  In societal-scale infrastructures, such as electric grids or transportation
networks, pricing mechanisms are often used as a way to shape users' demand in
order to lower operating costs and improve reliability. Existing approaches to
pricing design for safety-critical networks often require that users are
queried beforehand to negotiate prices, which has proven to be challenging to
implement in the real-world. To offer a more practical alternative, we develop
learning-based pricing mechanisms that require no input from the users. These
pricing mechanisms aim to maximize the utility of the users' consumption by
gradually estimating the users' price response over a span of $T$ time steps
(e.g., days) while ensuring that the infrastructure network's safety
constraints that limit the users' demand are satisfied at all time steps. We
propose two different algorithms for the two different scenarios when, 1) the
utility function is chosen by the central coordinator to achieve a social
objective and 2) the utility function is defined by the price response under
the assumption that the users are self-interested agents. We prove that both
algorithms enjoy $\tilde{O} (T^{2/3})$ regret with high probability. We then
apply these algorithms to demand response pricing for the smart grid and
numerically demonstrate their effectiveness.
