CPU branch prediction has hit a wall--existing techniques achieve near-perfect accuracy on 99% of static branches, and yet the mispredictions that remain hide major performance gains. In a companion report, we show that a primary source of mispredictions is a handful of systematically hard-to-predict branches (H2Ps), e.g. just 10 static instructions per SimPoint phase in SPECint 2017. The lost opportunity posed by these mispredictions is significant to the CPU: 14.0% in instructions-per-cycle (IPC) on Intel SkyLake and 37.4% IPC when the pipeline is scaled four-fold, on par with gains from process technology. However, up to 80% of this upside is unreachable by the best known branch predictors, even when afforded exponentially more resources. New approaches are needed, and machine learning (ML) provides a palette of powerful predictors. A growing body of work has shown that ML models are deployable within the microarchitecture to optimize hardware at runtime, and are one way to customize CPUs post-silicon by training to customer applications. We develop this scenario for branch prediction using convolutional neural networks (CNNs) to boost accuracy for H2Ps. Step-by-step, we (1) map CNNs to the global history data used by existing branch predictors; (2) show how CNNs improve H2P prediction in SPEC 2017; (3) adapt 2-bit CNN inference to the constraints of current branch prediction units; and (4) establish that CNN helper predictors are reusable across application executions on different inputs, enabling us to amortize offline training and deploy ML pattern matching to improve IPC.