Psychological Review Optimal Foraging in Semantic Memory Thomas T. Hills, Michael N. Jones, and Peter M. Todd Online First Publication, February 13, 2012. doi: 10.1037/a0027373 CITATION Hills, T. T., Jones, M. N., & Todd, P. M. (2012, February 13). Optimal Foraging in Semantic Memory. Psychological Review. Advance online publication. doi: 10.1037/a0027373 Psychological Review © 2012 American Psychological Association 2012, Vol. ●●, No. ●, 000 – 000 0033-295X/12/$12.00 DOI: 10.1037/a0027373 Optimal Foraging in Semantic Memory Thomas T. Hills Michael N. Jones and Peter M. Todd University of Warwick Indiana University Do humans search in memory using dynamic local-to-global search strategies similar to those that animals use to forage between patches in space? If so, do their dynamic memory search policies correspond to optimal foraging strategies seen for spatial foraging? Results from a number of fields suggest these possibilities, including the shared structure of the search problems—searching in patchy environments—and recent evidence supporting a domain-general cognitive search process. To investigate these questions directly, we asked participants to recover from memory as many animal names as they could in 3 min. Memory search was modeled over a representation of the semantic search space generated from the BEAGLE memory model of Jones and Mewhort (2007), via a search process similar to models of associative memory search (e.g., Raaijmakers & Shiffrin, 1981). We found evidence for local structure (i.e., patches) in memory search and patch depletion preceding dynamic local-to-global transitions between patches. Dynamic models also signif- icantly outperformed nondynamic models. The timing of dynamic local-to-global transitions was consistent with optimal search policies in space, specifically the marginal value theorem (Charnov, 1976), and partici- pants who were more consistent with this policy recalled more items. Keywords: optimal foraging, memory search, category fluency, semantic space models, marginal value theorem Supplemental materials: http://dx.doi.org/10.1037/a0027373.supp Animals often search for resources that occur in spatial patches, when the rate of finding new targets within the patch falls below the such as the berries on separate bushes or nuts beneath a cluster of long-term average rate achieved by following the optimal strategy. trees. Humans also search for cognitive resources that can be seen as We explore whether a version of this principle applies to search for patchy with respect to some metric other than space, such as memory items in semantic memory. Specifically, do humans move between representations of words grouped by semantic categories (Bousfield memory patches when global opportunities outweigh local benefits, & Sedgewick, 1944; Raaijmakers & Shiffrin, 1981; Romney, Brewer, just as bumblebees forage between flower patches in an open field? & Batchelder, 1993), or sets of solutions that can be navigated by To investigate the parallel between spatial and memory search, we working memory processes in a problem-solving task (Hills, Todd, & built models of participants’ search through semantic memory when Goldstone, 2008; Payne, Duggan, & Neth, 2007; Wilke, Hutchinson, engaged in a fluency task (e.g., “name all the animals you can think Todd, & Czienskowski, 2009). In spatial environments, adaptive of”; Lezak, 1995; Thurstone, 1938), and compared model fit to a foraging involves making appropriate global transitions between lo- classic model of optimal foraging in space—the marginal value the- cally exploited resource clusters: decisions that prevent animals from orem (Charnov, 1976). Two sources are used to represent the seman- staying too long in overexploited patches and from giving up too early tic space searched in the fluency task: hand-coded categorizations on patches full of resources yet to be found (Stephens & Krebs, 1987). from Troyer, Moscovitch, and Winocur (1997) and lexical semantic A classic model of optimal foraging theory (Charnov, 1976) predicts representations from BEAGLE (bound encoding of the aggregate that the overall rate of return is optimized if the forager leaves a patch language environment), a corpus-based semantic space model (Jones & Mewhort, 2007). This approach allowed us to address two ques- tions: (a) Does search in semantic memory involve switching between local exploitation of specific memory patches and global exploration Thomas T. Hills, Department of Psychology, University of Warwick, between patches, and, if so, (b) is the switching between local and Coventry, England; Michael N. Jones and Peter M. Todd, Cognitive global search in semantic memory consistent with optimal search Science Program and Department of Psychological and Brain Sciences, policies defined for animals foraging in space? In what follows we Indiana University. first explore the structural and neural parallels between spatial search This research was supported by Swiss National Science Foundation and memory search that motivate this study. Then we develop the two Grant 100014 130397/1 to Thomas T. Hills, U.S. National Science Foun- questions above, before describing our data collection and modeling dation Grant BCS-1056744 to Michael N. Jones, and Indiana University’s efforts to assess semantic foraging. Faculty Research Support Program. We thank Roxana Dietrich, Jason Dawson, Mark Steyvers, Bettina Helverson, Hansjeorg Neth, Ralph Hertwig, Richard Shiffrin, and Michael Dougherty for their insights during Structural and Neural Parallels Between Search in the development of this work. Space and Memory Correspondence concerning this article should be addressed to Thomas T. Hills, University of Warwick, CV4 7AL Coventry, England. E-mail: Structural similarities between spatial and nonspatial environ-

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ments have motivated a number of studies on human search 1 2 HILLS, JONES, AND TODD behavior. The key assumption underlying these investigations is work on free recall from natural categories and list learning has that when information is distributed in clusters, or patches (local consistently found that groups of semantically similar words are high-density areas of resources separated by regions with little produced together (Bousfield & Barclay, 1950; Gruenewald & resource availability), the optimal foraging policy of humans Lockhead, 1980; Howard, Jing, Addis, & Kahana, 2007; Romney searching for information should share features associated with et al., 1993). animals foraging for food in space. Research has demonstrated This grouping of semantically similar words in recall is consis- these parallels across spatial and cognitive search in tasks involv- tent with a cognitive foraging process that modulates between ing finding fish in virtual ponds (where patches are ponds and local and global memory cues, with the former producing clusters items are fish caught in each pond; Hutchinson, Wilke, & Todd, and the latter producing transitions between clusters. This dynamic 2008), search for words in multiword anagrams (where patches are search strategy is common to several different models of long-term sets of random letters and items are words created from subsets of memory retrieval (Gronlund & Shiffrin, 1986; Metcalfe & Mur- those letters; Hills et al., 2008; Wilke et al., 2009), and search for dock, 1981). One of the best known is the search of associative information on the web and in other naturalistic environments memory (SAM) model (Raaijmakers & Shiffrin, 1980, 1981). In (where patches can be sets of similar web pages; Pirolli, 2007; SAM, memory is probed with cues that lead to activation and Pirolli & Card, 1999). retrieval of memory items. Sets of cues make up the memory Critical to the success of the forager in all these cases is the probe, which can change over the course of the retrieval period in appropriate modulation between local and global search behav- a fashion similar to that outlined for patch-based foraging policies iors— deciding when to continue exploiting the current resource like area-restricted search. Initially, the probe consists of a global patch versus when to leave that patch and explore to find a new retrieval cue, related to the context and the category cue (i.e., the one. One particularly common strategy for making these ongoing superordinate category that defines the boundaries of the search trade-offs, observed in a wide range of animal species, is called space; e.g., “animals”). Following successful item recovery, the area-restricted search in the ecological literature (Gru¨nbaum, probe is modified to include the most recently recovered item as a 1998; Karieva & Odell, 1987). This strategy involves restricting cue (e.g., DOG), which is a form of local information. This one’s search to the local neighborhood for as long as resources increases activation for items that are semantically proximal to the continue to be found there and then at some point moving away most recent cue (e.g., CAT). Following failures to retrieve an item, from that area (sometimes gradually, and typically after the rate of the memory probe eventually loses its local cue, and returns to its finding resources falls off). global form. This is area-restricted search in memory, dynamically A comparative analysis of the underlying neural and molecular moving between local and global search efforts. architectures guiding area-restricted search (Hills, 2006) gives rise The cluster–switching hypothesis is a similar but less formal to the second reason for the proposed parallel between spatial model that has been investigated in the clinical literature (Troyer et search and memory search: evidence for a generalized cognitive al., 1997). This process involves “clustering” (the production of search process. Research from a number of fields has demonstrated words in a semantic subcategory) and “switching” (making the that molecular and neural mechanisms that appear to have evolved transition from one subcategory to another; Robert et al., 1998; initially for the purpose of area-restricted search in external envi- Troyer et al., 1997; Troyer, Moscovitch, Winocur, Leach, & ronments have subsequently been exapted in later species for the Freedman, 1998). The cluster–switching model defines patches purpose of modulating attention and search in internal environ- based on shared category membership (provided by hand-coded ments (Hills, 2006). This exaptation hypothesis is supported by the categorizations from Troyer et al., 1998), and offers a preliminary observation that, across species, neural processes similar to those means for evaluating patch structure in memory. As we show next, generally devoted to area-restricted search in space now modulate this allowed us to map hand-coded categorizations directly to goal-directed behaviors and attention in search for information semantic similarity. (e.g., Dulawa, Grandy, Low, Paulus, & Geyer, 1999; Floresco, When and how does memory search transition from local Seamans, & Phillips, 1996; Hills, Brockie, & Maricq, 2004; Sawa- within-patch search to global between-patch search? Recent re- guchi & Goldman-Rakic, 1991; Schultz, 2004), including search in search has investigated the algorithm-level question of what cues human memory (Berke & Hyman, 2000; Kischka et al., 1996; can lead to this transition, by modifying the standard free-recall Newman, Weingartner, Smallberg, & Calne, 1984; Wittmann et paradigm, allowing participants to determine when to terminate al., 2005). Thus, both shared environmental structure and shared memory search for items from a learned list (Dougherty & Har- mechanisms suggest the possibility of shared adaptive foraging bison, 2007; Harbison, Dougherty, Davelaar, & Fayyad, 2009). In policies for search in space and memory. particular, Harbison et al.’s (2009) results suggest that when par- ticipants begin primarily to recover items that have already been Dynamic Search in Semantic Memory retrieved, they are more likely to terminate the search process and revert to a global cue. Similar processes may drive patch switching Giving up on one patch to move to another assumes that the prior to search termination (as proposed by Raaijmakers & Shif- memory search space is distributed in a patchy way, analogous to frin, 1981). However, optimal foraging theory also focuses on a the distribution of many resources in the spatial environment. The different level of description—rather than just the cues used to patchiness of memory is evident in a variety of contexts including decide when to make a transition (the mechanistic or algorithmic lexical decision tasks and, more importantly for our purposes, level), it emphasizes the costs and benefits of deciding when to studies of free recall from natural categories, with clustered recall abandon a patch (the computational level)—that is, what opportu- of related items noted in the earliest such studies (Bousfield & nity costs are associated with staying or abandoning a given patch Sedgewick, 1944; Johnson, Johnson, & Mark, 1951). More recent in memory. Thus, we focus here on asking the question of how the OPTIMAL FORAGING AND SEMANTIC MEMORY 3 memory system should make local– global transitions and whether BEAGLE, the lexical semantic memory model of Jones and Me- people’s search patterns are consistent with optimal foraging whort (2007). BEAGLE provides measures of semantic proximity theory. between words based on their distributional regularities in a nat- ural language corpus, with a level of local structural detail not Optimal Foraging in Semantic Memory possible with the nominal category-based representations of Troyer et al. Having a formal model of semantic proximity among In the animal foraging literature, dynamic responses to the animal names offers a quantification of the semantic search space, environment are often assessed with respect to an optimal model which we can then use to predict the retrieval of specific animals representing a hypothesis about the trade-offs that must be nego- from memory and compare this with the search data we collected tiated in a given behavior– environment relationship. One of the from people; the same approach can also be directly extrapolated first and most successful models of optimal patch foraging at this to other categories (which is not the case for hand-coded subcat- level is the marginal value theorem (Charnov, 1976). The marginal egorization schemes). value theorem assumes that resources are distributed in patches By using both types of semantic representation, we extend prior that are monotonically depleted during foraging. The animal seeks work in memory search by making item-specific predictions, to maximize the gain per unit time of foraging defined as the rather than merely recording number of items produced or retrieval average resource intake, R, over all patches: time. Furthermore, these representations solve many of the tech- nical difficulties previously associated with characterizing item g共tW兲 similarity in memory (Bousfield & Sedgewick, 1944; Romney et R⫽ , (1) tW ⫹ tB al., 1993; for a similar approach, see Howard et al., 2007) or with using a random memory structure (Raaijmakers & Shiffrin, 1981). where tW is the time spent foraging within each resource patch, tB Semantic representations based on human coding or statistical is the average time spent traveling between patches, and g(tW) is regularities in language offer considerably more constraint to a the cumulative gain within a patch. model compared with randomly generated structures, which often Equation 1 provides a measure of resources per time unit, as a allow excessive freedom for an incorrect process model to fit the function of an individual’s control over their time tW within a data when it would have been rejected if the correct representa- patch. This is subject to patch quality, reflected by g(tW), and travel tional structure were used (Johns & Jones, 2010). time tB between patches. The organism is predicted to spend the To model the search over these representational spaces, we optimal amount of time in a patch (tⴱ) such that R is maximized: applied a generic model of memory retrieval common to the Rⴱ ⫽ g⬘共tⴱ 兲. (2) frameworks of SAM (Raaijmakers & Shiffrin, 1981) and ACT–R (adaptive control of thought–rational; Anderson, 1993; Anderson To maximize this resource intake, the optimal foraging policy is to & Lebiere, 1998). We then used various versions of these models leave a patch at time tⴱ when the instantaneous rate (or marginal to evaluate retrieval patterns and assess the dynamics of memory value) of resource gain, g⬘(tⴱ), is equal to the long-term average search and their correspondence with the marginal value theorem. resource intake over the entire environment (patches and time between), Rⴱ. In other words, the organism will switch to between- patch search when the within-patch rate (which usually starts high Method in a new, undepleted patch) drops to Rⴱ. With respect to memory, the corresponding prediction is that individuals should leave the Participants current memory patch when the benefits associated with searching further locally within it fall to the level of the expected benefits of Participants were 141 undergraduates (46 men and 95 women) searching elsewhere in memory. Indeed, the evidence for stopping at Indiana University, Bloomington, who received partial course rules in SAM based on failed retrievals (i.e., Harbison et al., 2009) credit. Participants were seated at a computer and followed in- suggests that patch depletion does lead to departure from local structions on-screen. memory patches, but it is unclear whether such patch departures are consistent with optimal foraging theory. In the rest of this Procedure article, we test this prediction. Participants were asked to produce items from each of seven The Present Study categories (animals, foods, vehicles, occupations, sports, cities, and movie titles), which were presented one at a time in a random To test more directly the applicability of the marginal value order. Participants typed as many items in a given category as they theorem to human memory search, we had participants produce could in 3 min. Entries were later corrected for spelling. Here we items from the category of “animals.” We analyzed the search focus solely on the category “animals,” for which we have the paths taken through memory in terms of the sequences of items predetermined subcategories from Troyer et al. (1997). Some produced. Search paths were assessed with two semantic repre- entries for this category were nonanimal items (e.g., “paw”), and sentations. We first evaluated patch boundaries with the hand- these were omitted from the analyses. Together, participants pro- coded subcategorization of animal terms (into specific subsets like duced 5,187 valid animal entries, consisting of 369 unique animal “pets” and “water animals”) derived by Troyer et al. (1997). We names. The mean number of animals per participant was 36.8 then compared search paths to the results of dynamic search (SD ⫽ 8.5). There was no correlation between order of category models applied to a representation of the semantic space built by appearance and number of items produced (p ⫽ .32). 4 HILLS, JONES, AND TODD Modeling Search in Semantic Memory zation of the entire category of animals. In addition, it is expected that items will affect search in semantic space even if they are not To model search in semantic memory, a structural representa- produced by participants, just as berries on a bush affect foragers’ tion of the search space is required in addition to a model of the external search behavior even if not consumed (e.g., by attracting search process. To represent the structure of semantic memory, we the foragers to search in particular rich-looking areas of the bush). use both hand-coded (Troyer) and statistically derived (BEAGLE) Details of the corpus preprocessing are found in the supplemental schemes. We describe these two structural models next, followed materials (in Appendix 2, as well as BEAGLE code and the animal by a description of the process model that will be applied to these similarity matrix). structural representations. Modeling the search process. The model framework we Representing the structure of semantic memory. The used to simulate the process of search is common to both the SAM Troyer et al. (1997; see also Troyer, 2000) categorization scheme and ACT–R architectures (described in Anderson, 1993; Raaij- contains 22 nonexclusive animal categories (e.g., “African ani- makers & Shiffrin, 1981). The foundational assumption of our mals,” “water animals,” “beasts of burden”). Support for the model is that recall is achieved by probing retrieval structures in Troyer et al. categories comes via their usefulness in detecting memory with a specific cue set, that is, the memory probe. With I specific clinical conditions in individuals, such as Alzheimer’s representing a possible target item for recovery in the search space, disease, depression, and Parkinson’s disease (e.g., Fossati, Le the probability of retrieving I is computed as the product of the Bastard, Ergis, & Allilaire, 2003; Murphy, Rich, & Troyer, 2006; individual retrieval strengths for I across a probe set of M cues, Raoux et al., 2008; Troyer et al., 1998). The categorization scheme with S(Q, I) representing the semantic similarity between cue Q contains 155 unique animal names, which we supplemented with and item I. This is incorporated into an overall probability of 214 additional names to cover the 369 animals reported by our retrieval for item I via the ratio rule: participants. We classified the new animals according to the orig- 写 inal 22 categories found in Troyer et al., based on the descriptions M of the additional animals found on Wikipedia. Our additions thus S共Qj, Ii兲␤ j did not change Troyer et al.’s categorization coding scheme, so j⫽1 that our new investigations remain fully compatible with previous P(Ii兩Q1 ,Q2 , . . ., QM) ⫽ , (3) 冘写 N M results. Our extended categorization coding is available in the S共Qj, Ik兲␤ j supplemental materials (in Appendix 1). k⫽1 j⫽1 To compute more fine-grained semantic similarities between words, we used the lexical semantic representations from the where N represents the total number of items available in the BEAGLE model (Jones, Kintsch, & Mewhort, 2006; Jones & category for retrieval and ␤ represents the saliency (or attention Mewhort, 2007). BEAGLE representations have seen success at weight) assigned to a given cue. accounting for a variety of human semantic data including seman- We examined various static and dynamic models (defined next), tic typicality, categorization, and sentence completion (Jones & using either one or both of two possible cues: frequency and/or the Mewhort, 2007), as well as for a range of semantic priming data previous item recalled. Frequency represents a global search cue, (Jones et al., 2006). In the simulations here, we specifically used which generates a retrieval strength S(Q, I) for each item based on the version of BEAGLE that learns from only contextual informa- that item’s frequency of occurrence in the Wikipedia corpus. The tion, similar to other high-dimensional semantic space models previous-item cue represents a local search cue, which generates a (e.g., Landauer & Dumais, 1997; Lund & Burgess, 1996). retrieval strength for a new item based on its semantic similarity The model begins by assigning each word an initial vector with with that item— here the S(Q, I) value is the cosine similarity in vector elements sampled randomly from a Gaussian distribution BEAGLE between the previous item generated and item I. Using with ␮ ⫽ 0 and ␴ ⫽ 1/ 冑D, where D is the arbitrary vector the maximum likelihood method, we fit ␤ to each participant’s dimensionality (set to 1,000 in these simulations). As the text data, for both cue types, using the participant’s individually gen- corpus is processed, each time a particular word is encountered a erated sequence of items. This produced a log-likelihood fit, which second vector, its memory vector, is updated as the sum of the was penalized based on the number of free parameters via the initial vectors for the other words appearing in context with it. Bayesian information criterion. Results are presented as the me- When the entire corpus has been learned, a word’s memory rep- dian improvement in the Bayesian information criterion relative to resentation is then a vector pattern reflecting the word’s history of a random model specifying that all remaining items in the search co-occurrence with other words. By this method, words that fre- space are equally likely to be retrieved. Specific details of param- quently co-occur will develop similar vector patterns (e.g., bee and eter optimization and model comparison may be found in the honey), as will words that commonly occur in similar contexts, supplemental materials (in Appendix 3). even if they never directly co-occur (e.g., bee and wasp). For all In our terminology, the static models we tested use the same our comparisons, the similarity metric used is the vector cosine (a memory probe (i.e., set of cues) over the entire retrieval interval, normalized dot-product) between two word vectors. effectively ignoring the patchy structure of the environment. In BEAGLE was trained on a 400-million-word Wikipedia corpus contrast, dynamic models exploit that patchy structure, switching (Willits, D’Mello, Duran, & Olney, 2007), and its memory repre- from patch to patch by changing the contents of the memory probe sentations were used to compute the pairwise cosine similarity where local-to-global transitions occur. Specifically, when leaving matrix for a list of 765 animals. The additional 396 animals that a patch, dynamic models switch from the use of the previous-item were not produced by our participants were added to the list to cue (similarity-based local search) to the frequency cue (context- generate a richer memory space representing the semantic organi- based global search) to find a new appropriate patch, and then back OPTIMAL FORAGING AND SEMANTIC MEMORY 5 again to the previous-item cue as the new patch is entered. For backwards—averaging over all words—and indicates that words example, a sequence of DOG–CAT–HAMSTER–HORSE may tran- retrieved immediately preceding any word (“⫺1”) were signifi- sition from a local cue to a frequency cue following HAMSTER, and cantly more likely to be semantically similar to that word than thereby retrieve the high frequency HORSE, which is not semanti- words further away (e.g., the word two items prior, “⫺2”). A cally similar to HAMSTER. We first used the Troyer et al. (1997) one-way analysis of variance predicting the similarity of words as categorization scheme to determine where local-to-global transi- a function of their order relative to the most recent word reveals tions occurred in our participants’ item sequences. On the basis of that words are more similar to the most recent word the closer they these results, we introduced a second patch scheme that can be are produced to that word, F(4, 532) ⫽ 19.54, p ⬍ .001. extended to other environments without the use of hand coding. The importance of local structure is also evident in the static model fits shown in the upper portion of Table 1. All models are Results a significant improvement over the random model with equal weightings (all participants had positive-adjusted Bayesian infor- In this section we first check that the assumptions necessary to test our models of patchy memory search hold, and then examine mation criteria), and both global (frequency) and local (previous three hypotheses predicted by a relationship between search in item) cues are supported as being relevant to the retrieval process. space and search in semantic memory, culminating in an evalua- However, the best static model combines the two retrieval struc- tion of the marginal value theorem in semantic memory as a test of tures—via the integrated cue framework proposed by previous optimal memory foraging. memory models (Anderson, 1993; Raaijmakers & Shiffrin, 1981). As discussed in the introduction, the marginal value theorem Combining local and global cues fit 100% of the participants better assumes that there is local structure in semantic memory analogous than either cue fit alone. This strongly supports the assumption of to spatial allocation of resources. If search in semantic memory is local memory structure in our data. similar to search in space, retrieving an item from a specific Another assumption of a model of memory search through location in memory should increase the likelihood that nearby (i.e., patches in semantic space is that similarity between successively semantically similar) items are retrieved on subsequent trials. A produced items will be lowest at transition points between local recalled item should share the highest semantic similarity with the search and global search. Because local-to-global transition points item retrieved just prior to it and share lower similarity with items imply that a depleted patch is being left and a new patch is being retrieved further back in the sequence. Figure 1 demonstrates that entered—with the local similarity cue being temporarily dropped this assumption holds for our data. The figure shows the data from the memory probe—it follows that the semantic similarity 0.5 0.4 BEAGLE similarity 0.3 0.2 0.1 0.0 -5 -4 -3 -2 -1 Item’s position preceding most recent item Figure 1. The BEAGLE similarity between a word and the words preceding it in the same categorical patch produced by participants. For all figures, patch transitions are computed with the Troyer et al. (1997) categorization. Words that are produced just prior to the most recent word in a patch (Position ⫺1) are the most similar to it, with decreasing similarity for words produced earlier. Error bars are standard error of the mean. 6 HILLS, JONES, AND TODD Table 1 foraging, including patch depletion and optimal patch-leaving pol- Bayesian Information Criterion (BIC) Comparisons (Median icies. Improvement Relative to a Random Model) of Static and Dynamic Models Using a Combination of Frequency (Global) Hypothesis 2: Transition points occur when local semantic and Previous Item Similarity (Local) Cues, Fit to Participant patches are depleted. Memory Retrieval Data Semantic memory patch depletion occurs as words are retrieved, Model ␤ BIC improvement leaving fewer remaining similar words in the same patch left to be found. This implies that over time, retrieved items will have One cue static models reduced similarity to all other remaining (still unretrieved) items in Frequency (global) 8.47 (1.98) 75.5 (20.3) Previous item (local) 4.34 (0.91) 70.1 (24.3) the semantic search space. At some point, a transition to a new Combined cue static model 98.6 (28.3) patch will occur when the local patch is depleted. We call a word’s Global cue 5.80 (2.09) semantic nearness to all other words its residual proximity—this is Local cue 3.29 (1.10) the word’s retrieval strength calculated as the mean similarity Combined cue dynamic model (inverse distance) to all possible remaining (not yet produced) Troyer et al. (1997) categories 100.12 (28.29) Global cue 7.22 (2.18) words in the overarching category (here “animals”) in the Local cue 5.03 (1.67) BEAGLE semantic search space. Residual proximity is thus an Similarity drop model 104.82 (29.45) indication of the richness of a word’s remaining local neighbor- Global cue 6.64 (2.15) hood in semantic space, in terms of how distant the remaining Local cue 4.73 (1.36) unretrieved words are, and thus roughly how long it could take on Note. One parameter (␤) was fit to each cue (global and/or local) for each average to continue to retrieve them. participant. Standard deviations are shown in parentheses. Figure 3 displays the relationship between a word’s residual proximity and its position relative to the beginning of a patch defined by the Troyer et al. (1997) categorization scheme. Resid- between two successively produced items should be lowest where ual proximity was averaged across all words that appear in a local-to-global transitions occur. Figure 2 shows that this is the particular position with respect to any patch switch (e.g., over all case. Based on the Troyer et al. (1997) norms to classify transition words that immediately follow a patch switch, in Position 1, or are points, the semantic similarity between items that occur immedi- two positions before a patch switch, in Position ⫺2). The figure ately before and after a transition point is substantially lower than clearly shows that words produced just prior to a patch switch have the pairwise semantic similarities before or after this point. lower residual proximity to remaining items than items produced This observation suggests an additional memory search model immediately after a patch switch. Moreover, the items produced incorporating a new way of identifying local-to-global transitions, which we call the similarity drop model. This model identifies transitions by noting where similarities drop between words, in the Ratio of pairwise similarity over subject's mean similarity 1.4 following way: If S(A, B) represents the similarity between re- trieved words A and B, then a switch following B is identified in a series of retrievals A, B, C, D if S(A, B) ⬎ S(B, C) and S(B, 1.2 C) ⬍ S(C, D). The bottom of Table 1 shows that a dynamic model employing these transitions performs as well as the model with 1.0 Troyer et al. category transitions. Moreover, approximately 65% of similarity drop switches were also patch switches using the Troyer et al. categories. 0.8 Hypothesis 1: A dynamic model that makes local– global 0.6 transitions will outperform a static model. Appropriate search through patchy structures implies modulat- 0.4 ing adaptively between local patch exploitation and global explo- ration in a dynamic fashion. The bottom portion of Table 1 shows that a dynamic memory search model based on the transition 0.2 points defined by the Troyer et al. (1997) categorization—that is, a model that makes the local-to-global switches where we find 0.0 Troyer-based subcategory switches in the data—accounts better -2 -1 1 2 3 for participant behavior than a static model that does not make any Order of entry relative to patch switch such transitions. This represents an improvement in 85 of the participants (results of a sign test, p ⬍ .01). Moreover, the simi- Figure 2. Mean ratio (and standard error of the mean) of pairwise larity drop model fits 131 of the participants better than the static similarity between successive items produced by a participant to that model (results of a sign test, p ⬍ .001). We show next that these participant’s mean pairwise similarity over all item pairs, by patch entry transition points are appropriate and consistent with an underlying position. For example, the bar above “1” indicates the relative similarity dynamic search process that shares important aspects with spatial between the first item in a patch and the last item in the preceding patch. OPTIMAL FORAGING AND SEMANTIC MEMORY 7 lowed by a global between-patch search. The same pattern of results was found for similarity drop switches. With similarity drop, the item immediately following a patch switch takes signif- icantly longer to produce on average than the participant’s mean IRT over the entire production interval (M ⫽ 1.47 s longer), t(140) ⫽ ⫺14.86, p ⬍ .001, and the second item in a patch takes significantly less time, t(140) ⫽ 12.97, p ⬍ .001. Moreover, as with the Troyer et al. defined patches, as more items were pro- duced within a given patch, the IRTs to produce those items grew longer. To examine the optimal foraging model further, we tested the prediction from the marginal value theorem that each participant’s preswitch IRTs should be at or below his or her long-term average IRT. On a per-participant basis, we assessed whether the distribu- tion of IRTs for the single word immediately preceding a switch (Column ⫺1 in Figure 4) was significantly different (using a one-sample t test) from that participant’s own long-term average Figure 3. The residual proximity of an item in relation to an item’s IRT (the IRTs for the earlier words were shorter, so were not position before or after a patch transition. Only items not yet retrieved are checked individually). With the Troyer et al. (1997) defined included in the computation of an item’s residual proximity value. Error patches, for most participants (132 of 141) the two distributions bars are standard error of the mean. were not significantly different, and for the nine with a significant difference, all their preswitch IRT distributions were less than their long-term averages, again supporting optimal foraging in memory. immediately following a patch switch have the highest residual With similarity drop defined patches, 24 participants had preswitch proximity, indicating that they mark the entry into a relatively IRTs that were significantly different from their long-term average undepleted patch in semantic memory. This is consistent with the IRT, and all these preswitch IRTs were less than their long-term prediction that transitions occur when local patches are depleted. averages. Finally, if the marginal value theorem is a plausible description Hypothesis 3: Transitions occur at points predicted by the for optimal foraging in memory—as it is in space—then individ- marginal value theorem. The results above provide evidence for a dynamic memory search process that combines exploitation with exploration by transitioning between use of local and global cues. The marginal value theorem states that the optimal time for these switches should be when the current intake rate in the patch falls to the mean global intake rate for all patches (Equation 2). Does this optimal foraging policy hold for search in semantic memory? Here we take intake rate to be proportional to the inverse of the time between producing word items—that is, the interitem response times (IRTs)—and so evaluate this hypothesis in terms of the IRTs at patch switches, relative to the mean IRT across all items. Figure 4 shows mean ratios (across participants) of item IRTs to each participant’s long-term average IRT, at different retrieval positions defined relative to Troyer et al. (1997) category bound- aries. The word immediately following a local-to-global transition (i.e., patch switch) takes significantly longer to produce on average than the mean IRT over the entire 3-min production interval (results of a within-participant paired t test), t(140) ⫽ 13.14, p ⬍ .001. The second word in a patch, however, takes significantly less time than the mean IRT, t(140) ⫽ 11.92, p ⬍ .001. Furthermore, as apparent in Figure 4 and in line with the marginal value theorem, word IRTs increase toward the patch transition point but Figure 4. The mean ratio between the interitem retrieval time (IRT) for an item and the participant’s mean IRT over the entire task, relative to the do not exceed the long-term average IRT until the first word after order of entry for the item. For example, the bar above “1” indicates the the transition point (representing the first item in a new patch). The relative IRT between the first word in a patch (defined by Troyer et al., idea here is that as soon as the IRT following some word exceeds 1997, subcategories) and the last word in the preceding patch. The dotted the overall mean IRT, search switches from local to global cues. line shows where item IRTs would be the same as the participant’s mean The IRT from last item in the previous patch to first item in the IRT for the entire task (i.e., the inverse of the long-term average resource new patch includes a longer-than-average within-patch search fol- intake over all patches). Error bars are standard error of the mean. 8 HILLS, JONES, AND TODD uals who were more consistent with its patch departure policies interitem retrieval times via a synthetic “random” search environ- should have recovered more items from memory. One indication ment. By predicting retrieval patterns of memory items searched of consistency with the theorem’s patch-leaving policy is whether for over a structured representation of semantic space in patches, the last IRT in a patch (leading up to the last item) is close to the the work presented here shows how the dynamic local-to-global mean global IRT over all items. If that last IRT is much smaller, search process extends to patchy semantic space in a way that then it would indicate leaving the patch too soon, and if it is much parallels optimal foraging search in physical space. In particular, greater, then it means the individual stayed in the patch too long. we found evidence for local (i.e., patchy) structure in memory, Using Troyer et al. (1997) defined patch boundaries, we computed patch depletion preceding patch departures, and optimal timing of the absolute difference between the mean last-item IRT across patch departures—with participants who more closely adhered to patches and the mean IRT over the entire task, and used this in a optimal foraging theory (i.e., the marginal value theorem) produc- linear regression model to predict the number of items produced by ing more items. each participant (see Figure 5). We found a significant negative Two common underlying factors that motivated our comparison relationship between the two, B ⫽ ⫺5.35, t(139) ⫽ ⫺5.77, p ⬍ between foraging in space and in memory were the shared patchy .001. The same analysis with similarity drop defined patch bound- structure in both types of search environments and the shared aries found the same relationship, B ⫽ ⫺3.50, t(139) ⫽ ⫺6.04, mechanisms that may underlie search processes across environ- p ⬍ .001. Greater deviation from the optimal departure time led to ments. Shared patchy structure is seen in the clustering of animal- fewer items produced. This supports the idea that individuals who and plant-based food resources (Bell, 1991; Taylor, Woiwod, & leave memory patches too early or too late will retrieve fewer Perry, 1978), in the clustering of items recalled from memory (e.g., items from memory than those who follow a policy more consis- Bousfield & Barclay, 1950) and in the small-world network struc- tent with the marginal value theorem. ture of word co-occurrences in text (e.g., Ferrer i Cancho & Sole´, 2001). Shared underlying processes are supported by shared neural Discussion correlates of search (Hills, 2006) and the ability to prime search Semantic memory search appears to be similar to search in from spatial to semantic domains (Hills et al., 2008), which has physical space, involving a dynamic process of mediating between given rise to the theoretical notion that executive cognition is a local exploitation and global exploration of clusters of information domain-general search process (Hills, Todd, & Goldstone, 2010; in much the same way that animals forage among patches of food see also Rhodes & Turvey, 2007). Our results here, demonstrating in their environment. A dynamic process has been postulated that memory search functions similarly to spatial search with before for semantic memory search (Raaijmakers & Shiffrin, 1981; regard to the marginal value theorem, adds an important compo- Troyer et al., 1997), but had only been tested by predicting nent to this argument. 60 50 Number of words produced 40 30 20 0 1 2 3 4 5 Absolute difference between last item IRT and mean IRT (sec) Figure 5. The relationship between a participant’s deviation from the marginal value theorem policy for patch departures (x-axis) and his or her total number of words produced, showing lower performance with less consistency with the optimal foraging rule. Each circle corresponds to one participant; line is the best fitting linear regression. IRT ⫽ interitem retrieval time. OPTIMAL FORAGING AND SEMANTIC MEMORY 9 Together, this evidence supports a theory of semantic memory not delayed spatial win-shift-based foraging. 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