nips nips2008 nips2008-32 knowledge-graph by maker-knowledge-mining

32 nips-2008-Bayesian Kernel Shaping for Learning Control

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Author: Jo-anne Ting, Mrinal Kalakrishnan, Sethu Vijayakumar, Stefan Schaal

Abstract: In kernel-based regression learning, optimizing each kernel individually is useful when the data density, curvature of regression surfaces (or decision boundaries) or magnitude of output noise varies spatially. Previous work has suggested gradient descent techniques or complex statistical hypothesis methods for local kernel shaping, typically requiring some amount of manual tuning of meta parameters. We introduce a Bayesian formulation of nonparametric regression that, with the help of variational approximations, results in an EM-like algorithm for simultaneous estimation of regression and kernel parameters. The algorithm is computationally efﬁcient, requires no sampling, automatically rejects outliers and has only one prior to be speciﬁed. It can be used for nonparametric regression with local polynomials or as a novel method to achieve nonstationary regression with Gaussian processes. Our methods are particularly useful for learning control, where reliable estimation of local tangent planes is essential for adaptive controllers and reinforcement learning. We evaluate our methods on several synthetic data sets and on an actual robot which learns a task-level control law. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Previous work has suggested gradient descent techniques or complex statistical hypothesis methods for local kernel shaping, typically requiring some amount of manual tuning of meta parameters. [sent-3, score-0.217]

2 We introduce a Bayesian formulation of nonparametric regression that, with the help of variational approximations, results in an EM-like algorithm for simultaneous estimation of regression and kernel parameters. [sent-4, score-0.317]

3 It can be used for nonparametric regression with local polynomials or as a novel method to achieve nonstationary regression with Gaussian processes. [sent-6, score-0.661]

4 We evaluate our methods on several synthetic data sets and on an actual robot which learns a task-level control law. [sent-8, score-0.122]

5 Most algorithms start with parameterizations that are the same for all kernels, independent of where in data space the kernel is used, but later recognize the advantage of locally adaptive kernels [2, 3, 4]. [sent-10, score-0.263]

6 Such locally adaptive kernels are useful in scenarios where the data characteristics vary greatly in different parts of the workspace (e. [sent-11, score-0.126]

7 For instance, in Gaussian process (GP) regression, using a nonstationary covariance function, e. [sent-14, score-0.473]

8 Previous work has suggested gradient descent techniques with cross-validation methods or involved statistical hypothesis testing for optimizing the shape and size of a kernel in a learning system [6, 7]. [sent-18, score-0.137]

9 In this paper, we consider local kernel shaping by averaging over data samples with the help of locally polynomial models and formulate this approach, in a Bayesian framework, for both function approximation with piecewise linear models and nonstationary GP regression. [sent-19, score-1.085]

10 Our local kernel shaping algorithm is computationally efﬁcient (capable of handling large data sets), can deal with functions of strongly varying curvature, data density and output noise, and even rejects outliers automatically. [sent-20, score-0.679]

11 Our approach to nonstationary GP regression differs from previous work by avoiding Markov Chain Monte Carlo (MCMC) sampling [8, 9] and by exploiting the full nonparametric characteristics of GPs in order to accommodate nonstationary data. [sent-21, score-0.911]

12 One of the core application domains for our work is learning control, where computationally efﬁcient function approximation and highly accurate local linearizations from data are crucial for deriving controllers and for optimizing control along trajectories [10]. [sent-22, score-0.212]

13 Our ﬁnal evaluations illustrate such a scenario by learning an inverse kinematics model for a real robot arm. [sent-25, score-0.297]

14 2 Bayesian Local Kernel Shaping We develop our approach in the context of nonparametric locally weighted regression with locally linear polynomials [11], assuming, for notational simplicity, only a one-dimensional output— extensions to multi-output settings are straightforward. [sent-26, score-0.328]

15 We assume a training set of N samples, N D = {xi , yi }i=1 , drawn from a nonlinear function y = f (x) + that is contaminated with meanzero (but potentially heteroscedastic) noise . [sent-27, score-0.113]

16 We wish to approximate a locally linear model of this function at a query point xq ∈ d×1 in order to make a prediction yq = bT xq , where b ∈ d×1 . [sent-29, score-0.341]

17 We assume the existence of a spatially localized weighting kernel wi = K (xi , xq , h) that assigns a weight to every {xi , yi } according to its Euclidean distance in input space from the query point xq . [sent-30, score-0.67]

18 The bandwidth h ∈ d×1 of the kernel is the crucial parameter that determines the local model’s quality of ﬁt. [sent-32, score-0.287]

19 1 Model For the locally linear model at the query i = 1,. [sent-35, score-0.131]

20 z1 d model yi = bT xi so that yi = m=1 zim + zi1 zid zi2 bd b1 b2 É , where zim = bT xim + zm and zm ∼ m 2 Normal (0, ψzm ), ∼ Normal 0, σ are both additive noise terms. [sent-41, score-0.858]

21 Note that xim = wi1 É x wid xi1 xi2 wi2 id [xim 1]T and bm = [bm bm0 ]T , where xim is the mth coefﬁcient of xi , bm is the mth É hd h2 h1 coefﬁcient of b and bm0 is the offset value. [sent-42, score-1.153]

22 In contrast to classical treatments of Bayesian weighted regression [13] where the weights enter as a heteroscedastic correction on the noise variance of each data sample, we associate a scalar indicator-like weight, wi ∈ {0, 1}, with each sample {xi , yi } in D. [sent-51, score-0.335]

23 The sample is fully included in d the local model if wi = 1 and excluded if wi = 0. [sent-52, score-0.322]

24 We deﬁne the weight wi to be wi = m=1 wim , where wim is the weight component in the mth input dimension. [sent-53, score-1.02]

25 While previous methods model the weighting kernel K as some explicit function, we model the weights wim as Bernoulli-distributed random variables, i. [sent-54, score-0.544]

26 , p(wim ) ∼ Bernoulli(qim ), choosing a symmetric bell-shaped function for the parameter qim : qim = 1/(1 + (xim − xqm )2r hm ). [sent-56, score-0.552]

27 xqm is the mth coefﬁcient of xq , hm is the mth coefﬁcient of h, and r > 0 is a positive integer1 . [sent-57, score-0.531]

28 As pointed out in [11], the particular mathematical formulation of a weighting kernel is largely computationally irrelevant for locally weighted learning. [sent-58, score-0.346]

29 Our choice of function for qim was dominated by the desire to obtain analytically tractable learning updates. [sent-59, score-0.213]

30 We place a Gamma prior over the bandwidth hm , i. [sent-60, score-0.224]

31 , p(hm ) ∼ Gamma (ahm0 , bhm0 ) where ahm0 and bhm0 are parameters of the Gamma distribution, to ensure that a positive weighting kernel width. [sent-62, score-0.22]

32 We can maximize this incomplete log likelihood by maximizing the expected value of the complete log likelihood p(y, Z, b, w, h, σ 2 , ψz |X) = N 2 i=1 p(yi , zi , b, wi , h, σ , ψz |xi ). [sent-65, score-0.214]

33 To address this, we use a variational approach on concave/convex functions suggested by [16] to produce analytically tractable expressions. [sent-67, score-0.112]

34 We can ﬁnd a lower bound on the 2r term so that − log(1 + xim − xqm )2r ≥ −λim (xim − xqm ) hm , where λim is a variational parameter to be optimized in the M-step of our ﬁnal EM-like algorithm. [sent-68, score-0.705]

35 Our choice of weighting kernel allows us to ﬁnd a lower bound to L in this manner. [sent-69, score-0.22]

36 We explored the use of other weighting kernels (e. [sent-70, score-0.117]

37 , a quadratic negative exponential), but had issues with ﬁnding a lower bound to the problematic terms in log p(wim ) such that analytically tractable inference for hm could be done. [sent-72, score-0.232]

38 Since this is an analytically tractable expression, a lower bound can be formulated using a technique from variational calculus where we make a factorial approximation of the true posterior, e. [sent-75, score-0.112]

39 This is a result of wi being deﬁned as the product of weights in all dimensions. [sent-80, score-0.121]

40 The posterior mean of wim is then d p(wim = 1|yi , zi , xi , θ, wi,k=m ) , and wi = m=1 wim , where . [sent-81, score-0.931]

41 Adjusting r affects how long the tails of the kernel are. [sent-84, score-0.137]

42 Closer examination of the expression for bm shows that it is a standard Bayesian weighted regression update [13], i. [sent-87, score-0.352]

43 , a data sample i with lower weight wi will be downweighted in the regression. [sent-89, score-0.179]

44 Since the weights are inﬂuenced by the residual error at each data point (see posterior update for wim ), an outlier will be downweighted appropriately and eliminated from the local model. [sent-90, score-0.466]

45 2 shows how local kernel shaping is able to ignore outliers that a classical GP ﬁts. [sent-92, score-0.632]

46 5 noise variance ψzm,0 Finally, the initial h of the weighting kernel 0 should be set so that the kernel is broad and wide. [sent-99, score-0.393]

47 This efﬁciency arises from the introduction of the hidden random variables zi , which allows zi and Σzi |yi ,xi to be computed in O(d) and avoids a d × d matrix inversion which would typically require O(d3 ). [sent-106, score-0.186]

48 , [5], require O(N 3 ) + O(N 2 ) for training and prediction, while other more efﬁcient stationary GP methods, e. [sent-109, score-0.129]

49 3 Extension to Gaussian Processes We can apply the algorithm in section 2 not only to locally weighted learning with linear models, but also to derive a nonstationary GP method. [sent-114, score-0.538]

50 A GP is deﬁned by a mean and and a covariance function, where the covariance function K captures dependencies between any two points as a function of the corresponding inputs, i. [sent-115, score-0.122]

51 Standard GP models use a stationary covariance function, where the covariance between any two points in the training data is a function of the distances |xi − xj |, not of their locations. [sent-120, score-0.251]

52 Various methods have been proposed to specify nonstationary GPs. [sent-124, score-0.412]

53 Given the data set D drawn from the function y = f (x)+ , as previously introduced in section 2, we propose an approach to specify a nonstationary covariance function. [sent-126, score-0.473]

54 Assuming the use of a quadratic negative exponential covariance function, the covariance function of a stationary GP is k(xi , xj ) = d 2 v1 exp(−0. [sent-127, score-0.248]

55 5 m=1 hm (xim − xjm )2 ) + v0 , where the hyperparameters {h1 , h2 , . [sent-128, score-0.175]

56 In a nonstationary GP, the covariance function could then take the form2 k(xi , xj ) = d h h im jm 2 v1 exp −0. [sent-132, score-0.529]

57 5 m=1 (xim − xjm )2 (him +hjm ) +v0 , where him is the bandwidth of the local model centered at xim and hjm is the bandwidth of the local model centered at xjm . [sent-133, score-0.676]

58 , N using our proposed local kernel shaping algorithm and then optimize the hyperparameters v0 and v1 . [sent-137, score-0.581]

59 , the bandwidth of the local model centered at xq . [sent-140, score-0.255]

60 Importantly, since the covariance function of the GP is derived from locally constant models, we learn with locally constant, instead of locally linear, polynomials. [sent-141, score-0.337]

61 We use r = 1 for the weighting kernel in order keep the degree of nonlinearity consistent with that in the covariance function (i. [sent-142, score-0.281]

62 Even though the weighting kernel used in the local kernel shaping algorithm is not a quadratic negative exponential, it has a similar bell shape, but with a ﬂatter top and shorter tails. [sent-145, score-0.824]

63 Because of this, our augmented GP is an approximated form of a nonstationary GP. [sent-146, score-0.484]

64 Nonetheless, it is able to capture nonstationary properties of the function f without needing MCMC sampling, unlike previously proposed nonstationary GP methods [8, 9]. [sent-147, score-0.824]

65 1 Synthetic Data First, we show our local kernel shaping algorithm’s bandwidth adaptation abilities on several synthetic data sets, comparing it to a stationary GP and our proposed augmented nonstationary GP. [sent-149, score-1.238]

66 3025; the data set for function ii) consists of 250 training samples, 101 test inputs and an output signal-to-noise ratio (SNR) of 10; and the data set for function iii) has 50 training samples, 21 test inputs and an output SNR of 100. [sent-152, score-0.146]

67 3 shows the predicted outputs of a stationary GP, augmented nonstationary GP and the local kernel shaping algorithm for data sets i)-iii). [sent-154, score-1.168]

68 The local kernel shaping algorithm smoothes over regions where a stationary GP overﬁts and yet, it still manages to capture regions of highly varying curvature, as seen in Figs. [sent-155, score-0.73]

69 When the data looks linear, the algorithm opens up the weighting kernel so that all data samples are considered, as Fig. [sent-158, score-0.22]

70 Our proposed augmented nonstationary GP also can handle the nonstationary nature of the data sets as well, and its performance is quantiﬁed in Table 1. [sent-160, score-0.896]

71 Returning to our motivation to use these algorithms to obtain linearizations for learning control, it is important to realize that the high variations from ﬁtting noise, as shown by the stationary GP in Fig. [sent-161, score-0.153]

72 4 shows results of the local kernel shaping algorithm and the proposed augmented nonstationary GP on the “real-world” motorcycle data set [20] consisting of 133 samples (with 80 equally spaced input query points used for prediction). [sent-165, score-1.128]

73 We also show results from a previously proposed MCMC-based nonstationary GP method: an alternate inﬁnite mixture of GP experts [9]. [sent-166, score-0.486]

74 We can see that the augmented nonstationary GP and the local kernel shaping algorithm both capture the leftmost ﬂatter region of the function, as well as some of the more nonlinear and noisier regions after 30msec. [sent-167, score-1.088]

75 2 Robot Data Next, we move on to an example application: learning an inverse kinematics model for a 3 degree-offreedom (DOF) haptic robot arm (manufactured by SensAble, shown in Fig. [sent-169, score-0.35]

76 This will allow us to verify that the kernel shaping algo2 This is derived from the deﬁnition of K as a positive semi-deﬁnite matrix, i. [sent-171, score-0.501]

77 Figures on the bottom show the bandwidths learnt by local kernel shaping and the corresponding weighting kernels (in dotted black lines) for input query points (shown in red circles). [sent-175, score-0.813]

78 From ˙ ˙ this data, we ﬁrst learn a forward kinematics model: x = J(q)q, where J(q) is the Jacobian matrix. [sent-179, score-0.196]

79 ˙ ˙ The transformation from q to x can be assumed to be locally linear at a particular conﬁguration q of the robot arm. [sent-180, score-0.18]

80 We learn the forward model using kernel shaping, building a local model around each training point only if that point is not already sufﬁciently covered by an existing local model (e. [sent-181, score-0.372]

81 Using insights into robot geometry, we localize the models only with respect to q while the regression of each model is trained only on a mapping ˙ ˙ from q to x—these geometric insights are easily incorporated as priors in the Bayesian model. [sent-185, score-0.147]

82 We artiﬁcially introduce a redundancy in our inverse kinematics problem on the 3-DOF arm by ˙ specifying the desired trajectory (x, x) only in terms of x, z positions and velocities, i. [sent-187, score-0.33]

83 Analytically, the inverse kinematics ∂g ˙ ˙ equation is q = J# (q)x − α(I − J# J) ∂q , where J # (q) is the pseudo-inverse of the Jacobian. [sent-190, score-0.209]

84 To learn a model for J# , we can reuse the local regions of q from the forward model, where J# is also locally linear. [sent-192, score-0.244]

85 This ensures that the learnt inverse model chooses a solution which produces a y that pushes the y coordinate toward ydes . [sent-196, score-0.161]

86 We invert each ˙ forward local model using a weighted linear regression, where each data point is weighted by the weight from the forward model and additionally weighted by the reward. [sent-197, score-0.309]

87 Method Stationary GP Augmented nonstationary GP Local Kernel Shaping Function i) 0. [sent-201, score-0.412]

88 This demonstrates that kernel shaping is an effective learning algorithm for use in robot control learning applications. [sent-217, score-0.623]

89 Applying any arbitrary nonlinear regression method (such as a GP) to the inverse kinematics problem would, in fact, lead to unpredictably bad performance. [sent-218, score-0.268]

90 The inverse kinematics problem is a one-tomany mapping and requires careful design of a learning problem to avoid problems with non-convex solution spaces [22]. [sent-219, score-0.209]

91 Our suggested method of learning linearizations with a forward mapping (which is a proper function), followed by learning an inverse mapping within the local region of the forward mapping, is one of the few clean approaches to the problem. [sent-220, score-0.29]

92 Instead of using locally linear methods, one could also use density-based estimation techniques like mixture models [23]. [sent-221, score-0.116]

93 For these reasons, applying a MCMC-type approach or GP-based method to the inverse kinematics problem was omitted as a comparison. [sent-223, score-0.209]

94 5 Discussion We presented a full Bayesian treatment of nonparametric local multi-dimensional kernel adaptation that simultaneously estimates the regression and kernel parameters. [sent-224, score-0.441]

95 We show that our local kernel shaping method is particularly useful for learning control, demonstrating results on an inverse kinematics problem, and envision extensions to more complex problems with redundancy, Desired Analytical IK 0. [sent-226, score-0.79]

96 1 (b) Desired versus actual trajectories Figure 5: Desired versus actual trajectories for SensAble Phantom robot arm e. [sent-240, score-0.187]

97 In its current form, our Bayesian kernel shaping algorithm is built for high-dimensional inputs due to its low computational complexity— it scales linearly with the number of input dimensions. [sent-247, score-0.525]

98 Other future extensions include an online implementation of the local kernel shaping algorithm. [sent-250, score-0.581]

99 Data-driven bandwidth selection in local polynomial ﬁtting: Variable bandwidth and spatial adaptation. [sent-289, score-0.22]

100 Bayesian inference for nonstationary spatial covariance structure via spatial deformations. [sent-380, score-0.473]

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