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194 nips-2009-Predicting the Optimal Spacing of Study: A Multiscale Context Model of Memory

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Author: Harold Pashler, Nicholas Cepeda, Robert Lindsey, Ed Vul, Michael C. Mozer

Abstract: When individuals learn facts (e.g., foreign language vocabulary) over multiple study sessions, the temporal spacing of study has a signiﬁcant impact on memory retention. Behavioral experiments have shown a nonmonotonic relationship between spacing and retention: short or long intervals between study sessions yield lower cued-recall accuracy than intermediate intervals. Appropriate spacing of study can double retention on educationally relevant time scales. We introduce a Multiscale Context Model (MCM) that is able to predict the inﬂuence of a particular study schedule on retention for speciﬁc material. MCM’s prediction is based on empirical data characterizing forgetting of the material following a single study session. MCM is a synthesis of two existing memory models (Staddon, Chelaru, & Higa, 2002; Raaijmakers, 2003). On the surface, these models are unrelated and incompatible, but we show they share a core feature that allows them to be integrated. MCM can determine study schedules that maximize the durability of learning, and has implications for education and training. MCM can be cast either as a neural network with inputs that ﬂuctuate over time, or as a cascade of leaky integrators. MCM is intriguingly similar to a Bayesian multiscale model of memory (Kording, Tenenbaum, & Shadmehr, 2007), yet MCM is better able to account for human declarative memory. 1

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Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 , foreign language vocabulary) over multiple study sessions, the temporal spacing of study has a signiﬁcant impact on memory retention. [sent-8, score-0.627]

2 Behavioral experiments have shown a nonmonotonic relationship between spacing and retention: short or long intervals between study sessions yield lower cued-recall accuracy than intermediate intervals. [sent-9, score-0.449]

3 Appropriate spacing of study can double retention on educationally relevant time scales. [sent-10, score-0.517]

4 MCM’s prediction is based on empirical data characterizing forgetting of the material following a single study session. [sent-12, score-0.272]

5 MCM is intriguingly similar to a Bayesian multiscale model of memory (Kording, Tenenbaum, & Shadmehr, 2007), yet MCM is better able to account for human declarative memory. [sent-17, score-0.263]

6 This advice is based on a memory phenomenon known as the distributed practice or spacing effect (Cepeda, Pashler, Vul, Wixted, & Rohrer, 2006). [sent-20, score-0.497]

7 The spacing effect is typically studied via a controlled experimental paradigm in which participants are asked to study unfamiliar paired associates (e. [sent-21, score-0.49]

8 ” The lag between second session and the test is known as the retention interval or RI. [sent-28, score-0.18]

9 The solid line of Figure 1a sketches this curve, which we will refer to as the spacing function. [sent-30, score-0.348]

10 The left edge of the graph corresponds to massed practice, when session two immediately follows session one. [sent-31, score-0.18]

11 (2006) sug1 forgetting function spacing function (b) m2 % recall (a) m3 m4 . [sent-34, score-0.542]

12 pool 1 pool 2 pool 3 pool 4 pool N ISI Figure 1: (a) The spacing function (solid line) depicts recall at test following two study sessions separated by a given ISI; the forgetting function (dashed line) depicts recall as a function of the lag between study and test. [sent-40, score-1.225]

13 For educationally relevant RIs on the order of weeks and months, the effect of spacing can be tremendous: optimal spacing can double retention over massed practice (Cepeda et al. [sent-44, score-0.808]

14 The spacing function is related to another observable measure of retention, the forgetting function, which characterizes recall accuracy following a single study session as a function of the lag between study and test. [sent-46, score-0.82]

15 For example, suppose participants in the experiment described above learned material in study session 1, and were then tested on the material immediately prior to study session 2. [sent-47, score-0.41]

16 Typical forgetting functions follow a generalized power-law decay, of the form P (recall) = A(1 + Bt)−C , where A, B, and C are constants, and t is the study-test lag (Wixted & Carpenter, 2007). [sent-50, score-0.208]

17 Our goal is to develop a model of long-term memory that characterizes the memory-trace strength of items learned over two or more sessions. [sent-51, score-0.197]

18 The model predicts recall accuracy as a function of the RI, taking into account the study schedule—the ISI or set of ISIs determining the spacing of study sessions. [sent-52, score-0.561]

19 The spacing effect is among the best known phenomena in cognitive psychology, and many theoretical explanations have been suggested. [sent-54, score-0.351]

20 Two well developed computational models of human memory have been elaborated to explain the spacing effect (Pavlik & Anderson, 2005; Raaijmakers, 2003). [sent-55, score-0.549]

21 These models are necessarily complex: the brain contains multiple, interacting memory systems whose decay and interference characteristics depend on the speciﬁc content being stored and its relationship to other content. [sent-56, score-0.23]

22 Consequently, these computational theories are fairly ﬂexible and can provide reasonable post-hoc ﬁts to spacing effect data, but we question their predictive value. [sent-57, score-0.412]

23 Rather than developing a general theory of memory, we introduce a model that speciﬁcally predicts the shape of the spacing function. [sent-58, score-0.358]

24 Because the spacing function depends not only on the RI, but also on the nature of the material being learned, and the manner and amount of study, the model requires empirical constraints. [sent-59, score-0.363]

25 We propose a novel approach to obtaining a predictive model: we collect behavioral data to determine the forgetting function for the speciﬁc material being learned. [sent-60, score-0.216]

26 We then use the forgetting function, which is based on a single study session, to predict the spacing function, which is based on two or more study sessions. [sent-61, score-0.643]

27 2 Accounts of the spacing effect We review two existing theories proposed to explain the spacing effect, and then propose a synthesis of these theories. [sent-65, score-0.793]

28 However, after introducing our model and showing its predictive power, we discuss an intriguingly similar Bayesian theory of memory adaptation (Kording et al. [sent-68, score-0.186]

29 1 Encoding-variability theories One class of theories proposed to explain the spacing effect focuses on the notion of encoding variability. [sent-72, score-0.503]

30 According to these theories, when an item is studied, a memory trace is formed that incorporates the current psychological context. [sent-73, score-0.378]

31 Retrieval of a stored item depends at least in part on the similarity of the contexts at the study and test. [sent-75, score-0.276]

32 If psychological context is assumed to ﬂuctuate randomly over time, two study sessions close together in time will have similar contexts. [sent-76, score-0.23]

33 Consequently, at the time of a recall test, either both study contexts will match the test context or neither will. [sent-77, score-0.208]

34 Increasing the ISI can thus prove advantageous because the test context will have higher likelihood of matching one study context or the other. [sent-78, score-0.19]

35 Greater contextual variation enhances memory on this account by making for less redundancy in the underlying memory traces. [sent-79, score-0.349]

36 Raaijmakers (2003) developed an encoding variability theory by incorporating time-varying contextual drift into the well-known Search of Associative Memory (SAM) model (Raaijmakers & Shiffrin, 1981), and explained a range of data from the spacing literature. [sent-82, score-0.408]

37 The input layer to this neural net is a pool of binary valued neurons that represent the contextual state at the current time; the output layer consists of a set of memory elements, one per item to be stored. [sent-88, score-0.484]

38 To simplify notation throughout this paper, we’ll describe this model and all others in terms of a single-item memory, allowing us to avoid an explicit index term for the item being stored or retrieved. [sent-89, score-0.2]

39 The memory element for the item under consideration has an activation level, m, which is a linear function of the context unit activities: m = j wj cj , where cj is the binary activation level of context unit j and wj is the strength of connection from context j. [sent-90, score-0.725]

40 The probability of retrieval of the item is assumed to be monotonically related to m. [sent-91, score-0.219]

41 When an item is studied, its connection strengths are adjusted according to a Hebbian learning rule with an upper limit on the connection strength: ∆wj = min(1 − wj , cj m), ˆ (2) where m = 1 if the item was just presented for study, or 0 otherwise. [sent-92, score-0.392]

42 When an item is studied, the ˆ weights for all contextual features present at the time of study will be strengthened. [sent-93, score-0.331]

43 Later retrieval is more likely if the context at test matches the context at study: the memory element receives a contribution only when an input is active and its connection strength is nonzero. [sent-94, score-0.381]

44 When an item has been studied twice, retrieval will be more robust if the two study opportunities strengthen different weights, which occurs when the ISI is large and the contextual states do not overlap signiﬁcantly. [sent-96, score-0.385]

45 After an item has been studied at least once, SAM assumes that the memory trace resulting from further study is inﬂuenced by whether the item is accessible to retrieval at the time of study. [sent-98, score-0.701]

46 Other memory models similarly claim that memory traces are weaker if an item is inaccessible to retrieval at the time of study (e. [sent-100, score-0.611]

47 We have described the key components of SAM that explain the spacing effect, but the model has additional complexity, including a short-term memory store, inter-item interference, and additional 3 context based on associativity and explicit cues. [sent-103, score-0.562]

48 , hours), but the same model cannot explain spacing effects on a different time scale (e. [sent-109, score-0.406]

49 2 Predictive-utility theories We now turn to another class of theories that has been proposed to explain the spacing effect. [sent-114, score-0.481]

50 When an item is studied multiple times with a given ISI, the rational analysis suggests that the need probability drops off rapidly following the last study once an interval of time greater than the ISI has passed. [sent-118, score-0.307]

51 In MTS, each item to be stored is represented by a dedicated cascade of N leaky integrators. [sent-123, score-0.284]

52 The activation of integrator i, xi , decays over time according to: xi (t + ∆t) = xi (t) exp(−∆t/τi ), (3) where τi is the decay time constant. [sent-124, score-0.223]

53 The probability of retrieving the item is related to the total k trace strength, sN , where sk = j=1 xj . [sent-125, score-0.203]

54 When an item is repeatedly presented for study with short ISIs, the trace can successfully be represented by the integrators with short time constants, and consequently, the trace will decay rapidly. [sent-131, score-0.537]

55 Increasing the spacing shifts the representation to integrators with slower decay rates. [sent-132, score-0.534]

56 Essentially, we take from SAM the notion of contextual drift and retrieval-dependent update, and from MTS the multiscale representation and the cascaded error-correction memory update, and we obtain a new model which we call the Multiscale Context Model or MCM. [sent-138, score-0.31]

57 MCM can also be described in terms of N leaky integrators, where integrator i has time constant τi and activity scaled by γi . [sent-143, score-0.197]

58 As the reader might infer from our description of SAM and MTS, these parameters characterize memory decay, extending Equation 3 such that the total trace strength at time t is deﬁned as: N sN (t) = γi exp(− i=1 4 t )xi (0). [sent-146, score-0.25]

59 τi If xi (0) = 1 for all i—which is the integrator activity following the ﬁrst study in MTS—the trace strength as a function of time is a mixture of exponentials. [sent-147, score-0.297]

60 To match the form of human forgetting (Figure 1), this mixture must approximate a power function. [sent-148, score-0.184]

61 Given N and human forgetting data collected in an experiment, we can search for the parameters {µ, ν, ω, ξ} that obtain a least squares ﬁt to the data. [sent-157, score-0.184]

62 Given the human forgetting function function, then, we can completely determine the {τi } and {γi }. [sent-158, score-0.184]

63 1 Casting MCM as a cascade of leaky integrators Assume that—as in MTS—a dedicated set of N leaky integrators hold the memory of each item to be learned. [sent-161, score-0.754]

64 Let xi denote the activity of integrator i associated with the item, and let si be the average strength of the ﬁrst i integrators, weighted by the {γj } terms: si = 1 Γi i i γj xj , where Γi = j=1 γj . [sent-162, score-0.228]

65 When an item is studied, its integrators receive a boost in activity. [sent-164, score-0.34]

66 Integrator i receives a boost that depends on how close the average strength of the ﬁrst i integrators is to full strength, i. [sent-165, score-0.217]

67 We adopt the retrieval-dependent update assumption of SAM, and ﬁx = 1 for an item that is unsuccessfully recalled at the time of study, and = r > 1 for an item that is successfully recalled. [sent-168, score-0.401]

68 (1) MTS weighs all integrators equally when combining the individual integrator activities. [sent-170, score-0.229]

69 (2) MTS provides no guidance in setting the τ and γ constants; MCM constrains these parameters based on the human forgetting function. [sent-172, score-0.184]

70 The context units are binary valued and units in pool i ﬂip with time constant τi . [sent-180, score-0.231]

71 ) 5 As depicted in Figure 1b, the model also includes a set of N memory elements for each item to be learned. [sent-183, score-0.32]

72 Activation of memory element i, denoted mi , indicates strength of retrieval for the item based on context pools 1. [sent-185, score-0.562]

73 The activation function is cascaded such that memory element i receives input from context units in pool i as well as memory element i − 1: mi = mi−1 + wij cij + b, j where wij is the connection weight from context unit j to memory element i, m0 ≡ 0, and b = −β/(1 − β) is a bias weight. [sent-189, score-0.901]

74 The bias simply serves to offset spurious activity reaching the memory elements, activity that is unrelated to the fact that the item was previously studied and stored. [sent-190, score-0.455]

75 The probability of recalling the item is related to the activity of memory element N : P (recall) = min(1, mN ). [sent-192, score-0.38]

76 When the item is studied, the weights from context units in pool i are adjusted according to an update rule that performs gradient descent in an error measure Ei = ei 2 , where ei = 1 − mi /Γi . [sent-193, score-0.472]

77 This error is minimized when the memory element i reaches activation level Γi (deﬁned earlier as the proportion of units in the entire context pool that contributes to activity at stage i). [sent-194, score-0.412]

78 (2) SAM’s memory update rule can be interpreted as Hebbian learning; MCM’s update can be interpreted as error-correction learning. [sent-203, score-0.204]

79 3 Relating leaky integrator and neural net characterizations of MCM To make contact with MTS, we have described MCM as a cascade of leaky integrators, and to make contact with SAM, we have described MCM as a neural net. [sent-205, score-0.251]

80 , in press) have recently conducted well-controlled experimental manipulations of spacing involving RIs on educationally relevant time scales of days to months. [sent-209, score-0.46]

81 Most research in the spacing literature involves brief RIs, on the scale of minutes to an hour, and methodological concerns have been raised with the few well-known studies involving longer RIs (Cepeda et al. [sent-210, score-0.349]

82 Recall accuracy at the start of the second session provides the basic forgetting function, and recall accuracy at test provides the spacing function. [sent-216, score-0.622]

83 In panel (e), the peaks of the model’s spacing functions are indicated by the triangle pointers. [sent-222, score-0.349]

84 These four model parameters determine the time constants and weighting coefﬁcients of the mixture-of-exponentials approximation to the forgetting function (Equation 5). [sent-224, score-0.186]

85 The model has only one other free parameter, r , the magnitude of update on a trial when an item is successfully recalled (see Equation 6). [sent-225, score-0.203]

86 With r , MCM is fully constrained and can make strong predictions regarding the spacing function. [sent-227, score-0.351]

87 For each experiment, MCM’s prediction of the peak of the spacing function is entirely consistent with the data, and for the most part, MCM’s quantiative predictions are excellent. [sent-231, score-0.394]

88 (It would be extremely surprising to psychologists if the peak were in general independent of the material, as content effects pervade the memory literature. [sent-236, score-0.212]

89 Each red circle represents a single spacing experiment in which the ISI was varied for a given RI. [sent-241, score-0.329]

90 MCM predicts the spacing functions with absolutely spectacular precision, considering the predictions are fully constrained and parameter free. [sent-249, score-0.38]

91 Moreover, MCM anticipates the peaks of the spacing functions, with the curvature of the peak decreasing with the RI, and the optimal ISI increasing with the RI. [sent-250, score-0.392]

92 In addition to these results, MCM also predicts the probability of recall at test conditional on successful or unsuccessful recall during the test at the start of the second study session. [sent-251, score-0.207]

93 Finally, MCM is able to post-hoc ﬁt classic studies from the spacing literature (for which forgetting functions are not available). [sent-254, score-0.491]

94 5 Discussion MCM’s blind prediction of 7 different spacing functions is remarkable considering that the domain’s complexity (the content, manner and amount of study) is reduced to four parameters, which are fully determined by the forgetting function. [sent-255, score-0.491]

95 The state predicts the appearance of an item in the temporal stream of experience. [sent-264, score-0.203]

96 (Two parameters determine the range of time scales; two specify internal and observation noise levels; and two perform an afﬁne transform from internal memory strength to recall probability. [sent-270, score-0.332]

97 ) In terms of sum-squared error, the model shows a reasonable ﬁt, but the model clearly misses the peaks of the spacing functions, and in fact predicts a peak that is independent of RI. [sent-271, score-0.421]

98 Notably, the KF model is a post-hoc ﬁt to the spacing functions, whereas MCM produces a true prediction of the spacing functions, i. [sent-272, score-0.658]

99 , parameters of MCM are determined without peeking at the spacing function. [sent-274, score-0.329]

100 Practice and forgetting effects on vocabulary memory: An activation-based model of the spacing effect. [sent-331, score-0.533]

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