arxiv.org/abs/1807.02582\n\n[18] S. Hubbert and T. M. Morton, \u201cLp-Error Estimates for Radial Basis Function Interpolation on\n\nthe Sphere,\u201d Journal of Approximation Theory, vol. 129, no. 1, pp. 58\u201377, 2004.\n\n[19] F. J. Narcowich, J. D. Ward, and H. Wendland, \u201cSobolev Error Estimates and a Bernstein Inequal-\nity for Scattered Data Interpolation via Radial Basis Functions,\u201d Constructive Approximation,\nvol. 24, no. 2, pp. 175\u2013186, 2006.\n\n[20] R. Beatson, O. Davydov, and J. Levesley, \u201cError Bounds for Anisotropic RBF Interpolation,\u201d\n\nJournal of Approximation Theory, vol. 162, no. 3, pp. 512\u2013527, 2010.\n\n[21] A. M. Stuart and A. L. Teckentrup, \u201cPosterior Consistency for Gaussian Process Approximations\nof Bayesian Posterior Distributions,\u201d Mathematics of Computation, vol. 87, no. 310, pp. 721\u2013\n753, 2018.\n\n[22] S. Mendelson, \u201cImproving the Sample Complexity using Global Data,\u201d IEEE Transactions on\n\nInformation Theory, vol. 48, no. 7, pp. 1977\u20131991, 2002.\n\n10\n\n\f[23] T. Zhang, \u201cLearning Bounds for Kernel Regression using Effective Data Dimensionality,\u201d\n\nNeural Computation, vol. 17, no. 9, pp. 2077\u20132098, 2005.\n\n[24] C. Cortes, M. Mohri, and A. Talwalkar, \u201cOn the Impact of Kernel Approximation on Learning\nAccuracy,\u201d Proceedings of 13th International Conference on Arti\ufb01cial Intelligece and Statistics,\nvol. 9, pp. 113\u2013120, 2010.\n\n[25] L. Shi, \u201cLearning Theory Estimates for Coef\ufb01cient-based Regularized Regression,\u201d Applied\n\nand Computational Harmonic Analysis, vol. 34, no. 2, pp. 252\u2013265, 2013.\n\n[26] L. H. Dicker, D. P. Foster, and D. Hsu, \u201cKernel Ridge vs. Principal Component Regression:\nMinimax Bounds and the Quali\ufb01cation of Regularization Operators,\u201d Electronic Journal of\nStatistics, vol. 11, no. 1, pp. 1022\u20131047, 2017.\n\n[27] F. Berkenkamp, A. P. Schoellig, M. Turchetta, and A. Krause, \u201cSafe Model-based Reinforcement\nLearning with Stability Guarantees,\u201d in Advances in Neural Information Processing Systems,\n2017.\n\n[28] A. van der Vaart and H. van Zanten, \u201cInformation Rates of Nonparametric Gaussian Process\n\nMethods,\u201d Journal of Machine Learning Research, vol. 12, pp. 2095\u20132119, 2011.\n\n[29] R. Adler and J. Taylor, Random Fields and Geometry. Springer Science & Business Media,\n\n2007.\n\n[30] W. Wang, R. Tuo, and C. F. J. Wu, \u201cOn Prediction Properties of Kriging: Uniform Error Bounds\n\nand Robustness,\u201d Journal of the American Statistical Society, pp. 1\u201338, 2019.\n\n[31] J. Mercer, \u201cFunctions of Positive and Negative Type, and their Connection with the Theory of\nIntegral Equations,\u201d Philosophical Transactions of the Royal Society A: Mathematical, Physical\nand Engineering Sciences, vol. 209, no. 441-458, pp. 415\u2013446, 1909.\n\n[32] I. Steinwart, \u201cOn the In\ufb02uence of the Kernel on the Consistency of Support Vector Machines,\u201d\n\nJournal of Machine Learning Research, vol. 2, pp. 67\u201393, 2001.\n\n[33] S. Ghosal and A. Roy, \u201cPosterior Consistency of Gaussian Process Prior for Nonparametric\n\nBinary Regression,\u201d The Annals of Statistics, vol. 34, no. 5, pp. 2413\u20132429, 2006.\n\n[34] R. M. Dudley, \u201cThe Sizes of Compact Subsets of Hilbert Space and Continuity of Gaussian\n\nProcesses,\u201d Journal of Functional Analysis, vol. 1, no. 3, pp. 290\u2013330, 1967.\n\n[35] M. Talagrand, \u201cSharper Bounds for Gaussian and Empirical Processes,\u201d The Annals of Proba-\n\nbility, vol. 22, no. 1, pp. 28\u201376, 1994.\n\n[36] J. Gonz\u00e1lez, Z. Dai, P. Hennig, and N. D. Lawrence, \u201cBatch Bayesian Optimization via Local\nPenalization,\u201d in Proceedings of the International Conference on Arti\ufb01cial Intelligence and\nStatistics, 2016, pp. 648\u2013657.\n\n[37] B. Laurent and P. Massart, \u201cAdaptive Estimation of a Quadratic Functional by Model Selection,\u201d\n\nThe Annals of Statistics, vol. 28, no. 5, pp. 1302\u20131338, 2000.\n\n[38] A. Lederer, J. Umlauft, and S. Hirche, \u201cPosterior Variance Analysis of Gaussian\nProcesses with Application to Average Learning Curves,\u201d 2019. [Online]. Available:\nhttp://arxiv.org/abs/1906.01404\n\n[39] J. Umlauft, T. Beckers, M. Kimmel, and S. Hirche, \u201cFeedback Linearization using Gaussian\nProcesses,\u201d in Proceedings of the IEEE Conference on Decision and Control, 2017, pp. 5249\u2013\n5255.\n\n[40] J. Umlauft, A. Lederer, and S. Hirche, \u201cLearning Stable Gaussian Process State Space Models,\u201d\n\nin Proceedings of the American Control Conference, 2017, pp. 1499\u20131504.\n\n[41] R. M. Murray, Z. Li, and S. Shankar Sastry, A Mathematical Introduction to Robotic Manipula-\n\ntion. CRC Press, 1994.\n\n[42] N. Srinivas, A. Krause, S. Kakade, and M. Seeger, \u201cGaussian Process Optimization in the Bandit\nSetting: No Regret and Experimental Design,\u201d in Proceedings of the International Conference\non Machine Learning, 2010, pp. 1015\u20131022.\n\n11\n\n\f", "award": [], "sourceid": 330, "authors": [{"given_name": "Armin", "family_name": "Lederer", "institution": "Technical University of Munich"}, {"given_name": "Jonas", "family_name": "Umlauft", "institution": "Technical University of Munich"}, {"given_name": "Sandra", "family_name": "Hirche", "institution": "Technische Universitaet Muenchen"}]}