A square root learning rate scheduler can be implemented from scratch as a custom class that acts as a callable function. When invoked with the current number of updates, it calculates and returns the decayed learning rate using the formula $$\eta_0 (t + 1)^{-\frac{1}{2}}$$. The following Python code demonstrates this implementation:

```python
class SquareRootScheduler:
    def __init__(self, lr=0.1):
        self.lr = lr

    def __call__(self, num_update):
        return self.lr * pow(num_update + 1.0, -0.5)
```

Claude

A simple learning rate scheduler can be defined to decay the learning rate proportionally to the inverse square root of the number of updates. Specifically, at step $$t$$, the learning rate is set to $$\eta = \eta_0 (t + 1)^{-\frac{1}{2}}$$, where $$\eta_0$$ is the initial learning rate. This formulation ensures that the step size gently and continuously decreases as training progresses.

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