When a company relies on distributed users' data to train a machine learning model, federated learning [37, 54, 25] promotes the idea that users/customers' data should be kept local, and only the locally held/learned hypothesis will be shared/contributed from each user. While federated learning has observed success in keyboard recognition [21] and in language modeling [8] , existing works have made an implicit assumption that participating users will be willing to contribute their local hypotheses to help the central entity to refine the model. Nonetheless, without proper incentives, agents can choose to opt out of the participation, to contribute either uninformative or outdated information, or to even contribute malicious model information. Though being an important question for federated learning [54, 34, 55, 56] , this capability of providing adequate incentives for user participation has largely been overlooked. In this paper we ask the questions that: Can a machine learning hypothesis be incentivized/elicited by a certain form of scoring rules from self-interested agents? The availability of a scoring rule will help us design a payment for the elicited hypothesis properly to motivate the reporting of high-quality ones. The corresponding solutions complement the literature of federated learning by offering a generic template for incentivizing users' participation.
We address the challenge via providing a scoring framework to elicit hypotheses truthfully from the self-interested agents/users 1 . More concretely, suppose an agent i has a locally observed hypothesis f * i . For instance, the hypothesis can come from solving a local problem: f * i = argmin fi∼Hi E (X,Y )∼D [ i (f i (X), Y )] according to a certain hypothesis class H i , a distribution D, a loss function i . The goal is to design a scoring function S(·) that takes a reported hypothesis f i , and possibly a second input argument (to be defined in the context) such that E [S(f * i , ·)] ≥ E [S(f i , ·)] , ∀f i , where the expectation is w.r.t. agent i's local belief, which is specified in context. If the above can be achieved, S(·) can serve as the basis of a payment system in federated learning such that agents paid by S(·) will be incentivized to contribute their local models truthfully. In this work, we primarily consider two settings, with arguably increasing difficulties in designing our mechanisms:
With ground truth verification (X, Y ) We will start with a relatively easier setting where we as the designer has access to a labeled dataset {(x n , y n )} N n=1 . We will demonstrate how this question is similar to the classical information elicitation problem with strictly proper scoring rule [16] , calibrated loss functions [2] and peer prediction (information elicitation without verification) [38] .
With only access to features X The second setting is when we only have X but not the ground truth Y . This case is arguably more popular in practice, since collecting label annotation requires a substantial amount of efforts. For instance, a company is interested in eliciting/training a classifier for an image classification problem. While it has access to images, it might not have spent efforts in collecting labels for the images. We will again present a peer prediciton-ish solution for this setting.
Besides establishing the desired incentive properties of the scoring rules, we will look into questions such as when the scoring mechanism is rewarding accurate classifiers, how to build a prediction market-ish solution to elicit improving classifiers, as well as our mechanism's robustness against possible collusion. Our work can be viewed both as a contribution to federated learning via providing incentives for selfish agents to share their hypotheses, as well as a contribution to the literature of information elicitation via studying the problem of hypothesis elicitation. We validate our claims via experiments using the MNIST and CIFAR-10 datasets.
All omitted proofs and experiment details can be found in the supplementary materials.
Due to space limit, we only briefly survey the related two lines of works:
Information elicitation Our solution concept relates most closely to the literature of information elicitation [5, 50, 46, 36, 24, 17] . Information elicitation primarily focuses on the questions of developing scoring rule to incentivize or to elicite self-interested agents' private probalistic beliefs about a private event (e.g., how likely will COVID-19 death toll reach 100K by May 1?). Relevant to us, [1] provides a market treatment to elicit more accurate classifiers but the solution requires the designer to have the ground truth labels and agents to agree on the losses. We provide a more generic solution without above limiations.
A more challenging setting features an elicitation question while there sans ground truth verification. Peer prediction [42, 39, 51, 43, 52, 12, 48, 44, 31, 27, 33] is among the most popular solution concept. The core idea of peer prediction is to score each agent based on another reference report elicited from the rest of agents, and to leverage on the stochastic correlation between different agents' information. Most relevant to us is the Correlated Agreement mechanism [12, 48, 27] . We provide a separate discussion of it in Section 2.1.
Federated learning Federated learning [37, 21, 54] arose recently as a promising architecture for learning from massive amounts of users' local information without polling their private data. The existing literature has devoted extensive efforts to make the model sharing process more secure [7, 40, 45, 23, 18, 4] , more efficient [35, 10, 15, 11, 26, 19, 49] , more robust [29, 9, 53, 41] to heterogeneity in the distributed data source, among many other works. For more detailed survey please refer to several thorough ones [54, 25] .
The incentive issue has been listed as an outstanding problem in federated learning [54] . There have been several very recent works touching on the challenge of incentive design in federated learning. [34] proposed a currency system for federated learning based on blockchain techniques. [55] describes a payoff sharing algorithm that maximizes system designer's utility, but the solution does not consider the agents' strategic behaviors induced by insufficient incentives. [56] further added fairness guarantees to an above reward system. We are not aware of a systematic study of the truthfulness in incentiving hypotheses in federated learning, and our work complements above results by providing an incentive-compatible scoring system for building a payment system for federated learning.
Consider the setting with a set K = {1, 2, ..., K} of agents, each with a hypothesis f * i ∈ H i which maps feature space X to label space Y ∈ {1, 2, ..., L} := [L]. The hypothesis space H i is the space of hypotheses accessible or yet considered by agent i, perhaps as a function of the subsets of X or Y which have been encountered by i or the agent's available computational power. f * i is often obtained following a local optimization process. For example, f * i can be defined as the function which minimizes a loss function over an agent's hypothesis space.
where in above D i is the local distribution that agent i has access to train and evaluate f * i . In the federated learning setting, note that f * i can also represent the optimal output from a private training algorithm and H i would denote a training hypothesis space that encodes a certainly level of privacy guarantees. In this paper, we do not discuss the specific ways to make a local hypothesis private 2 , but rather we focus on developing scoring functions to incentivize/elicit this "private" and ready-to-be shared hypothesis.
Suppose the mechanism designer has access to a dataset D: D can be a standard training set with pairs of features and labels D := {(x n , y n )} N n=1 , or we are in a unsupervised setting where we don't have labels associated with each sample x i : D := {x n } N n=1 . The goal of the mechanism designer is to collect f * i truthfully from agent i. Denote the reported/contributed hypothesis from agent i as f i 3 . Each agent will be scored using a function S that takes all reported hypotheses f j , ∀j and D as
inputs: S f i , {f j =i }, D such that it is "proper" at a Bayesian Nash Equilibrium:
is called inducing truthful reporting at a Bayesian Nash Equilibrium if for every agent i, assuming for all j = i, f j = f * j (i.e., every other agent is willing to report their hypotheses truthfully),
where the expectation encodes agent i's belief about {f * j =i } and D.
Peer prediction is a technique developed to truthfully elicit information when there is no ground truth verification. Suppose we are interested in eliciting private observations about a categorical event y ∈ [L] generated according to a random variable Y (in the context of a machine learning task, Y can be thought of as labels). Each of the K ≥ 2 agents holds a noisy observation of y, denoted as
. Again the goal of the mechanism designer is to elicit the y i s, but they are private and we do not have access to the ground truth Y to perform an evaluation. The scoring function S is designed so that truth-telling is a strict Bayesian Nash Equilibrium (implying other agents truthfully report their
Correlated Agreement Correlated Agreement (CA) [12, 47] is a recently established peer prediction mechanism for a multi-task setting. CA is also the core and the focus of our subsequent sections. This mechanism builds on a ∆ matrix that captures the stochastic correlation between the two sources of predictions y i and y j . ∆ ∈ R L×L is then defined as a squared matrix with its entries defined as follows:
∆(k, l) = P y i = k, y j = l − P y i = k P y j = l , k, l ∈ [L].
The intuition of above ∆ matrix is that each (i, j) entry of ∆ captures the marginal correlation between the two predictions. Sgn(∆) denotes the sign matrix of ∆: where Sgn(x) = 1, x > 0; Sgn(x) = 0, o.w.
CA requires each agent i to perform multiple tasks: denote agent i's observations for the N tasks as y i,1 , ..., y i,N . Ultimately the scoring function S(·) for each task k that is shared between i, j is defined as follows: randomly draw two other tasks k 1 , k 2 , k 1 = k 2 = k,
It was established in [47] that CA is truthful and proper (Theorem 5.2, [47] ) 4 . P(y j = y |y i = y) < P(y j = y ), ∀i, j ∈ [K], y = y then S(·) is strictly truthful (Theorem 4.4, [47] ).
We start by considering the setting where the mechanism designer has access to ground truth labels, i.e., D = {(x n , y n )} N n=1 . 2 There exists a variety of definitions of privacy and their corresponding solutions for achieving so. Notable solutions include output perturbation [6] or output sampling [3] to preserve privacy when differential privacy [14] is adopted to quantify the preserved privacy level. 3 fi can be none if users chose to not contribute. 4 To be precise, it is an informed truthfulness. We refer interested readers to [47] for the detailed differences.
As a warm-up, we start with the question of eliciting the Bayes optimal classifier:
It is straightforward to observe that, by definition using −1(·) (negative sign changes a loss to a reward (score)) and any affine transformation of it a1(·) + b, a < 0 will be sufficient to incentivize truthful reporting of hypothesis. Next we are going to show that any classification-calibrated loss function [2] can serve as a proper scoring function for eliciting hypothesis. 5 Theorem 1. Any classification calibrated loss function (·) (paying agents − (f i (X), Y )) induces truthful reporting of the Bayes optimal classifier.
3.2 Eliciting "any-optimal" classifier: a peer prediction approach
Now consider the case that an agent does not hold an absolute Bayes optimal classifier. Instead, in practice, agent's local hypothesis will depend on the local observations they have, the privacy level he desired, the hypothesis space and training method he is using. Consider agent i holds the following hypothesis f * i , according to a loss function i , and a hypothesis space
By definition, each specific i will be sufficient to incentivize a hypothesis. However, it is unclear how f * i trained using i would necessarily be optimal according to a universal metric/score. We aim for a more generic approach to elicit different f * i s that are returned from different training procedure and hypothesis classes. In the following sections, we provide a peer prediction approach to do so.
We first state the hypothesis elicitation problem as a standard peer prediction problem. The connection is made by firstly rephrasing the two data sources, the classifiers and the labels, from agents' perspective. Let's re-interpret the ground truth label Y as an "optimal" agent who holds a hypothesis f * (X) = Y . We denote this agent as A * . Each local hypothesis f * i agent i holds can be interpreted as the agent that observes
. Then a peer prediction mechanism induces truthful reporting if:
Correlated Agreement for hypothesis elicitation To be more concrete, consider a specific implementation of peer prediction mechanism, the Correlated Agreement (CA). Recall that the mechanism builds on a correlation matrix ∆(f * i (X), f * (X)) defined as follows:
. Then the CA for hypothesis elicitation is summarized in Algorithm 1.
1: For each sample x n , randomly sample two other tasks x p1 = x p2 = x n to pair with. 2: Pay a reported hypothesis f (·) for x n according to
3: Total payment to a reported hypothesis f :
We reproduce the incentive guarantees and required conditions: Theorem 2. CA mechanism induces truthful reporting of a hypothesis at a Bayesian Nash Equilibrium.
Knowledge requirement of ∆ * We'd like to note that knowing the sign of ∆ * matrix between f * i and f * is a relatively weak assumption to have to run the mechanism. For example, for a binary classification task L = 2, define the following accuracy measure,
We offer the following: 5 We provide details of the calibration in the proof. Classical examples include cross-entropy loss, squared loss, etc.
. Similar conditions can be derived for L > 2 to guarantee an identify Sgn(∆ * ). With identifying a simple structure of Sgn(∆ * ), the CA mechanism for hypothesis elicitation runs in a rather simple manner.
When do we reward accuracy The elegance of the above CA mechanism leverages the correlation between a classifier and the ground truth label. Ideally we'd like a mechanism that rewards the accuracy of the contributed classifier. Consider the binary label case: Theorem 3. When P(Y = 1) = 0.5 (uniform prior), and let Sgn(∆ * = I 2×2 ) be the identity matrix, the more accurate classifier within each H i receives a higher score.
Note that the above result does not conflict with our incentive claims. In an equal prior case, misreporting can only reduce a believed optimal classifier's accuracy instead of the other way. It remains an interesting question to understand a more generic set of conditions under which CA will be able to incentivize contributions of more accurate classifiers.
A market implementation The above scoring mechanism leads to a market implementation [20] that incentivizes improving classifiers. In particular, suppose agents come and participate at discrete time step t. Denote the hypothesis agent contributed at time step t as f * t (and his report f t ). Agent at time t will be paid according to
is an incentive-compatible scoring function that elicits f t truthfully using Y . The incentivecompatibility of the market payment is immediate due to S(·). The above market implementation incentivizes improving classifiers with bounded budget 6 .
Calibrated CA scores When ∆ * is the identity matrix, the CA mechanism reduces to:
That is the reward structure of CA builds on 0-1 loss function. We ask the question of can we extend the CA to a calibrated one? We define the following loss-calibrated scoring function for CA:
Here again we negate the loss to make it a reward (agent will seek to maximize it instead of minimizing it). If this extension is possible, not only we will be able to include more scoring functions, but also we are allowed to score/verify non-binary classifiers directly. Due to space limit, we provide positive answers and detailed results in Appendix, while we will present empirical results on the calibrated scores of CA in Section 5.
Now we move on to a more challenging setting where we do not have ground truth label Y to verify the accuracy, or the informativeness of f (X), i.e., the mechanism designer only has access to a D = {x n } N n=1 . The main idea of our solution from this section follows straight-forwardly from the previous section, but instead of having a ground truth agent f * , for each classifier f * i we only have a reference agent f * j drawn from the rest agents j = i to score with. The corresponding scoring rule takes the form of S(f i (X), f j (X)), and similarly the goal is to achieve the following:
As argued before, if we treat f i and f j as two agents A i and A j holding private information, a properly defined peer prediction scoring function that elicits A i using A j will suffice to elicit f i using f j . Again we will focus on using Correlated Agreement as a running example. Recall that the mechanism builds on a correlation matrix
The mechanism then operates as follows: For each task x n , randomly sample two other tasks x p1 , x p2 . Then pay a reported hypothesis according to
We reproduce the incentive guarantees: 6 Telescoping returns:
Theorem 4. CA mechanism induces truthful reporting of a hypothesis at a Bayesian Nash Equilibrium.
The proof is similar to the proof of Theorem 2 so we will not repeat the details in the Appendix.
To enable a clean presentation of analysis, the rest of this section will focus on using/applying CA for the binary case L = 2. First, as an extension to Lemma 1, we have:
When do we reward accuracy As mentioned earlier that in general peer prediction mechanisms do not incentivize accuracy. Nonetheless we provide conditions under which they do. The result below holds for binary classifications.
are conditional independent of Y , the more accurate classifier within each H i receives a higher score in expectation.
Implementing the above peer prediction setting in a market setting is hard, due to again the challenge of no ground truth verification. The use of reference answers collected from other peers to similarly close a market will create incentives for further manipulations.
Our first attempt is to crowdsource to obtain an independent survey answer and use the survey answer to close the market. Denote the survey hypothesis as f and use f to close the market:
Theorem 6. When the survey hypothesis f (x) is (i) conditionally independent from the market contributions, and (ii) Bayesian informative, then closing the market using the crowdsourcing survey hypothesis is incentive compatible.
The above mechanism is manipulable in several aspects. Particularly, the crowdsourcing process needs to be independent from the market, which implies that the survey participant will need to stay away from participating in the marketbut it is unclear whether this will be the case. In the Appendix we show that by maintaining a survey process that elicits C > 1 hypotheses, we can further improve the robustness of our mechanisms against agents performing a joint manipulation on both surveys and markets.
Remark Before we conclude this section, we remark that the above solution for the without verification setting also points to an hybrid solution when the designer has access to both sample points with and without ground truth labels. The introduction of the pure peer assessment solution helps reduce the variance of payment.
Running a peer prediction mechanism with verifications coming only from peer agents is vulnerable when facing collusion. In this section we answer the question of how robust our mechanisms are when facing a γ-fraction of adversary in the participating population. To instantiate our discussion, consider the following setting
• There are 1 − γ fraction of agents who will act truthfully if incentivized properly. Denote the randomly drawn classifier from this 1 − γ population as f * 1−γ . • There are γ fraction of agents are adversary, whose reported hypotheses can be arbitrary and are purely adversarial.
Denote the following quantifies α := P(f * 1−γ (X) = 2|Y = 1), β := P(f * 1−γ (X) = 1|Y = 2) α * := P(f * (X) = 2|Y = 1), β * := P(f * (X) = 1|Y = 2), that is α, β are the error rates for the eliciting classifier f * 1−γ while α * , β * are the error rates for the Bayes optimal classifier. We prove the following Theorem 7. CA is truthful in eliciting hypothesis when facing γ-fraction of adversary when γ satisfies:
When the agent believes that the classifier the 1 − γ crowd holds is as accurate as the Bayes optimal classifier we have
, then a sufficient condition for eliciting truthful reporting is γ < 50%, that is our mechanism is robust up to half of the population manipulating. Clearly the more accurate the reference classifier is, the more robust our mechanism is. 
In this section, we implement two reward structures of CA: 0-1 score and Cross-Entropy (CE) score as mentioned at the end of Section 3.2. We experiment on two image classification tasks: MNIST [30] and CIFAR-10 [28] in our experiments. For agent A W (weak agent), we choose LeNet [30] and ResNet34 [22] for MNIST and CIFAR-10 respectively. For A S (strong agent), we use a 13-layer CNN architecture for both datasets.
Either of them is trained on random sampled 25000 images from each image classification training task. A W and A S receive hypothesis scores based on the test data X test (10000 test images) of MNIST or CIFAR-10. For elicitation with verification, we use ground truth labels to calculate the hypothesis score. For elicitation without verification, we replace the ground truth labels with the other agent's prediction -A W will serve as A S 's peer reference hypothesis and vice versa.
Statistically, an agent i's mis-reported hypothesis can be expressed by a misreport transition matrix T . Each element T j,k represents the probability of flipping the truthfully reported label f * i (x) = j to the misreported labelf i (x) = k: T j,k = P(f i (X) = k|f * i (X) = j). Random flipping predictions will degrade the quality of a classifier. When there is no adversary attack, we focus on two kinds of misreport transition matrix: a uniform matrix or a sparse matrix. For the uniform matrix, we assume the probability of flipping from a given class into other classes to be the same: T i,j = T i,k = e, ∀i = j = k. e changes gradually from 0 to 0.56 after 10 increases, which results in a 0%-50% misreport rate. The sparse matrix focuses on particular 5 pairs of classes which are easily mistaken between each pair. Denote the corresponding transition matrix elements of class pair (i, j) to be: (T ij , T ji ), i = j, we assume that T ij = T ji = e, ∀(i, j). e changes gradually from 0 to 0.5 after 10 increases, which results in a 0%-50% misreport rate.
Every setting is simulated 5 times. The line in each figure consists of the median score of 5 runs as well as the corresponding "deviation interval", which is the maximum absolute score deviation. The y axis symbolizes the averaged score of all test images.
As shown in Figure 1 , 2, in most situations, 0-1 score and CE score of both A W and A S keep on decreasing while the misreport rate is increasing. As for 0-1 score without ground truth verification, the score of either agent begins to fluctuate more when the misreport rate in sparse misreport model is > 35%. Our results conclude that both the 0-1 score and CE score induce truthful reporting of a hypothesis and will penalize misreported agents whether there is ground truth for verification or not.
We test the robustness of our mechanism when facing a 0.3-fraction of adversary in the participating population. We introduce an adversarial agent, LinfPGDAttack, introduced in AdverTorch [13] to influence the labels for verification when there is no ground truth. In Figure 3 , both the 0-1 score and CE score induce truthful reporting of a hypothesis for MNIST.
However, for CIFAR-10, with the increasing of misreport rate, the decreasing tendency fluctuates more often. Two factors attribute to this phenomenon: the agents' abilities as well as the quality of generated "ground truth" labels. When the misreport rate is large and generated labels are of low quality, the probability of successfully matching the misreported label to an incorrect generated label can be much higher than usual. But in general, these two scoring structures incentivize agents to truthfully report their results. 
This paper provides an elicitation framework to incentivize contribution of truthful hypotheses in federated learning. We have offered a scoring rule based solution template which we name as hypothesis elicitation. We establish the incentive property of the proposed scoring mechanisms and have tested their performance with real-world datasets extensively. We have also looked into the accuracy, robustness of the scoring rules, as well as market approaches for implementing them.
Proof for Theorem 1 Definition 2.
[2] A classification-calibrated loss function is defined as the follows: there exists a non-decreasing convex function Ψ(·) that satisfies: We prove by contradiction. Suppose that reporting a hypothesis f returns higher payoff (or a smaller risk) than reporting f * i under . Then
which is a contradiction. In above, the first equality (1) is by definition, (2) is due to the calibration condition, (3) is due to the definition of R * .
Proof. The proof builds essentially on the law of total probability (note f * and Y are the same):
Now consider ∆ * (1, 2):
The second row of ∆ * involving ∆ * (2, 1), ∆ * (2, 2) can be symmetrically argued.
Proof. Note the following fact:
Therefore we can focus on the expected score of individual sample X. The incentive compatibility will then hold for the sum.
The below proof is a rework of the one presented in [48] :
Note that truthful reporting returns the following expected payment:
∆ * (k, l) · Sgn(∆ * (k, l)) (Only the corresponding k survives the 3rd summation) = k,l:∆ * (k,l)>0 1] we conclude that for any other reporting strategy:
completing the proof.
Proof. For any classifier f , the expected score is
(independence between x p1 and x p2 ) = P(f (X) = Y ) − P(f (X) = 1) · 0.5 − P(f (X) = 2) · 0.5 (equal prior)
The last equality indicates that higher than accuracy, the higher the expected score given to the agent, completing the proof.
We start with extending the definition for calibration for CA: Definition 3. We call S w.r.t. original CA scoring function S if the following condition holds:
Since S induces f * i as the maximizer, if S satisfies the calibration property as defined in Definition 2, we will establish similarly the incentive property of S(·) Theorem 8. If for a certain loss function such that S satisfies the calibration condition, then S induces truthful reporting of f * i .
The proof of the above theorem repeats the one for Theorem 1, therefore we omit the details.
Denote by f * = argmin f R (f ). Sufficient condition for CA calibration was studied in [32] : Theorem 9 (Theorem 6 of [32] ). Under the following conditions, S is calibrated if P(Y = 1) = 0.5, and f * satisfies the following:
Proof. Denote by
i.e., the correlation matrix defined between f * i and ground truth label Y (f * ). And
If k = 1, l = 1, the above becomes
− FPR(f * j ) ∆(2, 1) < 0 If k = 2, l = 2, the above becomes
Proof. Shorthand the following error rates: α = P(f (X) = 2|Y = 1), β = P(f (X) = 1|Y = 2)
When two classifiers f, f are conditionally independent given Y , next we establish the following fact:
(definition of CA on f (X) and Y ) By condition (iii), replace f, f above with
When condition (ii) considering an identify ∆ * we know that f * j is Bayesian informative, which states exactly that [31] 
Further we have proven that E[S(f (X), Y )] rewards accuracy under Condition (i): 
Since f is Bayesian informative we know that 1 − α − β > 0 (Definition 4). Using the incentive-compatibility property of S when the market is closed with ground truth Y (note that S(f t−1 (X), Y ) is independent from reporting f t ) we complete the proof.
Our idea of making a robust extension is to pair the market with a separate "survey" elicitation process that elicits redundant C hypotheses:
Algorithm 2 Market for Hypothesis Elicitation 1: Pay crowdsourcing survey participants using surveys and the CA mechanism; 2: Randomly draw a hypothesis from the surveys (from the C collected ones) to close the market according to Eqn.
(3), up to a scaling factor λ > 0.
Assuming the survey hypotheses are conditional independent w.r.t. the ground truth Y . Denote by f a randomly drawn hypothesis from the surveys, and
For an agent who participated both in survey and market will receive the following:
Below we establish its incentive property: Theorem 10. For any δ that δ := λ (1−α −β )·(C+λ(C−1)) −→ C→∞ 0, agents have incentives to report a hypothesis that is at most δ less accurate than the truthful one.
Proof. The agent can choose to participate in either the crowdsourcing survey or the prediction market, or both. For agents who participated only in the survey, there is clearly no incentive to deviate. This is guaranteed by the incentive property of a proper peer prediction mechanism (in our case it is the CA mechanism that we use to guarantee the incentive property.).
For agents who only participated in the market at time t, we have its expected score as follows:
where in above we have used Theorem 6, Eqn. (5) . We then have proved the incentive compatibility for this scenario. Now consider an agent who participated both on the survey and market. Suppose the index of this particular agent is i in the survey pool and t in the market.
By truthfully reporting f * i , f * t the agent is scored as
Term II: 1 − 1/C probability of seeing another survey
Term III: 1/C probability of seeing their own survey hypothesis for closing the market.
We analyze each of the above terms:
Term I Due to the incentive-compatibility of CA, the first term E[S(f * i (X), f * −i (X))] is maximized by truthful reporting. And the decrease in score with deviation is proportional to 7
Term II The second term C−1 C (·) is also maximized by truthful reporting, because it is proportional to E[S(f * t (X), Y )] (application of Eqn. (5))
Term III The third 1/C(·) term is profitable via deviation. Suppose agent deviates to reporting some f that differs from f * i by δ amount in accuracy:
λ C Therefore the possible gain from the third term is bounded by λ C . On the other hand, again since E[S(f (X), Y )] = P(f (X) = Y ) − 0.5 (from the proof for Theorem 3), the loss via deviation (the first two terms in Eqn. (6) ) is lower bounded by
where 1 comes from the survey reward. Therefore when C is sufficiently large such that
Agent has no incentive to report such a f .
Proof. Denote byf (X) the reference classifier randomly drawn from the population. Then the expected score for reporting a hypothesis f is given by:
whereα,β are the error rates of the adversarial classifier f * γ :
α := P(f * γ (X) = 2|Y = 1),β := P(f * γ (X) = 1|Y = 2). The second equality is due to the application of Theorem 6, Eqn. (5) on f * 1−γ (X) and f * γ (X). Therefore
Due to the incentive property of E[S(f (X), Y )], a sufficient and necessary condition to remain truthful is
This is because the most adversarial classifier cannot be worse than reversing the Bayesian optimal classifier -otherwise if the error rate is higher than reversing the Bayes optimal classifier (with error rates 1 − α * , 1 − β * ):
we can reverse the adversarial classifier to obtain a classifier that performs better than the Bayes optimal one:
Therefore a sufficient condition is given by
Training hyper-parameters
We did not tune hyper-parameters for the training process, since we focus on hypothesis elicitation rather than improving the agent's ability/performance. We concentrate on different misreport transition matrices as well as the misreport rate which falls in the range of [0.0, 0.5] during the hypothesis elicitation stage. In the original submitted version, the mentioned machine learning architecture for agent A S and A W is a typo. The architecture that we use in our experiments are specified below.
• MNIST Agent A W is trained on uniformly sampled 25000 training images from MNIST training dataset. The architecture is LeNet. Agent A S is trained on uniformly sampled 25000 training images (with a different random seed) from MNIST training dataset. The architecture is a 13-layer CNN architecture. Both agents are trained for 100 epochs. The optimizer is SGD with momentum 0.9 and weight decay 1e-4. The initial learning rate is 0.1 and times 0.1 every 20 epochs.
Agent A W is trained on uniformly sampled 25000 training images from CIFAR-10 training dataset. The architecture is ResNet34. Agent A S is trained on uniformly sampled 25000 training images (with a different random seed) from CIFAR-10 training dataset. The architecture is a 13-layer CNN architecture. Both agents are trained for 180 epochs. The optimizer is SGD with momentum 0.9 and weight decay 1e-4. The initial learning rate is 0.1 and times 0.1 every 40 epochs.
We use LinfPGDAttack, introduced in AdverTorch [13] to simulate the adversary untargeted attacks. We adopt an example parameter setting provided by AdverTorch: cross-entropy loss function, eps is 0.15, number of iteration is 40, maximum clip is 1.0 and minimum clip is 0.0.
• Uniform misreport model In certain real world scenarios, an agent refuses to truthfully report the prediction by randomly selecting another different class as the prediction. We use the uniform misreport transition matrix to simulate this case.
In our experiments, we assume that the probability of flipping from a given class into other classes to be the same: T i,j = T i,k = e, ∀i = j = k. Mathematically, the misreport transition matrix can be expressed as: Thus, he may purposely report the other one which is not the original prediction. Even if an expert agent is able to distinguish confusing classes, it may still choose not to truthfully report the prediction since many other agents can not classify the image successfully. Without ground truth for verification, reporting truthfully and giving the correct prediction may result in a lower score than misreporting. For the sparse misreport transition matrix, we assume T i,j = T j,i = e, ∀(i, j), i = j. Mathematically, the misreport transition matrix can be expressed as: 
In the training stage, we use categorical cross-entropy as our loss function for evaluation. In the hypothesis elicitation stage, we choose two kinds of reward structure in the CA mechanism: 0-1 score and CE score. Let f * denote the "optimal" agent, f i denote the agent waiting to be scored. Mathematically, 0-1 score (one-hot encode the model output probability list) can be expressed as:
CE score can be expressed as:
, f * (x p2 ))).
• Ground truth for verification When there are ground truth labels for verification, f * (x) is equal to the corresponding ground truth label for a test image x.
When there aren't ground truth labels for verification, we substitute the other agent's prediction for the ground truth label. Thus, f * (x) := f * j (x), j = i for a test image x.
To simulate the case when facing a 0.3-fraction of adversary in the participating population and without ground truth for verification, we use LinfPGDAttack to influence the labels for verification. Specifically, given a test image x, LinfPGDAttack will attack x and generate a noisy image x attack , we replace the ground truth labels with weighted agents' prediction. Mathematically, f * (x) := 0.7 · f * j (x) + 0.3 · f * j (x attack ), j = i for the test image x. The weighting procedure is implemented before one-hot encoding stage for f * j (x) and f * j (x attack ). After that, the one-hot encoded label is considered as the ground truth label for verification.
The statistics "misreport rate" in Figure 1 ,2 symbolizes the proportion of misreported labels. For example, in a uniform transition matrix, T i,j = T i,k = e, ∀i = j = k, the misreport rate is 9e. While in a sparse transition matrix setting, given T i,j = T j,i = e, ∀(i, j), i = j, the misreport rate is exactly e. To see how adversarial attacks will affect the elicitation, in Figure 3 , we choose the same misreport transition matrices used in Figure 1 ,2 and calculate the misreport rate before applying the adversarial attacks.
We run 5 times for each experiment setting as shown in Figure 1 ,2,3. The central line is the mean of 5 runs. The "deviation interval" (error bars) is the maximum absolute score deviation. For example, suppose in the elicitation with ground truth verification setting, we have 5 runs/scores for a uniform transition matrix with a 0.25 misreport rate: [0.5, 0.5, 0.2, 0.1, 0.7], the mean is 0.4, the corresponding absolution score deviation is: [0.1, 0.1, 0.2, 0.3, 0.3]. Then the "deviation interval" comes to: 0.4 ± 0.3. Since the number of runs is not a large number, absolute deviation is no less than the standard deviation in our experiment setting.
In our experiments, we use a GPU cluster (8 TITAN V GPUs and 16 GeForce GTX 1080 GPUs) for training and evaluation.