A Mechanism Design Approach for Multi-party Machine Learning

Abstract

In a multi-party machine learning system, different parties cooperate on optimizing towards better models by sharing data in a privacy-preserving way. A major challenge in learning is the incentive issue. For example, if there is competition among the parties, one may strategically hide his data to prevent other parties from getting better models.

In this paper, we study the problem through the lens of mechanism design and incorporate the features of multi-party learning in our setting. First, each agent’s valuation has externalities that depend on others’ types and actions. Second, each agent can only misreport a type lower than his true type, but not the other way round. We call this setting interdependent value with type-dependent action spaces. We provide the optimal truthful mechanism in the quasi-monotone utility setting. We also provide necessary and sufficient conditions for truthful mechanisms in the most general case. We show the existence of such mechanisms is highly affected by the market growth rate. Finally, we devise an algorithm to find the desirable mechanism that is truthful, individually rational, efficient and weakly budget-balance.

Appendices

Appendix

A Proof of Theorem 1

Proof

Intuitively, the MEP rule charges agent i the profit he gets from an model that the mechanism allocates to him. If the mechanism charges higher than the MEP, an agent would have negative utility after taking part in. The IR constraint would then be violated. So it’s easy to see that the MEP is the maximal payment among all IR mechanisms.

Then we prove that this payment rule also guarantees the IC condition. It suffices to show that if an agent hides some data, no matter which model he chooses to use, he would never get more utility than that of truthful reporting. We suppose that agent i’s type is $t_i'$ and he untruthfully reports $t_i'$.

Suppose that the agent i truthfully reports the type $t_i'=t_i$, since the payment function is defined to charge this agent until he reaches the valuation when he does not take part in the mechanism, the utility of this honest agent would be

$$\begin{aligned} u_i^{0}(t') = F_i(Q(t_i))+\theta _i(q_{-i}(\emptyset ,t_{-i}')). \end{aligned}$$

If the agent does not report truthfully, we suppose that the agent reports $t_i'$ where $t_i' \le t_i$. According to the MEP, the payment function for agent i would be

$$\begin{aligned} p_i(t'_i,t_{-i}')=F_i(q_i(t_i',t_{-i}'))+\theta _i(q_{-i}(t_i',t_{-i}')) -F_i(Q(t_i'))-\theta _i(q_{-i}(\emptyset ,t_{-i}')). \end{aligned}$$

It can be seen that the mechanism would never give an agent a worse model than the model trained by its reported data, otherwise the agents would surely select their private data to train models. Hence it is without loss of generality to assume that the allocation $x_i(t'_i,t_{-i}') \ge Q(t'_i)$, $\forall t'_i,t_{-i}', \forall i$. Thus we have $q_{-i}(t_i',t_{-i}')=x_{-i}(t_i',t_{-i}')$. We discuss the utility of agent i by two cases of choosing models.

Case 1: the agent chooses the allocation $x_i$. Since agent i selects the allocated model, we have $q_i=x_i(t_i', t_{-i}')$. Then the utility of agent i would be

$$\begin{aligned} u_i^{1}=&v_i(t_i', t_{-i}')-p_i(t_i', t_{-i}')\\ =&F_i(x_i(t_i', t_{-i}')) + \theta _i(x_{-i}(t_i',t_{-i}'))+F_i(Q(t_i')) \\&+\theta _i(x_{-i}(\emptyset ,t_{-i}'))-F_i(x_i(t_i',t_{-i}'))-\theta _i(x_{-i}(t_i',t_{-i}'))\\ =&F_i(Q(t_i'))+\theta _i(x_{-i}(\emptyset ,t_{-i}')). \end{aligned}$$

Because both $F_i$ and Q are monotone increasing functions and $t_i\ge t_i'$, we have

$$\begin{aligned}\begin{gathered} u_i^{1} \le F_i(Q(t_i))+\theta _i(x_{-i}(\emptyset ,t_{-i}')) = u_i^{0}. \end{gathered}\end{aligned}$$

Case 2: the agent chooses $Q(t_i)$. Since agent i selects the model trained by his private data, we have $q_i = Q(t_i)$. The final utility of agent i would be

$$\begin{aligned} u_i^{2}=&v_i(t_i', t_{-i}')-p_i(t_i', t_{-i}')\\ =&F_i(Q(t_i)) + \theta _i(x_{-i}(t_i',t_{-i}'))+F_i(Q(t_i'))\\&+\theta _i(x_{-i}(\emptyset ,t_{-i}'))-F_i(x_i(t_i',t_{-i}'))-\theta _i(x_{-i}(t_i',t_{-i}'))\\ =&F_i(Q(t_i)) + F_i(Q(t_i')) + \theta _i(x_{-i}(\emptyset ,t_{-i}')) - F_i(x_i(t_i',t_{-i}')). \end{aligned}$$

Subtract the original utility from the both sides, then we have

$$\begin{aligned} u_i^2 - u_i^0 =&F_i(Q(t_i)) + F_i(Q(t_i')) + \theta _i(x_{-i}(\emptyset ,t_{-i}'))\\&- F_i(x_i(t_i',t_{-i}')) - F_i(Q(t_i))-\theta _i(x_{-i}(\emptyset ,t_{-i}'))\\ =&F_i(Q(t_i')) - F_i(x_i(t_i',t_{-i}')). \end{aligned}$$

Because $x_i(t'_i,t_{-i}') \ge Q(t'_i)$, $\forall t'_i,t_{-i}', \forall i$ and because $F_i$ is a monotonically increasing function, we can get $u_i^2 - u_i^0 \le 0$. Therefore $\max \{u_i^1, u_i^2\}$ $\le u_i^0$, lying would not bring more benefits to any agent, and the mechanism is IC.

B Proof of Corollary 1

Proof

In Theorem 1 we know that the MEP mechanism is IR and IC. Since the linear coefficients are all positive and the externality setting is linear, any efficient mechanism would allocate the best model to all the agents. Since each agent gets a model with no less quality than his reported one and the payment is equal to the value difference between the case an agent truthfully report and the case he exit the mechanism. The agent’s value is always larger than the value when he exits the mechanism. Then the payment is always positive and the mechanism should satisfy all of the four properties.

C Proof of Theorem 2

Proof

We first prove that Eq. (1) holds. Observe that

$$\begin{aligned}&u_i(x(t_i,t'_{-i}),t_i,t_{-i})-u_i(x(t'_i,t'_{-i}),t'_i,t_{-i})\nonumber \\ =&[v_i(x(t_i,t'_{-i}),t_i,t_{-i})-p_i(t_i,t'_{-i})] -[v_i(x(t'_i,t'_{-i}),t'_i,t_{-i})-p_i(t'_i,t'_{-i})]\nonumber \\ \ge&[v_i(x(t_i,t'_{-i}),t_i,t_{-i})-p_i(t_i,t'_{-i})] -[v_i(x(t'_i,t'_{-i}),t_i,t_{-i})-p_i(t'_i,t'_{-i})]\nonumber \\ =&u_i(x(t_i,t'_{-i}),t_i,t_{-i})-u_i(x(t'_i,t'_{-i}),t_i,t_{-i})\nonumber \\ \ge&0, \end{aligned}$$

(5)

where the first inequality is because of Assumption 3, and the last inequality is because of the DSIC property.

Let $t'_i=0$ in Eq. (5). We have

$$\begin{aligned}\begin{gathered} u_i(x(t_i,t'_{-i}),t_i,t_{-i})\ge u_i(x(0,t'_{-i}),0,t_{-i}). \end{gathered}\end{aligned}$$

The IR property further requires that $u_i(x(0,t'_{-i}),0,t_{-i})\ge u_i(x(\emptyset , t'_{-i}),0,t_{-i})$, which Eq. (1) follows.

To show Eq. (2) must hold, we rewrite Eq. (5):

$$\begin{aligned} p_i(t_i,t'_{-i})-p_i(t'_i,t'_{-i})&\le v_i(x(t_i,t'_{-i}),t_i,t_{-i})-v_i(x(t'_i,t'_{-i}),t'_i,t_{-i})\nonumber \\&=\int _{t'_i}^{t_i}\frac{\textrm{d}v_i(x(s',t'_{-i}),s(s'),t_{-i})}{\textrm{d}s'}\,\textrm{d}s'. \end{aligned}$$

(6)

Fixing $t_{-i}$ and $t'_{-i}$, the total derivative of $v_i(x(s', t'_{-i}),s,t_{-i})$ is:

$$\begin{aligned}&\textrm{d}v_i(x(s',t'_{-i}),s,t_{-i})\\ =&\frac{\partial v_i(x(s', t'_{-i}),s,t_{-i})}{\partial s'}\,\textrm{d}s'+\frac{\partial v_i(x(s', t'_{-i}),s,t_{-i})}{\partial s}\,\textrm{d}s. \end{aligned}$$

View s as a function of $s'$ and let $s(s')=s'$:

$$\begin{aligned}&\frac{\textrm{d}v_i(x(s',t'_{-i}),s(s'),t_{-i})}{\textrm{d}s'}\\ =&\left. \frac{\partial v_i(x(s', t'_{-i}),s,t_{-i})}{\partial s'}\right| _{s=s'} +\frac{\partial v_i(x(s', t'_{-i}),s(s'),t_{-i})}{\partial s(s')}\frac{\textrm{d}s(s')}{\textrm{d}s'}. \end{aligned}$$

Plug into Eq. (6), and we obtain:

$$\begin{aligned}&p_i(t_i,t'_{-i})-p_i(t'_i,t'_{-i})\\ \le&\int _{t'_i}^{t_i}\left. \frac{\partial v_i(x(s', t'_{-i}),s,t_{-i})}{\partial s'}\right| _{s=s'} +\int _{t'_i}^{t_i}\frac{\partial v_i(x(s', t'_{-i}),s(s'),t_{-i})}{\partial s(s')}\,\textrm{d}s'. \end{aligned}$$

Since the above inequality holds for any valuation function with $v_i(x,t_i,t_{-i})\ge v_i(x,t'_i,t_{-i}), \forall x, \forall t_{-i},\forall t_i\ge t'_i$, we have:

$$\begin{aligned} p_i(t_i,t'_{-i})-p_i(t'_i,t'_{-i}) \le \int _{t'_i}^{t_i}\left. \frac{\partial v_i(x(s', t'_{-i}),s,t_{-i})}{\partial s'}\right| _{s=s'}\,\textrm{d}s'. \end{aligned}$$

D Proof of Theorem 3

Proof

Equation (3) indicates that the function $\frac{\partial v_i(x(t'_i, t'_{-i}),t_i,t_{-i})}{\partial t'_i}$ is minimized at $t'_i$:

$$\begin{aligned} \left. \frac{\partial v_i(x(t'_i, t'_{-i}),s,t_{-i})}{\partial t'_i}\right| _{s=t'_i}\le \frac{\partial v_i(x(t'_i, t'_{-i}),t_i,t_{-i})}{\partial t'_i} . \end{aligned}$$

(7)

Therefore, we have

$$\begin{aligned}&u_i(x(t_i,t'_{-i}),t_i,t_{-i})-u_i(x(t'_i,t'_{-i}),t_i,t_{-i})\nonumber \\ =&\int _{t'_i}^{t_i}\frac{\partial v_i(x(s',t'_{-i}),t_i,t_{-i})}{\partial s'}\,\textrm{d}s'-p_i(t_i,t'_{-i})+p_i(t'_i,t'_{-i})\nonumber \\ \ge&\int _{t'_i}^{t_i}\left. \frac{\partial v_i(x(s',t'_{-i}),s,t_{-i})}{\partial s'}\right| _{s=s'}\,\textrm{d}s' -p_i(t_i,t'_{-i})+p_i(t'_i,t'_{-i})\nonumber \\ \ge&\int _{t'_i}^{t_i}\frac{\partial v_i(x(\emptyset ,t'_{-i}),s,t_{-i})}{\partial s}\textrm{d}s , \end{aligned}$$

(8)

where the two inequalities are due to Eq. (7) and (4), respectively. Since $v_i(x,t_i,t_{-i})$ $\ge$ $v_i(x,t'_i,t_{-i})$, $\forall x,$ $\forall t_{-i},$ $\forall t_i\ge t'_i$ indicates $\frac{\partial v_i(x(\emptyset ,t'_{-i}),s,t_{-i})}{\partial s}\ge 0$, the above inequality shows that the mechanism guarantees the DSIC property.

To prove that the mechanism is IR, we first observe that

$$\begin{aligned}&[u_i(x(t_i,t'_{-i}),t_i,t_{-i})- v_i(x(\emptyset , t'_{-i}),t_i,t_{-i})]-[u_i(x(t'_i,t'_{-i}),t'_i,t_{-i}) \\&-v_i(\emptyset ,x(t_{-i}),t'_i,t_{-i})]\\ =&u_i(x(t_i,t'_{-i}),t_i,t_{-i}) - u_i(x(t'_i,t'_{-i}),t'_i,t_{-i}) - \int _{t'_i}^{t_i}\frac{\partial v_i(x(\emptyset ,t'_{-i}),s,t_{-i})}{\partial s}\textrm{d}s\\ \ge&u_i(x(t_i,t'_{-i}),t_i,t_{-i}) - u_i(x(t'_i,t'_{-i}),t_i,t_{-i}) -\int _{t'_i}^{t_i}\frac{\partial v_i(x(\emptyset ,t'_{-i}),s,t_{-i})}{\partial s}\textrm{d}s \\ \ge&0, \end{aligned}$$

where the two inequalities are Assumption 3 and Eq. (8). Letting $t'_i=0$ using Eq. (2), we get:

$$\begin{aligned}&u_i(x(t_i,t'_{-i}),t_i,t_{-i}) - v_i(x(\emptyset ,t'_{-i}),t_i,t_{-i})\\ \ge&u_i(x(0,t'_{-i}),0,t_{-i})- v_i(x(\emptyset ,t'_{-i}),0,t_{-i})\\ =&v_i(x(0,t'_{-i}),0,t_{-i})-p_i(0,t'_{-i})- v_i(x(\emptyset ,t'_{-i}),0,t_{-i})\\ \ge&0. \end{aligned}$$

E Proof of Theorem 4

Proof

Suppose that the platform uses the mechanism mentioned in the theorem. Then for each agent, contributing with more data increases all participants’ model qualities. By definition, in a non-competitive market, improving others’ models does not decrease one’s profit. Therefore, the optimal strategy for each participant is to contribute with all his valid data, making the mechanism truthful. Also because of the definition, entering the platform always weakly increases one’s model quality. Thus the mechanism is IR. With the IC and IR properties, it is easy to see that the mechanism is also efficient and weakly budget-balance.

F Proof of Theorem 5

Proof

Suppose that there is a larger payment for agent i such that $p_i(t')>p_i^{\max }(t')$ where $t'$ is the profile of reported types. In the process of our algorithm, the $p_i^{\max }(t')$ is the minimal path length from $VB_{-i}$ to $V_{t_i}{t_{-i}}$, denoted by $(VB_{-i}, V_{t_{i1}}{t_{-i}},V_{t_{i2}}{t_{-i}},\cdots ,$ $V_{t_{ik}=t_i'}{t_{-i}})$. By the definition of edge weight, we have the following inequalities:

$$\begin{aligned}\begin{gathered} p_i(t_{i1}, t_{-i})\le \overline{p_i(t_{i1},t_{-i})},\\ p_i(t_{i2},t_{-i}) - p_i(t_{i1},t_{-i})\le Gap_i(t_{i1}, t_{i2},t_{-i}),\\ \vdots \\ p_i(t_{ik},t_{-i}) - p_i(t_{i(k-1)},t_{-i})\le Gap_i(t_{i1}, t_{i2},t_{-i}).\\ \end{gathered}\end{aligned}$$

Adding these inequalities together, we get

$$p_i(t')\le \overline{p_i(t_{i1},t_{-i})} + \sum _{j=1}^{k-1} Gap_i(t_{ij}, t_{i(j+1)},t_{-i})=p_i^{\max }(t').$$

If $p_i(t')<p_i^{\max }(t')$ holds, this would violate at least 1 of the k inequalities above. If the first inequality is violated, the mechanism would not be IR, by the definition of $\overline{p_i(t_{i1},t_{-i})}$. If any other inequality is violated, the mechanism would not be IC, by the definition of $Gap_i(t_{ij}, t_{i(j+1)},t_{-i})$.

On the other hand, if we select $p_i^{\max }(t')$ to be payment of agent i, all the inequalities should be satisfied, otherwise the shortest path would be updated to a smaller length.

Therefore the $p_i^{\max }(t')$ must be the maximum payment for agent i. If the maximal payment sum up to less than 0, there would obviously be no mechanism that is IR, IC and weakly budget-balance under the efficient allocation function.

G Experiments

We design experiments to demonstrate the performance of our mechanism for practical use. We first show the mechanism with the maximal exploitation payments can guarantee a good quality of trained model and high revenues under the linear externality cases. Then we conduct simulations to exhibit the relation of the market growth of competitive markets to the existence of desirable mechanisms.

1.1 G.1 The MEP Mechanism

We consider the valuation with linear externalities setting where $\alpha _{ij}$’s (defined in Example 1) are generated uniformly in $[-1,1]$. Each agent’s type is drawn uniformly from [0, 1] independently and the Q(t) is $\frac{1-e^{-t}}{1+e^{-t}}$. The performance of a mechanism is measured by the platform’s revenue and its best quality of trained model under the mechanism. All the values of each instance are averaged over 50 samples. We both show the performance changes as the number of agents increases and as the agents’ type changes.

When the number of agents becomes larger, the platform can obtain more revenues and train better models (see Fig. 1). Particularly, the model quality is close to be optimal when the number of agents over 12. An interesting phenomenon is that the revenue may surpass the social welfare. This is because the average external effect of other agents on one agent i tends to be negative when agent i does not join in the mechanism, thus the second term in the MEP payment is averagely negative and revenue is larger than the welfare.

To see the influence of type on performance, we fix one agent’s type to be 1 and set the other agent’s type from 0 to 10. It can be seen in Fig. 2 that the welfare and opponent agent’s utility (uti_2) increase as the opponent’s type increases but the platform’s revenue and the utility of the static agent (uti_1) are almost not affected by the type. So we draw the conclusion that the most efficient way for the platform to earn more revenue is to attract more small companies to join the mechanism, since in the Fig. 1 the revenue obviously increases as the number of agents increases.

1.2 G.2 Existence of Desirable Mechanisms

We assume all the agents’ types lie in [0, D], and the type space can be discretized into intervals of length $\epsilon$, which can be viewed as the minimal size of a dataset. Thus each agent’s type is a multiple of $\epsilon$. The data disparity is defined as the ratio of the largest possible data size to the smallest possible data size, namely, $D/\epsilon$. We measure the condition for existence of desirable mechanisms by the maximal data disparity when the market growth rate is given.

Fig. 1.

Performance of MEP under different numbers of agents

Fig. 2.

Performance of MEP under different types

Fig. 3.

Data disparity vs. Market growth (Color figure online)

To describe the market growth, we use the following form of valuation function and model quality function:

$$\begin{aligned}\begin{gathered} Q(t)=t \text { and } v_i(q)=\left( \sum \nolimits _{j = 1}^{n}Q(q_j)\right) ^\gamma \cdot Q(q_{i}), \forall i, \end{gathered}\end{aligned}$$

where $\gamma$ indicates the market growth rate. We consider the competitive growing market case where $-1\le \gamma <0$.¹

The algorithm we use to find desirable mechanisms under different valuation functions is described in Sect. 6. We enumerate the value of $\gamma$ from −1 to −0.668 with step length 0.002 and run the algorithm to figure out the boundary of $D/\epsilon$ under different $\gamma$ in a market with 2 agents. The range of $\gamma$ is determined by our computing capability, and the disparity boundary has been over 10000 when $\gamma$ is near −0.66.

Figure 3 shows the boundary of data disparity for existence of desirable mechanisms under different market growth rates. For every fixed $\gamma$, there does not exist any desirable mechanism when the data disparity is larger than the point on the red line. It can be seen an obvious trend that when $\gamma$ becomes larger, the constraint on data size disparity would become looser. A desirable mechanism is more likely to exist in a market that grows faster. When the market is not growing, there would not be such a desirable mechanism at all. On the other hand, if the market grows so fast such that there does not exist any competition between the agents, the desirable mechanism always exists.