RMSprop optimizer

Background

RMSprop fixes Adagrad's vanishing learning rate by tracking an exponential moving average of squared gradients instead of their cumulative sum. The accumulator can shrink as well as grow, so the per-parameter step size adapts to recent gradient magnitude — a robust default for RNNs and online settings.

Problem statement

Implement rmsprop_optimizer(parameter, grad, G, learning_rate=0.01, beta=0.9, epsilon=1e-8) for one update step:

$$G \leftarrow \beta G + (1-\beta)g^2, \qquad \theta \leftarrow \theta - \frac{\eta\, g}{\sqrt{G} + \epsilon}$$

Return the updated parameter and accumulator, rounded to 5 decimals.

Input

• parameter, grad, G — current value(s), gradient, and EMA accumulator (same shape; G starts at 0).
• learning_rate — float, $\eta$.
• beta — float, the EMA decay (default 0.9).
• epsilon — float.

Output

Returns (updated_parameter, updated_G), each rounded to 5 decimals.

Examples

Example 1

Input:  parameter = 1.0, grad = 0.1, G = 1.0, learning_rate = 0.01, beta = 0.9
Output: (0.99895, 0.901)

Explanation: $G = 0.9(1.0) + 0.1(0.1^2) = 0.901$; the step is $0.01\cdot0.1/\sqrt{0.901}\approx0.001054$, so $\theta \approx 0.99895$.

Constraints

• Update $G$ as an EMA ($\beta G + (1-\beta)g^2$) before computing the step.
• Step $= \eta g/(\sqrt{G}+\epsilon)$.
• Round both outputs to 5 decimals.

Notes

• Versus Adagrad: replacing the cumulative sum with an EMA lets $G$ forget old gradients, so the effective rate doesn't decay to zero on long runs.
• Adam is essentially RMSprop's second-moment EMA plus a first-moment (momentum) EMA with bias correction.