Optimal policy computing for blockchain based smart contracts via federated learning

Abstract

In this paper, we develop a blockchain based decision-making system via federated learning along with an evolving convolution neural net, which can be applied to assemble-to-order services and Metaverses. The design and analysis of an optimal policy computing algorithm for smart contracts within the blockchain will be the focus. Inside the system, each order associated with a demand may simultaneously require multiple service items from different suppliers and the corresponding arrival rate may depend on blockchain history data represented by a long-range dependent stochastic process. The optimality of the computed dynamic policy on maximizing the expected infinite-horizon discounted profit is proved concerning both demand and supply rate controls with dynamic pricing and sequential packaging scheduling in an integrated fashion. Our policy is a pathwise oriented one and can be easily implemented online. The effectiveness of our optimal policy is supported by simulation comparisons.

Appendix: Demand and price functions

For reader’s convenience, we here introduce some assumptions concerning the relationship between the batch demand function and the price function. First, we assume that the vector form $\lambda (p(t),{\hat{\lambda }}(t))$ of the expression in (2.3) for each t has the first-order continuous derivative in terms of p(t) and the following Jacobian matrix is non-singular everywhere,

$$\begin{aligned}&\Bigg [\frac{\partial \lambda_{k}(p(t),{\hat{\lambda }}(t))}{\partial p_{m}(t)}\Bigg ]_{k,m\in \{1,\ldots ,K\}}. \end{aligned}$$

(5.1)

Second, we assume that the batch demand rate function $\lambda_{k}(\cdot ,\cdot )$ for each $k\in {{\mathcal{K}}}$ is strictly decreasing with respect to the price of the demand k while it is non-decreasing in the price of any other demand, i.e.,

$$\begin{aligned}&\frac{\partial \lambda_{k}(p(t),{\hat{\lambda }}(t))}{\partial p_{k}(t)}<0,\;\;\;\frac{\partial \lambda_{k}(p(t),{\hat{\lambda }}(t))}{\partial p_{m}(t)}\ge 0\;\;\hbox {for}\;\;m\ne k. \end{aligned}$$

(5.2)

Third, we assume that the quantity that the batch demand rate function k decreases is strictly smaller than the summation of the quantities that all other demands’ prices increase, i.e.,

$$\begin{aligned}&\sum \limits_{m=1}^{K}\frac{\partial \lambda_{k}(p(t),{\hat{\lambda }}(t))}{\partial p_{m}(t)}<0\;\;\hbox {for}\;\;k\in \{1,\ldots ,K\}. \end{aligned}$$

(5.3)

Hence, we can further assume that the following revenue rate is strictly concave in $\lambda (t)$,

$$\begin{aligned}&r(\lambda (t),{\hat{\lambda }}(t))\equiv \sum \limits_{k=1}^{K}\lambda_{k}(t)p_{k}(\lambda (t),{\hat{\lambda }}(t))e_{k}^{o}, \end{aligned}$$

(5.4)

where, $p(\lambda (t),{\hat{\lambda }}(t))$ is the unique inverse function of $\lambda (p(t),{\hat{\lambda }}(t))$, satisfying

$$\begin{aligned}&\frac{\partial p_{k}(\lambda (t),{\hat{\lambda }}(t))}{\partial \lambda (t)}<0,\;\;\;\frac{\partial p_{k}(\lambda (t),{\hat{\lambda }}(t))}{\partial \lambda_{m}(t)}\le 0\;\;\hbox {for}\;\;m\ne k. \end{aligned}$$

(5.5)