Face recognition (FR) using deep convolutional neural networks (DCNNs) has seen remarkable success in recent years. One key ingredient of DCNN-based FR is the design of a loss function that ensures discrimination between various identities. The state-of-the-art (SOTA) solutions utilise normalised Softmax loss with additive and/or multiplicative margins. Despite being popular and effective, these losses are justified only intuitively with little theoretical explanations. In this work, we show that under the LogSumExp (LSE) approximation, the SOTA Softmax losses become equivalent to a proxy-triplet loss that focuses on nearest-neighbour negative proxies only. This motivates us to propose a variant of the proxy-triplet loss, entitled Nearest Proxies Triplet (NPT) loss, which unlike SOTA solutions, converges for a wider range of hyper-parameters and offers flexibility in proxy selection and thus outperforms SOTA techniques. We generalise many SOTA losses into a single framework and give theoretical justifications for the assertion that minimising the proposed loss ensures a minimum separability between all identities. We also show that the proposed loss has an implicit mechanism of hard-sample mining. We conduct extensive experiments using various DCNN architectures on a number of FR benchmarks to demonstrate the efficacy of the proposed scheme over SOTA methods.