On the Efficiency of Data Collection for Crowdsourced Classification

1 Introduction

In the past decade crowdsourcing has emerged as an effective way to gather a temporary workforce, and execute large numbers of small and repetitive tasks at a competitive price. This allows an employer to complete large scale data-processing projects when alternative methods are either too expensive (e.g. hiring a team of full-time experts) or impractical (e.g. developing an ad hoc machine learning system of equivalent accuracy) [Lintott et al., 2011]. The applications of this approach range from image and video annotation [Vondrick et al., 2013], to speech recognition [Lasecki et al., 2013], language processing [Snow et al., 2008] and even research studies and surveys [Buhrmester et al., 2011].

Currently, the most popular way of crowdsourcing data involves an online platform like Amazon Mechanical Turk 1 or Crowdflower 2 , where workers from all around the world can login and execute the tasks submitted by the employers in exchange for a small payment [Ross et al., 2010]. Despite the efforts to screen the workers and introduce qualifications and reputation systems, the crowdsourced data tends to contain a sizeable amount of errors or random answers [Downs et al., 2010]. As cleaning the data by hand goes against the very reason behind using crowdsourcing in the first place, most of the literature focuses on automatically singling out the errors by collecting redundant data and aggregating it together [Whitehill et al., 2009; Liu et al., 2012; Augustin et al., 2017]. On the theoretical side, the main concern is uncovering the relationship between the number of available data points and the accuracy of the aggregated estimates. Specifically, Berend and Kontorovich [2014] derive bounds on the accuracy of the weighted majority voting rule given a fixed set of workers, Gao et al. [2016] compare the asymptotic performance of several probabilistic inference methods and Bonald and Combes [2017] propose an optimal algorithm to estimate the reliability of the individual workers.

1 mturk.com

2 www.crowdflower.com

However, the aforementioned research focuses solely on the aggregation phase of a crowdsourcing project (see Figure 1), which happens after all the data has been collected. Due to the iterative nature of crowdsourcing, where the data is collected over several days or weeks, there is also an opportunity to improve the efficiency of the collection phase. In this regard, the simplest collection strategy is the non-adaptive uniform allocation policy, i.e. always collect a fixed number of data points on each task [Karger et al., 2014]. Alternatively, some authors have proposed the use of adaptive policies, which use the data collected so far to inform their future decisions, in an attempt to optimise the subsequent aggregation phase. In particular, Barowy et al. [2012] suggest collecting more data on the tasks where a clear majority has not formed yet, Welinder and Perona [2010] propose retraining the aggregator on every new data point and collecting more data on the tasks with larger uncertainty, and Simpson and Roberts [2014] attempt to estimate the information gain of future data points.

While these adaptive policies have been shown to have an empirical advantage over non-adaptive ones [Welinder and Perona, 2010; Barowy et al., 2012; Simpson and Roberts, 2014], we are still missing a clear theoretical understanding of their performance. More specifically, a prominent result by Karger et al. [2014] states that every policy exhibits an exponential tradeoff between the number of data points R and the final accuracy in the form P(error) ≤ exp(-cR). However, the authors provide a value of the constant factor c only for non-adaptive policies. In contrast, other authors have proven the advantage of adaptive policies for some specific scenarios, but failed to address the general case above. In particular, Chen et al. [2013] assume that we can summon specific workers from the crowd at any time, while Ho et al. [2013] assume we can test the accuracy of each worker beforehand.

In this paper we provide the first theoretical explanation of the impact of the data collection process on the accuracy of a crowdsourcing system. Furthermore, we derive new bounds on the accuracy tradeoff of the most popular collection policies: uniform allocation, uncertainty sampling and information gain maximisation. More specifically, we make the following contributions to the state of the art. First, we propose a new way to represent the runtime behaviour of a collection policy in terms of a random walk in the log-odds domain. Second, we use this tool to analyse the tradeoff P(error) ≤ exp(-cR) of the existing collection policies under a weighted majority voting aggregator. In so doing, we are able to bound the performance of the uncertainty sampling policy proposed by Welinder and Perona [2010] from both sides, and show its equivalence with the information gain maximisation policy of Simpson et al. [2014]. Third, we repeat our analysis on the more challenging case of probabilistic inference aggregators, improve the bound of Karger et al. [2014] and derive new upper and lower bounds on the error rate of adaptive policies. Finally, with these bounds we are able to quantify the advantage that adaptive policies have over non-adaptive ones.

The paper is structured in the following way. In Section 2 we introduce all the relevant data aggregators and collection policies. In Section 3 we study the performance of the policies under weighted majority voting. In Section 4 we repeat our analysis under probabilistic inference. In Section 5 we conclude and outline possible future work. All the proofs of our theorems are collated in Appendix A.

2 Preliminaries

Among the models for crowdsourced classification, the onecoin Dawid-Skene model [Dawid and Skene, 1979] has received most attention from the theoretical literature [Liu et al., 2012; Karger et al., 2014; Bonald and Combes, 2017]. The reason for this lies in the simplicity of the model, coupled with its ability to capture all the major characteristics of the crowdsourcing scenario.

According to this model, we assume that the objective of the system is to recover the correct classification of M distinct tasks. We denote the underlying ground-truth vector as y, with yi ∈ {±1} being the true class of task i. Moreover, we assume the presence of a crowd of N workers who provide us with a set of labels X = {xij} over the course of the crowdsourcing effort. These workers become available one by one in random order, get assigned to a task i and provide a label xij ∈ {±1} in exchange for a unitary payment. We assume that a maximum budget B << MN can be spent on collecting new labels, and we set by convention xij = 0 for any missing task-worker pair. Furthermore, we assume that the probability of observing a correct label depends only on the accuracy of the individual worker, which we denote by P(xij = yi) = pj. In other words, we assume that the workers act independently from each other, and their accuracy is not affected by the task they are assigned to (we plan to extend our analysis to more complicated models as future work, see Section 5). Finally, we assume that the population of workers is extracted from a common distribution pj ∼ fp, and that the prior on the ground-truth labels is P(yi = +1) = 1/2 (it is trivial to extend our results to other priors).

2.1 Label AggregationMethods

Given a set of labels X, we need a way to aggregate them into a vector ˆy of predictions over the task classes (where ˆ yi is the prediction on task i). Here we consider two of the most common aggregators, one that assumes perfect knowledge over the workers' accuracy vector p and one that does not need this piece of information.

Weighted Majority Voting. The simple majority voting rule predicts the class of a task by ˆ yi = sign{∑Nj=0 xij}, where ties are broken randomly. However, the presence of workers with varying degrees of accuracy, particularly when pj < 1/2, may dramatically decrease the performance of this system. When the workers' accuracy p is known, it is possible to assign a larger weight to the more accurate workers, hence increasing the reliability of the system. This method is known as weighted majority voting. Nitzan and Paroush [1982] show that, by assigning each worker a weight wj = log(pj/(1 - pj)), the resulting aggregation method ˆ yi = sign{∑Nj=0 xijwj} achieves optimal accuracy. Notably, these weights stem from a probabilistic interpretation of the weighted majority rule. In particular, it is possible to show that the weighted sum zi = ∑Nj=0 xijwj is equal to the posterior log-odds as follows:

Probabilistic Inference. When the workers' accuracy p is not known, we can resort to several unsupervised probabilistic methods [Liu et al., 2012; Zhang et al., 2014; Gao et al., 2016; Bonald and Combes, 2017]. This family of methods infer an estimate ˆp ≈ p from the set of labels X itself, which can then be plugged into the weighted majority voting rule to compute the log-odds ˆ zi = ∑Nj=0 xij log(ˆ pj /(1 - ˆ pj)). In particular in Section 4 we present a result for the approximate variational inference method proposed by Liu et al. [2012] as it fits our theoretical framework well. This method computes ˆp in an expectation-maximisation fashion as follows:

where the prior distribution of the workers' accuracy is fp ∼ Beta(α, β) and the sigmoid function σ(z) = 1/(1+exp(-z)) is the inverse of the log-odds function.

2.2 Label Collection Policies

During the course of the crowdsourcing effort, we need to decide which task i we want to label next. The sequence of decisions is usually formalised in terms of a collection policy, a rule or heuristic that selects i given the incoming worker j t and the set of labels Xt collected so far. The existing literature provides us with the following three main collection policies.

Uniform Allocation (UNI). This non-adaptive policy assigns the same number R = B/M of labels to each task. Using a variant of this policy, Karger et al. [2014] are able to bound the probability of a classification error under probabilistic inference to:

where q = Efp {(2pj - 1)2 }. We will improve upon this bound in Section 4.1.

Uncertainty Sampling (US). This adaptive policy maintains a measure of uncertainty over the current set of predictions ˆy, and collects new labels over the most uncertain tasks. Proposed first in the active learning community [Lewis and Gale, 1994], uncertainty sampling has been empirically shown to be successful for crowdsourcing in a number of applications [Welinder and Perona, 2010; Barowy et al., 2012]. We derive the first bounds on the performance of this policy in Sections 3.3 and 4.2.

Information Gain Maximisation (IG). This adaptive policy always chooses the action that maximises the expected information gain on the current posterior distribution [MacKay, 1992]. The empirical performance of this policy in crowdsourcing applications is described by Simpson and Roberts [2014]. We provide the first theoretical analysis on this policy in Section 3.4.

3 Data Collection underWeightedMajority

In this section, we analyse the performance of the collection policies when the workers' accuracy p is known and weighted majority voting is used to aggregate the data. More specifically, we take advantage of the properties of the weighted majority aggregator to model the data collection process as a random walk. In turn, this enables us to bound the performance of the policies.

3.1 Modeling as a RandomWalk

From the point of view of a single task i, the collection policy selects a subset of workers Ni ⊆ N to work on i throughout the crowdsourcing process. The subset of labels Xi ⊆ X provided by these workers move the log-odds on task i from its starting point zi = 0 to its final value zi = ∑j∈Ni xijwj . Since the labels are collected one-by-one iteratively, we can model the evolution of zi as a random walk that is made of a sequence of independent steps sij = xijwj.

Key to our analysis is how these steps sij are distributed. First, let us derive the probability density function of the weights wj from that of the workers' accuracy pj ∼ fp:

Now, assume by convention that the true class of task i is yi = +1, and that the collection policy does not base its decisions on the accuracy of the incoming worker j t . Then, if we consider that a step sj can be taken either by workers with weight wj = sj or wj = -sj, we can write the common probability density function of the steps as:

In general, we have E{fs} ≥ 0 for any fp, with equality holding only if all the workers have pj = 1/2 or, in other terms, when the whole crowd provides only random answers. This is due to the fact that workers with pj < 1/2 get assigned a negative weight, so that even their labels move the weighted majority in the right direction. As a consequence, the random walk on the log-odds zi of each task i will always drift towards the true class yi. However, we show in the following sections how different policies capitalise on this drift at different rates.

3.2 Performance of the UNI Policy

The UNI policy always collects at least R = ⌊B/M⌋ labels on each task i. We can thus interpret its behaviour as a sum of R independent random steps extracted from the same distribution sij ∼ fs. Keeping the convention that yi = +1, we can compute the probability density function of the sum zi = ∑j∈Ni sij as the convolution between R copies of fs. The probability of a classification error is thus given by:

where ∗ is the convolution operator.

While Equation 6 provides us with the exact performance of the UNI policy, it suffers from two drawbacks. First, it might be difficult to compute it without access to the full probability density function fs. Second, its lack of a simple closed form makes it difficult to compare it theoretically with the performance of other policies. We can solve both drawbacks by measuring the concentration of zi around its expected value. This leads us to the following theorem:

Theorem 1. Assume that fs is subgaussian, with parameter γ such that E{exp(ts)} ≤ exp(γ 2 t2 /2) for all t ∈ R. Then, the probability of a classification error under the UNI policy is bounded by:

Equation 7 clearly exposes the exponential tradeoff between number of labels and accuracy achieved by the UNI policy. We use this result to compare it with the other policies in Section 3.5. At the same time, note that the bound in Equation 7 is only tight in an asymptotical sense. For small values of R there exists a tighter bound that, however, quickly becomes inefficient as R increases in value (see Appendix A).

3.3 Performance of the US Policy

We move now to our analysis of the performance of the US policy. This policy always chooses the task with the largest uncertainty, i.e. the one whose log-odds are closest to zero:

Due to this specific behaviour, the magnitude of the log-odds |zi| tends to increase at the same pace on all M tasks during the collection process. This phenomenon allows us to prove the following two bounds:

Theorem 2. The probability of a classification error under the US policy is upper bounded by:

Theorem 3. The probability of a classification error under the US policy is lower bounded by:

Notice that the bounds in Equations 9 and 10 match asymptotically and guarantee that the US policy has an exponential tradeoff with constant c = E{fs}. This means that the policy fully exploits the drift in the randomwalk over the log-odds zi to reduce the probability of an error. We compare this result with the other policies in Section 3.5.

3.4 Performance of the IG Policy

The IG policy always selects the task that yields the largest information gain. Since we cannot predict the value of the next label xij in advance, we evaluate the impact of adding it to our current set Xt in expectation:

where the information gain I is defined as the KullbackLeibler divergence between the future posterior and the current one. Given this definition, we can prove that the policies US and IG are in fact equivalent:

Theorem 4. Given the current log-odds zt and a worker with weight wj ̸= 0, the two policies US and IG select the same task i∗ , except in case of a tie.

3.5 Policy Comparison

We now have the tools to compare the performance of the non-adaptive UNI policy with the adaptive US and IG ones. On the one hand, all of them exhibit an asymptotic tradeoff between the number of labels R and the final accuracy in the form P(error) ≤ exp(-cR). However, the specific constant factor c varies from policy to policy. In fact, for the UNI policy we have cuni = E{fs}2 /2γ 2 (see Equation 7), whereas for the two equivalent policies US and IG we have cada = E{fs} (see Equation 9).

In general, a higher value of c means that the policy is more efficient in using additional labels to improve the accuracy. In this respect, we can prove that adaptive policies offer superior guarantees by examining the ratio cada/cuni = 2γ 2 /E{fs}:

Theorem 5. For any distribution fs bounded in [-a,+a] with 0 < a < +∞, the efficiency ratio between adaptive (US, IG) and non-adaptive (UNI) policies is cada/cuni ≥ 4.

On the other hand, the result in Theorem 5 is derived from upper bounds on the error rate of the policies, and thus it might not reflect the actual performance gap between the UNI, US and IG policies at runtime. For this reason, we run synthetic experiments with M = 10000 tasks and a uniform distribution of workers fp over the interval [0.4, 0.8] to simulate a mixed crowd (similar results can be obtained with different choices of fp). We report the results in Figure 2, where each point is the average of 100 runs and has standard error below 5×10-4 . As the figure shows, the error rate of the US and IG policies quickly diverges from that of the UNI policy as the average budget per task R increases. At the same time, the observed cada/cuni ratio has a value of 1.9. In order to overcome this discrepancy with the result in Theorem 5, tighter bounds on the UNI policy are needed.

4 Data Collection under Probabilistic Inference

We now remove the assumption that the workers' accuracy p is known, and repeat our analysis of the collection policies assuming that some unsupervised probabilistic method is employed to aggregate the crowdsourced data. In general, our results are valid for any state-of-the-art probabilistic method that is provably better than majority voting (see [Gao et al., 2016]).

4.1 Performance of the UNI Policy

Similarly to our analysis in Section 3.2, we take advantage of Hoeffding's concentration inequality to bound the accuracy of the UNI policy:

Theorem 6. The probability of a classification error under the UNI policy is bounded by

This bound is an improvement over the result of Karger et al. [2014] for two reasons. First it only depends on the number of labels R and the average accuracy of the workers ¯ p, instead of the quantity q = E{(2pj - 1)2 } which is more difficult to estimate in a practical scenario (see Equation 3). Second, for any distribution with the property (2¯ p - 1)2 > q/16 our bound is tighter.

4.2 Performance of the US and IG Policies

Let us move on to the performance of adaptive collection policies. It is worth noting that the equivalence result between the US and IG policies shown in Section 3.4 is not valid here because the estimates pt may change upon receiving a new label at time t. Nevertheless, the two following results still apply to both policies.

Theorem 7. The performance of the US and IG policies is upper bounded by

Theorem 8. The performance of the US and IG policies is lower bounded by

where fˆ s is the distribution of steps given the weight estimates ˆwj provided by the probabilistic inference method of choice at the end of the crowdsourcing process.

Theorem 8 can be adapted to any probabilistic method by providing a value for E{fˆ s} and E{|fˆ s|}. In most cases this can be done only by numerical estimation. However, for the approximate variational inference algorithm proposed by Liu et al. [2012] we can derive an upper bound as follows:

Theorem 9. Given a population of workers with accuracy pj ∼ Beta(α, β), the expected value of a step under the approximate variational inference algorithm in [Liu et al., 2012] is bounded by:

where Q is the number of labels provided by a single worker.

Finally, the value of E{|fˆ s|} can be computed from the result in Theorem 9 by taking the absolute value of the logarithm and discarding the 2c-Q Q term.

4.3 Policy Comparison

We now have the tools to compare the performance of the three policies UNI, US and IG. As for the case with known workers' accuracies p in Section 3, all the policies have an asymptotic tradeoff between the number of labels R and the expected accuracy in the form P(error) ≤ exp(-cR). However, the constant factor c differs between the non-adaptive (UNI) and adaptive (US, IG) policies. For the UNI policy Equation 12 yields a factor cuni = 1 2 (2¯ p - 1)2 , whereas for the UNI and IG policies Equation 13 yields a factor cada = ¯w(2¯ p-1) instead.

Given this, we can prove that both US and IG offer superior guarantees over UNI when used in conjunction with a probabilistic inference method. In particular, by comparing the ratio cada/cuni = 2 ¯w/(2¯ p - 1) we can state the following:

Theorem 10. For any distribution of the workers' accuracy fp with E{fp} ̸= 1/2, the efficieny ratio between adaptive (US, IG) and non-adaptive (UNI) policies is cada/cuni ≥ 4.

At the same time, the result in Theorem 10 is again derived from upper bounds on the error rate of the policies, and thus it might not reflect the actual performance gap between the UNI, US and IG policies at runtime. In order to rule out this eventuality, we run synthetic experiments with M = 200 tasks, Q = 10 labels per worker and fp ∼ Beta(α=4, β=3) to simulate a mixed crowd. Moreover, we use the approximate variational inference method in [Liu et al., 2012] to aggregate the labels, and average the results over 1000 runs, which yields a standard error below 2×10-3 . Note that similar results can be achieved with different values of Q, α and β, whereas the use of the Beta distribution is necessary to match the assumptions of Liu's algorithm. We report the results in Figure 3, which shows that the error rate of the US and IG policies becomes smaller than that of the UNI policy as the average budget per task R increases. On the other hand, the observed value of the cada/cuni ratio is 1.3, which suggests that the empirical advantage of the adaptive US and IG policies is smaller than the theoretical prediction in Theorem 10.

5 Conclusions

We have analysed the performance of a number of the most common data collection policies for crowdsourced classification. By representing them as random walks in the log-odds domain, we derived new upper and lower bounds on their accuracy. Consequently, we were able to quantify the advantage that some adaptive policies can have over non-adaptive ones for the first time.

We believe that the techniques presented here can be extended to additional scenarios, both in terms of data aggregators and crowdsourcing models. Among them, the case of tasks with varying degrees of difficulty is of particular interest to us. Under this model, Khetan [2016] have already shown an exponential tradeoff in the form P(error) ≤ exp(-cR), but the values of c for different policies are still unknown. We aim to compute them using our methods as future work.

Acknowledgments

This research is funded by the UK Research Council project ORCHID, grant EP/I011587/1. The authors acknowledge the use of the IRIDIS High Performance Computing Facility, and associated support services at the University of Southampton.

A Proofs

Proof of Theorem 1. Hoeffding's concentration inequality can be written as follows:

where RE{fs} = E{zi} since the R steps in the random walk are independent. Then, by substituting t = -E{zi} and noting that P(ˆ yi ̸= yi) = P(zi ≤ 0) we get the result in the theorem.

As mentioned in Section 3.2, for small values of R there exist a tighter (but non-exponential) bound:

Theorem 1 bis. Assume that fs has finite variance. Then, the probability of a classification error under the UNI policy is bounded by:

Proof. This result can be derived in the same way as Theorem 1, by using the Chebyshev-Cantelli inequality:

Proof of Theorem 2. From the perspective of a single task i, the US policy operates in short bursts of activity, as i keeps receiving new labels until it is no more the most uncertain one. We define zB = mini{|zi|} as the threshold that all tasks have crossed at some point of the crowdsourcing effort. In this respect, we can model the evolution of the log-odds zi as a bounded random walk, which starts in zi = 0 and ends when zi leaves the interval (-zB,+zB).

Given this, let us assume that we can fix the threshold zB and then collect as many labels as needed in order to cross it. We denote the log-odds after crossing the threshold as zr i , where zr i ̸∈ (-zB,+zB), and the log-odds at the step before as zr-1 i . According to this definition, r is a stopping time since it is uniquely defined by the information collected before step r. Thus, we can use Wald's equation [Wald, 1944] to link the expected value of zr i and the stopping time r:

Recall, however, that zr i is the sum of r i.i.d random variables, and that zr-1 i ∈ (-zB,+zB) by definition. As a consequence, we can further bound the expected value of zr i by:

Putting Equations 19 and 20 together, we can derive a bound for the threshold zB:

At the same time, we also know that the random walks on the M tasks are independent, and that the variance of r for a bounded random walk with i.i.d. steps is finite. Therefore, as M → ∞ the total number of steps required to cross the threshold will converge to its expected value, i.e. B → ME{r}. This property allows to substitute E{r} = R.

However, we also know that the confidence in our prediction is P(ˆ yi = yi) = σ(|zi|). Thus, we can bound the final accuracy as:

which yields Equation 9 after some simple algebraic manipulations.

Proof of Theorem 3. Similarly to the proof of Theorem 2, we can bound the value of the threshold xB as follows:

where

Now, we can compute maxz≥0{zσ(-z)} ≈ 0.28. Additionally, we know that E{zr i } = RE{fs}. Thus, the following is true:

By noting that P(ˆ yi ̸= yi) = σ(-zB) ≥ 12 exp(-zB) we obtain the result of the theorem.

Proof of Theorem 4. Let us write the information gain in closed form as follows:

since the log-odds vector zt changes on task i only. The expected value of Equation 26 is the following:

where

First, note that Equation 27 is even in zt i and wj (this can be easily proven by substitution). Given this, we can focus on the positive intervals zt i ,wj ∈ (0,∞) and show that the function is monotonically decreasing. We can do so by taking the derivative of Equation 27 and proving that it is negative by contradiction:

which yields:

which is false for any zt i ,wj ∈ (0,∞). Finally, we can see that both the objective function |zt i | for the US policy and Equation 27 for the IG policy are even in zt i , monotonically increasing (decreasing) for zt i → ∞ and have a minimum (maximum) in zt i = 0 for any wj ̸= 0. Hence, if a task i1 is preferred to i2 under one policy, it is also preferred under the other.

Proof of Theorem 5. The best-case scenario for the UNI policy is when fs is such that E{fs} has the largest possible value and γ has the smallest possible one. First, notice that the subgaussian parameter of fs is γ = a since fs is bounded. Second, from Equation 5 we can derive that fs(s) = fs(-s) exp(s), thus the best-case is achieved when all the mass of fs is concentrated in ±a and E{fs} = a(2σ(a) - 1). Under this scenario, the efficiency ratio becomes:

While numerator and denominator of Equation 31 are both zero for a = 0, the former has first-order derivative equal to 2 whereas the latter has derivative less than 1/2. Hence, the ratio is larger than 4.

Proof of Theorem 6. If we set our estimates of the workers' accuracy to ˆ pj = ¯ p = E{fp}, ∀j, the distribution of steps fs reduces to sij = + ¯w with probability ¯ p, and sij = - ¯w with probability 1 - ¯ p. Hence, fs has mean E{fs} = ¯w(2¯ p - 1) and is bounded in [-¯w,+¯w], yielding the subgaussian parameter γ = ¯w. By plugging these values into Equation 7 we get the result of the theorem. Finally, note that by setting ˆ pj = ¯ p, ∀j the aggregation method becomes a simple majority voting rule. Therefore, any provably superior probabilistic aggregation method (see [Gao et al., 2016]) must achieve a lower error rate and thus satisfy Equation 12.

Proof of Theorem 7. As per the proof of Theorem 6, we set our estimates of the worker's accuracy to ˆ pj = ¯ p = E{fp}, ∀j. In terms of the random walk interpretation, the presence of workers with the same weight ¯w = log(¯ p/(1-¯ p))

means that all the steps have the same length, i.e. sij ∈ {±¯w}. Thus, we can use a classic results on bounded discrete random walks [Feller, 1968] to derive the number of steps r required to reach one of the two boundaries ±K ¯w (with K ∈ N) from the initial starting point zi = 0:

By inverting this relationship, we can compute the minimum threshold ±K ¯w we can reach given a budget of R labels in expectation:

Thus, at the end of the crowdsourcing process the posterior on the majority class is, in expectation:

which by substituting E{fs} = ¯w(2¯ p - 1) and considering the posterior probability of an error ˆ yi ̸= yi yields the result in the theorem.

Proof of Theorem 8. Define ˆ zB = min{|ˆ zi| : |X| = B} as the minimum estimated log-odds at the end of the collection process. Also, note that any non-trivial probabilistic inference method updates its estimates ˆpt during the collection process as new information comes in. As a consequence, the US and IG policies may collect additional labels on a task i whose log-odds |ˆ zi| are already above ˆ zB (according to the final estimates ˆpB ). If the policies had access to ˆ zB from the beginning, they would avoid this inefficiency and achieve a larger ˆ zB in expectation. Thus, the error rate in this ideal scenario is a lower bound on the real error rate and can be computed according to Theorem 3.

Proof of Theorem 9. As the accuracy on the tasks increases to 1, i.e. ˆ zi → ∞, the estimates in Equation 2 converge to ˆ pj = (cj + α)/(Qj + α + β), where cj is the number of the worker's correct answers. Moreover, the probability of cj given that pj ∼ Beta(α, β) is (Qj cj )B(cj + α, Qj - cj + β)/B(α, β). Finally, a worker with cj correct answers has weight ˆwj = log((cj +α)/(Qj -cj +β)) and makes cj/Qj positive steps and (Qj - cj)/cj negative ones, which yields the equality in the theorem. For any ˆ zi < ∞ this result becomes an inequality.

Proof of Theorem 10. The theorem can be proven by taking the first-order approximation of both the numerator and denominator of cada/cuni centred in ¯ p = 1/2. For the former, notice that 2 ¯w = 2 log(¯ p/(1 - ¯ p), whose derivative in ¯ p = 1/2 yields the following value:

Also note that ¯w is monotonic in ¯ p and that ¯w = +∞ for ¯ p = 1 and ¯w = -∞ for ¯ p = 0. Thus in general the derivative of 2 ¯w is always greater than the value in Equation 35. Finally, the first-order derivative of the denominator is d(2¯ p - 1)/d¯ p = 2 and both cada and cuni are zero with ¯ p = 1/2. Hence, for any ¯ p ̸= 1/2 the ratio cada/cuni ≥ 8/2 which proves the theorem.