Attention mechanisms have been widely used in deep learning, and recent efforts have been devoted to incorporating attention modules into deep neural networks (DNNs) for person Re-Identification (Re-ID) to enhance their discriminative feature learning capabilities. Existing attention modules, including self-attention and channel attention, learn attention weights that quantify the importance of feature tokens or feature channels. However, existing attention methods do not explicitly ensure that the attention weights are informative for predicting the identity of the person in the input image, and may consequently introduce noisy information from the input image. To address this issue, we propose a novel method termed Reduction of Information Bottleneck loss (RIB), motivated by the principle of the Information Bottleneck (IB). A novel distribution-free and efficient variational upper bound for the IB loss (IBB), which can be optimized by standard SGD, is derived and incorporated into the training loss of the RIB models. RIB is applied to DNNs with self-attention modules through a novel Differentiable Channel Selection Attention module, or DCS-Attention, that selects the most informative channels for computing attention weights, leading to competitive models termed RIB-DCS. RIB is also incorporated into DNNs with existing channel attention modules to promote the learning of informative channel attention weights, leading to models termed RIB-CA. Both RIB-DCS and RIB-CA are applied to fixed neural network backbones and learnable backbones with Differentiable Neural Architecture Search (DNAS). Extensive experiments on multiple person Re-ID benchmarks show that RIB significantly enhances the prediction accuracy of DNNs for person Re-ID, even for the occluded person Re-ID.