SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Yiding Chen
UW-Madison
ychen695@wisc.edu
Xiaojin Zhu
jerryzhu@cs.wisc.edu
Kirthevasan Kandasamy
kandasamy@cs.wisc.edu
We study collaborative normal mean estimation, where  $m$  strategic agents collect i.i.d samples from a normal distribution  $\mathcal{N}(\mu, \sigma^2)$  at a cost. They all wish to estimate the mean  $\mu$ . By sharing data with each other, agents can obtain better estimates while keeping the cost of data collection small. To facilitate this collaboration, we wish to design mechanisms that encourage agents to collect a sufficient amount of data and share it truthfully, so that they are all better off than working alone. In naive mechanisms, such as simply pooling and sharing all the data, an individual agent might find it beneficial to under-collect and/or fabricate data, which can lead to poor social outcomes. We design a novel mechanism that overcomes these challenges via two key techniques: first, when sharing the others' data with an agent, the mechanism corrupts this dataset proportional to how much the data reported by the agent differs from the others; second, we design minimax optimal estimators for the corrupted dataset. Our mechanism, which is Nash incentive compatible and individually rational, achieves a social penalty (sum of all agents' estimation errors and data collection costs) that is at most a factor 2 of the global minimum. When applied to high dimensional (non-Gaussian) distributions with bounded variance, this mechanism retains these three properties, but with slightly weaker results. Finally, in two special cases where we restrict the strategy space of the agents, we design mechanisms that essentially achieve the global minimum.
With the rise in popularity of machine learning, data is becoming an increasingly valuable resource for businesses, scientific organizations, and government institutions. However, data collection is often costly. For instance, to collect data, businesses may need to carry out market research, scientists may need to conduct experiments, and government institutions may need to perform surveys on public services. However, once data has been generated, it can be freely replicated and used by many organizations [20]. Hence, instead of simply collecting and learning from their own data, by sharing data with each other, organizations can mutually reduce their own data collection costs and improve the utility they derive from data [21]. In fact, there are already several platforms to facilitate data sharing among businesses [1, 40], scientific organizations [2, 3], and public institutions [16, 34].
However, simply pooling everyone's data and sharing with each other can lead to free-riding [23, 35]. For instance, if an agent (e.g. an organization) sees that other agents are already contributing a large amount of data, then, the cost she incurs to collect her own dataset may not offset the marginal improvement in her own learned model due to diminishing returns of increasing dataset sizes (we describe this rigorously in §2). Hence, while she benefits from others' data, she has no incentive to collect and contribute data to the pool. A seemingly simple fix to this free-riding problem is to only return the datasets of the others if an agent submits a large enough dataset herself. However, this can be easily manipulated by a strategic agent who submits a large fabricated (fake) dataset without incurring any cost, receives the others' data, and then discards her fabricated dataset when learning. While the agent has benefited by this bad behavior, other agents who may use this fabricated dataset
are worse off. Moreover, a naive test by the mechanism to check if the agent has fabricated data can be sidestepped by agents who collect only a small dataset and fabricate a larger dataset using this small dataset (e.g. by fitting a model to the small dataset and then sampling from this fitted model).
In this work, we study these challenges in data sharing in one of the most foundational statistical problems, normal mean estimation, where the goal is to estimate the mean  $\mu$  of a normal distribution  $\mathcal{N}(\mu, \sigma^2)$  with known variance  $\sigma^2$ . We wish to design mechanisms for data sharing that satisfy the three fundamental desiderata of mechanism design; Nash incentive compatibility (NIC): agents have incentive to collect a sufficiently large amount of data and share it truthfully provided that all other agents are doing so; individual rationality (IR): agents are better off participating in the mechanism than working on their own; and efficiency: the mechanism leads to outcomes with small estimation error and data collection costs for all agents.
Contributions: (i) In §2, we formalize collaborative normal mean estimation in the presence of strategic agents. (ii) In §3, we design an NIC and IR mechanism for this problem to prevent free-riding and data fabrication and show that its social penalty, i.e. sum of all agents' estimation errors and data collection costs, is at most twice that of the global minimum. (iii) In Appendix E, we study the same mechanism in high dimensional settings and relax the Gaussian assumption to distributions with bounded variance. We show that the mechanism retains its properties, with only a slight weakening of the NIC and efficiency guarantees. (iv) In §4, we consider two special cases where we impose natural restrictions on the agents' strategy space. We show that it is possible to design mechanisms which essentially achieve the global minimum social penalty in both settings. Next, we will summarize our primary mechanism and the associated theorem in §3.
1.1 Summary of main results
Formalism: We assume that all agents have a fixed cost for collecting one sample, and define an agent's penalty (negative utility) as the sum of her estimation error and the cost she incurred to collect data. To make the problem well-defined, for the estimation error, we find it necessary to consider the maximum risk, i.e. maximum expected error over all  $\mu \in \mathbb{R}$ . A mechanism asks agents to collect data, and then shares the data among the agents in an appropriate manner to achieve the three desiderata. An agent's strategy space consists of three components: how much data she wishes to collect, what she chooses to submit after collecting the data, and how she estimates the mean  $\mu$  using the dataset she collected, the dataset she submitted, and the information she received from the mechanism.
Mechanism and theoretical result: In our mechanism, which we call C3D (Cross-Check and Corrupt based on Difference), each agent  $i$  collects a dataset  $X_{i}$  and submits a possibly fabricated or altered version  $Y_{i}$  to the mechanism. The mechanism then determines agent  $i$ 's allocation in the following manner. It pools the data from the other agents and splits them into two subsets  $Z_{i}, Z_{i}^{\prime}$ . Then,  $Z_{i}$  is returned as is, while  $Z_{i}^{\prime}$  is corrupted by adding noise that is proportional to the difference between  $Y_{i}$  and  $Z_{i}$ . If an agent collects less or fabricates, she risks looking different to the others, and will receive a dataset  $Z_{i}^{\prime}$  of poorer quality. We show that this mechanism has a Nash equilibrium where all agents collect a sufficiently large amount of data, submit it truthfully, and use a carefully weighted average of the three datasets  $X_{i}, Z_{i}$ , and  $Z_{i}^{\prime}$  as their estimate for  $\mu$ . The weighting uses some additional side information that the mechanism provides to each agent. Below, we state an informal version of the main theoretical result of this paper, which summarizes the properties of our mechanism.
Theorem 1 (informal): The above mechanism is Nash incentive compatible, individually rational, and achieves a social penalty that is at most twice the globally minimum social penalty.
Corruption is the first of two ingredients to achieving NIC. The second is the design of the weighted average estimator which is (minimax) optimal after corruption. To illustrate why this is important, say that the mechanism had assumed that the agents will use any other sub-optimal estimator (e.g a simple average). Then it will need to lower the amount of corruption to ensure IR and efficiency. However, a strategic agent will realize that she can achieve a lower maximum risk with a better estimator (instead of collecting more data herself and/or receiving less corrupt data from the mechanism). She can leverage this insight to collect less data and lower her overall penalty.
Proof techniques: The most challenging part of our analysis is to show NIC, First, to show minimax optimality of our estimator, we construct a sequence of normal priors for  $\mu$  and show that the minimum Bayes' risk converges to the maximum risk of the weighted average estimator. However, when compared to typical minimax proofs, we face more significant challenges. The first of these
is that the combined dataset  $X_{i} \cup Z_{i} \cup Z_{i}'$  is neither independent nor identically distributed as the corruption is data-dependent. The second is that the agent's submission  $Y_{i}$  also determines the degree of corruption, so we cannot look at the estimator in isolation when computing the minimum Bayes' risk; we should also consider the space of functions an agent may use to determine  $Y_{i}$  from  $X_{i}$ . The third is that the expressions for the minimum Bayes' risk do not have closed form solutions and require non-trivial algebraic manipulations. To complete the NIC proof, we show that due to the carefully chosen amount of corruption, the agent should collect a sufficient amount of data to avoid excessive corruption, but not too much so as to increase her data collection costs.
Mechanism design is one of the core areas of research in game theory [13, 18, 36]. Our work here is more related to mechanism design without payments, which has seen applications in fair division [31], matching markets [32], and kidney exchange [33] to name a few. There is a long history of work in the intersection of machine learning and mechanism design, although the overwhelming majority apply learning techniques when there is incomplete information about the mechanism or agent preferences, (e.g [6, 8, 22, 28, 30]). On the flip side, some work have designed data marketplaces, where customers may purchase data from contributors [4, 5, 19, 38]. These differ from our focus where we wish to incentivize agents to collaborate without payments.
Due to the popularity of shared data platforms [1, 2, 16, 34] and federated learning [21], there has been a recent interest in designing mechanisms for data sharing. Sim et al. [35] and Xu et al. [39] study fairness in collaborative data sharing, where the goal is to reward agents according to the amount of data they contribute. However, their mechanisms do not apply when strategic agents may try to manipulate a mechanism. Blum et al. [9] and Karimireddy et al. [23] study collaboration in federated learning. However, the strategy space of an agent is restricted to how much data they collect and their mechanism rewards each agent according to the quantity of the data she submitted. The above four works recognize that free-riding can be detrimental to data sharing, but assume that agents will not fabricate data. As discussed above, if this assumption is not true, agents can easily manipulate such mechanisms. Fraboni et al. [17] and Lin et al. [25] study federated learning settings where free-riders may send in fabricated gradients without incurring the computational cost of computing the gradients. However, their focus is on designing gradient descent algorithms that are robust to such attacks and not on incentivizing agents to perform the gradient computations. Some work have designed mechanisms for federated learning so as to elicit private information (such as data collection costs), but their focus is not on preventing free-riding or fabrication [15, 26]. Miller et al. [29] uses scoring systems to develop mechanisms that prevent signal fabrication. However, the agents in their settings can only choose to report either their true signal or something else but can not freely choose how much data to collect. Cai et al. [11] study mechanism design where a learner incentivizes agents to collect data via payments. Their mechanism, which also cross-checks the data submitted by the agents, has connections to our setting in §4.2 where we consider a restricted strategy space for the agents.
Our approach of using corruption to engender good behaviour draws inspiration from the robust estimation literature, which design estimators that are robust to data from malicious agents [12, 14, 27]. However, to the best of our knowledge, the specific form of corruption and the subsequent design of the minimax optimal estimator are new in this work, and require novel analysis techniques.
We will now formally define our problem. We have  $m$  agents, who are each able to collect i.i.d samples from a normal distribution  $\mathcal{N}(\mu, \sigma^2)$ , where  $\sigma^2$  is known. They wish to estimate the mean  $\mu$  of this distribution. To collect one sample, the agent has to incur a cost  $c$ . We will assume that  $\sigma^2$ ,  $c$ , and  $m$  are public information. However,  $\mu \in \mathbb{R}$  is unknown, and no agent has auxiliary information, such as a prior, about  $\mu$ . An agent wishes to minimize her estimation error, while simultaneously keeping the cost of data collection low. While an agent may collect data on her own to manage this trade-off, by sharing data with other agents, she can reduce costs while simultaneously improving her estimate. We wish to design mechanisms to facilitate such sharing of data.
**Mechanism:** A mechanism receives a dataset from each agent, and in turn returns an allocation  $A_{i}$  to each agent. An agent will use her allocation to estimate  $\mu$ . This allocation could be, for instance, a larger dataset obtained with other agents' datasets. The mechanism designer is free to choose a space of allocations  $\mathcal{A}$  to achieve the desired goals. Formally, we define a mechanism as a tuple  $M = (\mathcal{A}, b)$  where  $\mathcal{A}$  denotes the space of allocations, and  $b$  is a procedure to map the datasets collected from the  $m$  agents to  $m$  allocations. Denoting the universal set by  $\mathcal{U}$ , we write the space of mechanisms  $\mathcal{M}$  as
$$
\mathcal {M} = \left\{M = (\mathcal {A}, b): \mathcal {A} \subset \mathcal {U}, b: \left(\bigcup_ {n \geq 0} \mathbb {R} ^ {n}\right) ^ {m} \rightarrow \mathcal {A} ^ {m} \right\}. \tag {1}
$$
As is customary, we will assume that the mechanism designer will publish the space of allocations  $\mathcal{A}$  and the mapping  $b$  (the procedure used to obtain the allocations) ahead of time, so that agents can determine their strategies. However, specific values computed/realized during the execution of the mechanism are not revealed, unless the mechanism chooses to do so via the allocation  $A_{i}$ .
Agents' strategy space: Once the mechanism is published, the agent will choose a strategy. In our setting, this will be the tuple  $(n_i, f_i, h_i)$ , which determines how much data she wishes to collect, what she chooses to submit, and how she wishes to estimate the mean  $\mu$ . First, the agent samples  $n_i$  points to collect her initial dataset  $X_i = \{x_{i,j}\}_{j=1}^{n_i}$ , where  $x_{i,j} \sim \mathcal{N}(\mu, \sigma^2)$ , incurring  $cn_i$  cost. She then submits  $Y_i = \{y_{i,j}\}_j = f_i(X_i)$  to the mechanism. Here  $f_i$  is a function which maps the collected dataset to a possibly fabricated or falsified dataset of a potentially different size. In particular, this fabrication can depend on the data she has collected. For instance, the agent could collect only a small dataset, fit a Gaussian, and then sample from it.
Finally, the mechanism returns the agent's allocation  $A_{i}$ , and the agent computes an estimate  $h_i(X_i,Y_i,A_i)$  for  $\mu$  using her initial dataset  $X_{i}$ , the dataset she submitted  $Y_{i}$ , and the allocation she received  $A_{i}$ . We include  $Y_{i}$  as part of the estimate since an agent's submission may affect the allocation she receives. Consequently, agents could try to elicit additional information about  $\mu$  via a carefully chosen  $Y_{i}$ . We can write the strategy space of an agent as  $S = \mathbb{N}\times \mathcal{F}\times \mathcal{H}$ , where  $\mathcal{F}$  is the space of functions mapping the dataset collected to the dataset submitted, and  $\mathcal{H}$  is the space of all estimators using all the information she has. We have:
$$
\mathcal {F} = \left\{f: \bigcup_ {n \geq 0} \mathbb {R} ^ {n} \rightarrow \bigcup_ {n \geq 0} \mathbb {R} ^ {n} \right\}, \quad \mathcal {H} = \left\{h: \bigcup_ {n \geq 0} \mathbb {R} ^ {n} \times \bigcup_ {n \geq 0} \mathbb {R} ^ {n} \times \mathcal {A} \rightarrow \mathbb {R} \right\}. \tag {2}
$$
One element of interest in  $\mathcal{F}$  is the identity  $\mathbf{I}$  which maps a dataset to itself. A mechanism designer would like an agent to use  $f_{i} = \mathbf{I}$ , i.e. to submit the data that she collected as is, so that other agents can benefit from her data.
Going forward, when  $s = \{s_i\}_i \in S^m$  denotes the strategies of all agents, we will use  $s_{-i} = \{s_j\}_{j \neq i}$  to denote the strategies of all agents except  $i$ . Without loss of generality, we will assume that agent strategies are deterministic. If they are stochastic, our results will carry through for every realization of any external source of randomness that the agent uses to determine  $(n_i, f_i, h_i)$ .
Agent penalty: The agent's penalty  $p_i$  (i.e. negative utility) is the sum of her squared estimation error and the cost  $cn_i$  incurred to collect her dataset  $X_i$  of  $n_i$  points. The agent's penalty depends on the mechanism  $M$  and the strategies  $s = \{s_j\}_{j}$  of all the agents. Making this explicit,  $p_i$  is defined as:
$$
p _ {i} (M, s) = \sup  _ {\mu \in \mathbb {R}} \mathbb {E} \left[ \left(h _ {i} \left(X _ {i}, Y _ {i}, A _ {i}\right) - \mu\right) ^ {2} \mid \mu \right] + c n _ {i} \tag {3}
$$
The term inside the expectation is the squared difference between the agent's estimate and the true mean (conditioned on the true mean  $\mu$ ). The expectation is with respect to the randomness of all agents' data and possibly any randomness in the mechanism. We consider the maximum risk, i.e. supremum over  $\mu \in \mathbb{R}$ , since the true mean  $\mu$  is unknown to the agent a priori, and their strategy should yield good estimates, and hence small penalty, over all possible values  $\mu$ . To illustrate this further, note that when the value of true mean  $\mu$  is  $\mu'$ , the optimal strategy for an agent will always be to not collect any data and choose the estimator  $h_i(\cdot, \cdot, \cdot) = \mu'$  leading to 0 penalty. However, this strategy can be meaningfully realized by an agent only if she knew that  $\mu = \mu'$  a priori which renders the problem meaningless<sup>1</sup>. Considering the maximum risk accounts for the fact that  $\mu$  is unknown and makes the problem well-defined.
This is akin to the reason why it is customary to study the maximum risk in frequentist statistics [24, 37]. An alternative approach is to take a Bayesian view, considering a prior on  $\mu$  and using the Bayes' risk  $\mathbb{E}_{\mu}[\mathbb{E}[(h_i(X_i,Y_i,A_i) - \mu)^2 |\mu ]]$  instead of the maximum risk in  $p_i$ . While we have adopted a frequentist formalism here, our main proof ideas can be ported over to the Bayesian setting as well (See Appendix F for more details)
Recommended strategies: In addition to publishing the mechanism, the mechanism designer will recommend strategies  $s^{\star} = \{s_{i}^{\star}\}_{i} \in S^{m}$  for the agents so as to incentivize collaboration and induce optimal social outcomes.
Desiderata: We can now define the three desiderata for a mechanism:
1. Nash Incentive compatibility (NIC): A mechanism  $M = (\mathcal{A}, b)$  is said to be NIC at the recommended strategy profile  $s^{\star}$  if, for each agent  $i$ , and for every other alternative strategy  $s_i \in S$  for that agent, we have  $p_i(M, s^{\star}) \leq p_i(M, (s_i, s_{-i}^{\star}))$ . That is,  $s^{\star}$  is a Nash equilibrium so no agent has incentive to deviate if all other agents are following  $s^{\star}$ .
2. Individual rationality (IR): We say that a mechanism  $M$  is IR at  $s^{\star}$  if no agent suffers from a higher penalty by participating in the mechanism than the lowest possible penalty she could achieve on her own when all other agents are following  $s^{\star}$ . If an agent does not participate, she does not submit nor receive any data from the mechanism; she will simply choose how much data to collect and design the best possible estimator. Formally, we say that a mechanism  $M$  is IR if the following is true for each agent  $i$ :
$$
p _ {i} (M, s ^ {\star}) \leq \inf  _ {n _ {i} \in \mathbb {N}, h _ {i} \in \mathcal {H}} \left\{\sup  _ {\mu \in \mathbb {R}} \mathbb {E} \left[ \left(h _ {i} \left(X _ {i}, \varnothing , \varnothing\right) - \mu\right) ^ {2} \mid \mu \right] + c n _ {i} \right\}. \tag {4}
$$
3. Efficiency: The social penalty  $P(M, s)$  of a mechanism  $M$  when agents follow strategies  $s$ , is the sum of agent penalties (defined below). We define  $\mathrm{PR}(M, s^{\star})$  to be the ratio between the social penalty of a mechanism at the recommended strategies  $s^{\star}$ , and the lowest possible social penalty among all possible mechanisms and strategies (without NIC or IR constraints). We have:
$$
P (M, s) = \sum_ {i \in [ m ]} p _ {i} (M, s), \quad \operatorname {P R} (M, s ^ {\star}) = \frac {P (M , s ^ {\star})}{\inf  _ {M ^ {\prime} \in \mathcal {M} , s \in \mathcal {S} ^ {m}} P \left(M ^ {\prime} , s\right)} \tag {5}
$$
Note that  $\mathrm{PR} \geq 1$ . We say that a mechanism is efficient if  $\mathrm{PR}(M, s^{\star}) = 1$  and that it is approximately efficient if  $\mathrm{PR}(M, s^{\star})$  is bounded by some constant that does not depend on  $m$ . If  $s^{\star}$  is a Nash equilibrium, then  $\mathrm{PR}(M, s^{\star})$  can be viewed as an upper bound on the price of stability [7].
For what follows, we will discuss optimal strategies for agents working on her own and present a simple mechanism which minimizes the social penalty, but has a poor Nash equilibrium.
Optimal strategies for an agent working on her own: Recall that, given  $n$  samples  $\{x_i\}_{i=1}^n$  from  $\mathcal{N}(\mu, \sigma^2)$ , the sample mean is a minimax optimal estimator [24]; i.e. among all possible estimators  $h$ , the sample mean minimizes the maximum risk  $\sup_{\mu \in \mathbb{R}} \mathbb{E}[(\mu - h(\{x_i\}_{i=1}^n, \emptyset, \emptyset))^2 | \mu]$  (note that the agent only has the dataset she collected). Moreover, its mean squared error is  $\sigma^2 / n$  for all  $\mu \in \mathbb{R}$ . Hence, an agent acting on her own will choose the sample mean and collect  $n_i = \sigma / \sqrt{c}$  samples so as to minimize their penalty; as long as the amount of data is less than  $\sigma / \sqrt{c}$ , an agent has incentive to collect more data since the cost of collecting one more point is offset by the marginal decrease in estimation error. This can be seen via the following simple calculation:
$$
\inf_{\substack{n_{i}\in \mathbb{R}\\ h_{i}\in \mathcal{H}}}\Big(\sup_{\mu}\mathbb{E}\left[(h_{i}(X_{i},\varnothing ,\varnothing) - \mu)^{2}  \bigg|  \mu \right] +   cn_{i}\Big) = \min_{n_{i}\in \mathbb{R}}\left(\frac{\sigma^{2}}{n_{i}} +cn_{i}\right) = 2\sigma \sqrt{c} \triangleq p_{\min}^{\mathrm{IR}}. \tag{6}
$$
Let  $p_{\mathrm{min}}^{\mathrm{IR}} = 2\sigma \sqrt{c}$  denote the lowest achievable penalty by an agent working on her own. If all  $m$  agents work independently, then the total social penalty is  $mp_{\mathrm{min}}^{\mathrm{IR}} = 2\sigma m\sqrt{c}$ . Next, we will look at a simple mechanism and an associated set of strategies which achieve the global minimum penalty. This will show that it is possible for all agents to achieve a significantly lower penalty via collaboration.