This paper introduces a novel deep learning-based user-side feedback reduction framework, termed self-nomination. The goal of self-nomination is to reduce the number of users (UEs) feeding back channel state information (CSI) to the base station (BS), by letting each UE decide whether to feed back based on its estimated likelihood of being scheduled and its potential contribution to precoding in a multiuser MIMO (MU-MIMO) downlink. Unlike SNR- or SINR-based thresholding methods, the proposed approach uses rich spatial channel statistics and learns nontrivial correlation effects that affect eventual MU-MIMO scheduling decisions. To train the self-nomination network under an average feedback constraint, we propose two different strategies: one based on direct optimization with gradient approximations, and another using policy gradient-based optimization with a stochastic Bernoulli policy to handle non-differentiable scheduling. The framework also supports proportional-fair scheduling by incorporating dynamic user weights. Numerical results confirm that the proposed self-nomination method significantly reduces CSI feedback overhead. Compared to baseline feedback methods, self-nomination can reduce feedback by as much as 65%, saving not only bandwidth but also allowing many UEs to avoid feedback altogether (and thus, potentially enter a sleep mode). Self-nomination achieves this significant savings with negligible reduction in sum-rate or fairness.