Deep Structured Teams with Linear Quadratic Model: Partial Equivariance and Gauge Transformation

Motivated by the recent developments in artificial intelligence, we introduce linear quadratic deep structured teams in this paper. Two notions of equivariant and partially equivariant systems are defined, and it is shown that such systems can be partitioned into a few sub-populations of decision makers, where every decision maker in each sub-population is coupled in both dynamics and cost function through a set of linear regressions of the states and actions of all decision makers. Two non-classical information structures are considered: deep-state sharing and partial deep-state sharing, where deep state refers to the linear regression of the states of the decision makers in each sub-population. For a risk-sensitive cost function with deep-state sharing structure, a closed-form low-complexity representation of the globally optimal strategy is obtained, whose computational complexity is independent of the number of decision makers in each sub-population. In addition, it is shown that the risk-sensitive solution converges to the risk-neutral one as the number of decision makers increases to infinity. Moreover, two sub-optimal sequential strategies under partial deep-state sharing information structure are proposed by introducing two Kalman-like filters, one based on the finite-population model and the other one based on the infinite-population model. It is proved that the prices of information associated with the above sub-optimal solutions converge to zero as the number of decision makers goes to infinity. Furthermore, a class of feed-forward deep neural networks with multiple layers of weighted sums and products is introduced wherein the optimal weights and biases are explicitly obtained. A supply-chain management example is presented to demonstrate the efficacy of the obtained results.