Spiking neural networks (SNNs) have been proposed as energy-efficient alternatives to deep neural networks. Their training is non-convex and involves a non-differentiable spike function, so existing methods offer no guarantee of reaching a global optimum. We address this issue by formulating SNN optimization as an equivalent convex optimization problem. Specifically, we show that the weight-decay-regularized training problem for a class of threshold RNNs, including leaky integrate-and-fire SNNs, has zero duality gap and the same optimal value as a convex program. Since the resulting convex programs can have exponentially many variables in the amount of data, we introduce a practical randomized surrogate and establish a high-probability guarantee on its approximation quality. Empirically, the best convex variant outperforms surrogate-gradient training on addition and first-last-XOR, matches its performance on sequential MNIST, and trails a reservoir baseline on base-3 and base-5 addition.