TheAlgorithms
diff --git a/‎machine_learning/rmsprop.py‎
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+"""
+RMSprop (Root Mean Square Propagation) optimizer implementation.
+
+RMSprop is an adaptive learning rate optimizer that maintains a moving
+average of squared gradients to normalize the gradient. It was proposed
+by Geoffrey Hinton in his Coursera course on Neural Networks.
+
+Key idea: Instead of using a fixed learning rate, RMSprop adapts the
+learning rate for each parameter by dividing by a running average of
+recent gradient magnitudes.
+
+Update rules:
+    v(t) = rho * v(t-1) + (1 - rho) * gradient^2
+    param = param - (learning_rate / sqrt(v(t) + epsilon)) * gradient
+
+Where:
+    v(t)          = moving average of squared gradients
+    rho           = decay factor (typically 0.9)
+    learning_rate = step size
+    epsilon       = small value to avoid division by zero
+
+Reference: https://en.wikipedia.org/wiki/Stochastic_gradient_descent#RMSProp
+
+>>> rmsprop([0.0], [0.1], 0.01)  # doctest: +ELLIPSIS
+[-0.031...]
+>>> rmsprop([1.0, -1.0], [0.5, -0.5], 0.01)  # doctest: +ELLIPSIS
+[0.968..., -0.968...]
+"""
+
+import math
+
+
+def rmsprop(
+    params: list[float],
+    gradients: list[float],
+    learning_rate: float,
+    rho: float = 0.9,
+    epsilon: float = 1e-8,
+    moving_avg: list[float] | None = None,
+) -> list[float]:
+    """
+    Perform one step of the RMSprop optimization algorithm.
+
+    :param params: Current parameter values to be updated.
+    :param gradients: Gradients of the loss with respect to each parameter.
+    :param learning_rate: Step size for the update (must be positive).
+    :param rho: Decay factor for the moving average (default 0.9).
+    :param epsilon: Small constant to avoid division by zero (default 1e-8).
+    :param moving_avg: Running average of squared gradients. Updated in
+                       place each call. Initialized to zeros if not provided.
+    :return: Updated parameter values after one RMSprop step.
+
+    :raises ValueError: If params and gradients have different lengths.
+    :raises ValueError: If learning_rate, rho, or epsilon are out of range.
+
+    >>> rmsprop([0.0], [0.0], 0.01)
+    [0.0]
+    >>> rmsprop([1.0], [0.0], 0.01)
+    [1.0]
+    >>> len(rmsprop([1.0, 2.0, 3.0], [0.1, 0.2, 0.3], 0.01)) == 3
+    True
+    >>> rmsprop([1.0], [0.5], learning_rate=0.01)  # doctest: +ELLIPSIS
+    [0.968...]
+    """
+    if len(params) != len(gradients):
+        msg = (
+            f"params and gradients must have the same length, "
+            f"got {len(params)} and {len(gradients)}"
+        )
+        raise ValueError(msg)
+    if learning_rate <= 0:
+        msg = f"learning_rate must be positive, got {learning_rate}"
+        raise ValueError(msg)
+    if not 0.0 <= rho < 1.0:
+        msg = f"rho must be in [0, 1), got {rho}"
+        raise ValueError(msg)
+    if epsilon <= 0:
+        msg = f"epsilon must be positive, got {epsilon}"
+        raise ValueError(msg)
+
+    # Initialize moving average of squared gradients to zeros
+    if moving_avg is None:
+        moving_avg = [0.0] * len(params)
+
+    updated_params = []
+    for i, (param, grad) in enumerate(zip(params, gradients)):
+        # Update moving average IN PLACE so state persists across calls
+        moving_avg[i] = rho * moving_avg[i] + (1 - rho) * grad**2
+        # Compute adaptive update
+        param = param - (learning_rate / math.sqrt(moving_avg[i] + epsilon)) * grad
+        updated_params.append(param)
+
+    return updated_params
+
+
+if __name__ == "__main__":
+    import doctest
+
+    doctest.testmod()
+
+    print("RMSprop Optimizer Demo")
+    print("=" * 40)
+    print("Minimizing f(x) = x^2  (minimum at x = 0)")
+    print("Gradient: f'(x) = 2x | Learning rate: 0.1\n")
+
+    param = [5.0]
+    moving_avg = [0.0]
+
+    print(f"{'Step':>6} | {'param':>12} | {'f(x)':>12}")
+    print("-" * 38)
+    print(f"{'0':>6} | {param[0]:>12.6f} | {param[0] ** 2:>12.6f}")
+
+    for step in range(1, 101):
+        gradient = [2 * param[0]]
+        param = rmsprop(param, gradient, learning_rate=0.1, moving_avg=moving_avg)
+        if step % 20 == 0:
+            print(f"{step:>6} | {param[0]:>12.6f} | {param[0] ** 2:>12.8f}")
+
+    print(f"\nConverged to x = {param[0]:.8f}  (true minimum = 0.0)")