A resource-rational theory of set size effects in human visual working memory
To test the predictions of the model against empirical data , we apply it to the delayed-estimation task ( Wilken and Ma , 2004; Blake et al . , 1997; Prinzmetal et al . , 1998 ) , which is currently one of the most widely used paradigms in VWM research . In this task , the observer briefly holds a set of items in memory and then reports their estimate of a randomly probed target item ( Figure 2A ) . Set size effects manifest as a widening of the estimation error distribution as the number of items is increased ( Figure 2B ) , which suggests a decrease in the amount of resource per item ( Figure 2C ) . To apply our model to this task , we express the expected local behavioral cost as an expected value of the behavioral cost with respect to the error distribution , ( 12 ) c¯behavioral , i ( J¯i;τ ) =∫cbehavioral , i ( ε ) p ( ε;J¯i , τ ) dε , where the behavioral cost function cbehavioral , i ( ε ) maps an encoding error ε to a cost and p ( ε;J¯i , τ ) is the predicted distribution of ε for an item encoded with resource J¯i . We first specify p ( ε;J¯i , τ ) and then turn to cbehavioral , i ( ε ) . As the task-relevant feature in delayed-estimation experiments is usually a circular variable ( color or orientation ) , we make the common assumption that ε follows a Von Mises distribution . We denote this distribution by VM ( ε;J ) , where J is one-to-one related to the distribution’s concentration parameter κ ( Appendix 1 ) . The distribution of ε for a stimulus encoded with resource J¯i is found by integrating over J , p ( ε;J¯i , τ ) =∫VM ( ε;J ) Gamma ( J;J¯i , τ ) dJ Finally , we specify the behavioral cost function cbehavioral , i ( ε ) in Equation 12 , which maps an estimation error ε to a behavioral cost . As in most psychophysical experiments , human subjects tend to perform well on delayed-estimation tasks even when the reward is independent of their performance . This suggests that the behavioral cost function is strongly determined by internal incentives . A recent paper ( Sims , 2015 ) has attempted to measure this mapping and proposed a two-parameter function . We will test that proposal later , but for the moment we assume a simpler , one-parameter power-law function , cbehavioral , i ( ε;β ) =| ε |β , where power β is a free parameter . To obtain an intuition for the predictions of this model , we plot in Figure 2D for a specific set of parameters the two expected costs per item and their sum , Equation 11 , as a function of J¯ . The expected behavioral cost per item depends on pi and decreases with J¯ ( colored curves in left panel ) , while the expected neural cost per item is independent of pi and increases ( black line in left panel ) . The expected total cost per item has a unique minimum ( middle panel ) . The value of J¯ corresponding to this minimum , J¯optimal , increases with pi ( Figure 2D , right ) . Hence , in this example , the optimal amount of resource per item is an increasing function of its probing probability . We next consider the special case in which each item is equally likely to be probed , that is , pi = 1/N . The values of pi in Figure 2D then correspond to set sizes 1 , 2 , 4 , and 8 . When replotting J¯optimal as a function of N , we find a set size effect ( Figure 2E , right panel ) that is qualitatively similar to the empirical result in Figure 2C . An alternative way to understand this predicted set size effect is by considering how the three expected costs across all items , Equation 3 , depend on J¯ . Substituting pi = 1/N in Equation 9 , we find that the expected behavioral cost across all items is independent of set size ( Figure 2E , left panel , black curve ) . Moreover , when all items are encoded with the same amount of resource ( which is necessarily the optimal solution when pi is identical across items ) , the expected neural cost across all items equals NJ¯ and therefore scales linearly with set size ( Figure 2E , left panel , colored lines ) . The sum of these terms has a unique minimum J¯optimal ( Figure 2E , center panel ) , which monotonically decreases with set size ( Figure 2E , right panel ) . The costs plotted in Figure 2E can be considered as obtained by multiplying the corresponding costs in Figure 2D by N . The model thus predicts set size effects in delayed-estimation tasks that are fully mediated by individual-item probing probability . The latter notion is consistent with empirical observations . Palmer et al . ( 1993 ) reported that "relevant set size" ( where irrelevance means pi = 0 ) acts virtually identically to actual set size . Emrich et al . ( 2017 ) independently varied probing probability and set size in their experiment , and found that the former was a better predictor of performance than the latter . Based on this , they hypothesized that set size effects are mediated by probing probability . The predictions of our model are qualitatively consistent with these findings . To examine how well the model accounts for set size effects in empirical data , we fit it to data from six experiments that are part of a previously published benchmark set ( E1-E6 in Table 1 ) . We use a Bayesian optimization method ( Acerbi and Ma , 2017 ) to estimate the maximum-likelihood parameter values , separately for each individual data set ( see Table 2 for a summary of these estimates ) . The model accounts well for the subject-level error distributions ( Figure 3A ) and the two statistics that summarize these distributions ( Figure 3B ) . The original benchmark set ( van den Berg et al . , 2014 ) contained four more data sets , but three of those were published in papers that were later retracted and another one contains data at only two set sizes . Although we decided to leave those four datasets out of our main analyses , the model accounts well for them too ( Figure 2—figure supplement 1 ) . We next compare the goodness of fit of the resource-rational model to that of a descriptive variant in which the amount of resource per item , J¯ , is assumed to be a power-law function of set size ( all other aspects of the model are kept the same ) . This variant is identical to the VP-A model in our earlier work , which is one of the most accurate descriptive models currently available ( van den Berg et al . , 2014 ) . Model comparison based on the Akaike Information Criterion ( AIC ) ( Akaike , 1974 ) indicates that the data provide similar support for both models , with a small advantage for the resource-rational model ( ΔAIC = 5 . 27 ± 0 . 70; throughout the paper , X ± Y indicates mean ±s . e . m . across subjects ) . Hence , the resource-rational model provides a principled explanation of set size effects without sacrificing quality of fit compared to one of the best available descriptive models of VWM . We find that the resource-rational model also fits better than a model in which the total amount of resource is fixed and divided equally across items ( ΔAIC = 13 . 9 ± 1 . 4 ) . So far , we have assumed that there is random variability in the actual amount of resource assigned to an item . Next , we test an equal-precision variant of the resource-rational model , by fixing parameter τ to a very small value ( 10−3 ) . Consistent with the results obtained with the variable-precision model , we find that the rational model has a substantial AIC advantage over a fixed-resource model ( ΔAIC = 43 . 0 ± 6 . 8 ) and is on equal footing with the power-law model ( ΔAIC = 2 . 0 ± 1 . 7 in favor of the power-law model ) . However , all three equal-precision models ( fixed resource , power law , rational ) are outperformed by their variable-precision equivalents by over 100 AIC points . Therefore , we will only consider variable-precision models in the remainder of the paper . To get an indication of the absolute goodness of fit of the resource-rational model , we next examine how much room for improvement there is in the fits . We do this by fitting a non-parametric model variant in which resource J¯ is a free parameter at each set size , while keeping all other aspects of the model the same . We find a marginal AIC difference , suggesting that the fits of the rational model cannot be improved much further without overfitting the data ( ΔAIC = 3 . 49 ± 0 . 93 , in favor of the non-parametric model ) . An examination of the fitted parameter values corroborates this finding: the estimated resource values in the non-parametric model closely match the optimal values in the rational model ( Figure 3C ) . So far , we have assumed that behavioral cost is a power-law function of the absolute estimation error , cbehavioral ( ε ) =|ε|β . To evaluate the necessity of a free parameter in this function , we also test three parameter-free choices: |ε| , ε2 , and −cos ( ε ) . Model comparison favors the original model with AIC differences of 14 . 0 ± 2 . 8 , 24 . 4 ± 4 . 1 , and 19 . 5 ± 3 . 5 , respectively . While there may be other parameter-free functions that give better fits , we expect that a free parameter is unavoidable here , as the error-to-cost mapping may differ across experiments ( because of differences in external incentives ) and also across subjects within an experiment ( because of differences in intrinsic motivation ) . Finally , we also test a two-parameter function that was proposed recently ( Equation ( 5 ) in Sims [2015] ) . The main difference with our original choice is that this alternative function allows for saturation effects in the error-to-cost mapping . However , this extra flexibility does not increase the goodness of fit sufficiently to justify the additional parameter , as the original model outperforms this variant with an AIC difference of 5 . 3 ± 1 . 8 . Finally , we use five-fold cross validation to verify the AIC-based results reported in this section . We find that they are all consistent ( Table 3 ) . One quantitative feature that sets the resource-rational theory apart from previous theories is its predicted relation between set size and the total amount of invested resource , J¯total=∑i=1N J¯i . This quantity is by definition constant in fixed-resource models , and in power-law models it varies monotonically with set size . By contrast , we find that in the fits to several of the experiments , J¯total varies non-monotonically with set size ( Figure 3D , gray curves ) . To examine whether there is evidence for non-monotonic trends in the subject data , we next compute an "empirical" estimate J¯total=∑i=1N J¯^i , where J¯^i are the best-fitting resource estimates in the non-parametric model . We find that these estimates show evidence of similar non-monotonic relations in some of the experiments ( Figure 3D , black circles ) . To quantify this evidence , we perform Bayesian paired t-tests in which we compare the estimates of J¯total at set size 3 with the estimates at set sizes 1 and 6 in the experiments that included these three set sizes ( E2 and E4-E6 ) . These tests reveal strong evidence that the total amount of resource is higher at set size 3 than at set sizes 1 ( BF+0=1 . 05·107 ) and 6 ( BF+0=4 . 02·102 ) . We next compute for each subject the set size at which J¯total is largest , which we denote by Npeak , and find a subject-averaged value of 3 . 52 ± 0 . 18 . Altogether , these findings suggest that the total amount of resource that subjects spend on item encoding varies non-monotonically with set size , which is consistent with predictions from the resource-rational model , but not with any of the previously proposed models . To the best of our knowledge , evidence for a possible non-monotonicity in the relation between set size and total encoding resource has not been reported before . As we noted before , the model predictions do not explicitly depend on set size , N . Yet , we found that the model accounts well for set size effects in the experiments that we considered so far ( E1-E6 ) . This happens because in all those experiments , N was directly coupled with probing probability pi , through pi = 1/N . This coupling makes it impossible to determine whether changes in subjects’ encoding precision are the result of changes in N or changes in pi . Therefore , we will next consider experiments in which individual probing probabilities and set size were varied independently of each other ( E7-E9 in Table 1 ) . According to our model , the effects of N that we found in E1-E6 were really effects of pi . Therefore , we should be able to make predictions about effects of pi in E7-E9 by recasting the effects of N in E1-E6 as effects of pi = 1/N . Given that the amount of resource per item in E1-E6 decreases with N , a first prediction is that it should increase as a function of pi in E7-E9 . A second and particularly interesting prediction is that the estimated total amount of invested resource should vary non-monotonically with pi and peak at a value ppeak that is close to 1/Npeak found in E1-E6 ( see previous section ) . Based on the values of Npeak in experiments E1-E6 , we find a prediction ppeak = 0 . 358 ± 0 . 026 . To test the predictions presented in the previous section and , more generally , to evaluate how well our model accounts for effects of pi on encoding precision , we fit it to data from three experiments in which probing probability was varied independently of set size ( E7-E9 in Table 1 ) . In the first of these experiments ( E7 ) , seven subjects performed a delayed-estimation task at set sizes 2 , 4 , and 8 . On each trial , one of the items – indicated with a cue – was three times more likely to be probed than any of the other items . Hence , the probing probabilities for the cued and uncued items were 3/4 and 1/4 at N = 2 , respectively , 1/2 and 1/6 at N = 4 , and 3/10 and 1/10 at N = 8 . The subject data show a clear effect of pi: the higher the probing probability of an item , the more precise the subject responses ( Figure 4A , top row , black circles ) . We find that the resource-rational model , Equation ( 11 ) , accounts well for this effect ( Figure 4A , top row , curves ) and does so by increasing the amount of resource as a function of probing probability pi ( Figure 4B , left panel , red curves ) . In the other two experiments ( E8 and E9 ) , the number of cued items and cue validity were varied between conditions , while set size was kept constant at 4 or 6 . For example , in one of the conditions of E8 , three of the four items were cued with 100% validity , such that pi was 1/3 for each cued item and 0 for the uncued item; in another condition of the same experiment , two of the four items were cued with 66 . 7% validity , meaning that pi was 1/3 for each cued item and 1/6 for each uncued item . The unique values of pi across all conditions were {0 , 1/6 , 2/9 , 1/4 , 1/3 , 1/2 , 1} in E8 and {0 , 1/12 , 1/10 , 2/15 , 1/6 , 1/3 , 1/2 , and 1} in E9 . As in E7 , responses become more precise with increasing pi and the model accounts well for this ( Figure 4A ) , again by increasing the amount of resource assigned to an item with pi ( Figure 4B ) . We next examine how our model compares to the models proposed in the papers that originally published these three data sets . In contrast to our model , both Bays ( 2014 ) and Emrich et al . ( 2017 ) proposed that the total amount of invested resource is fixed . However , while Bays proposed that the distribution of this resource is in accordance with minimization of a behavioral cost function ( as in our model ) , Emrich et al . postulated that the resource is distributed in proportion to each item’s probing probability . Hence , while our model optimizes both the amount of invested resource and its distribution , Bays’ model only optimizes the distribution , and Emrich et al . ’s model does not explicitly optimize anything . To examine how the three proposals compare in terms of how well they account for the data , we fit two variants of our model that encapsulate the main assumptions of these two earlier proposals . In the first variant , we compute J¯optimal as argminbfJ¯ [ ∑i=1N pic¯behavioral ( J¯i;β , τ ) ] under the constraint ∑i=1N J¯i=J¯total , which is consistent with Bays’ proposal . Hence , in this variant , the neural cost function is removed and parameter λ is replaced by a parameter J¯total – otherwise , all aspects of the model are the same as in our main model . In the variant that we use to test Emrich et al . ’s proposal , we compute J¯i for each item as piJ¯total , where pi is the probing probability and J¯total is again a free parameter that represents the total amount of resource . Fitting the models to the data from all 47 subjects in E7-E9 , we find a substantial advantage of our model over the proposal by Emrich et al . , with an AIC difference of 18 . 0 ± 3 . 9 . However , our model cannot reliably be distinguished from the proposal by Bays: either model is preferred in about half of the subjects ( our model: 27; Bays: 20 ) and the subject-averaged AIC difference is negligible ( 1 . 8 ± 2 . 5 in favor of our model ) . Hence , the model comparison suggests quite convincingly that subjects distribute their resource near-optimally across items with unequal probing probabilities , but it is inconclusive regarding the question of whether the total amount of invested resource is fixed or optimized . As an alternative way to address the question of whether the total amount of resource is fixed , we again fit a non-parametric model to obtain “empirical” estimates of the total amount of invested resource . To this end , we define J¯^total=J¯^i/pi , where J¯^i are the best-fitting values in a non-parametric model , such that J¯^total represents the estimated total amount of resource that a subject would invest to encode a display filled with items that all have probing probability pi . We find that these estimates show signs of a non-monotonicity as a function of pi ( Figure 4B , black points ) , which are captured reasonably well by the resource-rational model ( Figure 4B , black curves ) . Averaged across all subjects in E7-E9 , the value of pi at which J¯^total is largest is 0 . 384 ± 0 . 037 , which is close to the predicted value of 0 . 358 ± 0 . 026 ( see previous section ) . Indeed , a Bayesian independent-samples t-test supports the null hypothesis that there is no difference ( BF01 = 4 . 27 ) . Hence , while the model comparison results in the previous paragraph were inconclusive regarding the question of whether the total amount of invested resource is fixed or optimized , the present analysis provides evidence against fixed-resource models and confirms a prediction made by our own model . In summary , the results in this section show that effects of probing probability in E7-E9 are well accounted for by the same model as we used to explain effects of set size in E1-E6 . Regardless of whether total resource is fixed or optimized , this finding provides further support for the suggestion that set size effects are mediated by probing probability ( Emrich et al . , 2017 ) or , more generally , by item relevance ( Palmer et al . , 1993 ) . There is an ongoing debate about the question of whether a task-relevant item is sometimes completely left out of working memory ( Adam et al . , 2017; Luck and Vogel , 2013; Ma et al . , 2014; Rouder et al . , 2008 ) . Specifically , slot models predict that this happens when set size exceeds the number of slots ( Zhang and Luck , 2008 ) . In resource models , the possibility of complete forgetting has so far been an added ingredient separate from the core of the model ( van den Berg et al . , 2014 ) . Our normative theory allows for a reinterpretation of this question: are there situations in which it is optimal to assign zero resource to the encoding of an item ? We already established that this could happen in delayed-estimation tasks: whenever the probing probability is lower than a threshold value p0=λ| c¯behavioral ( 0 ) | , the optimal amount of resource to invest on encoding the item is zero ( see Theory ) . But what values does p0 take in practice ? Considering the expected behavioral cost function of a fixed-precision model ( a variable-precision model with τ↓0 ) , we can prove that p0 = 0 , that is , it is never optimal to invest no resource ( Appendix 1 ) . For the expected behavioral cost function of the variable-precision model , however , simulations indicate that p0 can be greater than 0 ( we were not able to derive this result analytically ) . We next examine whether this ever happens under parameter values that are representative for human subjects . Using the maximum-likelihood parameters obtained from the data in E7-E9 , we estimate that p0 ( expressed as a percentage ) equals 8 . 86 ± 0 . 54% . Moreover , we find that for 8 of the 47 subjects , p0 is larger than the lowest probing probability in the experiment , which suggests that these subjects sometimes entirely ignored one or more of the items . For these subjects , the error distributions on items with pi<p0 look uniform ( see Figure 4C for an example ) and Kolmogorov-Smirnov tests for uniformity did not reject the null hypothesis in any of these cases ( p>0 . 05 in all tests ) . These results suggest that there might be a principled reason why people sometimes leave task-relevant items out of visual working memory in delayed-estimation experiments . However , our model cannot explain all previously reported evidence for this . In particular , when probing probabilities are equal for all items , the model makes an "all or none" prediction: all items are encoded when pi>p0 and none are encoded otherwise . Hence , the model cannot explain why subjects in tasks with equal probing probabilities sometimes seem to encode a subset of task-relevant items . For example , a recent study reported that in a whole-report delayed-estimation experiment ( pi = 1 for all items ) , subjects encoded about half of the six presented items on each trial ( Adam et al . , 2017 ) . Unless additional assumptions are made , our model cannot account for this finding . The results so far show that the resource-rational model accounts well for data in a variety of delayed-estimation experiments . To examine how its predictions generalize to other tasks , we next consider a change detection task , which is another widely used paradigm in research on VWM . In this task , the observer is sequentially presented with two sets of items and reports if any one of them changed ( Figure 5A ) . In the variant that we consider here , a change is present on exactly half of the trials and is equally likely to occur in any of the items . We construct a model for this task by combining Equations 3 , 4 , and 8 with an expected behavioral cost function based on the Bayesian decision rule for this task ( see Appendix 1 ) , which yields ( 14 ) J¯optimal=argminJ¯ [ p ( error|J¯ ) +λ∑i=1N J¯i ] , where p ( error|J¯ ) is the expected behavioral cost function , which in this case specifies the probability of an error response when a set of items is encoded with resource J¯ . In contrast to local tasks , the expected total cost in global tasks cannot be written as a sum of expected costs per item , because the expected behavioral cost – such as p ( error|J¯ ) in Equation ( 14 ) – can only be computed globally , not per item . Consequently , the elements of J¯optimal in global tasks cannot be computed separately for each item . This makes resource optimization computationally much more demanding , because it requires solving an N-dimensional minimization problem instead of N one-dimensional problems . We perform a simulation at N = 2 ( which is still tractable ) to get an intuition of the predictions that follow from Equation ( 14 ) . For practical convenience , we assume in this simulation that there is no variability in precision , τ↓0 , such that λ is the only model parameter . The results ( Figure 5B ) show that the cost-minimizing strategy is to encode neither of the items when the amount of reward per correct trial is very low ( left panel ) and encode them both when reward is high ( right panel ) . However , interestingly , there is also an intermediate regime in which the optimal strategy is to encode one of the two items , but not the other one ( Figure 5B , central panel ) . Hence , just as in the delayed-estimation task , there are conditions in which it is optimal to encode only a subset of items . An important difference , however , is that in the delayed-estimation task this only happens when items have unequal probing probabilities , while in this change detection task it even happens when all items are equally likely to change . Simulations at larger set sizes quickly become computationally intractable , because of the reason mentioned above . However , the results at N = 2 suggest that if two items are encoded , the optimal solution is to encode them with the same amount of resource ( Figure 5C ) . Therefore , we conjecture that all non-zero values in J¯optimal are identical , which would mean that the entire vector can be summarized by two values: the number of encoded items , which we denote by Koptimal , and the amount of resource assigned to each encoded item , which we denote by J¯optimal . Using this conjecture ( which we have not yet been able to prove ) , we are able to efficiently compute predictions at an arbitrary set size . Simulation results show that the model then predicts that both Koptimal and J¯optimal depend on set size ( Figure 5D , left ) and produces response data that are qualitatively similar to human data ( Figure 5D , right ) .
Descriptive models of visual working memory ( VWM ) have evolved to a point where there is little room for improvement in how well they account for experimental data . Nevertheless , the basic finding that VWM precision depends on set size still lacks a principled explanation . Here , we examined a normative proposal in which expected task performance is traded off against the cost of spending neural resource on encoding . We used this principle to construct a resource-rational model for "local" VWM tasks and found that set size effects in this model are fully mediated by the probing probabilities of the individual items; this is consistent with suggestions from earlier empirical work ( Emrich et al . , 2017; Palmer et al . , 1993 ) . From the perspective of our model , the interpretation is that as more items are added to a task , the relevance of each individual item decreases , which makes it less cost-efficient to spend resource on its encoding . We also found that in this model it is sometimes optimal to encode only a subset of task-relevant items , which implies that resource rationality could serve as a principled bridge between resource and slot-based models of VWM . We tested the model on data from nine previous delayed-estimation experiments and found that it accounts well for effects of both set size and probing probability , despite having relatively few parameters . Moreover , it accounts for a non-monotonicity that appears to exist between set size and the total amount of resource that subjects invest in item encoding . The broader implication of our findings is that VWM limitations – and cognitive limitations in general – may be driven by a mechanism that minimizes a cost , instead of by a fixed constraint on available encoding resource . Our theory makes a number of assumptions that need further investigation . First , we have assumed that the expected behavioral cost decreases indefinitely with the amount of invested resource , such that in the limit of infinite resource there is no encoding error and no behavioral cost . However , encoding precision in VWM is fundamentally limited by the precision of the sensory input , which is itself limited by irreducible sources of neural noise – such as Johnson noise and Poisson shot noise ( Faisal et al . , 2008; Smith , 2015 ) – and suboptimalities in early sensory processing ( Beck et al . , 2012 ) . One way to incorporate this limitation is by assuming that there is a resource value J¯input beyond which the expected behavioral cost no longer decreases as a function of J¯ . In this variant , J¯input represents the quality of the input and J¯optimal will never exceed this value , because any additional resource would increase the expected neural cost without decreasing the expected behavioral cost . Moreover , our theory assumes that there is no upper limit on the total amount of resource available for encoding: cost is the only factor that matters . However , as the brain is a finite entity , the total amount of resource must obviously have an upper limit . This limit can be incorporated by optimizing Joptimal under the constraint ∑i=1N J¯optimal , i≤J¯max , where J¯max represents the maximum amount of resource that can be invested . While an upper limit certainly exists , it may be much higher than the average amount of resource needed to encode information with the same fidelity as the sensory input . If that is the case , then J¯input would be the constraining factor and J¯max would have no effect . Similarly , our theory assumes that there is no lower limit on the amount of resource available for encoding . However , there is evidence that task-irrelevant stimuli are sometimes automatically encoded ( Yi et al . , 2004; Shin and Ma , 2016 ) , perhaps because in natural environments few stimuli are ever completely irrelevant . This would mean that there is a lower limit to the amount of resource spent on encoding . In contradiction to the predictions of our model , such a lower limit would prevent subjects from sometimes encoding nothing at all . For local tasks , such a lower limit can be incorporated by assuming that probing probability pi is never zero . We have fitted our model only to data from delayed-estimation experiments . However , it applies without modification to other local tasks , such as single-probe change detection ( Luck and Vogel , 1997; Todd and Marois , 2004 ) and single-probe change discrimination ( Klyszejko et al . , 2014 ) . Further work is needed to examine how well the model accounts for empirical data of such tasks . Moreover , it should further examine how the theory generalizes to global tasks . One such task could be whole-report change detection; we presented simulation results for this task but the theory remains to be further worked out and fitted to the data . A final limitation is that our theory assumes that items are uniformly distributed and uncorrelated . Although this is correct for most experimental settings , items in more naturalistic settings are often correlated and can take non-uniform distributions . In such environments , the expected total cost can probably be further minimized by taking into account statistical regularities ( Orhan et al . , 2014 ) . Moreover , recent work has suggested that even when items are uncorrelated and uniformly distributed , the expected estimation error can sometimes be reduced by using a "chunking" strategy , that is , encoding similar items as one ( Nassar et al . , 2018 ) . However , as Nassar et al . assumed a fixed total resource and did not take neural encoding cost into account in their optimization , it remains to be seen whether chunking is also optimal in the kind of model that we proposed . We speculate that this is likely to be the case , because encoding multiple items as one will reduce the expected neural cost ( fewer items to encode ) , while the increase in expected behavioral cost will be negligible if the items are very similar . Hence , it seems worthwhile to examine models that combine resource rationality with chunking . Throughout the paper , we have assumed that there is variability in resource assignment . Part of this variability is possibly a result of stochastic factors , but part of it may also be systematic – for example , particular colors and orientations may be encoded with higher precision than others ( Bae et al . , 2014; Girshick et al . , 2011 ) . Whereas the systematic component could have a rational basis ( e . g . higher precision for colors and orientations that occur more frequently in natural scenes [Ganguli and Simoncelli , 2010; Wei and Stocker , 2015] ) , this is unlikely to be true for the random component . Indeed , when we jointly optimize J¯and τ in Equation 11 , we find estimates of τ that consistently approach 0 , meaning that any variability in encoding precision is suboptimal under our proposed cost function . One way to reconcile this apparent suboptimality with the otherwise normative theory is to postulate that maintaining exactly equal resource assignment across cortical regions may itself be a costly process; under such a cost , it could be optimal to allow for some variability in resource assignment . Another possibility is that there are unavoidable imperfections in mental inference ( Drugowitsch et al . , 2016 ) that make it impossible to compute J¯optimal without error , such that the outcome of the computation will vary from trial to trial even when the stimuli are identical . In the present study , we have focused on effects of set size and probing probability on encoding precision . However , our theory also makes predictions about effects of incentive manipulations on encoding precision , because such manipulations affect the expected behavioral cost function . Incentives can be experimentally manipulated in a variety of ways . One method used in at least two previously published delayed-estimation experiments is to make the feedback binary ( "correct , " "error" ) and vary the value of the maximum error allowed to receive positive feedback ( Zhang and Luck , 2011; Nassar et al . , 2018 ) . In both studies , subjects in a "low precision" condition received positive feedback whenever their estimation error was smaller than a threshold value of π/3 . Subjects in the "high precision" condition , however , received positive feedback only when the error was smaller than π/12 ( Zhang and Luck , 2011 ) or π/8 ( Nassar et al . , 2018 ) . Neither of the two studies found evidence for a difference in encoding precision between the low- and high-precision conditions . At first , this may seem to be at odds with the predictions of our model , as one may expect that it should assign more resource to items in the high-precision condition . To test whether this is the case , we simulated this experimental manipulation using a behavioral cost function cbehavioral , i ( ε ) that maps values of |ε| smaller than the feedback threshold to 0 and larger values to 1 . The results reveal that the model predictions are not straightforward and that it can actually account for the absence of an effect ( Figure 6 ) . In particular , the simulation results suggest that the experimental manipulations in the studies by Zhang and Luck and Nassar et al . may not have been strong enough to measure an effect . Indeed , another study has criticized the study by Zhang and Luck on exactly this point and did find an effect when using an experimental design with stronger incentives ( Fougnie et al . , 2016 ) . Another method to manipulate incentives is to vary the amount of potential reward across items within a display . For example , Klyszejko and colleagues performed a local change discrimination experiment in which the monetary reward for a correct response depended on which item was probed ( Klyszejko et al . , 2014 ) . They found a positive relation between the amount of reward associated with an item and response accuracy , which indicates that subjects spent more resource on encoding items with larger potential reward . This incentive manipulation can be implemented by multiplying the behavioral cost function with an item-dependent factor ui , which modifies Equation ( 11 ) to J¯optimal , i ( ri;λ , τ ) =argminJ¯ ( uipic¯behavioral ( J¯;τ ) +λJ¯ ) . The coefficients ui and pi can be combined into a single "item relevance" coefficient ri = uipi , and all theoretical results and predictions that we derived for pi now apply to ri . A difference between the two discussed methods is that the former varied incentives within a trial and the latter across trials . However , both methods can be applied in both ways . A within-trial variant of the experiments by Zhang and Luck ( 2011 ) and Nassar et al . ( 2018 ) would be a N = 2 task in which one of the items always has a low positive feedback threshold and the other a high one . Similarly , a between-trial variant of the experiment by Klyszejko et al . ( 2014 ) would be to scale the behavioral cost function of items with a factor that varies across trials or blocks , but is constant within a trial . Our model can be used to derive predictions for these task variants , which to our knowledge have not been previously reported in the published literature . Our results raise the question of what neural mechanism could implement the optimal allocation policy that forms the core of our theory . Some form of divisive normalization ( Bays , 2014; Carandini and Heeger , 2012 ) would be a likely candidate , which is already a key operation in neural models of attention ( Reynolds and Heeger , 2009 ) and visual working memory ( Bays , 2014; Wei et al . , 2012 ) . The essence of this mechanism is that it lowers the gain when set size is larger , without requiring explicit knowledge of the set size prior to the presentation of the stimuli . Consistent with the predictions of this theory , empirical work has found that the neural activity associated with the encoding of an item decreases with set size , as observed in for example the lateral intraparietal cortex ( Churchland et al . , 2008; Balan et al . , 2008 ) and superior colliculus ( Basso and Wurtz , 1998 ) . Moreover , the work by Bays ( 2014 ) has shown that a modified version of divisive normalization can account for the near-optimal distribution of resources across items with unequal probing probabilities . As set size effects in our model are mediated by probing probability , its predicted set size effects can probably be accounted for by a similar mechanism . Another question concerns the timescale at which the optimization takes place . In all experimental data that we considered here , the only factors that changed from trial to trial were set size ( E1-E7 ) and probing probability ( E7-E9 ) . When we fitted the model , we assumed that the expected total cost in these experiments was minimized on a trial-by-trial basis: whenever set size or probing probability changed from one trial to the next , the computation of Joptimal followed this change . This assumption accounted well for the data and , as discussed above , previous work has shown that divisive normalization can accommodate trial-by-trial changes in set size and probing probability . However , can the same mechanism also accommodate changes in the optimal resource policy changes driven by other factors , such as the behavioral cost function , cbehavioral ( ε ) ? From a computational standpoint , divisive normalization is a mapping from an input vector of neural activities to an output vector , and the shape of this mapping depends on the parameters of the mechanism ( such as gain , weighting factors , and a power on the input ) . As the mapping is quite flexible , we expect that it can accommodate a near-optimal allocation policy for most experimental conditions . However , top-down control and some form of learning ( e . g . reinforcement learning ) are likely required to adjust the parameters of the normalization mechanism , which would prohibit instantaneous optimality after a change in the experimental conditions . The total amount of resource that subjects spend on item encoding may vary non-monotonically with set size in our model . At the neural level , this translates to a prediction of a non-monotonic relation between population-level spiking activity and set size . We are not aware of any studies that have specifically addressed this prediction , but it can be tested using neuroimaging experiments similar to previously conducted experiments . For example , Balan et al . used single-neuron recording to estimate neural activity per item for set sizes 2 , 4 , and 6 in a visual search task ( Balan et al . , 2008 ) . To test for the existence of the predicted non-monotonicity , the same recoding techniques can be used in a VWM task with a more fine-grained range of set sizes . Even though it is practically impossible to directly measure population-level activity , reasonable estimates may be obtained by multiplying single-neuron recordings with set size ( under the assumption that an increase in resource translates to an increase in firing rate and not an increase of neurons used to encode an item ) . A similar method can also assess the relation between an item’s probing probability and the spiking activity related to its neural encoding . Our theory might apply beyond working memory tasks . In particular , it has been speculated that the selectivity of attention arises from a need to balance performance against the costs associated with spiking ( Pestilli and Carrasco , 2005; Lennie , 2003 ) . Our theory provides a normative formalism to test this speculation and may thus explain set size effects in attention tasks ( Lindsay et al . , 1968; Shaw , 1980; Ma and Huang , 2009 ) . Furthermore , developmental studies have found that that working memory capacity estimates change with age ( Simmering and Perone , 2012; Simmering , 2012 ) . Viewed from the perspective of our proposed theory , this raises the question of why the optimal trade-off between behavioral and neural cost would change with age . A speculative answer is that a subject's coding efficiency – formalized by the reciprocal of parameter α in Equation 7 – may improve during childhood: an increase in coding efficiency reduces the neural cost per unit of precision , which shifts the optimal amount of resource to use for encoding to larger values . Neuroimaging studies might provide insight into whether and how coding efficiency changes with age , for example by estimating the amount of neural activity required per unit of precision in memory representations . Our work fits into a broader tradition of normative theories in psychology and neuroscience ( Table 4 ) . The main motivation for such theories is to reach a deeper level of understanding by analyzing a system in the context of the ecological needs and constraints under which it evolved . Besides work on ideal-observer decision rules ( Green and Swets , 1966; KordingKörding , 2007; Geisler , 2011; Shen and Ma , 2016 ) and on resource-limited approximations to optimal inference ( Gershman et al . , 2015; Griffiths et al . , 2015; Vul and Pashler , 2008; Vul , 2009 ) , normative approaches have also been used at the level of neural coding . For example , properties of receptive fields ( Vincent et al . , 2005; Liu et al . , 2009; Olshausen and Field , 1996 ) , tuning curves ( Attneave , 1954; Barlow , 1961; Ganguli and Simoncelli , 2010 ) , neural architecture ( Cherniak , 1994; Chklovskii et al . , 2002 ) , receptor performance ( Laughlin , 2001 ) , and neural network modularity ( Clune et al . , 2013 ) have been explained as outcomes of optimization under either a cost or a hard constraint ( on total neural firing , sparsity , or wiring length ) , and are thus mathematically closely related to the theory presented here . However , a difference concerns the timescale at which the optimization takes place: while optimization in the context of neural coding is typically thought to take place at the timescale over which the statistics of the environment change or a developmental timescale , the theory that we presented here could optimize on a trial-by-trial basis to follow changes in task properties . We already mentioned the information-theory models of working memory developed by Chris R . Sims et al . A very similar framework has been proposed by Chris A . Sims in behavioral economics , who used information theory to formalize his hypothesis of "rational inattention , " that is , the hypothesis that consumers make optimal decisions under a fixed budget of attentional resources that can be allocated to process economic data ( Sims , 2003 ) . The model presented here differs from these two approaches in two important ways . First , similar to early models of visual working memory limitations , they postulate a fixed total amount of resources ( formalized as channel capacity ) , which is a constraint rather than a cost . Second , even if it had been a cost , it would have been the expected value of a log probability ratio . Unlike neural spike count , a log probability ratio does not obviously map to a biologically meaningful cost on a single-trial level . Nevertheless , recent work has attempted to bridge rational inattention and attention in a psychophysical setting ( Caplin et al . , 2018 ) .
Data from experiments E1-E7 ( Table 1 ) and Matlab code for model fitting and simulations are available at http://dx . doi . org/10 . 5061/dryad . nf5dr6c . Bayesian t-tests were performed using the JASP software package ( JASP Team , 2017 ) with the scale parameter of the Cauchy prior set to its default value of 0 . 707 . We used a Bayesian optimization method ( Acerbi and Ma , 2017 ) to find the parameter vector θ={ β , λ , τ } that maximizes the log likelihood function , ∑i=1n logp ( εi;pi , θ ) , where n is the number of trials in the subject’s data set , εi the estimation error on the ith trial , and pi the probing probability of the probed item on that trial . To reduce the risk of converging into a local maximum , initial parameter estimates were chosen based on a coarse grid search over a large range of parameter values . The predicted estimation error distribution for a given parameter vector θ and probing probability pi was computed as follows . First , J¯optimal was computed by applying Matlab's fminsearch function to Equation 11 . Thereafter , the gamma distribution over J ( with mean J¯optimal and shape parameter τ ) was discretized into 50 equal-probability bins . The predicted ( Von Mises ) estimation error distribution was then computed under the central value of each bin . Finally , these 50 predicted distributions were averaged . We verified that increasing the number of bins used in the numerical approximation of the integral over J did not substantially affect the results . In the cross-validation analysis , we fitted the models in the same way as described above , but using only 80% of the data . We did this five times , each time leaving out a different subset of 20% of the data ( in the first run we left out trials 1 , 6 , 11; in the second run we left out trials 2 , 7 , 12 , etc . ) . At the end of each run , we used the maximum-likelihood parameter estimates to compute the log likelihood of the 20% of trials that were left out . These log likelihood values were then combined across the five runs to give an overall cross-validated log likelihood value for each model .
Encoding precision in visual working memory decreases with the number of encoded items . Here , we propose a normative theory for such set size effects: the brain minimizes a weighted sum of an error-based behavioral cost and a neural encoding cost . We construct a model from this theory and find that it predicts set size effects . Notably , these effects are mediated by probing probability , which aligns with previous empirical findings . The model accounts well for effects of both set size and probing probability on encoding precision in nine delayed-estimation experiments . Moreover , we find support for the prediction that the total amount of invested resource can vary non-monotonically with set size . Finally , we show that it is sometimes optimal to encode only a subset or even none of the relevant items in a task . Our findings raise the possibility that cognitive "limitations" arise from rational cost minimization rather than from constraints .
You can read this sentence from beginning to end without losing track of its meaning thanks to your working memory . This system temporarily stores information relevant to whatever task you are currently performing . However , the more items you try to hold in working memory at once , the poorer the quality of each of the resulting memories . It has long been argued that this phenomenon – known as the set size effect – occurs because the brain devotes a fixed amount of neural resources to working memory . But this theory struggles to account for certain experimental results . It also fails to explain why the brain would not simply recruit more resources whenever it has more items to remember . After all , your heart does something similar by beating faster whenever you increase your physical activity . Van den Berg and Ma break with the idea that working memory resources are fixed . They propose that resource allocation is flexible and driven by two conflicting goals: maximize memory performance , but use as few neural resources as necessary . Indeed , a computer simulation that follows this strategy mimics the set size effects seen in healthy volunteers . In the model , the items most relevant for a task are stored more accurately than less important ones , a phenomenon also observed in participants . Lastly , the simulation predicts that the total amount of resources devoted to working memory will vary with the number of items to be remembered . This too is consistent with the results of previous experiments . Working memory thus appears to be more flexible than previously thought . The amount of resources that the brain allocates to working memory is not fixed but could be the result of balancing resource cost against cognitive performance . If this is confirmed , it may be possible to improve working memory by offering rewards , or by increasing the perceived importance of a task .