nips nips2013 nips2013-54 knowledge-graph by maker-knowledge-mining

54 nips-2013-Bayesian optimization explains human active search

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Author: Ali Borji, Laurent Itti

Abstract: Many real-world problems have complicated objective functions. To optimize such functions, humans utilize sophisticated sequential decision-making strategies. Many optimization algorithms have also been developed for this same purpose, but how do they compare to humans in terms of both performance and behavior? We try to unravel the general underlying algorithm people may be using while searching for the maximum of an invisible 1D function. Subjects click on a blank screen and are shown the ordinate of the function at each clicked abscissa location. Their task is to ﬁnd the function’s maximum in as few clicks as possible. Subjects win if they get close enough to the maximum location. Analysis over 23 non-maths undergraduates, optimizing 25 functions from different families, shows that humans outperform 24 well-known optimization algorithms. Bayesian Optimization based on Gaussian Processes, which exploits all the x values tried and all the f (x) values obtained so far to pick the next x, predicts human performance and searched locations better. In 6 follow-up controlled experiments over 76 subjects, covering interpolation, extrapolation, and optimization tasks, we further conﬁrm that Gaussian Processes provide a general and uniﬁed theoretical account to explain passive and active function learning and search in humans. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 Bayesian optimization explains human active search Ali Borji Department of Computer Science USC, Los Angeles, 90089 borji@usc. [sent-1, score-0.463]

2 Many optimization algorithms have also been developed for this same purpose, but how do they compare to humans in terms of both performance and behavior? [sent-5, score-0.303]

3 Their task is to ﬁnd the function’s maximum in as few clicks as possible. [sent-8, score-0.509]

4 Analysis over 23 non-maths undergraduates, optimizing 25 functions from different families, shows that humans outperform 24 well-known optimization algorithms. [sent-10, score-0.333]

5 Bayesian Optimization based on Gaussian Processes, which exploits all the x values tried and all the f (x) values obtained so far to pick the next x, predicts human performance and searched locations better. [sent-11, score-0.371]

6 Here, inspired by the optimization literature, we design and conduct a series of experiments to understand human search and active learning behavior. [sent-19, score-0.444]

7 We compare and contrast humans with standard optimization algorithms, to learn how well humans perform 1D function optimization and to discover which algorithm best approaches or explains human search strategies. [sent-20, score-0.948]

8 We aim to decipher how humans choose the next x to be queried when attempting to locate the maximum of an unknown 1D function. [sent-22, score-0.301]

9 We focus on the following questions: Do humans perform local search (for instance by randomly choosing a location and following the gradient of the function, e. [sent-23, score-0.405]

10 Do the sets of sample x locations queried by humans resemble those of some algorithms more than others? [sent-28, score-0.342]

11 Can Gaussian processes [9] offer a unifying theory of human function learning and active search? [sent-31, score-0.335]

12 1 2 Experiments and Results We seek to study human search behavior directly on 1D function optimization, for the ﬁrst time systematically and explicitly. [sent-32, score-0.323]

13 8 Original function Histogram of clicks Histogram of first clicks 0. [sent-49, score-0.906]

14 During search for the function’s maximum, a red was drawn for each cases and to investigate the generaliza- Right: A sample dot at (x, f (x))pdf of human clicks. [sent-56, score-0.323]

15 In the majority of cases, the distribution of clicks starts with a strong leftward bias for the ﬁrst clicks, then progressively focusing around the true function maximum as subjects make more clicks and approach the maximum. [sent-68, score-1.22]

16 They were asked to “ﬁnd the maximum value (highest point) of a function in as few clicks as possible”. [sent-73, score-0.522]

17 Starting from a blank screen, subjects could click on any abscissa x location, and we would show them the corresponding f (x) ordinate. [sent-75, score-0.391]

18 , as opposed to automatically terminating a trial if humans happened to click near the maximum). [sent-81, score-0.434]

19 We highlighted to subjects that they should decide carefully where to click next, to minimize the number of clicks before hitting the maximum location. [sent-88, score-0.843]

20 Before the experiment, we had a training session in which subjects completed 5 trials on a different set of functions than those used in the real experiment. [sent-90, score-0.302]

21 The purpose was for subjects to understand the task and familiarize themselves to the general complexity and shapes of functions (i. [sent-91, score-0.299]

22 To prohibit subjects from using the vertical extent of the screen to guess the maximum location, we randomly elevated or lowered the function plot. [sent-96, score-0.337]

23 To measure to what degree human search behavior deviates from a random process, we devise a Random search algorithm which chooses the next point uniformly random from [−300 300] without replacement. [sent-176, score-0.416]

24 To generate a starting point for algorithms, we randomly sampled from 3 the distribution of human ﬁrst clicks (over all subjects and functions, p(x1 ); see Fig. [sent-192, score-0.928]

25 As in the behavioral experiment, after termination of an algorithm, a hit is declared when: ∃ xi ∈ B : |xi − argmaxx f (x)| ≤ xTolh , where set B includes the history of searched locations in a search trial. [sent-194, score-0.311]

26 As shown, humans are better than all algorithms tested, if hit rate and function calls are weighed equally (i. [sent-197, score-0.391]

27 It can achieve better-than-human hit rate, with a number of function calls which is smaller than BO algorithms, though still higher than humans (it was not able to reach human performance with equal number of function calls). [sent-204, score-0.621]

28 These algorithms were chosen because their performance curve in the ﬁrst analysis intersected a window where accuracy is half of humans and function call is twice as humans (black rectangle in Fig. [sent-210, score-0.538]

29 , point matching), 3) agreement between probability distributions of searched locations by all humans and an algorithm, and 4) agreement between pdfs of normalized step sizes 1 Human 10 (to [0 1] on each trial). [sent-216, score-0.499]

30 These two algorithms also show higher agreement to humans in terms of searched locations (Fig. [sent-240, score-0.446]

31 The clearest pattern happens over step size agreement with BO methods (except GP-MV) being closest to humans (Fig. [sent-243, score-0.305]

32 GP-MPI and GP-UCB show higher resemblance to human search behavior over all scores. [sent-246, score-0.323]

33 Further, we measure the regret of algorithms and humans deﬁned as fmax (·) − f ∗ where f ∗ is the best value found so far for up to 15 function calls averaged over all trials. [sent-247, score-0.359]

34 Furthermore, the sequential nature of the BO and updating the GP posterior after each function call seems a natural strategy humans might be employing, and 3) GP models explain function learning in humans over simple functional relationships (linear and quadratic) [24]. [sent-254, score-0.538]

35 Panel (e) shows average regret of algorithms and humans (f ∗ is normalized to fmax for each trial separately). [sent-282, score-0.366]

36 We thus designed 6 additional controlled experiments to further explore GP as a uniﬁed computational principle guiding human function learning and active search. [sent-283, score-0.317]

37 In experiments 2 to 5, subjects performed interpolation and extrapolation tasks, as well as active versions of these tasks by choosing points to help them learn about the functions. [sent-291, score-0.607]

38 In experiments 6 and 7, we then return to the optimization task, for a detailed model-based analysis of human search behavior over functions from the same family. [sent-292, score-0.387]

39 In experiments 2 to 6, we ﬁtted a GP to different types of functions using the same set of (x, y) points shown to subjects during training (Fig. [sent-301, score-0.297]

40 In the interpolation task, on each function, subjects were shown 4 points x ∈ {−300, a, b, 300} along with their f (x) values. [sent-310, score-0.392]

41 In the active interpolation task, the same 4 points as in interpolation were shown to subjects. [sent-313, score-0.359]

42 a shows mean distance of human clicks from the GP mean at x = 0 over test trials (averaged over absolute pairwise distances between clicks and the GP) in the interpolation task. [sent-321, score-1.422]

43 Distances of the GP and the actual function from humans are the same over Deg2 and Deg3 functions (no signiﬁcant difference in medians, Wilcoxon signed-rank test, p > 0. [sent-323, score-0.329]

44 Interestingly, on Deg5 functions, GP is closer to human clicks than the actual function (signed-rank test, p = 0. [sent-325, score-0.733]

45 053) implying that GP captures clicks well in this case. [sent-326, score-0.453]

46 GP did ﬁt the human data even better than the actual function, thereby lending support to our hypothesis that GP may be a reasonable approximation to human function estimation. [sent-327, score-0.49]

47 Could it be that subjects locally ﬁt a line to the two middle points to guess f (0)? [sent-328, score-0.314]

48 To evaluate this hypothesis, we measured the distance, at x = 0, from human clicks to a line passing through (a, f (a)) and (b, f (b)). [sent-329, score-0.702]

49 We then calculate the average of the pairwise distances between human clicks and 200 draws from each random model. [sent-342, score-0.701]

50 Both random models fail to predict human answers on all types of functions (signiﬁcantly worse than GP, signed-rank test ps < 0. [sent-343, score-0.3]

51 a (inset) demonstrates similar uncertainty patterns for humans and the GP, showing that both uncertainties (at x = 0) rise as functions become more complicated. [sent-348, score-0.317]

52 If this is correct, we expect that humans will tend to click on high uncertainty regions (according to GP std) in the active interpolation task (see Fig. [sent-350, score-0.636]

53 b shows the average of GP standard deviation at locations of human selections. [sent-354, score-0.322]

54 Another possibility is because subjects had slight preference to click toward center. [sent-359, score-0.358]

55 However, GP std at human-selected locations was signiﬁcantly higher than the GP std at random and SRand points, over all types of functions (signed-rank test, ps < 1e–4; non-signiﬁcant on Deg2 vs. [sent-360, score-0.6]

56 This result suggests that since humans did not know in advance where a follow-up query might happen, they chose high-uncertainty locations according to GP, as clicking at those locations would most shrink the overall uncertainty about the function. [sent-363, score-0.433]

57 In the extrapolation task, subjects were asked to guess the function value at x = 200 as accurately as possible (Fig. [sent-372, score-0.438]

58 In the active extrapolation task, subjects were asked to choose the most informative 4th point in [−300 100] regarding estimating f (200). [sent-374, score-0.497]

59 c, in alignment with interpolation results, humans are good at Deg2 and Deg3 but fail on Deg5, and so does the GP model. [sent-379, score-0.394]

60 Here again, with Deg5, GP and humans are closer to each other than to the actual function, further suggesting that their behaviors and errors are similar. [sent-380, score-0.319]

61 SRand performs better than uniform random, indicating 16 14 human random shuffled Random max std min std 12 10 8 6 4 2 0 Deg2 Deg3 Deg5 Figure 7: a) Mean distance of human clicks from models. [sent-387, score-1.496]

62 6 a) distance of human clicks from location of max Q b) distance of random clicks from location of max Q c) left: mean selected function values right: mean selected GP values 0. [sent-395, score-1.386]

63 8 100 100 180 d) mean selected GP std values 180 rand − mean human1 − mean 140 human1 − std rand − std 140 0. [sent-396, score-0.871]

64 a and b) distance of human and random clicks from locations of max Q (i. [sent-403, score-0.802]

65 As in the interpolation task, GP and human standard deviations rise as functions become more complex (Fig. [sent-414, score-0.385]

66 d), similar to active interpolation, shows that humans tended to choose locations with signiﬁcantly higher uncertainty than uniform and SRand points, for all function types (ps < 0. [sent-419, score-0.469]

67 Some subjects in this task tended to click toward the right (close to 100), maybe to obtain a better idea of the curvature between 100 and 200. [sent-421, score-0.404]

68 This is perhaps why the ratio of human std to max std is lower in active extrapolation compared to active interpolation (0. [sent-422, score-1.15]

69 They were allowed to make two equally important clicks and were shown the function value after each one. [sent-431, score-0.453]

70 In the ﬁrst one, we measure the mean distance of human clicks (1st and 2nd clicks) from the location of the maximum GP mean and maximum GP standard deviation (Fig. [sent-436, score-0.908]

71 We hypothesized that the human ﬁrst click would be at a location of high GP variance (to reduce uncertainty about the function), while the second click would be close to the location of highest GP mean (estimated function maximum). [sent-440, score-0.596]

72 However, results showed that human 1st clicks were close to the max GP mean and not very close to the max GP std. [sent-441, score-0.757]

73 Human 2nd clicks were even closer (signedrank test, p < 0. [sent-442, score-0.473]

74 001) to the max GP mean and further away from the max GP std (p < 0. [sent-443, score-0.311]

75 Repeating this analysis for random clicks (uniform and SRand) shows quite the opposite trend (Fig. [sent-446, score-0.453]

76 Random locations are further apart from maximum of the GP mean (compared to human clicks) while being closer to the maximum of the GP std point (compared to human clicks). [sent-449, score-0.89]

77 This cross pattern between human and random clicks (wrt. [sent-450, score-0.683]

78 Distances of human clicks from the max GP mean and max GP std rise as functions become more complicated. [sent-452, score-1.024]

79 d shows that humans chose points with signiﬁcantly less std (normalized to the entire function) in their 2nd clicks compared to random and ﬁrst clicks. [sent-462, score-0.981]

80 Human 1st clicks have higher std than uniform random clicks. [sent-463, score-0.69]

81 7 mean distance from max GP mean and std 1 b) Results. [sent-475, score-0.355]

82 8 subjects and functions, and average clicks 200 of 8. [sent-481, score-0.698]

83 4 −2 clicks on each trial, and exploited this GP 100 10 0 5 10 15 to evaluate the next click of the same sub0. [sent-488, score-0.566]

84 Renormalized GP std human− max GP std 0 0 sults are shown in Fig. [sent-491, score-0.493]

85 The regret of the 1 3 5 7 9 11 13 15 0 5 10 15 number of clicks number of clicks GP model and humans decline with more Figure 9: Exploration vs. [sent-493, score-1.192]

86 clicks, implying that humans chose informative clicks regarding optimization (ﬁgure inset). [sent-495, score-0.756]

87 a shows that subjects get closer to the location of maximum GP mean and further away from max GP std (for 15 clicks). [sent-499, score-0.632]

88 b shows the normalized mean and standard deviation of human clicks (from the GP model), averaged over all trials. [sent-502, score-0.738]

89 Interestingly, we observe that humans tended to click on higher uncertainty regions (according to GP) in their ﬁrst 6 clicks (average over all subjects and functions), then gradually relying more on the GP mean (i. [sent-505, score-1.156]

90 Results of optimization tasks suggest that human clicks during search for a maximum of a 1D function can be predicted by a Gaussian process model. [sent-509, score-0.842]

91 3 regret human random shuffled random GP Discussion and Conclusion Our contributions are twofold: First, we found a striking capability of humans in 1D function optimization. [sent-510, score-0.579]

92 In spite of the relative naivety of our subjects (not maths or engineering majors), the high human efﬁciency in our search task does open the challenge that even more efﬁcient optimization algorithms must be possible. [sent-511, score-0.647]

93 Additional pilot investigations not shown here suggest that humans may perform even better in optimization when provided with ﬁrst and second derivatives. [sent-512, score-0.303]

94 Second, we showed that Gaussian processes provide a reasonable (though not perfect) unifying theoretical account of human function learning, active learning, and search (GP plus a selection strategy). [sent-515, score-0.428]

95 Results of experiments 2 to 5 lead to an interesting conclusion: In interpolation and extrapolation tasks, subjects try to minimize the error between their estimation and the actual function, while in active tasks they change their objective function to explore uncertain regions. [sent-516, score-0.615]

96 In the optimization task, subjects progressively sample the function, update their belief and use this belief again to ﬁnd the location of maximum. [sent-517, score-0.359]

97 Yet, while they showed that Gaussian processes can predict human errors and difﬁculty in function learning, here we focused on explaining human active behavior with GP, thus extending explanatory power of GP one step ahead. [sent-523, score-0.565]

98 Najemnik and Geisler [6, 28, 29] proposed a Bayesian ideal-observer model to explain human eye movement strategies during visual search for a small Gabor patch hidden in noise. [sent-525, score-0.374]

99 [30] studied human active learning on the well-understood problem of ﬁnding the threshold in a binary search problem. [sent-531, score-0.41]

100 Geisler, “Eye movement statistics in humans are consistent with an optimal search strategy,” Journal of Vision, vol. [sent-693, score-0.379]

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