You double the batch size on a training run. What do you do to the learning rate, and why?

Scale the learning rate up, roughly linearly with batch size for moderate increases, or by the square root under some analyses. A larger batch gives a lower variance gradient estimate, so you can afford a bigger step. In practice you also need warmup, since the linear rule breaks down early in training, and very large batches eventually stop improving generalization.

Default is linear scaling: double the batch, double the learning rate. The logic is that the gradient you compute on a batch is an estimate of the true gradient, and its variance falls with batch size, so a bigger batch gives you a more trustworthy direction and you can take a proportionally bigger step in it. What people skip is warmup. Right at the start the weights are random and the loss surface is not friendly, so a large learning rate applied immediately after a batch size jump will blow the run up in the first few hundred steps. So I ramp linearly from near zero to the target over a warmup window, then decay. Beyond a certain batch size the rule stops paying, both because the gradient estimate is already low variance so you gain nothing, and because you take far fewer optimizer steps per epoch, which seems to hurt generalization. I have hit that ceiling around a few thousand samples per batch on the models I have worked on. So my routine after any batch size change is scale, warm up, then watch the loss curve for the first thousand steps rather than walking away.