  Grant-free random access (GFRA) is now a popular protocol for large-scale
wireless multiple access systems in order to reduce control signaling. Resource
allocation in GFRA can be viewed as a form of frame slotted ALOHA, where a
ubiquitous design assumption is device homogeneity. In particular, the
probability that a device seeks to transmit data is common to all devices.
Recently, there has been an interest in designing frame slotted ALOHA
algorithms for networks with heterogeneous activity probabilities. These works
have established that the throughput can be significantly improved over the
standard uniform allocation. However, the algorithms for optimizing the
probability a device accesses each slot require perfect knowledge of the active
devices within each frame. In practice, this assumption is limiting as device
identification algorithms in GFRA rarely provide activity estimates with zero
errors. In this paper, we develop a new algorithm based on stochastic gradient
descent for optimizing slot allocation probabilities in the presence of
activity estimation errors. Our algorithm exploits importance weighted bias
mitigation for stochastic gradient estimates, which is shown to provably
converge to a stationary point of the throughput optimization problem. In
moderate size systems, our simulations show that the performance of our
algorithm depends on the type of error distribution. We study symmetric bit
flipping, asymmetric bit flipping and errors resulting from a generalized
approximate message passing (GAMP) algorithm. In these scenarios, we observe
gains up to 40\%, 66\%, and 19\%, respectively.
