A resource-rational theory of set size effects in human visual working memory.

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.

Introduction

A well-established property of visual working memory (VWM) is that the precision with which items are encoded decreases with the number of encoded items (Ma et al., 2014; Luck and Vogel, 2013). A common way to explain this set size effect has been to assume that there is a fixed amount of resource available for encoding: the more items, the less resource per item and, therefore, the lower the precision per item. Different forms have been proposed for this encoding resource, such as samples (Palmer, 1994; Sewell et al., 2014), Fisher information (van den Berg et al., 2012; Keshvari et al., 2013), and neural firing rate (Bays, 2014). Models with a fixed amount of resource generally predict that the encoding precision per item (defined as inverse variance of the encoding error) is inversely proportional to set size. This prediction is often inconsistent with empirical data, which is the reason that more recent studies instead use a power law to describe set size effects (Bays et al., 2009; Bays and Husain, 2008; van den Berg et al., 2012; van den Berg et al., 2014; Devkar et al., 2015; Elmore et al., 2011; Mazyar et al., 2012; Wilken and Ma, 2004; Donkin et al., 2016; Keshvari et al., 2013). In these power-law models, the total amount of resource across all items is no longer fixed, but instead decreases or increases monotonically with set size. These models tend to provide excellent fits to experimental data, but they have been criticized for lacking a principled motivation (Oberauer et al., 2016; Oberauer and Lin, 2017): they accurately describe how memory precision depends on set size, but not why these effects are best described by a power law – or why they exist at all. In the present study, we seek a normative answer to these fundamental questions.

While previous studies have used normative theories to account for certain aspects of VWM, none of them has accounted for set size effects in a principled way. Examples include our own previous work on change detection (Keshvari et al., 2012; Keshvari et al., 2013), change localization (van den Berg et al., 2012), and visual search (Mazyar et al., 2012). In those studies, we modelled the decision stage using optimal-observer theory, but assumed an ad hoc power law to model the relation between encoding precision and set size. Another example is the work by Sims and colleagues, who developed a normative framework in which working memory is conceptualized as an optimally performing information channel (Sims, 2016; Sims et al., 2012). Their information-theoretic framework offers parsimonious explanations for the relation between stimulus variability and encoding precision (Sims et al., 2012) and the non-Gaussian shape of encoding noise (Sims, 2015). However, it does not offer a normative explanation of set size effects. In their early work (Sims et al., 2012), they accounted for these effects by assuming that total information capacity is fixed, which is similar to other fixed-resource models and predicts an inverse proportionality between encoding precision and set size. In their later work (Orhan et al., 2014; Sims, 2016), they add to this the assumption that there is an inefficiency in distributing capacity across items and fit capacity as a free parameter at each set size. Neither of these assumptions has a normative motivation. Finally, Nassar and colleagues have proposed a normative model in which a strategic trade-off is made between the number of encoded items and their precision: when two items are very similar, they are encoded as a single item, such that there is more resource available per encoded item (Nassar et al., 2018). They showed that this kind of "chunking" is rational from an information-theoretical perspective, because it minimizes the observer’s expected estimation error. However, just as in much of the work discussed above, this theory assumes a fixed resource budget for item encoding, which is not necessarily optimal when resource usage is costly.

The approach that we take here aligns with the recent proposal that cognitive systems are "resource-rational," that is, trade off the cost of using resources against expected task performance (Griffiths et al., 2015). The starting point of our theory is the principle that neural coding is costly (Attwell and Laughlin, 2001; Lennie, 2003; Sterling and Laughlin, 2015), which may have pressured the brain to trade off the behavioral benefits of high precision against the cost of the resource invested in stimulus encoding (Pestilli and Carrasco, 2005; Lennie, 2003; Ma and Huang, 2009; Christie and Schrater, 2015). We hypothesize that set size effects – and limitations in VWM in general – may be the result of making this trade-off near-optimally. We next formalize this hypothesis in a general model that can be applied to a broad range of tasks, analyze the theoretical predictions of this model, and fit it to data from nine previous delayed-estimation experiments.

Theory

General theoretical framework: trade-off between behavioral and neural cost

We define a vector Q={Q1,…, Q} that specifies the amount of resource with which each of N task-relevant items is encoded. We postulate that Q affects two types of cost: an expected behavioral cost induced by task errors and an expected neural cost induced by spending neural resources on encoding. The expected total cost is a weighted combination, where the weight λ≥0 represents the importance of the neural cost relative to the behavioral cost. Generally, increasing the amount of resource spent on encoding will reduce the expected behavioral cost, but simultaneously increase the expected neural cost.

The key novelty of our theory is that instead of assuming that there is a fixed resource budget for stimulus encoding (a hard constraint), we postulate that the brain – possibly on a trial-by-trial basis – chooses its resource vector Q in a manner that minimizes the expected total cost. We denote the vector that yields this minimum by Qoptimal:

Under this policy, the total amount of invested resource – the sum of the elements of Qoptimal – does not need to be fixed: when it is "worth it" (i.e. when investing more resource reduces the expected behavioral cost more than it increases the expected neural cost), more resource may be invested.

Equations (1) and (2) specify the theory at the most general level. To derive testable predictions, we next propose specific formalizations of resource and of the two expected cost functions.

Formalization of resource

As in our previous work (Keshvari et al., 2012; Keshvari et al., 2013; Mazyar et al., 2012; van den Berg et al., 2012; van den Berg et al., 2014), we quantify encoding precision as Fisher information, J. This measure provides a lower bound on the variance of any unbiased estimator (Cover and Thomas, 2005; Ly et al., 2017) and is a common tool in the study of theoretical limits on stimulus coding and discrimination (Abbott and Dayan, 1999). Moreover, we assume that there is item-to-item and trial-to-trial variation in precision (Fougnie et al., 2012; van den Berg et al., 2012; van den Berg et al., 2014; Keshvari et al., 2013; van den Berg et al., 2017). Following our previous work, we model this variability using a gamma distribution with a mean and shape parameter τ ≥0 (larger τ means more variability); we denote this distribution by gamma .

We specify resource vector Q as the vector with mean encoding precisions, , such that the general theory specified by Equations (1) and (2) modifies toand

In this formulation, it is assumed that the brain has control over resource vector , but not over the variability in how much resource is actually assigned to an item. It should be noted, however, that our choice to incorporate variability in J is empirically motivated and not central to the theory: parameter τ mainly affects the kurtosis of the predicted estimation error distributions, not their variance or the way that the variance depends on set size (which is the focus of this paper). We will show that the theory also predicts set size effects when there is no variability in J.

Formalization of expected neural cost

To formalize the neural cost function, we make two general assumptions. First, we assume that the expected neural cost induced by encoding a set of N items is the sum of the expected neural cost associated with each of the individual items. Second, we assume that each of these “local” neural costs has the same functional dependence on the amount of allocated resource: if two items are encoded with the same amount of resource, they induce equal amounts of neural cost. Combining these assumptions, the expected neural cost induced by encoding a set of N items with resource takes the formwhere we introduced the convention to denote local costs (associated with a single item) with small c, to distinguish them from the global costs (associated with the entire set of encoded items), which we denote with capital C.

We denote by cneural(J) the neural cost induced by investing an amount of resource J. The expected neural cost induced by encoding an item with resource is obtained by integrating over J,

The theory is agnostic about the exact nature of the cost function : it could include spiking and non-spiking components (Lennie, 2003), be associated with activity in both sensory and non-sensory areas, and include other types of cost that are linked to “mental effort” in general (Shenhav et al., 2017).

To motivate a specific form of this function, we consider the case that the neural cost is incurred by spiking activity. For many choices of spike variability, including the common one of Poisson-like variability (Ma et al., 2006), Fisher information J of a stimulus encoded in a neural population is proportional to the trial-averaged neural spiking rate (Paradiso, 1988; Seung and Sompolinsky, 1993). If we further assume that each spike has a fixed cost, we find that the local neural cost induced by each item is proportional to J,where α is the amount of neural cost incurred by a unit increase in resource. Combining Equations (5–7) yields

Hence, the global expected neural cost is proportional to the total amount of invested resource and independent of the amount of variability in J. Although we use this linear expected neural cost function throughout the paper, we show in Appendix 1 that the key model prediction – a decrease of the optimal resource per item with set size – generalizes to a broad range of choices.

Formalization of expected behavioral cost for local tasks

Before we specify the expected behavioral cost function, we introduce a distinction between two classes of tasks. First, we define a task as "local" if the observer’s response depends on only one of the encoded items. Examples of local tasks are single-probe delayed-estimation (Blake et al., 1997; Prinzmetal et al., 1998; Wilken and Ma, 2004), single-probe change detection (Todd and Marois, 2004; Luck and Vogel, 1997), and single-probe change discrimination (Klyszejko et al., 2014). By contrast, when the task response depends on all memorized items, we define the task as "global." Examples of global tasks are whole-display change detection (Luck and Vogel, 1997; Keshvari et al., 2013), change localization (van den Berg et al., 2012), and delayed visual search (Mazyar et al., 2012). The theory that we developed up to this point – Equations (1–8) – applies to both global and local tasks. However, from here on, we develop our theory in the context of local tasks only; we will come back to global tasks at the end of the Results.

As in local tasks only one item gets probed, the expected behavioral cost across all items is a weighted average,where p is the experimentally determined probing probability of the ith item and is the local expected behavioral cost associated with reporting the ith item. We will refer to the product as the 'expected behavioral cost per item'. The only remaining step is to specify . This function is task-specific and we will specify it after we have described the task to which we apply the model.

A resource-rational model for local tasks

Combining Equations 3, 8, and 9 yields the following expected total cost function for local tasks:

As parameters α and λ have interchangeable effects on the model predictions, we will fix α = 1 and only treat λ as a free parameter.

We recognize that the right-hand side of Equation 10 is a sum of independent terms. Therefore, each element of , Equation 4, can be computed independently of the other elements, by minimizing the expected total cost per item,

This completes the specification of the general form of our resource-rational model for local tasks. Its free parameters are λ and τ.

Set size effects result from cost minimization and are mediated by probing probability

To obtain an understanding of the model predictions, we analyze how depends on probing probability and set size. We perform this analysis under two general assumptions about the local expected behavioral cost function: first, that it monotonically decreases with (i.e. increasing resource reduces the expected behavioral cost) and, second, that it satisfies a law of diminishing returns (i.e. the reductions per unit increase of resource decrease with the total amount of already invested resource). It can be proven (see Appendix 1) that under these assumptions, the domain of probing probability p consists of three potential regimes, each with a different optimal encoding strategy (Figure 1A). First, there might exist a regime 0≤pexpected behavioral cost by more than it increases the expected neural cost. Second, there might exist a regime p0≤pexpected behavioral cost evaluated at ; specifically, . The threshold p∞ depends on λ and on the derivative of the local expected behavioral cost evaluated at ; specifically, . If p∞>1, then the third regime does not exist, whereas if p0 >1, only the first regime exists.

Figure 1.

Effects of probing probability and set size on in the resource-rational model for local tasks.

(A) The model has three different optimal solutions depending on probing probability p: invest no resource when p is smaller than some threshold value p0, invest infinite resource when p is larger than p∞, and invest a finite amount of resource when p0 1, then only the first regime exists; if p0 <1 < p∞ then only the first two regimes exist. (B) If, in addition, p = 1/N, then the domain of N partitions in a similar manner.

(A) The model has three different optimal solutions depending on probing probability p: invest no resource when p is smaller than some threshold value p0, invest infinite resource when p is larger than p∞, and invest a finite amount of resource when p0 expected behavioral cost function evaluated at 0 and ∞, respectively. If p0 >1, then only the first regime exists; if p0 <1 < p∞ then only the first two regimes exist. (B) If, in addition, p = 1/N, then the domain of N partitions in a similar manner.

We next turn to set size effects. An interesting property of the model is that depends only on the probing probability, p, and on the model parameters – it does not explicitly depend on set size, N. Therefore, the only way in which the model can predict set size effects is through a coupling between N and p. Such a coupling exists in most studies that use a local task. For example, in delayed-estimation tasks, each item is usually equally likely to be probed such that p = 1/N. For those experiments, the above partitioning of the domain of p translates to a similar partitioning of the domain of N (Figure 1B). Then, a set size N∞≥0 may exist below which it is optimal to encode items with infinite resource, a region N∞≤N < N0 in which it is optimal to encode items with a finite amount of resource, and a region N>N0 in which it is optimal to not encode items at all.

Results

Model predictions for delayed-estimation tasks

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).

Figure 2.

A resource-rational model for delayed-estimation tasks.

(A) Example of a trial in a delayed-estimation experiment. The subject is briefly presented with a set of stimuli and, after a short delay, reports the value of the item at a randomly chosen location (here indicated with thick circle). (B) The distribution of estimation errors in delayed-estimation experiments typically widens with set size (data from Experiment E5 in Table 1). (C) This suggests that the amount of resource per encoded item decreases with set size. The estimated amount of resource per item was computed using the same non-parametric model as the one underlying Figure 3C. (D) Expected cost per item as a function of the amount of invested resource (model parameters: λ = 0.01, β = 2, τ↓0). Left: The expected behavioral cost per item (colored curves) decreases with the amount of invested resource, while the expected neural cost per item increases (black line). Center: The sum of these two costs has a unique minimum, whose location (arrows) depends on probing probability p: The optimal amount of resource per item increases with the probability that the item will be probed. (E) Expected cost across all items, when each item is probed with a probability p = 1/N; the model parameters are the same as in D and the set sizes correspond with the values of p in D. The predicted set size effect (right panel) is qualitatively similar to set size effects observed in empirical data (cf. panel C). (D) and (E) are alternative illustrations of the same optimization problem; the right panel of (E) could also be obtained by replotting the right panel of (D) as a function of N = 1/p.

Circular variance (top) and circular kurtosis (bottom) of the estimation error distributions as a function of set size, split by experiment. Error bars and shaded areas represent 1 s.e.m. of the mean across subjects. The first three data sets were excluded from the main analyses on the grounds that they were published in papers that were later retracted (Anderson and Awh, 2012; Anderson et al., 2011). The data set from the study by Rademaker et al. (2012) was excluded from the main analyses because it contains only two set sizes, which makes it less suitable for a fine-grained study of the relationship between encoding precision and set size.

Table 1.

Overview of experimental datasets.

Experiments E5 and E6 differed in the way that subjects provided their responses (E5: color wheel; E6: scroll).

Exp. ID	Reference	Feature	Set size(s)	Probing probability	Number of subjects
E1	Wilken and Ma (2004)	Color	1, 2, 4, 8	Equal	15
E2	Zhang and Luck (2008)	Color	1, 2, 3, 6	Equal	8
E3	Bays et al. (2009)	Color	1, 2, 4, 6	Equal	12
E4	van den Berg et al. (2012)	Orientation	1-8	Equal	6
E5	van den Berg et al. (2012)	Color	1-8	Equal	13
E6	van den Berg et al. (2012)	Color	1-8	Equal	13
E7	Bays et al. (2009)	Orientation	2,4,8	Unequal	7
E8	Emrich et al. (2017)	Color	4	Unequal	20
E9	Emrich et al. (2017)	Color	6	Unequal	20

Figure 3.

Model fits to data from six delayed-estimation experiments with equal probing probabilities.

(A) Maximum-likelihood fits to raw data of the worst-fitting and best-fitting subjects (subjects S10 in E6 and S4 in E4, respectively). Goodness of fit was measured as R2, computed for each subject by concatenating histograms across set sizes. (B) Subject-averaged circular variance and kurtosis of the estimation error, as a function of set size and split by experiment. The maximum-likelihood fits of the model account well for the trends in these statistics. (C) Estimated amounts of resource per item in the resource-rational model scattered against the estimates in the non-parametric model. Each dot represents estimates from a single subject. (D) Estimated amount of resource per item (red) and total resource (black) plotted against set size. Here and in subsequent figures, error bars and shaded areas represent 1 s.e.m. of the mean across subjects.

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,where the behavioral cost function cbehavioral(ε) maps an encoding error ε to a cost and is the predicted distribution of ε for an item encoded with resource . We first specify and then turn to cbehavioral(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 is found by integrating over J,

Finally, we specify the behavioral cost function 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, , 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 . The expected behavioral cost per item depends on p and decreases with (colored curves in left panel), while the expected neural cost per item is independent of p and increases (black line in left panel). The expected total cost per item has a unique minimum (middle panel). The value of corresponding to this minimum, , increases with p (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, p = 1/N. The values of p in Figure 2D then correspond to set sizes 1, 2, 4, and 8. When replotting 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 . Substituting p = 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 p is identical across items), the expected neural cost across all items equals and therefore scales linearly with set size (Figure 2E, left panel, colored lines). The sum of these terms has a unique minimum (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 p = 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.

Model fits to data from delayed-estimation experiments with equal probing probabilities

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).

Table 2.

Subject-averaged parameter estimates of the resource-rational model fitted to data from nine previously published experiments.

See Table 1 for details about the experiments.

Experiment	β	λ	τ
E1	1.87 ± 0.29	(4.8 ± 1.2)·10−2	17.9±2.5
E2	(1.33 ± 0.30)·10−2	(4.27 ± 0.83)·10−4	14.8±1.1
E3	0.138 ± 0.042	(2.78 ± 0.87) ·10−3	19.1±2.6
E4	0.106 ± 0.052	(3.2 ± 1.4)·10−3	8.2±1.8
E5	0.356 ± 0.085	(5.8 ± 1.1)·10−3	18.1±2.8
E6	0.61 ± 0.15	(8.8 ± 1.5)·10−3	7.4±1.3
E7	1.19 ± 0.51	(9.5 ± 6.6)·10−2	5.7±1.5
E8	0.58 ± 0.19	(1.58 ± 0.66)·10−2	27.0±3.7
E9	0.93 ± 0.25	(3.0 ± 1.0)·10−2	23.7±2.3

Figure 2—figure supplement 1.

Fits to the three delayed-estimation benchmark data sets that were excluded from the main analyses.

Circular variance (top) and circular kurtosis (bottom) of the estimation error distributions as a function of set size, split by experiment. Error bars and shaded areas represent 1 s.e.m. of the mean across subjects. The first three data sets were excluded from the main analyses on the grounds that they were published in papers that were later retracted (Anderson and Awh, 2012; Anderson et al., 2011). The data set from the study by Rademaker et al. (2012) was excluded from the main analyses because it contains only two set sizes, which makes it less suitable for a fine-grained study of the relationship between encoding precision and set size.

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, , 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 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|. 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).

Table 3.

Comparing two metrics for model comparison: AIC and five-fold cross-validated log likelihood.

Each comparison is between the main version of the resource-rational model, Equation (11), and the model listed in the first column of the table. Negative AIC differences and positive cross-validated log likelihood differences indicate an advantage of the resource-rational model over the alternative model. In all comparisons, these differences have opposite signs, which means that the AIC-based results are consistent with the cross-validation results.

Model with which the main model is compared	AIC difference	Cross-validation log likelihood difference
Descriptive power-law model	−5.27±0.70	2.59±0.39
Descriptive fixed-resource model	−13.9±1.4	8.4±1.0
Descriptive unconstrained model	3.49±0.93	−1.26±0.49
Rational model variant: equal precision	−110±10	56±4.7
Rational model variant: cbehavioral=|ε|	−14±2.8	7.1±1.4
Rational model variant: cbehavioral=ε2	−24.4±4.1	12.2±2.0
Rational model variant: cbehavioral=−cos(ε)	−19.5±3.5	9.8±1.8
Rational model variant: cbehavioral as in Sims (2015)	−5.3±1.8	4.7±0.74

Non-monotonic relation between total resource and set size

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, . 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, 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 , where 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 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 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.

Predicted effects of probing probability

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 p, through p = 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 p. 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 p. Therefore, we should be able to make predictions about effects of p in E7-E9 by recasting the effects of N in E1-E6 as effects of p = 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 p in E7-E9. A second and particularly interesting prediction is that the estimated total amount of invested resource should vary non-monotonically with p 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.

Model fits to data from delayed-estimation experiments with unequal probing probabilities

To test the predictions presented in the previous section and, more generally, to evaluate how well our model accounts for effects of p 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 p: 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 p (Figure 4B, left panel, red curves).

Figure 4.

Model fits to data from three delayed-estimation experiments with unequal probing probabilities.

(A) Fits of the resource-rational model (curves) to the data (black circles) of experiments E7-E9. (B) Estimated amount of resource per item as a function of probing probability (red) and the corresponding estimated total amount of resource that the subject would spend on encoding a display filled with items with equal probing probabilities (black). (C) Error histograms and a plot of as a function of p for a single subject (S4 in E9). The estimated value of p0 was 0.18 for this subject, which was larger than the smallest probing probability in the experiment. The error histograms for items with the four lowest probing probabilities appear to be uniform for this subject, which is indicative of guessing (p>0.23 in Kolgomorov-Smirnov tests for uniformity on these four distributions).

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 p 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 p was 1/3 for each cued item and 1/6 for each uncued item. The unique values of p 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 p and the model accounts well for this (Figure 4A), again by increasing the amount of resource assigned to an item with p (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 as under the constraint , which is consistent with Bays’ proposal. Hence, in this variant, the neural cost function is removed and parameter λ is replaced by a parameter – 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 for each item as , where p is the probing probability and 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 , where are the best-fitting values in a non-parametric model, such that represents the estimated total amount of resource that a subject would invest to encode a display filled with items that all have probing probability p. We find that these estimates show signs of a non-monotonicity as a function of p (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 p at which 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).

Is it ever optimal to not encode an item?

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 , 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 ), 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 p0.05 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 p>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 (p = 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.

Predictions for a global task: whole-display change detection

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

Figure 5.

A resource-rational model for change detection tasks.

(A) Example of a trial in a change detection task with a set size of 2. The subject is sequentially presented with two sets of stimuli and reports whether there was a change at any of the item locations. (B) Simulated expected total cost in the resource-rational cost function applied to a task with a set size of 2 and a reward of 0.05 (left), 0.20 (center), or 0.35 (right) units per correct trial. The red dot indicates the location of minimum cost, that is the resource-optimal combination of and (note that the expected cost function in the central panel has a minimum at two distinct locations). When reward is low (left), the optimal strategy is to encode neither of the two stimuli. When reward is high (right), the optimal strategy is to encode both stimuli with equal amounts of resource. For intermediate reward (center), the optimal strategy is to encode one of the two items, but not the other one. (C) Model predictions as a function of trial rewards at N = 2. Left: The amount of resource assigned to the two items for a range of reward values. Right: the corresponding optimal number of encoded items (top) and optimal amount of resource per encoded item (bottom) as a function of reward. (D) Model predictions as a function of set size (trial reward = 1.5). The model predicts set size effects in both the number of encoded items (left, top) and the amount of resource with which these items are encoded (left, bottom). Moreover, the model produces response data (right) that are qualitatively similar to human data (see, for example, Figure 2C in Keshvari et al., 2013). The parameter values used in all simulations were λ = 0.01 and τ↓0.

where 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 .

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 in Equation (14) – can only be computed globally, not per item. Consequently, the elements of 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 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 . 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 depend on set size (Figure 5D, left) and produces response data that are qualitatively similar to human data (Figure 5D, right).

Discussion

Summary

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.

Limitations

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 beyond which the expected behavioral cost no longer decreases as a function of . In this variant, represents the quality of the input and 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 under the constraint , where 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 would be the constraining factor and 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 p 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.

Variability in resource assignment

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 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 without error, such that the outcome of the computation will vary from trial to trial even when the stimuli are identical.

Experimental predictions of incentive manipulations

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,(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).

Figure 6.

Model predictions for a delayed-estimation task with binary feedback (N = 5).

In this experiment, the observer receives positive feedback (e.g. "correct") when their estimation error is smaller than the positive feedback threshold and negative feedback (e.g. "error") otherwise. We modelled this using a behavioral cost function that maps errors below the feedback threshold to a cost of 0 and errors larger than this threshold to a cost equal to 1. The model predicts that subjects do not invest any resource when the feedback threshold is very small (extremely difficult tasks) or very large (extremely easy tasks), such that the expected absolute estimation error is π/2 (guessing). In an intermediate regime, the prediction is U-shaped and contains a region in which the predicted estimation error barely changes as a function of feedback threshold. In this region, any performance benefit from increasing the amount of invested resource is almost exactly outdone by the added neural cost. The dashed lines show the feedback thresholds corresponding to the "high precision" and "low precision" conditions in the experiment by Nassar et al. (2018). Under the chosen parameter settings (λ = 0.08, τ = 30), the model predicts that the average absolute estimation errors in these two conditions (black circles) are very similar to each other.

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 u, which modifies Equation (11) to . The coefficients u and p can be combined into a single "item relevance" coefficient r = u, and all theoretical results and predictions that we derived for p now apply to r.

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.

Neural mechanisms and timescale of optimization

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 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.

Neural prediction

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.

Extensions to other domains

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.

Broader context

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.

Table 4.

Examples of resource-rationality concepts in neuroscience, psychology, and economics.

Study	Optimized quantity	Performance term	Resource cost/constraint
Efficient coding in neural populations
Ganguli and Simoncelli (2010)	Tuning curve spacing and width	Fisher information or discriminability	Neural activity (constraint)
Olshausen and Field (1996)	Receptive field specificity	Information	Sparsity
Capacity “limitations” in attention and memory
Sims et al. (2012)	Information channel bit allocation	Channel distortion (e.g. squared error)	Channel capacity (constraint)
Van den Berg and Ma (present study)	Mean encoding precision	Behavioral task accuracy	Neural activity (cost)
Rational inattention in consumer choice
Sims (2003)	Distribution of attention	Channel distortion (e.g. squared error)	Channel capacity (constraint)

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).

Materials and methods

Data and code sharing

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.

Statistical analyses

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.

Model fitting

We used a Bayesian optimization method (Acerbi and Ma, 2017) to find the parameter vector that maximizes the log likelihood function, , where n is the number of trials in the subject’s data set, ε the estimation error on the ith trial, and p 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 p was computed as follows. First, was computed by applying Matlab's fminsearch function to Equation 11. Thereafter, the gamma distribution over J (with mean 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.

Model comparison using cross-validation

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.

In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.

Thank you for submitting your article "Ecological rationality in human working memory and attention" for consideration by eLife. Your article has been reviewed by three peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Timothy Behrens as the Senior Editor. The following individual involved in review of your submission has agreed to reveal her identity: Jacqueline Gottlieb (Reviewer #2).

The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.

Summary:

In this manuscript, van den Berg and Ma propose a normative theory of a phenomenon called set-size effect – the fact that attention and working memory performance degrades as a function of the number of items. The set-size effect is fundamental in attention and working memory research, and many previous models have successfully described it by using ad hoc assumptions about the relation between precision and set-size. The authors' main innovation is to explain this relation in a normative framework. Specifically, they propose that the degradation in precision as a function of set size reflects a tradeoff between the benefits of precise encoding and the neural costs that this encoding demands. The authors develop the model, use it to fit several existing data sets, and offer an extensive discussion of the limitations of their model and its relation to the previous theoretical and empirical literatures.

The paper incorporates a genuinely new idea and is clearly written and quite thorough in its analysis and discussion. The Results section details a substantive amount of work, in which the model was fit to data from several experiments where working memory precision was measured in the context of local delayed-estimation, global estimation, change detection, change localization and present/absent visual search, and was quantitatively evaluated against other (often equally good) prediction schemes.

Some major revisions are requested to clarify situations in which not all items are stored, to answer questions about the model parameter τ, and to expand the discussion about how the flexibility of resource allocation in the model could be mechanistically realized in the brain.

Essential revisions:

1) The number of encoded items: The paper mentions a possible hard constraint on the number of items encoded. Depending on set size and cost functions, when (if ever) is it "optimal" not to encode some of the items? Some new modeling results should be shown here to shore this up and a longer discussion of this point should be added to the Results and Discussion.

Furthermore, please expand the discussion of the costs of encoding more items versus fewer. Intuitively, it is obvious that it is more costly to encode 8 items than 2, but there are numerous reasons why this could be the case. Are there any experimental data in support of this assumption? In particular, are there any experimental data showing that encoding more items results in higher firing rates (at the population level)?

Michael Frank and his group have recently made a very principled attempt to characterize the nature of the cost in these tasks, in terms of how participants might group or chunk the items. It may be beyond the scope of the current paper to attempt to outline a theory that explains these results, but perhaps readers would enjoy some more elaborate discussion of this issue.

2) τ: Please provide additional model results that show how the fits look when τ = 0. In particular, please show the goodness of fit of the rational model (with τ = 0) as compared to the fit of a model with a "hard" constraint on resources? In general, it was confusing that the theory is initially described in absence of τ, while τ is used for actual fits to data. It would have been easier to understand if the theory had been evaluated in presence of τ, and its effect studied within that theoretical framework.

3) The speed of policy update: An assumption of the model that may be problematic is that people must almost instantaneously optimize their encoding precision when set-sizes change unpredictably from trial to trial. The theory predicts that, when a trial contains 2 stimuli and the next trial contains 4 stimuli, the participant instantaneously lowers the encoding precision to the new (near-) optimal level. This process sounds pretty demanding itself, especially considering that humans may also play an active role in determining their intrinsic motivation or deciding which items to memorize, which may further slow down the adjustment process. The flexibility in allocating resources that is implied by this model seems to be at odds with the slowness of cognitive control, well-documented by task switching costs. The authors touch on this point in the very last line of the paper, where they note that divisive normalization can provide a rapid adjustment mechanism. Even though the discussion is already long, it would be good to hear more about this point, and a comparison between a hardwired allocation mechanism and slower but more flexible cognitive control strategies.

Furthermore, in the Frank paper described above, they use a task in which there is binary feedback that depends on the liberal vs. conservative error criterion (and they don't report major differences in performance as a function of this). One might suspect that participants would fail to adapt their policy even in an incentive-compatible version of this task which systematically varied these behavioral costs, and this would present a challenge to the authors' theory as described here. Please add some text to the Discussion addressing this point.

4) Total precision vs. set-size: A novel and interesting prediction of the model is the non-monotonic relation between set-size and total precision (Figure 3B). Although the authors state that this point requires more empirical documentation, are the model results consistent with a non-monotonic encoding of target location that was reported by Balan et al. (2008) in monkey area LIP? That study found that, in a covert visual search task with different set sizes, the fidelity of target location encoding by area LIP was higher at set size 4 than at set size 2 or set size 6 (see Figure 5) – a non-linearity that was puzzling at the time but may gain new significance in light of this paper. Please add some discussion of this point to the manuscript.

Essential revisions:

1) The number of encoded items: The paper mentions a possible hard constraint on the number of items encoded. Depending on set size and cost functions, when (if ever) is it "optimal" not to encode some of the items? Some new modeling results should be shown here to shore this up and a longer discussion of this point should be added to the Results and Discussion.

This is an interesting question, because its answer can possibly provide a principled bridge between slot-based and resource models of VWM. We now address this question in three different places. First, a mathematical analysis of the general conditions under which it is optimal to not encode an item is provided in Appendix 1. Second, the question is addressed in the context of delayed-estimation tasks in the new Results section “Is it ever optimal to not encode an item?”.Third, for the change-detection task, the question is addressed in the new section “Predictions for a global task: whole-display change detection”.

Furthermore, please expand the discussion of the costs of encoding more items versus fewer. Intuitively, it is obvious that it is more costly to encode 8 items than 2, but there are numerous reasons why this could be the case. Are there any experimental data in support of this assumption? In particular, are there any experimental data showing that encoding more items results in higher firing rates (at the population level)?

For many choices of spike variability, the total precision of a set of stimuli encoded in a neural population is proportional to the trial-averaged neural spiking rate (e.g., Paradiso, 1988; Seung and Sompolinsky, 1993; Ma et al., 2006). Based on this theoretical argument, it is expected that it is more costly (in terms of neural spiking) to encode 8 items compared to 2, if they are encoded with the same precision.

However, it is important to keep in mind that our model does not predict that the total spiking rate will increase with set size, because it generally predicts the precision per item (i.e., spike rate per item) to decrease with set size, which is consistent with physiological evidence (e.g., Churchland et al., 2008; Balan et al., 2008; Basso and Wurtz, 1998). The maximum-likelihood fits suggest that the total amount of invested resource varies non-monotonically with set size, which predicts that the population-level spiking activity also varies non-monotonically with set size. We are not aware of any work that strongly supports or rejects this prediction (see also our response below to the point about the Balan et al. paper). We address this point as follows in a new discussion section “Neural prediction”.

Michael Frank and his group have recently made a very principled attempt to characterize the nature of the cost in these tasks, in terms of how participants might group or chunk the items. It may be beyond the scope of the current paper to attempt to outline a theory that explains these results, but perhaps readers would enjoy some more elaborate discussion of this issue.

We assume that this comment refers to the recent paper by Nassar, Helmers, and Frank. If we understand correctly, this paper is currently in press, so we base our response to this comment on the preprint that is available on bioRxiv.

We agree that this paper has several connections with our own study, and we now refer to it at two different places. First, in the Introduction:

“Finally, Nassar and colleagues have proposed a normative model in which a strategic trade-off is made between the number of encoded items and their precision: when two items are very similar, they are encoded as a single item, such that there is more resource available per encoded item (Nassar et al., 2018). […] However, just as in much of the work discussed above, this theory assumes a fixed resource budget for item encoding, which is not necessarily optimal when resource usage is costly.”

And then again in the “Limitations” section in the Discussion:

“A final limitation is that our theory assumes that items are uniformly distributed and uncorrelated. […] Hence, it seems worthwhile to examine models that combine resource rationality with chunking.”

2) τ: Please provide additional model results that show how the fits look when τ = 0. In particular, please show the goodness of fit of the rational model (with τ = 0) as compared to the fit of a model with a "hard" constraint on resources?

We have added this analysis:

“So far, we have assumed that there is random variability in the actual amount of resource assigned to an item. […] Therefore, we will only consider variable-precision models in the remainder of the paper.”

As in the variable-precision model, the optimal amount of resource per item decreases with set size in the equal-precision variant of the rational model:

Since we think that focusing too much on equal-precision results distracts a bit from the main story, we decided not to include this plot in the paper. As we explain in response to a later comment, the difference between the equal-precision and variable-precision models is mainly in the predicted kurtosis (“peakiness”) of the error distribution, not in the variance of these distributions (let alone in how the variance changes with set size). Hence, the equal-precision vs.variable-precision question is orthogonal to our main question.

In general, it was confusing that the theory is initially described in absence of τ, while τ is used for actual fits to data. It would have been easier to understand if the theory had been evaluated in presence of τ, and its effect studied within that theoretical framework.

Sorry, this was indeed confusing. In the rewritten “Theory” section, we explicitly indicate which equations depend on τ, by including it in the function arguments.

3) The speed of policy update: An assumption of the model that may be problematic is that people must almost instantaneously optimize their encoding precision when set-sizes change unpredictably from trial to trial. The theory predicts that, when a trial contains 2 stimuli and the next trial contains 4 stimuli, the participant instantaneously lowers the encoding precision to the new (near-) optimal level. This process sounds pretty demanding itself, especially considering that humans may also play an active role in determining their intrinsic motivation or deciding which items to memorize, which may further slow down the adjustment process. The flexibility in allocating resources that is implied by this model seems to be at odds with the slowness of cognitive control, well-documented by task switching costs. The authors touch on this point in the very last line of the paper, where they note that divisive normalization can provide a rapid adjustment mechanism. Even though the discussion is already long, it would be good to hear more about this point, and a comparison between a hardwired allocation mechanism and slower but more flexible cognitive control strategies.

This is an important issue, which we now discuss in the new Discussion section “Neural mechanisms and timescale of optimization”.

Furthermore, in the Frank paper described above, they use a task in which there is binary feedback that depends on the liberal vs. conservative error criterion (and they don't report major differences in performance as a function of this). One might suspect that participants would fail to adapt their policy even in an incentive-compatible version of this task which systematically varied these behavioral costs, and this would present a challenge to the authors' theory as described here. Please add some text to the Discussion addressing this point.

The experiment by Frank et al. used feedback threshold of π/3 (“low precision” condition) and π/8 (“high precision” condition) and found no difference in absolute estimation error between these two conditions. This would be at odds with any model that predicts that encoding precision is higher in the “high precision” condition, which is what one may expect to happen in our model. However, it turns out that the predictions for this experiment are not that straightforward and that the model can actually account for the lack of an effect. The short explanation is that there is a threshold region in which the prediction barely changes as a function of threshold, due to the performance benefit of adding extra resource is almost exactly outdone by the added neural cost. For a more detailed explanation, we refer to the new Figure 6.

This point is now also discussed in a new Discussion section “Experimental predictions of incentive manipulations”.

4) Total precision vs. set-size: A novel and interesting prediction of the model is the non-monotonic relation between set-size and total precision (Figure 3B). Although the authors state that this point requires more empirical documentation, are the model results consistent with a non-monotonic encoding of target location that was reported by Balan et al. (2008) in monkey area LIP? That study found that, in a covert visual search task with different set sizes, the fidelity of target location encoding by area LIP was higher at set size 4 than at set size 2 or set size 6 (see Figure 5) – a non-linearity that was puzzling at the time but may gain new significance in light of this paper. Please add some discussion of this point to the manuscript.

We thank the reviewer for the reference, as we were not aware of that paper. However, after a careful study of the results reported in that paper, we don’t see how the non-monotonic trend in Figure 3 can be linked to the predicted non-monotonicity in the total amount of invested resource. The non-monotonicity in the Balan paper shows that the stimulus identity (target/distractor) can be decoded more accurately from neural data in N=4 trials compared to N=2 and N=6 trials. However, we do not see how decoding accuracy of a single item relates to the total amount of resource invested in all items. Although it would have been nice if the Balan paper backs up the non-monotonicity prediction, we believe that linking our prediction to their result would be a bit misleading, so we decided to not include this point. (However, if we misunderstood the reviewer’s suggestion, we would of course be happy to have another look at it after some clarification).

Nevertheless, the Balan paper is relevant to our work for other reasons and we now cite it at two different places in the Discussion. First, in the Discussion section about experimental predictions and, second, in the rewritten part about Neural mechanisms (see responses to previous comments).