A factor learning rate scheduler applies multiplicative decay to the learning rate while enforcing a lower-bound constraint, mathematically expressed as $\eta_{t+1} \leftarrow \mathop{\mathrm{max}}(\eta_{\mathrm{min}}, \eta_t \cdot \alpha)$. This can be implemented from scratch in Python by defining a custom FactorScheduler class. The __call__ method updates the learning rate at each step by multiplying the base learning rate by a specified decay factor, ensuring the rate does not fall below the minimum threshold (stop_factor_lr). Alternatively, deep learning frameworks like MXNet provide built-in solutions such as lr_scheduler.FactorScheduler, which offer additional hyperparameters like warmup periods and maximum update limits.

class FactorScheduler:
    def __init__(self, factor=1, stop_factor_lr=1e-7, base_lr=0.1):
        self.factor = factor
        self.stop_factor_lr = stop_factor_lr
        self.base_lr = base_lr

    def __call__(self, num_update):
        self.base_lr = max(self.stop_factor_lr, self.base_lr * self.factor)
        return self.base_lr