nips nips2011 nips2011-97 knowledge-graph by maker-knowledge-mining

97 nips-2011-Finite Time Analysis of Stratified Sampling for Monte Carlo

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Author: Alexandra Carpentier, Rémi Munos

Abstract: We consider the problem of stratiﬁed sampling for Monte-Carlo integration. We model this problem in a multi-armed bandit setting, where the arms represent the strata, and the goal is to estimate a weighted average of the mean values of the arms. We propose a strategy that samples the arms according to an upper bound on their standard deviations and compare its estimation quality to an ideal allocation that would know the standard deviations of the strata. We provide two regret analyses: a distribution� dependent bound O(n−3/2 ) that depends on a measure of the disparity of � the strata, and a distribution-free bound O(n−4/3 ) that does not. 1

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Summary: the most important sentenses genereted by tfidf model

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1 fr Abstract We consider the problem of stratiﬁed sampling for Monte-Carlo integration. [sent-5, score-0.059]

2 We model this problem in a multi-armed bandit setting, where the arms represent the strata, and the goal is to estimate a weighted average of the mean values of the arms. [sent-6, score-0.295]

3 We propose a strategy that samples the arms according to an upper bound on their standard deviations and compare its estimation quality to an ideal allocation that would know the standard deviations of the strata. [sent-7, score-0.908]

4 We provide two regret analyses: a distribution� dependent bound O(n−3/2 ) that depends on a measure of the disparity of � the strata, and a distribution-free bound O(n−4/3 ) that does not. [sent-8, score-0.431]

5 1 Introduction Consider a polling institute that has to estimate as accurately as possible the average income of a country, given a ﬁnite budget for polls. [sent-9, score-0.282]

6 The institute has call centers in every region in the country, and gives a part of the total sampling budget to each center so that they can call random people in the area and ask about their income. [sent-10, score-0.206]

7 A naive method would allocate a budget proportionally to the number of people in each area. [sent-11, score-0.302]

8 However some regions show a high variability in the income of their inhabitants whereas others are very homogeneous. [sent-12, score-0.139]

9 Now if the polling institute knew the level of variability within each region, it could adjust the budget allocated to each region in a more clever way (allocating more polls to regions with high variability) in order to reduce the ﬁnal estimation error. [sent-13, score-0.323]

10 This example is just one of many for which an eﬃcient method of sampling a function with natural strata (i. [sent-14, score-0.421]

11 Note that even in the case that there are no natural strata, it is always a good strategy to design arbitrary strata and allocate a budget to each stratum that is proportional to the size of the stratum, compared to a crude Monte-Carlo. [sent-17, score-0.874]

12 There are many good surveys on the topic of stratiﬁed sampling for Monte-Carlo, such as (Rubinstein and Kroese, 2008)[Subsection 5. [sent-18, score-0.059]

13 The main problem for performing an eﬃcient sampling is that the variances within the strata (in the previous example, the income variability per region) are usually unknown. [sent-20, score-0.605]

14 One possibility is to estimate the variances online while sampling the strata. [sent-21, score-0.136]

15 The work of Etor´ and Jourdain (2010) matches e e exactly our problem of designing an eﬃcient adaptive sampling strategy. [sent-23, score-0.087]

16 In this article they propose to sample according to an empirical estimate of the variance of the strata, whereas Kawai (2010) addresses a computational complexity problem which is slightly diﬀerent from ours. [sent-24, score-0.093]

17 (2011) describes a strategy that enables to sample e asymptotically according to the (unknown) standard deviations of the strata and at the same time adapts the shape (and number) of the strata online. [sent-26, score-0.998]

18 1 These works provide asymptotic convergence of the variance of the estimate to the targeted stratiﬁed variance1 divided by the sample size. [sent-28, score-0.093]

19 They also prove that the number of pulls within each stratum converges to the desired number of pulls i. [sent-29, score-0.466]

20 the optimal allocation if the variances per stratum were known. [sent-31, score-0.449]

21 Our contribution is to design a sampling strategy for which we can derive a ﬁnite-time analysis (where ’time’ refers to the number of samples). [sent-33, score-0.162]

22 This enables us to predict the quality of our estimate for any given budget n. [sent-34, score-0.152]

23 We model this problem using the setting of multi-armed bandits where our goal is to estimate a weighted average of the mean values of the arms. [sent-35, score-0.066]

24 Although our goal is diﬀerent from a usual bandit problem where the objective is to play the best arm as often as possible, this problem also exhibits an exploration-exploitation trade-oﬀ. [sent-36, score-0.43]

25 The arms have to be pulled both in order to estimate the initially unknown variability of the arms (exploration) and to allocate correctly the budget according to our current knowledge of the variability (exploitation). [sent-37, score-0.798]

26 The authors present an algorithm, called GAFS-MAX, that allocates samples proportionally to the empirical variance of the arms, √ while imposing that each arm is pulled at least n times to guarantee a suﬃciently good estimation of the true variances. [sent-40, score-0.601]

27 it targets an allocation which is proportional to the standard deviation (and not to the variance) of the strata times their size2 . [sent-44, score-0.677]

28 Some questions remain open in this work, notably that no distribution independent regret bound is provided for GAFS-WL. [sent-45, score-0.215]

29 The algorithm, called MC-UCB, samples the arms proportionally to an UCB3 on the standard deviation times the size of the stratum. [sent-51, score-0.359]

30 Our contributions are the following: • We derive a ﬁnite-time analysis for the stratiﬁed sampling for Monte-Carlo setting by using an algorithm based on upper conﬁdence bounds. [sent-54, score-0.091]

31 � • We provide two regret analysis: (i) a distribution-dependent bound O(n−3/2 )4 that depends on the disparity of the stratas (a measure of the problem complexity), and which corresponds to a stationary regime where the budget n is large compared to � this complexity. [sent-56, score-0.606]

32 (ii) A distribution-free bound O(n−4/3 ) that does not depend on the the disparity of the stratas, and corresponds to a transitory regime where n is small compared to the complexity. [sent-57, score-0.345]

33 3 Note that we consider a sampling strategy based on UCBs on the standard deviations of the arms whereas the so-called UCB algorithm of Auer et al. [sent-66, score-0.519]

34 (2002), in the usual multi-armed bandit setting, computes UCBs on the mean rewards of the arms. [sent-67, score-0.1]

35 2 illustrate our method on the problem of pricing Asian options as introduced in (Glasserman et al. [sent-69, score-0.101]

36 2 Preliminaries The allocation problem mentioned in the previous section is formalized as a K-armed bandit problem where each arm (stratum) k = 1, . [sent-72, score-0.64]

37 At each round t ≥ 1, an allocation strategy (or algorithm) A selects an arm kt and receives a sample drawn from νkt independently of the past samples. [sent-76, score-0.796]

38 , the arm selected at round t may depend on past observed samples. [sent-79, score-0.387]

39 The goal is to deﬁne a strategy that estimates as precisely �K as possible µ = k=1 wk µk using a total budget of n samples. [sent-85, score-0.397]

40 �t Let us write Tk,t = s=1 I {ks = k} the number of times arm k has been pulled up to time Tk,t 1 � Xk,s the empirical estimate of the mean µk at time t, where Xk,s t, and µk,t = ˆ Tk,t s=1 denotes the sample received when pulling arm k for the s-th time. [sent-86, score-0.99]

41 Note ˆ that in the case of a deterministic strategy, the expected quadratic estimation error of the �K ˆ weighted mean µ as estimated by the weighted average µn = k=1 wk µk,n satisﬁes: ˆ �� 2 � �2 � �K �2 � K 2 =E = k=1 wk Eνk µk,n − µk . [sent-88, score-0.416]

42 ˆ wk (ˆk,n − µk ) µ E µn − µ ˆ k=1 We thus use the following measure for the performance of any algorithm A: � � �K 2 2 Ln (A) = k=1 wk E (µk − µk,n ) . [sent-89, score-0.348]

43 ˆ (1) The goal is to deﬁne an allocation strategy that minimizes the global loss deﬁned in Equation 1. [sent-90, score-0.339]

44 If the variance of the arms were known in advance, one could design an optimal static5 allocation strategy A∗ by pulling each arm k proportionally to the quantity wk σk . [sent-91, score-1.267]

45 ∗ Indeed, if arm k is pulled a deterministic number of times Tk,n , then 2 �K 2 σ (2) Ln (A∗ ) = k=1 wk T ∗k . [sent-92, score-0.639]

46 In the following, we write λk = k,n = wΣwk the optimal allocation n proportion for arm k and λmin = min1≤k≤K λk . [sent-94, score-0.59]

47 Note that a small λmin means a large disparity of the wk σk and, as explained later, provides for the algorithm we build in Section 3 a characterization of the hardness of a problem. [sent-95, score-0.37]

48 However, in the setting considered here, the σk are unknown, and thus the optimal allocation u is out of reach. [sent-96, score-0.236]

49 A possible allocation is the uniform strategy Au , i. [sent-97, score-0.339]

50 Its performance is i=1 wi �K w σ 2 �K Σ Ln (Au ) = k=1 wk k=1 k k = w,2 , n n 5 Static means that the number of pulls allocated to each arm does not depend on the received samples. [sent-100, score-0.819]

51 In addition, since i wi = 1, we have Σ2 − Σw,2 = − k wk (σk − Σw )2 . [sent-104, score-0.233]

52 The diﬀerence w between those two quantities is the weighted quadratic variation of the σk around their weighted mean Σw . [sent-105, score-0.068]

53 As a result the gain of A∗ compared to Au grow with the disparity of the σk . [sent-107, score-0.148]

54 We would like to do better than the uniform strategy by considering an adaptive strategy A that would estimate the σk at the same time as it tries to implement an allocation strategy as close as possible to the optimal allocation algorithm A∗ . [sent-108, score-0.841]

55 This introduces a natural trade-oﬀ between the exploration needed to improve the estimates of the variances and the exploitation of the current estimates to allocate the pulls nearly-optimally. [sent-109, score-0.339]

56 In order to assess how well A solves this trade-oﬀ and manages to sample according to the true standard deviations without knowing them in advance, we compare its performance to that of the optimal allocation strategy A∗ . [sent-110, score-0.533]

57 For this purpose we deﬁne the notion of regret of an adaptive algorithm A as the diﬀerence between the performance loss incurred by the algorithm and the optimal algorithm: Rn (A) = Ln (A) − Ln (A∗ ). [sent-111, score-0.175]

58 (5) The regret indicates how much we loose in terms of expected quadratic estimation error Σ2 w by not knowing in advance the standard deviations (σk ). [sent-112, score-0.348]

59 Note that since Ln (A∗ ) = n , a consistent strategy i. [sent-113, score-0.103]

60 , asymptotically equivalent to the optimal strategy, is obtained whenever its regret is neglectable compared to 1/n. [sent-115, score-0.147]

61 1 The algorithm In this section, we introduce our adaptive algorithm for the allocation problem, called Monte Carlo Upper Conﬁdence Bound (MC-UCB). [sent-117, score-0.264]

62 The algorithm computes a high-probability bound on the standard deviation of each arm and samples the arms proportionally to their bounds times the corresponding weights. [sent-118, score-0.805]

63 , n do � � � wk 1 ˆ Compute Bk,t = Tk,t−1 σk,t−1 + b Tk,t−1 for each arm 1 ≤ k ≤ K Pull an arm kt ∈ arg max1≤k≤K Bk,t end for Output: µk,t for each arm 1 ≤ k ≤ K ˆ Figure 1: The pseudo-code of the MC-UCB algorithm. [sent-130, score-1.28]

64 The empirical standard deviations σk,t−1 are computed using Equation 6. [sent-131, score-0.145]

65 ˆ The algorithm starts by pulling each arm twice in rounds t = 1 to 2K. [sent-132, score-0.478]

66 From round t = 2K+1 on, it computes an upper conﬁdence bound Bk,t on the standard deviation σk , for each arm k, and then pulls the one with largest Bk,t . [sent-133, score-0.705]

67 The upper bounds on the standard deviations are built by using Theorem 10 in (Maurer and Pontil, 2009)6 and based on the empirical standard deviation σk,t−1 : ˆ σk,t−1 = ˆ2 6 1 Tk,t−1 − 1 Tk,t−1 � i=1 (Xk,i − µk,t−1 )2 , ˆ We could also have used the variant reported in (Audibert et al. [sent-134, score-0.255]

68 4 (6) where Xk,i is the i-th sample received when pulling arm k, and Tk,t−1 is the number of pulls allocated to arm k up to time t − 1. [sent-136, score-1.051]

69 After n rounds, MC-UCB returns the empirical mean µk,n for each arm 1 ≤ k ≤ K. [sent-137, score-0.354]

70 A distribution-dependent result: We now report the ﬁrst bound on the regret of MCUCB algorithm. [sent-148, score-0.215]

71 , the diﬀerence in the number of pulls of each arm compared to the optimal allocation (see Lemma 3). [sent-152, score-0.739]

72 Theorem 1 Under Assumption 1 and if we choose c2 such that c2 ≥ 2Kn−5/2 , the regret of MC-UCB run with parameter δ = n−7/2 with n ≥ 4K is bounded as � � log(n)c1 (c2 + 2) � 19 � 112Σw + 6K + 3 KΣ2 + 720c1 (c2 + 1) log(n)2 . [sent-153, score-0.147]

73 Rn (AM C−U CB ) ≤ w 3/2 λmin n2 n3/2 λmin Note that this result crucially depends on the smallest proportion λmin which is a measure of the disparity of the standard deviations times their weight. [sent-154, score-0.327]

74 A distribution-free result: Now we report our second regret bound that does not depend on λmin but whose rate is poorer. [sent-156, score-0.215]

75 Theorem 2 Under Assumption 1 and if we choose c2 such that c2 ≥ 2Kn−5/2 , the regret of MC-UCB run with parameter δ = n−7/2 with n ≥ 4K is bounded as √ � 200 c1 (c2 + 2)Σw K 365 � Rn (AM C−U CB ) ≤ log(n) + 3/2 129c1 (c2 + 2)2 K 2 log(n)2 + KΣ2 . [sent-158, score-0.147]

76 Note that the bound is not entirely distribution free since Σw appears. [sent-160, score-0.068]

77 1 Discussion on the results Distribution-free versus distribution-dependent � −5/2 Theorem 1 provides a regret bound of order O(λmin n−3/2 ), whereas Theorem 2 provides a � bound in O(n−4/3 ) independently of λmin . [sent-164, score-0.283]

78 , a given λmin , the distribution-free result of Theorem 2 is more informative than the distribution-dependent result of Theorem 1 in the transitory regime, that is to say when n is small compared to λ−1 . [sent-167, score-0.084]

79 The distribution-dependent result of Theorem 1 is better in the stationary regime i. [sent-168, score-0.074]

80 7 The distribution dependent bound is in O(K log n/Δ), where Δ is the diﬀerence between the √ mean value of the two best arms, and the distribution-free bound is in O( nK log n) as explained in (Auer et al. [sent-172, score-0.285]

81 5 Although we do not have a lower bound on the regret yet, we believe that the rate n−3/2 cannot be improved for general distributions. [sent-174, score-0.215]

82 As explained in the proof in Appendix B of (Carpentier and Munos, 2011), this rate is a direct consequence of the high probability √ bounds on the estimates of the standard deviations of the arms which are in O(1/ n), and those bounds are tight. [sent-175, score-0.372]

83 A natural question is whether there exists an algorithm with a regret � of order O(n−3/2 ) without any dependence in λ−1 . [sent-176, score-0.147]

84 The problem dependent upper bound is similar to the one provided for GAFS-WL in (Grover, 2009). [sent-181, score-0.1]

85 Note however that when there is an arm with 0 standard deviation, GAFS-WL is likely to perform better than MC-UCB, as it will only sample this √ � arm O( n) times while MC-UCB samples it O(n2/3 ) times. [sent-184, score-0.768]

86 By the choice of the value assigned to δ in the two theorems, b should be chosen of order c log(n), where c can be interpreted as a high probability bound on the range of the samples. [sent-188, score-0.068]

87 Note that in the case of bounded distributions, b can be chosen as � � b = 4 5 c log(n) where c is a true bound on the variables. [sent-190, score-0.068]

88 5 Numerical experiment: Pricing of an Asian option We consider the pricing problem of an Asian option introduced in (Glasserman et al. [sent-192, score-0.177]

89 The discounted payoﬀ of the Asian option is deﬁned as a function of W , by: � � � � � √ d 1 (8) F (W ) = exp(−rT ) max d i=1 S0 exp (r − 1 s2 ) iT + s0 T Wi − C, 0 , 0 d 2 where S0 , r, and s0 are constants, and the price is deﬁned by the expectation p = EW F (W ). [sent-196, score-0.09]

90 We want to estimate the price p by Monte-Carlo simulations (by sampling on W = (Wi )1≤i≤d ). [sent-197, score-0.143]

91 In order to reduce the variance of the estimated price, we can stratify the space of W . [sent-198, score-0.109]

92 (1999) suggest to stratify according to a one dimensional projection of W , i. [sent-200, score-0.074]

93 , by choosing a projection vector u ∈ Rd and deﬁne the strata as the set of W such that u · W lies in intervals of R. [sent-202, score-0.362]

94 , to stratify according to the last component Wd of W . [sent-205, score-0.074]

95 e Like in (Kawai, 2010) we consider 5 strata of equal weight. [sent-212, score-0.362]

96 Since Wd follows a N (0, 1), the strata correspond to the 20-percentile of a normal distribution. [sent-213, score-0.362]

97 The left plot of Figure 2 represents the cumulative distribution function of Wd and shows the strata in terms of 6 percentiles of Wd . [sent-214, score-0.387]

98 The right plot represents, in dot line, the curve E[F (W )|Wd = x] versus P(Wd < x) parameterized by x, and the box plot represents the expectation and standard deviations of F (W ) conditioned on each stratum. [sent-215, score-0.145]

99 We observe that this stratiﬁcation produces an important heterogeneity of the standard deviations per stratum, which indicates that a stratiﬁed sampling would be proﬁtable compared to a crude Monte-Carlo sampling. [sent-216, score-0.244]

100 Expectation of the payoff in every strata for W with C=90 d 1000 900 E[F(W)|W =x] 800 E[F(W)|W ∈ strata] d d 700 E[F(W)|Wd=x] 600 500 400 300 200 100 0 −100 0 0. [sent-217, score-0.362]

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LGMD computations with the ψ-function: No multiplication, no exponentiation 2 A circular object which starts its approach at distance x0 and with speed v projects a visual angle Θ(t) = 2 arctan[l/(x0 − vt)] on the retina [34, 9]. The kinematics is hence entirely speciﬁed by the ˙ half-size-to-velocity ratio l/|v|, and x0 . Furthermore, Θ(t) = 2lv/((x0 − vt)2 + l2 ). In order to deﬁne ψ, we consider at ﬁrst the LGMD neuron as an RC-circuit with membrane potential3 V [17] dV Cm = β (Vrest − V ) + gexc (Vexc − V ) + ginh (Vinh − V ) (1) dt 4 Cm = membrane capacity ; β ≡ 1/Rm denotes leakage conductance across the cell membrane (Rm : membrane resistance); gexc and ginh are excitatory and inhibitory inputs. Each conductance gi (i = exc, inh ) can drive the membrane potential to its associated reversal potential Vi (usually Vinh ≤ Vexc ). Shunting inhibition means Vinh = Vrest . Shunting inhibition lurks “silently” because it gets effective only if the neuron is driven away from its resting potential. With synaptic input, the neuron decays into its equilibrium state Vrest β + Vexc gexc + Vinh ginh V∞ ≡ (2) β + gexc + ginh according to V (t) = V∞ (1 − exp(−t/τm )). Without external input, V (t 1) → Vrest . The time scale is set by τm . Without synaptic input τm ≡ Cm /β. Slowly varying inputs gexc , ginh > 0 modify the time scale to approximately τm /(1 + (gexc + ginh )/β). For highly dynamic inputs, such as in late phase of the object approach, the time scale gets dynamical as well. The ψ-model assigns synaptic inputs5 ˙ ˙ ˙ ˙ gexc (t) = ϑ(t), ϑ(t) = ζ1 ϑ(t − ∆tstim ) + (1 − ζ1 )Θ(t) (3a) e ginh (t) = [γϑ(t)] , ϑ(t) = ζ0 ϑ(t − ∆tstim ) + (1 − ζ0 )Θ(t) 1 (3b) This linear approximation gets worse with increasing Θ, but turns out to work well until short before ttc (τ adopts a minimum at tc − 0.428978 · l/|v|). 2 LGMD activity is usually monitored via its postsynaptic neuron, the Descending Contralateral Movement Detector (DCMD) neuron. This represents no problem as LGMD spikes follow DCMD spikes 1:1 under visual stimulation [22] from 300Hz [21] to at least 400Hz [24]. 3 Here we assume that the membrane potential serves as a predictor for the LGMD’s mean ﬁring rate. 4 Set to unity for all simulations. 5 LGMD receives also inhibition from a laterally acting network [21]. The η-function considers only direct feedforward inhibition [22, 6], and so do we. 2 Θ ∈ [7.63Â°, 180.00Â°[ temporal resolution ∆ tstim=1.0ms l/|v|=20.00ms, β=1.00, γ=7.50, e=3.00, ζ0=0.90, ζ1=0.99, nrelax=25 0.04 scaled dΘ/dt continuous discretized 0.035 0.03 Θ(t) (input) ϑ(t) (filtered) voltage V(t) (output) t = 56ms max t =300ms c 0.025 0 10 2 η(t): α=3.29, R =1.00 n =10 → t =37ms log Θ(t) amplitude relax max 0.02 0.015 0.01 0.005 0 −0.005 0 50 100 150 200 250 300 −0.01 0 350 time [ms] 50 100 150 200 250 300 350 time [ms] (b) ψ versus η (a) discretized optical variables Figure 1: (a) The continuous visual angle of an approaching object is shown along with its discretized version. Discretization transforms angular velocity from a continuous variable into a series of “spikes” (rescaled). (b) The ψ function with the inputs shown in a, with nrelax = 25 relaxation time steps. Its peak occurs tmax = 56ms before ttc (tc = 300ms). An η function (α = 3.29) that was ﬁtted to ψ shows good agreement. For continuous optical variables, the peak would occur 4ms earlier, and η would have α = 4.44 with R2 = 1. For nrelax = 10, ψ is farther away from its equilibrium at V∞ , and its peak moves 19ms closer to ttc. t =500ms, dia=12.0cm, ∆t c =1.00ms, dt=10.00µs, discrete=1 stim 250 n relax = 50 2 200 α=4.66, R =0.99 [normal] n = 25 relax 2 α=3.91, R =1.00 [normal] n =0 relax tmax [ms] 150 2 α=1.15, R =0.99 [normal] 100 50 0 β=1.00, γ=7.50, e=3.00, V =−0.001, ζ =0.90, ζ =0.99 inh −50 5 10 15 20 25 30 0 35 1 40 45 50 l/|v| [ms] (a) different nrelax (b) different ∆tstim ˆ ˆ Figure 2: The ﬁgures plot the relative time tmax ≡ tc − t of the response peak of ψ, V (t), as a function of half-size-to-velocity ratio (points). Line ﬁts with slope α and intercept δ were added (lines). The predicted linear relationship in all cases is consistent with experimental evidence [9]. (a) The stimulus time scale is held constant at ∆tstim = 1ms, and several LGMD time scales are deﬁned by nrelax (= number of intercalated relaxation steps for each integration time step). Bigger values of nrelax move V (t) closer to its equilibrium V∞ (t), implying higher slopes α in turn. (b) LGMD time scale is ﬁxed at nrelax = 25, and ∆tstim is manipulated. Because of the discretization of optical variables (OVs) in our simulation, increasing ∆tstim translates to an overall smaller number of jumps in OVs, but each with higher amplitude. Thus, we say ψ(t) ≡ V (t) if and only if gexc and ginh are deﬁned with the last equation. The time ˙ scale of stimulation is deﬁned by ∆tstim (by default 1ms). The variables ϑ and ϑ are lowpass ﬁltered angular size and rate of expansion, respectively. The amount of ﬁltering is deﬁned by memory constants ζ0 and ζ1 (no ﬁltering if zero). The idea is to continue with generating synaptic input ˙ after ttc, where Θ(t > tc ) = const and thus Θ(t > tc ) = 0. Inhibition is ﬁrst weighted by γ, and then potentiated by the exponent e. Hodgkin-Huxley potentiates gating variables n, m ∈ [0, 1] instead (potassium ∝ n4 , sodium ∝ m3 , [12]) and multiplies them with conductances. Gabbiani and co-workers found that the function which transforms membrane potential to ﬁring rate is better described by a power function with e = 3 than by exp(·) (Figure 4d in [8]). 3 Dynamics of the ψ-function 3 Discretization. In a typical experiment, a monitor is placed a short distance away from the insect’s eye, and an approaching object is displayed. Computer screens have a ﬁxed spatial resolution, and as a consequence size increments of the displayed object proceed in discrete jumps. The locust retina is furthermore composed of a discrete array of ommatidia units. We therefore can expect a corresponding step-wise increment of Θ with time, although optical and neuronal ﬁltering may ˙ smooth Θ to some extent again, resulting in ϑ (ﬁgure 1). Discretization renders Θ discontinuous, ˙ For simulating the dynamics of ψ, we discretized angular size what again will be alleviated in ϑ. ˙ with ﬂoor(Θ), and Θ(t) ≈ [Θ(t + ∆tstim ) − Θ(t)]/∆tstim . Discretized optical variables (OVs) were re-normalized to match the range of original (i.e. continuous) OVs. To peak, or not to peak? Rind & Simmons reject the hypothesis that the activity peak signals impending collision on grounds of two arguments [28]: (i) If Θ(t + ∆tstim ) − Θ(t) 3o in consecutively displayed stimulus frames, the illusion of an object approach would be lost. Such stimulation would rather be perceived as a sequence of rapidly appearing (but static) objects, causing reduced responses. (ii) After the last stimulation frame has been displayed (that is Θ = const), LGMD responses keep on building up beyond ttc. This behavior clearly depends on l/|v|, also according to their own data (e.g. Figure 4 in [26]): Response build up after ttc is typically observed for sufﬁ˙ ciently small values of l/|v|. Input into ψ in situations where Θ = const and Θ = 0, respectively, ˙ is accommodated by ϑ and ϑ, respectively. We simulated (i) by setting ∆tstim = 5ms, thus producing larger and more infrequent jumps in discrete OVs than with ∆tstim = 1ms (default). As a consequence, ϑ(t) grows more slowly (deˆ layed build up of inhibition), and the peak occurs later (tmax ≡ tc − t = 10ms with everything else ˆ ˆ identical with ﬁgure 1b). The peak amplitude V = V (t) decreases nearly sixfold with respect to default. Our model thus predicts the reduced responses observed by Rind & Simmons [28]. Linearity. Time of peak ﬁring rate is linearly related to l/|v| [10, 9]. The η-function is consistent ˆ with this experimental evidence: t = tc − αl/|v| + δ (e.g. α = 4.7, δ = −27ms). The ψ-function reproduces this relationship as well (ﬁgure 2), where α depends critically on the time scale of biophysical processes in the LGMD. We studied the impact of this time scale by choosing 10µs for the numerical integration of equation 1 (algorithm: 4th order Runge-Kutta). Apart from improving the numerical stability of the integration algorithm, ψ is far from its equilibrium V∞ (t) in every moment ˙ t, given the stimulation time scale ∆tstim = 1ms 6 . Now, at each value of Θ(t) and Θ(t), respectively, we intercalated nrelax iterations for integrating ψ. Each iteration takes V (t) asymptotically closer to V∞ (t), and limnrelax 1 V (t) = V∞ (t). If the internal processes in the LGMD cannot keep up with stimulation (nrelax = 0), we obtain slopes values that underestimate experimentally found values (ﬁgure 2a). In contrast, for nrelax 25 we get an excellent agreement with the experimentally determined α. This means that – under the reported experimental stimulation conditions (e.g. [9]) – the LGMD would operate relatively close to its steady state7 . Now we ﬁx nrelax at 25 and manipulate ∆tstim instead (ﬁgure 2b). The default value ∆tstim = 1ms corresponds to α = 3.91. Slightly bigger values of ∆tstim (2.5ms and 5ms) underestimate the experimental α. In addition, the line ﬁts also return smaller intercept values then. We see tmax < 0 up to l/|v| ≈ 13.5ms – LGMD activity peaks after ttc! Or, in other words, LGMD activity continues to increase after ttc. In the limit, where stimulus dynamics is extremely fast, and LGMD processes are kept far from equilibrium at each instant of the approach, α gets very small. As a consequence, tmax gets largely independent of l/|v|: The activity peak would cling to tmax although we varied l/|v|. 4 Freeze! Experimental data versus steady state of “psi” In the previous section, experimentally plausible values for α were obtained if ψ is close to equilibrium at each instant of time during stimulation. In this section we will thus introduce a steady-state 6 Assuming one ∆tstim for each integration time step. This means that by default stimulation and biophysical dynamics will proceed at identical time scales. 7 Notice that in this moment we can only make relative statements - we do not have data at hand for deﬁning absolute time scales 4 tc=500ms, v=2.00m/s ψ∞ → (β varies), γ=3.50, e=3.00, Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, γ=3.50, (e varies), Vinh=−0.001 300 tc=500ms, v=2.00m/s ψ∞ → β=2.50, (γ varies), e=3.00, Vinh=−0.001 350 300 β=10.00 β=5.00 norm. rmse = 0.058...0.153 correlation (β,α)=−0.90 (n=4) ∞ β=1.00 e=4.00 norm. |η−ψ | = 0.009...0.114 e=3.00 300 norm. rmse = 0.014...0.160 correlation (e,α)=0.98 (n=4) ∞ e=2.50 250 250 norm. |η−ψ | = 0.043...0.241 ∞ norm. rmse = 0.085...0.315 correlation (γ,α)=1.00 (n=5) 150 tmax [ms] 200 tmax [ms] 200 tmax [ms] γ=5.00 γ=2.50 γ=1.00 γ=0.50 γ=0.25 e=5.00 norm. |η−ψ | = 0.020...0.128 β=2.50 250 200 150 100 150 100 100 50 50 50 0 5 10 15 20 25 30 35 40 45 0 5 50 10 15 20 l/|v| [ms] 25 30 35 40 45 0 5 50 10 15 20 l/|v| [ms] (a) β varies 25 30 35 40 45 50 l/|v| [ms] (b) e varies (c) γ varies ˆ ˆ Figure 3: Each curve shows how the peak ψ∞ ≡ ψ∞ (t) depends on the half-size-to-velocity ratio. In each display, one parameter of ψ∞ is varied (legend), while the others are held constant (ﬁgure title). Line slopes vary according to parameter values. Symbol sizes are scaled according to rmse (see also ﬁgure 4). Rmse was calculated between normalized ψ∞ (t) & normalized η(t) (i.e. both functions ∈ [0, 1] with original minimum and maximum indicated by the textbox). To this end, the ˆ peak of the η-function was placed at tc , by choosing, at each parameter value, α = |v| · (tc − t)/l (for determining correlation, the mean value of α was taken across l/|v|). tc=500ms, v=2.00m/s ψ∞ → (β varies), γ=3.50, e=3.00, Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, γ=3.50, (e varies), Vinh=−0.001 tc=500ms, v=2.00m/s ψ∞ → β=2.50, (γ varies), e=3.00, Vinh=−0.001 0.25 β=5.00 0.12 β=2.50 β=1.00 0.1 0.08 (normalized η, ψ∞) 0.12 β=10.00 (normalized η, ψ∞) (normalized η, ψ∞) 0.14 0.1 0.08 γ=5.00 γ=2.50 0.2 γ=1.00 γ=0.50 γ=0.25 0.15 0.06 0.04 0.02 0 5 10 15 20 25 30 35 40 45 50 meant |η(t)−ψ∞(t)| meant |η(t)−ψ∞(t)| meant |η(t)−ψ∞(t)| 0.06 0.04 e=5.00 e=4.00 e=3.00 0.02 e=2.50 10 l/|v| [ms] 15 20 25 30 35 40 45 50 l/|v| [ms] (a) β varies (b) e varies 0.1 0.05 0 5 10 15 20 25 30 35 40 45 50 l/|v| [ms] (c) γ varies Figure 4: This ﬁgure complements ﬁgure 3. It visualizes the time averaged absolute difference between normalized ψ∞ (t) & normalized η(t). For η, its value of α was chosen such that the maxima of both functions coincide. Although not being a ﬁt, it gives a rough estimate on how the shape of both curves deviate from each other. The maximum possible difference would be one. version of ψ (i.e. equation 2 with Vrest = 0, Vexc = 1, and equations 3 plugged in), ψ∞ (t) ≡ e ˙ Θ(t) + Vinh [γΘ(t)] e ˙ β + Θ(t) + [γΘ(t)] (4) (Here we use continuous versions of angular size and rate of expansion). The ψ∞ -function makes life easier when it comes to ﬁtting experimental data. However, it has its limitations, because we brushed the whole dynamic of ψ under the carpet. Figure 3 illustrates how the linˆ ear relationship (=“linearity”) between tmax ≡ tc − t and l/|v| is inﬂuenced by changes in parameter values. Changing any of the values of e, β, γ predominantly causes variation in line slopes. The smallest slope changes are obtained by varying Vinh (data not shown; we checked Vinh = 0, −0.001, −0.01, −0.1). For Vinh −0.01, linearity is getting slightly compromised, as slope increases with l/|v| (e.g. Vinh = −1 α ∈ [4.2, 4.7]). In order to get a notion about how well the shape of ψ∞ (t) matches η(t), we computed timeaveraged difference measures between normalized versions of both functions (details: ﬁgure 3 & 4). Bigger values of β match η better at smaller, but worse at bigger values of l/|v| (ﬁgure 4a). Smaller β cause less variation across l/|v|. As to variation of e, overall curve shapes seem to be best aligned with e = 3 to e = 4 (ﬁgure 4b). Furthermore, better matches between ψ∞ (t) and η(t) correspond to bigger values of γ (ﬁgure 4c). And ﬁnally, Vinh marches again to a different tune (data not shown). Vinh = −0.1 leads to the best agreement (≈ 0.04 across l/|v|) of all Vinh , quite different from the other considered values. For the rest, ψ∞ (t) and η(t) align the same (all have maximum 0.094), 5 ˙ (a) Θ = 126o /s ˙ (b) Θ = 63o /s Figure 5: The original data (legend label “HaGaLa95”) were resampled from ref. [10] and show ˙ DCMD responses to an object approach with Θ = const. Thus, Θ increases linearly with time. The η-function (ﬁtting function: Aη(t+δ)+o) and ψ∞ (ﬁtting function: Aψ∞ (t)+o) were ﬁtted to these data: (a) (Figure 3 Di in [10]) Good ﬁts for ψ∞ are obtained with e = 5 or higher (e = 3 R2 = 0.35 and rmse = 0.644; e = 4 R2 = 0.45 and rmse = 0.592). “Psi” adopts a sigmoid-like curve form which (subjectively) appears to ﬁt the original data better than η. (b) (Figure 3 Dii in [10]) “Psi” yields an excellent ﬁt for e = 3. RoHaTo10 gregarious locust LV=0.03s Θ(t), lv=30ms e011pos014 sgolay with 100 t =107ms max ttc=5.00s ψ adj.R2 0.95 (LM:3) ∞ η(t) adj.R2 1 (TR::1) 2 ψ : R =0.95, rmse=0.004, 3 coefficients ∞ → β=2.22, γ=0.70, e=3.00, V =−0.001, A=0.07, o=0.02, δ=0.00ms inh η: R2=1.00, rmse=0.001 → α=3.30, A=0.08, o=0.0, δ=−10.5ms 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5 5.2 time [s] (b) α versus β (a) spike trace Figure 6: (a) DCMD activity in response to a black square (l/|v| = 30ms, legend label “e011pos14”, ref. [30]) approaching to the eye center of a gregarious locust (ﬁnal visual angle 50o ). Data show the ﬁrst stimulation so habituation is minimal. The spike trace (sampled at 104 Hz) was full wave rectiﬁed, lowpass ﬁltered, and sub-sampled to 1ms resolution. Firing rate was estimated with Savitzky-Golay ﬁltering (“sgolay”). The ﬁts of the η-function (Aη(t + δ) + o; 4 coefﬁcients) and ψ∞ -function (Aψ∞ (t) with ﬁxed e, o, δ, Vinh ; 3 coefﬁcients) provide both excellent ﬁts to ﬁring rate. (b) Fitting coefﬁcient α (→ η-function) inversely correlates with β (→ ψ∞ ) when ﬁtting ﬁring rates of another 5 trials as just described (continuous line = line ﬁt to the data points). Similar correlation values would be obtained if e is ﬁxed at values e = 2.5, 4, 5 c = −0.95, −0.96, −0.91. If o was determined by the ﬁtting algorithm, then c = −0.70. No clear correlations with α were obtained for γ. despite of covering different orders of magnitude with Vinh = 0, −0.001, −0.01. Decelerating approach. Hatsopoulos et al. [10] recorded DCMD activity in response to an ap˙ proaching object which projected image edges on the retina moving at constant velocity: Θ = const. ˙ This “linear approach” is perceived as if the object is getting increasingly implies Θ(t) = Θ0 + Θt. slower. But what appears a relatively unnatural movement pattern serves as a test for the functions η & ψ∞ . Figure 5 illustrates that ψ∞ passes the test, and consistently predicts that activity sharply rises in the initial approach phase, and subsequently declines (η passed this test already in the year 1995). 6 Spike traces. We re-sampled about 30 curves obtained from LGMD recordings from a variety of publications, and ﬁtted η & ψ∞ -functions. We cannot show the results here, but in terms of goodness of ﬁt measures, both functions are in the same ballbark. Rather, ﬁgure 6a shows a representative example [30]. When α and β are plotted against each other for ﬁve trials, we see a strong inverse correlation (ﬁgure 6b). Although ﬁve data points are by no means a ﬁrm statistical sample, the strong correlation could indicate that β and α play similar roles in both functions. Biophysically, β is the leakage conductance, which determines the (passive) membrane time constant τm ∝ 1/β of the neuron. Voltage drops within τm to exp(−1) times its initial value. Bigger values of β mean shorter τm (i.e., “faster neurons”). Getting back to η, this would suggest α ∝ τm , such that higher (absolute) values for α would possibly indicate a slower dynamic of the underlying processes. 5 Discussion (“The Good, the Bad, and the Ugly”) Up to now, mainly two classes of LGMD models existed: The phenomenological η-function on the one hand, and computational models with neuronal layers presynaptic to the LGMD on the other (e.g. [25, 15]; real-world video sequences & robotics: e.g. [3, 14, 32, 2]). Computational models predict that LGMD response features originate from excitatory and inhibitory interactions in – and between – presynaptic neuronal layers. Put differently, non-linear operations are generated in the presynaptic network, and can be a function of many (model) parameters (e.g. synaptic weights, time constants, etc.). In contrast, the η-function assigns concrete nonlinear operations to the LGMD [7]. The η-function is accessible to mathematical analysis, whereas computational models have to be probed with videos or artiﬁcial stimulus sequences. The η-function is vague about biophysical parameters, whereas (good) computational models need to be precise at each (model) parameter value. The η-function establishes a clear link between physical stimulus attributes and LGMD activity: It postulates what is to be computed from the optical variables (OVs). But in computational models, such a clear understanding of LGMD inputs cannot always be expected: Presynaptic processing may strongly transform OVs. The ψ function thus represents an intermediate model class: It takes OVs as input, and connects them with biophysical parameters of the LGMD. For the neurophysiologist, the situation could hardly be any better. Psi implements the multiplicative operation of the η-function by shunting inhibition (equation 1: Vexc ≈ Vrest and Vinh ≈ Vrest ). The η-function ﬁts ψ very well according to our dynamical simulations (ﬁgure 1), and satisfactory by the approximate criterion of ﬁgure 4. We can conclude that ψ implements the η-function in a biophysically plausible way. However, ψ does neither explicitly specify η’s multiplicative operation, nor its exponential function exp(·). Instead we have an interaction between shunting inhibition and a power law (·)e , with e ≈ 3. So what about power laws in neurons? Because of e > 1, we have an expansive nonlinearity. Expansive power-law nonlinearities are well established in phenomenological models of simple cells of the primate visual cortex [1, 11]. Such models approximate a simple cell’s instantaneous ﬁring rate r from linear ﬁltering of a stimulus (say Y ) by r ∝ ([Y ]+ )e , where [·]+ sets all negative values to zero and lets all positive pass. Although experimental evidence favors linear thresholding operations like r ∝ [Y − Ythres ]+ , neuronal responses can behave according to power law functions if Y includes stimulus-independent noise [19]. Given this evidence, the power-law function of the inhibitory input into ψ could possibly be interpreted as a phenomenological description of presynaptic processes. The power law would also be the critical feature by means of which the neurophysiologist could distinguish between the η function and ψ. A study of Gabbiani et al. aimed to provide direct evidence for a neuronal implementation of the η-function [8]. Consequently, the study would be an evidence ˙ for a biophysical implementation of “direct” multiplication via log Θ − αΘ. Their experimental evidence fell somewhat short in the last part, where “exponentation through active membrane conductances” should invert logarithmic encoding. 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