nips nips2011 nips2011-56 knowledge-graph by maker-knowledge-mining

56 nips-2011-Committing Bandits

Source: pdf

Author: Loc X. Bui, Ramesh Johari, Shie Mannor

Abstract: We consider a multi-armed bandit problem where there are two phases. The ﬁrst phase is an experimentation phase where the decision maker is free to explore multiple options. In the second phase the decision maker has to commit to one of the arms and stick with it. Cost is incurred during both phases with a higher cost during the experimentation phase. We analyze the regret in this setup, and both propose algorithms and provide upper and lower bounds that depend on the ratio of the duration of the experimentation phase to the duration of the commitment phase. Our analysis reveals that if given the choice, it is optimal to experiment Θ(ln T ) steps and then commit, where T is the time horizon.

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 The ﬁrst phase is an experimentation phase where the decision maker is free to explore multiple options. [sent-2, score-0.931]

2 In the second phase the decision maker has to commit to one of the arms and stick with it. [sent-3, score-0.611]

3 Cost is incurred during both phases with a higher cost during the experimentation phase. [sent-4, score-0.664]

4 We analyze the regret in this setup, and both propose algorithms and provide upper and lower bounds that depend on the ratio of the duration of the experimentation phase to the duration of the commitment phase. [sent-5, score-1.152]

5 1 Introduction In a range of applications, a dynamic decision making problem exhibits two distinctly different kinds of phases: experimentation and commitment. [sent-7, score-0.638]

6 However, eventually the decision maker must commit to a choice, and use that decision for the duration of the problem horizon. [sent-9, score-0.496]

7 A notable feature of these phases in the models we study is that costs are incurred during both phases; that is, experimentation is not carried out “ofﬂine,” but rather is run “live” in the actual system. [sent-10, score-0.683]

8 However, such testing is not carried out without consequences; the retailer might lose potential rewards if experimentation leads to suboptimal recommendations. [sent-13, score-0.64]

9 Eventually, the recommendation engine must be stabilized (both from a software development standpoint and a customer expectation standpoint), and when this happens the retailer has effectively committed to one strategy moving forward. [sent-14, score-0.281]

10 The process of experimentation during design entails costs to the producer, but eventually the experimentation must stop and the design must be committed. [sent-18, score-1.257]

11 If the problem consists of only experimentation, then we have the classical multi-armed bandit problem, where the decision maker is interested in minimizing the expected total regret against the best arm [9, 2]. [sent-22, score-0.875]

12 il † 1 exploration or budgeted learning problem, where the goal is to output the best arm at the end of an experimentation phase [13, 6, 4]; no costs are incurred for experimentation, but after ﬁnite time a single decision must be chosen (see [12] for a review). [sent-29, score-1.206]

13 Formally, in a committing bandit problem, the decision maker can experiment without constraints for the ﬁrst N of T periods, but must commit to a single decision for the last T − N periods, where T is the problem horizon. [sent-30, score-0.823]

14 We ﬁrst consider the soft deadline setting where the experimentation deadline N can be chosen by the decision maker, but there is a cost incurred per experimentation period. [sent-31, score-1.771]

15 We divide this setting into two regimes depending on how N is chosen: the non-adaptive regime (Section 3) in which the decision maker has to choose N before the algorithm begins running, and the adaptive regime (Section 4) in which N can be chosen adaptively as the algorithm runs. [sent-32, score-0.692]

16 We obtain two main results for the soft deadline setting. [sent-33, score-0.272]

17 Finally, we demonstrate that if the algorithm has no initial distributional information, adaptivity is beneﬁcial: we demonstrate an adaptive algorithm that achieves Θ(ln T /T ) regret in this case. [sent-36, score-0.31]

18 We then study the hard deadline regime where the value of N is given to the decision maker in advance (Section 5). [sent-37, score-0.589]

19 This is a sensible assumption for problems where the decision maker cannot control how long the experimentation period is; for example, in the product design example above, the release date is often ﬁxed well in advance, and the engineers are not generally free to alter it. [sent-38, score-0.853]

20 We propose the UCB-poly(δ) algorithm for this setting, where the parameter δ ∈ (0, 1) reﬂects the tradeoff between experimentation and commitment. [sent-39, score-0.613]

21 During the ﬁrst N periods the tradeoff between exploration and exploitation exists bearing in mind that the last T − N periods will be used solely for exploitation. [sent-42, score-0.247]

22 2 The committing bandit problem We ﬁrst describe the setup of the classical stochastic multi-armed bandit problem, as it will serve as background for the committing bandit problem. [sent-44, score-0.944]

23 In a stochastic multi-armed bandit problem, there are K independent arms; each arm i, when pulled, returns a reward which is independently and identically drawn from a ﬁxed Bernoulli distribution1 with unknown parameter θi ∈ [0, 1]. [sent-45, score-0.615]

24 Let It denote the index of the arm pulled at time t (It ∈ {1, 2, . [sent-46, score-0.389]

25 i:∆i >0 An allocation policy is an algorithm that chooses the next arm to pull based on the sequence of past pulled arms and obtained rewards. [sent-52, score-0.881]

26 The cumulative regret of an allocation policy A after time n is: n ∗ (Xt − Xt ) , Rn = t=1 ∗ Xt where is the reward that the algorithm would have received at time t if it had pulled the optimal arm i∗ . [sent-53, score-0.997]

27 In other words, Rn is the cumulative loss due to the fact that the allocation policy does not always pull the optimal arm. [sent-54, score-0.355]

28 Let Ti (n) be the number of times that arm i is pulled up to time n. [sent-55, score-0.389]

29 A recommendation policy is an algorithm that tries to recommend the “best” arm based on the sequence of past pulled arms and obtained rewards. [sent-62, score-0.808]

30 Suppose that after time n, a recommendation policy R recommends the arm Jn as the “best” arm. [sent-63, score-0.544]

31 Then the regret of recommendation policy R after time n, called the simple regret in [4], is deﬁned as rn = θ∗ − θJn = ∆Jn . [sent-64, score-0.64]

32 The committing bandit problem considered in this paper is a version of the stochastic multi-armed bandit problem in which the algorithm is forced to commit to only one arm after some period of time. [sent-68, score-1.078]

33 From time 1 to some time N (N < T ), the algorithm can pull any arm in {1, 2, . [sent-71, score-0.401]

34 Then, from time N + 1 to the end of the horizon (time T ), it must commit to pull only one arm. [sent-75, score-0.275]

35 The ﬁrst phase (time 1 to N ) is called the experimentation phase, and the second phase (time N + 1 to T ) is called the commitment phase. [sent-76, score-0.93]

36 We refer to time N as the experimentation deadline. [sent-77, score-0.571]

37 An algorithm for the committing bandit problem is a combination of an allocation and a recommendation policy. [sent-78, score-0.679]

38 That is, the algorithm has to decide which arm to pull during the ﬁrst N slots, and then choose an arm to commit to during the remaining T − N slots. [sent-79, score-0.89]

39 Because we consider settings where the algorithm designer can choose the experimentation deadline, we also assume a cost is imposed during the experimentation phase; otherwise, it is never optimal to be forced to commit. [sent-80, score-1.238]

40 In particular, we assume that the reward earned during the experimentation phase is reduced by a constant factor γ ∈ [0, 1). [sent-81, score-0.767]

41 Thus the expected regret E[Reg] of such an algorithm is the average regret across both phases, i. [sent-82, score-0.357]

42 1 Committing bandit regimes We focus on three distinct regimes, that differ in the level of control given to the algorithm designer in choosing the experimentation deadline. [sent-86, score-0.866]

43 That is, the algorithm cannot control the experimentation deadline N . [sent-94, score-0.825]

44 2 Known lower-bounds As mentioned in the Introduction section, the experimentation and commitment phases have each been extensively studied in isolation. [sent-97, score-0.813]

45 In this subsection, we only summarize brieﬂy the known lower bounds on cumulative regret and simple regret that will be used in the paper. [sent-98, score-0.47]

46 Result 1 (Distribution-dependent lower bound on cumulative regret [9]). [sent-99, score-0.311]

47 For any allocation policy, and for any set of reward distributions such that their parameters θi are not all equal, there exists 3 an ordering of (θ1 , . [sent-100, score-0.363]

48 , θK ) such that  E[Rn ] ≥  pi p∗ p∗ pi ∗ i=i∗  ∆i + o(1) ln n, D(pi p∗ ) where D(pi p∗ ) = pi log + p∗ log is the Kullback-Leibler divergence between two Bernoulli reward distributions pi (of arm i) and p (of the optimal arm), and o(1) → 0 as n → ∞. [sent-103, score-0.853]

49 Result 2 (Distribution-free lower bound on cumulative regret [13]). [sent-104, score-0.311]

50 There exist positive constants c and N0 such that for any allocation policy, there exists a set of Bernoulli reward distributions such that E[Rn ] ≥ cK(ln n − ln K), ∀n ≥ N0 . [sent-105, score-0.523]

51 The difference between Result 1 and Result 2 is that the lower bound in the former depends on the parameters of reward distributions (hence, called distribution-dependent), while the lower bound in the latter does not (hence, called distribution-free). [sent-106, score-0.356]

52 For any pair of allocation and recommendation policies, if the allocation policy can achieve an upper bound such that for all (Bernoulli) reward distributions θ1 , . [sent-110, score-0.749]

53 , θK , there exists a constant C ≥ 0 with E[Rn ] ≤ Cf (n), then for all sets of K ≥ 3 Bernoulli reward distributions with parameters θi that are all distinct and all different from 1, there exists an ordering (θ1 , . [sent-113, score-0.263]

54 Result 4 (Distribution-free lower bound on simple regret [4]). [sent-124, score-0.253]

55 mendation policies, there exists a set of Bernoulli reward distributions such that E[rn ] ≥ 20 n In the subsequent sections we analyze each of the committing bandit regimes in detail; in particular, we provide constructive upper bounds and matching lower bounds on the regret in each regime. [sent-126, score-0.934]

56 3 Regime 1: Soft experimentation deadline, non-adaptive In this regime, for a given value of T , the value of N can be chosen freely between 1 and T − 1, but only before the algorithm begins pulling arms. [sent-128, score-0.651]

57 (1) Distribution-dependent lower bound: In Regime 1, for any algorithm, and any set of K ≥ 3 Bernoulli reward distributions such that θi are all distinct and all different from 1, there exists an ordering (θ1 , . [sent-131, score-0.272]

58 (2) Distribution-free lower bound: Also, for any algorithm in Regime 1, there exists a set of Bernoulli reward distributions such that ln K ln T E[Reg] ≥ cK 1 − , ln T T where c is the constant in Result 2. [sent-135, score-0.811]

59 Commit to the arm with maximum empirical average reward for the remaining periods. [sent-145, score-0.444]

60 E[Reg] ≤ 2 (1 − γ)θ∗ + ∆ K ln T T ∗ i=i This matches the lower bounds in Theorem 1 to the correct order in T . [sent-148, score-0.286]

61 Observe that in this regime, both distribution-dependent and distribution-free lower bounds have the same asymptotic order of ln T /T. [sent-149, score-0.284]

62 If ∆ is unknown, a low regret algorithm that matches the lower bound does not seem to be possible in this regime, because of the relative nature of the regret. [sent-151, score-0.297]

63 An algorithm may be unable to choose an N that explores sufﬁciently long when arms are difﬁcult to distinguish, and yet commits quickly when arms are easy to distinguish. [sent-152, score-0.389]

64 4 Regime 2: Soft experimentation deadline, adaptive The setting in this regime is the same as the previous one, except that the algorithm is not required to choose N before it runs, i. [sent-153, score-0.775]

65 We ﬁrst present the lower bounds on regret for any algorithm in this regime. [sent-157, score-0.265]

66 (1) Distribution-dependent lower bound: In Regime 2, for any algorithm, and any set of K ≥ 3 Bernoulli reward distribution such that θi are all distinct and all different from 1, there exists an ordering (θ1 , . [sent-159, score-0.242]

67 (2) Distribution-free lower bound: Also, for any algorithm in Regime 2, there exists a set of Bernoulli reward distributions such that ln K ln T E[Reg] ≥ cK 1 − , ln T T where c is the constant in Result 2. [sent-163, score-0.811]

68 For the SEC1 algorithm (Algorithm 2),   K  γ ln T E[Reg] ≤ 2 (1 − γ)θ∗ + , ∆i + b ∆ K T ∗ i=i where b = 2 + ∆2 (K+2) (1−e−∆2 /2 )2 1 ln T → 0 as T → ∞. [sent-167, score-0.395]

69 Let Sm be the total reward obtained from arm i so far. [sent-176, score-0.444]

70 for i ∈ Bm do i if m ≤ α ln T and |mθ∗ − Sm | > 1 ln T then Delete arm i from Bm . [sent-178, score-0.694]

71 end if i if m > α ln T and |mθ∗ − Sm | > 2 m then Delete arm i from Bm . [sent-179, score-0.524]

72 end if end for until there is only one arm in Bm , then commit to that arm or the horizon T is reached. [sent-180, score-0.854]

73 We note that when N can be chosen adaptively, both distribution-dependent and distribution-free lower bounds have the same asymptotic order of ln T /T as the ones in the non-adaptive regime. [sent-182, score-0.307]

74 The reason is rather intuitive: due to its adaptive nature, SEC1 is able to eliminate poor arms much earlier than the ln T /∆2 threshold, while Non-adaptive Unif-EBA has to wait until that point to make decisions. [sent-188, score-0.384]

75 , log2 (T /e)/2 do if |Bm | > 1 then ˜ ˜ Sample each arm in Bm until each arm has been chosen nm = 2 ln(T ∆2 )/∆2 times. [sent-204, score-0.682]

76 m m i Let Sm be the total reward obtained from arm i so far. [sent-205, score-0.444]

77 5 Regime 3: Hard experimentation deadline We now investigate the third regime where, in contrast to the previous two, the experimentation deadline N is ﬁxed exogenously together with T . [sent-213, score-1.748]

78 Note that since in this case the experimentation deadline is outside the algorithm designer’s control, we set the cost of experimentation γ = 1 for this section. [sent-215, score-1.396]

79 We know from Result 3 that for any pair of allocation and recommendation policies, if E[RN ] ≤ C1 f (N ), then E[rN ] ≥ (∆/2)e−Df (N ) . [sent-218, score-0.285]

80 In other words, given an allocation policy A that has a cumulative regret bound C1 f (N ) (for some constant C1 ), the best (distribution-dependent) upper bound that any recommendation policy can achieve is C2 e−C3 f (N ) (for some constants C2 and C3 ). [sent-219, score-0.803]

81 Assuming that there exists a recommendation policy RA that achieves such an upper bound, we have the following upper bound on regret when applying [A, RA ] to the committing bandit problem: f (N ) T − N E[Reg] ≤ C1 + C2 e−C3 f (N ) . [sent-220, score-0.901]

82 (1) T T One can clearly see the trade-off between experimentation and commitment in (1): the smaller the ﬁrst term, the larger the second term, and vice versa. [sent-221, score-0.754]

83 In particular, we focus on ﬁnding a pair of allocation and recommendation policies which can simultaneously achieve the allocation bound C1 N δ and the recommendation δ bound C2 e−C3 N where 0 < δ < 1. [sent-226, score-0.695]

84 Let us consider a modiﬁcation of the UCB allocation policy called UCB-poly(δ) (for 0 < δ < 1), ˆ where for t > K, with θi,Ti (t−1) be the empirical average of rewards from arm i so far, It = arg max 1≤i≤K ˆ θi,Ti (t−1) + 2(t − 1)δ Ti (t − 1) . [sent-227, score-0.592]

85 In particular, if T (N ) is super-exponential in N we get an optimal δ of 1 representing pure exploration in the experimentation phase. [sent-233, score-0.657]

86 If T (N ) is sub-exponential we get an optimal δ of 0 representing a standard UCB during the experimentation phase. [sent-234, score-0.571]

87 The simulation setting includes K arms with Bernoulli reward distributions, the time horizon T , and the values of γ and ∆. [sent-240, score-0.329]

88 5, 1] interval, and the second best arm reward is set as θ2 = θ∗ − ∆. [sent-243, score-0.444]

89 These two values are then assigned to two randomly chosen arms, and the rest of arm rewards are generated ∗ independently and uniformly in [0, θ2 ]. [sent-244, score-0.362]

90 Particularly, the performance of Non-adaptive Unif-EBA is quite poor when the experimentation deadline is roughly equal to T , since the algorithm does not commit before the experimentation deadline. [sent-250, score-1.545]

91 7 Extensions and future directions Our work is a ﬁrst step in the study of the committing bandit setup. [sent-253, score-0.373]

92 First, an extension of the basic committing bandits setup to the case of contextual bandits [10, 11] is natural. [sent-255, score-0.315]

93 In this setup before choosing an arm an additional “context” is provided to the decision maker. [sent-256, score-0.414]

94 The problem is to choose a decision rule from a given class that prescribes what arm to choose for every context. [sent-257, score-0.427]

95 This setup is more realistic when the decision maker has to commit to such a rule after some exploration time. [sent-258, score-0.457]

96 Second, models with many arms (structured as in [8, 5]) or even inﬁnitely arms (as in [1, 7, 14]) are of interest here as they may lead to different regimes and results here. [sent-259, score-0.397]

97 Third, our models assumed that the commitment time is either predetermined or according to the decision maker’s will. [sent-260, score-0.25]

98 Finally, a situation where the exploration and commitment phases alternate (randomly or according to a given schedule or at a cost) is of practical interest. [sent-262, score-0.307]

99 UCB revisited: Improved regret bounds for the stochastic multi-armed bandit problem. [sent-277, score-0.373]

100 The sample complexity of exploration in the multi-armed bandit problem. [sent-335, score-0.236]

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Moreover, we do not need to know anything about the approaching object: The ttc estimation computed by τ is practically independent of object size and velocity. Neurons with τ -like responses were indeed identiﬁed in the nucleus retundus of the pigeon brain [34]. In humans, only fast interceptive actions seem to rely exclusively on τ [37, 35]. Accurate ttc estimation, however, seems to involve further mechanisms (rate of disparity change [31]). ˙ Another function of OVs with biological relevance is η ≡ Θ exp(−αΘ), with α = const. [10]. While η-type neurons were found again in pigeons [34] and bullfrogs [20], most data were gathered from the LGMD2 in locusts (e.g. [10, 9, 7, 23]). The η-function is a phenomenological model for the LGMD, and implies three principal hypothesis: (i) An implementation of an exponential function exp(·). Exponentation is thought to take place in the LGMD axon, via active membrane conductances [8]. Experimental data, though, seem to favor a third-power law rather than exp(·). (ii) The LGMD carries out biophysical computations for implementing the multiplicative operation. It has been suggested that multiplication is done within the LGMD itself, by subtracting the loga˙ rithmically encoded variables log Θ − αΘ [10, 8]. (iii) The peak of the η-function occurs before ˆ ttc, at visual angle Θ(t) = 2 arctan(1/α) [9]. It follows ttc for certain stimulus conﬁgurations (e.g. ˆ l/|v| 5ms). In principle, t > tc can be accounted for by η(t + δ) with a ﬁxed delay δ < 0 (e.g. −27ms). But other researchers observed that LGMD activity continuous to rise after ttc even for l/|v| 5ms [28]. These discrepancies remain unexplained so far [29], but stimulation dynamics perhaps plays a role. We we will address these three issues by comparing the novel function “ψ” with the η-function. 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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