Enabling low power wireless devices to adopt Nyquist sampling at high carriers is prohibitive. In spectrum sensing, this limit calls for an analog front-end that can sweep different bands quickly, in order to use the available spectrum opportunistically. In this paper we propose a new framework that allows to optimize a sub-Nyquist sampling front-end to combine the benefits of optimum sequential sensing with those of Compressive Spectrum Sensing (CSS). The sensing strategy we propose is formulated as a sequential utility optimization problem. The utility function decreases linearly with the number of measurements (accrued to make a decision) and increases as a function of the signal components binary states. The optimization selects the best linear combinations of sub-bands to mix, in order to accrue maximum utility. The structure of the utility represents the trade-off between exploration, exploitation and risk of making an error, that is characteristic of the spectrum sensing problem. We first present the analog front-end architecture, and then map the measurement model into the abstract optimization problem proposed, and analyzed, in the remainder of the paper. We characterize the optimal policy under constraints on the sensing matrix and derive the approximation factor of the greedy approach. Numerical simulations showcase the benefits of combining opportunistic spectrum sensing with sub-Nyquist sampling.