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268 nips-2010-The Neural Costs of Optimal Control

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Author: Samuel Gershman, Robert Wilson

Abstract: Optimal control entails combining probabilities and utilities. However, for most practical problems, probability densities can be represented only approximately. Choosing an approximation requires balancing the beneﬁts of an accurate approximation against the costs of computing it. We propose a variational framework for achieving this balance and apply it to the problem of how a neural population code should optimally represent a distribution under resource constraints. The essence of our analysis is the conjecture that population codes are organized to maximize a lower bound on the log expected utility. This theory can account for a plethora of experimental data, including the reward-modulation of sensory receptive ﬁelds, GABAergic effects on saccadic movements, and risk aversion in decisions under uncertainty. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 We propose a variational framework for achieving this balance and apply it to the problem of how a neural population code should optimally represent a distribution under resource constraints. [sent-7, score-0.574]

2 The essence of our analysis is the conjecture that population codes are organized to maximize a lower bound on the log expected utility. [sent-8, score-0.422]

3 This theory can account for a plethora of experimental data, including the reward-modulation of sensory receptive ﬁelds, GABAergic effects on saccadic movements, and risk aversion in decisions under uncertainty. [sent-9, score-0.206]

4 Typically one has a choice of approximation, with more exact approximations demanding more computational resources, a penalty that can be naturally incorporated into the utility function. [sent-12, score-0.429]

5 The question we address in this paper is: given a family of approximations and their associated resource demands, what approximation will lead as close as possible to the optimal control policy? [sent-13, score-0.209]

6 However, maximizing information transfer is only one component of adaptive behavior; the utility of information must be taken into account when choosing a code [15], and this may interact in complicated ways with the computational costs of approximate inference. [sent-16, score-0.619]

7 Central to our analysis is the observation that while this expected utility cannot be maximized directly, it is possible to maximize a variational lower bound on log expected utility (see also [17, 5] for related approaches). [sent-18, score-1.274]

8 We study the properties of this lower bound and show how it accounts for some intriguing empirical properties of neural codes. [sent-19, score-0.158]

9 1 2 Optimal control with approximate densities Let a denote an action and s denote a hidden state variable drawn from some probability density p(s). [sent-20, score-0.548]

10 1 Given a utility function U (a; s), the optimal action ap is the one that maximizes expected utility Vp (a): ap = argmax Vp (a), (1) a where Vp (a) = Ep [U (a; s)] = p(s)U (a; s)ds. [sent-21, score-0.933]

11 (2) s Computing the expected utility for each action requires solving a possibly intractable integral. [sent-22, score-0.543]

12 An approximation of expected utility can be obtained by substituting an alternative density q(s) for which the expected utility is tractable. [sent-23, score-1.122]

13 Using an approximate density presents the “meta-decision” of which density to use. [sent-25, score-0.496]

14 If one chooses optimally under q(s), then the expected utility is given by Ep [U (aq ; s)] = Vp (aq ), therefore the optimal density q ∗ should be chosen according to q ∗ = argmax Vp (aq ), (3) q∈Q where Q is some family of densities. [sent-26, score-0.711]

15 3, consider the optimization as consisting of two parts: ﬁrst, select an approximate density q(s) and choose the optimal action with respect to this density; then evaluate the true value of that action under the target density. [sent-28, score-0.556]

16 In general, we cannot optimize this function directly because it requires solving precisely the integral we are trying to avoid: the expected utility under p(s). [sent-30, score-0.451]

17 Examining the utility lower bound, we see that the terms exert conceptually distinct inﬂuences: 1. [sent-32, score-0.49]

18 A utility component, Eq [log U (a; s)], the expected log utility under the approximate density. [sent-33, score-0.961]

19 A cross-entropy component, −Eq [log p(s)], reﬂecting the mismatch between the approximate density and the target density. [sent-35, score-0.372]

20 An entropy component, −Eq [log q(s)], embodying a maximum entropy principle [8]: for a ﬁxed utility and cross-entropy, choose the distribution with maximal entropy. [sent-38, score-0.39]

21 Intuitively, a more accurate approximate density q(s) should incur a larger computational cost. [sent-39, score-0.276]

22 One way to express this notion of cost is to incorporate it directly into the utility function. [sent-40, score-0.437]

23 That is, we consider an augmented utility function U (a, q; s) that depends on the approximate density. [sent-41, score-0.446]

24 We also refer throughout this paper to probability densities over a multimdensional, continuous state variable, but our results still apply to one dimensional and discrete variables (in which case the probability densities are replaced with probability mass functions). [sent-44, score-0.256]

25 2 The assumption that the log utility decomposes into additive reward and cost components is intuitive: it implies that reward is measured relative to the computational cost of earning it. [sent-45, score-0.78]

26 In summary, the utility lower bound L(q, a) provides an objective function for simultaneously choosing an action and choosing an approximate density over hidden states. [sent-46, score-0.873]

27 Whereas in classical decision theory, optimization is performed over the action space, in variational decision theory optimization is performed over the joint space of actions and approximate densities. [sent-47, score-0.341]

28 3 Choosing a probabilistic population code While the theory developed in the previous section applies to any representation scheme, in this section, for illustrative purposes, we focus on one speciﬁc family of approximate densities deﬁned by the ﬁring rate of neurons in a network. [sent-49, score-0.549]

29 Speciﬁcally, we consider a population of N neurons tasked with encoding a probability density over s. [sent-50, score-0.501]

30 We assume that the kernel density functions are Gaussian, parameterized by a preferred stimulus (mean) sn and a standard deviation σn : fn (s) = √ 1 (s − sn )2 exp − 2 2σn 2πσn (7) For simplicity, in this paper we will focus on the limiting case in which σ ⇒ 0. [sent-52, score-0.661]

31 2 In this case q(s) degenerates onto a collection of delta functions: q(s) = 1 Z N exn δ(s − sn ), (8) n=1 where δ(·) is the Dirac delta function. [sent-53, score-0.399]

32 This density corresponds to a collection of sharply tuned neurons; provided that the preferred values {s1 , . [sent-54, score-0.305]

33 Technically, the lower bound is not well deﬁned in the limit because the target density is non-atomic (i. [sent-63, score-0.431]

34 5 by N Eq [g(s)] ≈ Z −1 n=1 exn g(sn ), as we do above, can be justiﬁed in terms of ﬁrst-order Taylor series expansions around the preferred stimuli, which will be arbitrarily accurate as σ → 0. [sent-67, score-0.223]

35 3 (b) convolutional coding (c) gain coding (d) exponential coding firing rate probability density (a) probability distributions 10 30 50 s 70 10 30 50 70 neuron number 10 30 50 neuron number 70 10 30 50 70 neuron number Figure 1: Comparison between coding schemes. [sent-69, score-1.542]

36 We next seek a neuronal update rule that performs gradient ascent on the utility lower bound. [sent-74, score-0.44]

37 3 This update rule deﬁnes an attractor network whose Lyapunov function is the (negative) utility lower bound. [sent-76, score-0.44]

38 1 Relation to other probability coding schemes Exponential, convolutional and gain coding The probability coding scheme proposed in Eq. [sent-80, score-0.774]

39 8 is closely related to the exponential coding described in [16]. [sent-81, score-0.176]

40 Other related schemes include convolutional coding [28], in which a distribution is encoded by convolving it with a neural tuning function, and gain coding [11, 27], in which the variance of the distribution is inversely proportional to the gain of the neural response. [sent-83, score-0.71]

41 Convolutional coding (Figure 1b) is characterized by a neural response pattern that gets broader as the distribution gets broader. [sent-85, score-0.277]

42 In contrast, gain coding schemes (Figure 1c) posit that changes in uncertainty only change the overall gain, and not the shape, of the neural response. [sent-89, score-0.325]

43 This point is crucial for the biological plausibility of this scheme, as it seems unlikely that these minute differences in population response width would be easily measured experimentally. [sent-94, score-0.175]

44 It is also important to note that both the convolutional and gain coding schemes ignore the utility function in constructing probabilistic representations. [sent-95, score-0.836]

45 As we explore in later sections, rewards and costs place strong constraints on the types of codes that are learned by the variational objective, and the available experimental data is congruent with this view. [sent-96, score-0.368]

46 Psychological phenomena like perceptual multistability [6] and speech perception [21] are parsimoniously explained by a model in which a density over the complete hypothesis space is replaced by a small set of discrete samples. [sent-102, score-0.319]

47 Thus, it is reasonable to speculate whether our theory of population coding relates to these at the neural level. [sent-103, score-0.336]

48 In fact, we can make this correspondence precise: for any population code of the form in Eq. [sent-105, score-0.18]

49 The corresponding proposal density takes the form: p(sn ) π(s) ∝ δ(s − sn ). [sent-107, score-0.434]

50 (13) exn n This means that optimizing the bound with respect to x is equivalent to selecting a proposal density so as to maximize utility under resource constraints. [sent-108, score-1.042]

51 [26], though in a more restricted setting, showing that maximal utility is achieved with very few samples when sampling is costly. [sent-110, score-0.39]

52 Similarly, π(s) will be sensitive to the computational costs inherent in the utility lower bound, favoring a small number of samples. [sent-111, score-0.55]

53 In that treatment, the proposal density was assumed to be the prior, leading to the prediction that neurons with preferred stimulus s∗ should occur with frequency proportional to the prior probability of s∗ . [sent-113, score-0.534]

54 One source of evidence for this prediction comes from the oblique effect: the observation that more V1 neurons are tuned to cardinal orientations than to oblique orientations [3], consistent with the statistics of the natural visual environment. [sent-114, score-0.291]

55 In contrast, our model predicts that the proposal density will be sensitive to rewards in addition to the prior; as we argue in the section 5. [sent-115, score-0.293]

56 5 Results In the following sections, we examine some of the neurophysiological and psychological implications of the variational objective. [sent-117, score-0.237]

57 Tying these diverse topics together is the central idea that utilities, costs and probabilistic beliefs exert a synergistic effect on neural codes and their behavioral outputs. [sent-118, score-0.383]

58 One consequence of the variational objective is that a clear separation of these components in the brain may not exist: rewards and costs inﬁltrate very early sensory areas. [sent-119, score-0.517]

59 Accumulating evidence indicates that perceptual representations in the brain are modulated by reward expectation. [sent-123, score-0.275]

60 For example, Shuler and Bear [23] paired retinal stimulation of the left and right 5 Natural sounds Neural code 0. [sent-124, score-0.193]

61 Probability density of natural sounds and the optimized approximate density, with black lines demarcating the region of behaviorally relevant sounds. [sent-135, score-0.471]

62 eyes with reward after different delays and recorded neurons in primary visual cortex that switched from representing purely physical attributes of the stimulation (e. [sent-136, score-0.343]

63 Similarly, Serences [20] showed that spatially selective regions of visual cortex are biased by the prior reward associated with different spatial locations. [sent-139, score-0.184]

64 These studies raise the possibility that the brain does not encode probabilistic beliefs separately from reward; indeed, this idea has been enshrined by a recent theoretical account [4]. [sent-140, score-0.232]

65 On the other hand, the variational framework we have described accounts for these ﬁndings by showing that decision-making using approximate densities leads automatically to reward-modulated probabilistic beliefs. [sent-142, score-0.439]

66 [12] recorded the responses of grasshopper auditory neurons to different stimulus ensembles and found that the ensembles that elicited the optimal response differed systematically from the natural auditory statistics of the grasshopper’s environment. [sent-146, score-0.522]

67 In particular, the optimal ensembles were restricted to a region of stimulus space in which behaviorally important sounds live, namely species-speciﬁc mating signals. [sent-147, score-0.31]

68 , “an organism may seek to distribute its sensory resources according to the behavioral relevance of the natural stimuli, rather than according to purely statistical principles. [sent-149, score-0.161]

69 ” We modeled this phenomenon by constructing a relatively wide density of natural sounds with a narrow region of behaviorally relevant sounds (in which states are twice as rewarding). [sent-150, score-0.509]

70 Figure 2 shows the results, conﬁrming that maximizing the utility lower bound selects a kernel density estimate that is narrower than the target density of natural sounds. [sent-151, score-1.041]

71 , using injections of muscimol, a GABA agonist) and hence increasing the metabolic requirements for action potential generation. [sent-156, score-0.165]

72 A second method is by manipulating the availability of glucose [7], either by making the subject hypoglycemic or by administering local infusions of glucose directly into the brain. [sent-157, score-0.202]

73 We predict that increasing spiking costs (either by reducing glucose levels or increasing GABAergic transmission) will result in a diminished ability to detect weak signals embedded in noise. [sent-158, score-0.271]

74 These predictions have received a more direct test in a recent visual search experiment by McPeek and Keller [14], in which muscimol was injected into local regions of the superior colliculus, a brain area known to control saccadic target selection. [sent-160, score-0.494]

75 In the absence of distractors, response latencies to the target were increased when it appeared in the receptive ﬁelds of the inhibited neurons. [sent-161, score-0.354]

76 In the presence of distractors, response latencies increased and choice accuracy decreased when the target appeared in the receptive ﬁelds of the inhibited neurons. [sent-162, score-0.354]

77 We simulated these ﬁndings by constructing a cost-ﬁeld γ(n) to represent the amount of GABAergic transmission at different neurons induced by muscimol injections. [sent-163, score-0.238]

78 05 0 50 0 −50 0 s s 50 4 2 0 0 50 100 neuron number Figure 3: Spiking cost in the superior colliculus. [sent-180, score-0.177]

79 (Left column) Target density, with larger bump in the top panel representing the target; (Center column) neural code under different settings of cost-ﬁeld γ(n); (Right column) ﬁring rates under different cost-ﬁelds. [sent-183, score-0.153]

80 decreases because the increased cost of spiking in the neurons representing the target location dampens the probability density in that location. [sent-184, score-0.546]

81 Increasing spiking cost also reduces the overall ﬁring rate in the target-representing neurons relative to the distractor-representing neurons. [sent-185, score-0.23]

82 3 Non-linear probability weighting In this section, we show that the variational objective provides a new perspective on some wellknown peculiarities of human probabilistic judgment. [sent-189, score-0.255]

83 In particular, the ostensibly irrational nonlinear weighting of probabilities in risky choice emerges naturally from optimization of the variational objective under a natural assumption about the ecological distribution of rewards. [sent-190, score-0.193]

84 The variational objective explains these phenomena by virtue of the fact that under neural resource constraints, the approximate density will be biased towards high reward regions of the state space. [sent-193, score-0.746]

85 6 Discussion We have presented a variational objective function for neural codes that balances motivational, statistical and metabolic demands in the service of optimal behavior. [sent-198, score-0.374]

86 The essential idea is that the intractable problem of computing expected utilities can be ﬁnessed by instead computing expected utilities under an approximate density that optimizes a variational lower bound on log expected utility. [sent-199, score-1.005]

87 This lower bound captures the neural costs of optimal control: more accurate approximations will require more metabolic resources, whereas less accurate approximations will diminish the amount of earned reward. [sent-200, score-0.419]

88 sensory neurons have repeatedly been found to be sensitive to reward contingencies. [sent-211, score-0.336]

89 Intuitively, expending more resources on accurately approximating the complete density of natural sensory statistics is inefﬁcient (from an optimal control perspective) if the behaviorally relevant signals live in a compact subspace. [sent-212, score-0.534]

90 We showed that the approximation that maximizes the utility lower bound concentrates its density within this subspace. [sent-213, score-0.725]

91 Our variational framework differs in important ways from the one recently proposed by Friston [4]. [sent-214, score-0.193]

92 In his treatment, utilities are not represented explicitly at all; rather, they are implicit in the probabilistic structure of the environment. [sent-215, score-0.149]

93 Based on an evolutionary argument, Friston suggests that high utility states are precisely those that have high probability, since otherwise organisms who ﬁnd themselves frequently in low utility states are unlikely to survive. [sent-216, score-0.78]

94 Thus, adopting a control policy that minimizes a variational upper bound on surprise will lead to optimal behavior. [sent-217, score-0.346]

95 In contrast, our variational framework is motivated by quite different considerations arising from the computational constraints of the brain’s architecture. [sent-221, score-0.193]

96 Nonetheless, these approaches have in common the idea that probabilistic beliefs will be shaped by the utility structure of the environment. [sent-222, score-0.505]

97 The variational framework offers a rather different perspective on bounded rationality; it asserts that humans are indeed trying to ﬁnd optimal solutions, but subject to certain computational resource constraints. [sent-224, score-0.311]

98 By making explicit what these constraints are, and how they interact at a neural level, our work provides a foundation upon which to develop a more complete neurobiological theory of optimal control under resource constraints. [sent-225, score-0.259]

99 Role of glucose in regulating the brain and cognition. [sent-279, score-0.218]

100 Testing the efﬁciency of sensory coding with optimal stimulus ensembles. [sent-321, score-0.344]

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