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Title: Credible Sample Elicitation by Deep Learning, for Deep Learning

Paper content:
It is important to collect credible training samples $(x,y)$ for building data-intensive learning systems (e.g., a deep learning system).
In the literature, there is a line of studies on eliciting distributional information from self-interested agents who hold a relevant information.
Asking people to report complex distribution $p(x)$, though theoretically viable, is challenging in practice.
This is primarily due to the heavy cognitive loads required for human agents to reason and report this high dimensional information.
Consider the example where we are interested in building an image classifier via first collecting a certain category of  high-dimensional image data.
While classical elicitation results apply to eliciting a complex and generative (and continuous) distribution $p(x)$ for this image data, we are interested in eliciting samples $x_i \sim p(x)$ from agents.
This paper introduces a deep learning aided method to incentivize credible sample contributions from selfish and rational agents.
The challenge to do so is to design an incentive-compatible score function to score each reported sample to induce truthful reports, instead of an arbitrary or even adversarial one.
We show that with accurate estimation of a certain $f$-divergence function we are able to achieve approximate incentive compatibility in eliciting truthful samples.
We then present an efficient estimator with theoretical guarantee via studying the variational forms of $f$-divergence function.
Our work complements the literature of information elicitation via introducing the problem of \emph{sample elicitation}.  We also show a connection between this sample elicitation problem and $f$-GAN, and how this connection can help reconstruct an estimator of the distribution based on collected samples.
The availability of a large quantity of credible samples is crucial for building high-fidelity machine learning models.
This is particularly true for deep learning systems that are data-hungry.
Arguably, the most scalable way to collect a large amount of training samples is to crowdsource from a decentralized population of agents who hold relevant sample information.
The most popular example is the build of ImageNet (Deng et al., 2009 ).
The main challenge in eliciting private information is to properly score reported information such that the self-interested agent who holds a private information will be incentivized to report truthfully.
At a first look, this problem of eliciting quality data is readily solvable with the seminal solution for eliciting distributional information, called the strictly proper scoring rule (Brier, 1950; Winkler, 1969; Savage, 1971; Matheson & Winkler, 1976; Jose et al., 2006; Gneiting & Raftery, 2007) : suppose we are interested in eliciting information about a random vector X = (X 1 , ..., X d−1 , Y ) ∈ Ω ⊆ R d , whose probability density function is denoted by p with distribution P. As the mechanism designer, if we have a sample x drawn from the true distribution P, we can apply strictly proper scoring rules to elicit p: the agent who holds p will be scored using S (p, x) .
S is called strictly proper if it holds for any p and q that E x∼P [S(p, x) ] > E x∼P [S(q, x) ].
The above elicitation approach has two main caveats that limited its application:
• When the outcome space |Ω| is large and is even possibly infinite, it is practically impossible for any human agents to report such a distribution with reasonable efforts.
This partially inspired a line of follow-up works on eliciting property of the distributions, which we will discuss later.
In this work we aim to collect credible samples from self-interested agents via studying the problem of sample elicitation.
Instead of asking each agent to report the entire distribution p, we hope to elicit samples drawn from the distribution P truthfully.
We consider the samples x p ∼ P and x q ∼ Q. In analogy to strictly proper scoring rules 1 , we aim to design a score function S s.t.
for any q ̸ = p, where x ′ is a reference answer that can be defined using elicited reports.
Often, this scoring procedure requires reports from multiple peer agents, and x ′ is chosen as a function of the reported samples from all other agents (e.g., the average across all the reported xs, or a randomly selected x).
This setting will relax the requirements of high reporting complexity, and has wide applications in collecting training samples for machine learning tasks.
Indeed our goal resembles similarity to property elicitation (Lambert et al., 2008; Steinwart et al., 2014; Frongillo & Kash, 2015b ), but we emphasize that our aims are different -property elicitation aims to elicit statistical properties of a distribution, while ours focus on eliciting samples drawn from the distributions.
In certain scenarios, when agents do not have the complete knowledge or power to compute these properties, our setting enables elicitation of individual sample points.
Our challenge lies in accurately evaluating reported samples.
We first observe that the f -divergence function between two properly defined distributions of the samples can serve the purpose of incentivizing truthful report of samples.
We proceed with using deep learning techniques to solve the score function design problem via a data-driven approach.
We then propose a variational approach that enables us to estimate the divergence function efficiently using reported samples, via a variational form of the f -divergence function, through a deep neutral network.
These estimation results help us establish an approximate incentive compatibility in eliciting truthful samples.
It is worth to note that our framework also generalizes to the setting where there is no access to ground truth samples, where we can only rely on reported samples.
There we show that our estimation results admit an approximate Bayesian Nash Equilibrium for agents to report truthfully.
Furthermore, in our estimation framework, we use a generative adversarial approach to reconstruct the distribution from the elicited samples.
We want to emphasize that the deep learning based estimators considered above are able to handle complex data.
And with our deep learning solution, we are further able to provide estimates for the divergence functions used for our scoring mechanisms with provable finite sample complexity.
In this paper, we focus on developing theoretical guarantees -other parametric families either can not handle complex data, e.g., it is hard to handle images using kernel methods, or do not have provable guarantees on the sample complexity.
Our contributions are three-folds.
(1) We tackle the problem of eliciting complex distribution via proposing a sample elicitation framework.
Our deep learning aided solution concept makes it practical to solicit complex sample information from human agents.
(2) Our framework covers the case when the mechanism designer has no access to ground truth information, which adds contribution to the peer prediction literature.
(3) On the technical side, we develop estimators via deep learning techniques with strong theoretical guarantees.
This not only helps us establish approximate incentive-compatibility, but also enables the designer to recover the targeted distribution from elicited samples.
Our contribution can therefore be summarized as "eliciting credible training samples by deep learning, for deep learning".

assistant: text: This paper proposes a deep learning aided method to elicit credible samples from self-interested agents.