On iteration t:
         Compute dW, db on the current mini-batch
                $S_{dW} = \beta S_{dW} + (1-\beta) dW^2$
                $S_{db} = \beta S_{db} + (1-\beta) db^2$
                $W := W - \alpha \frac{dW}{\sqrt{S_{dW}}}, b := b - \alpha \frac{db}{\sqrt{S_{db}}}$


University of Michigan - Ann Arbor

- Stands for Root Mean Square Propagation
- RMSProp is an optimization algorithm closely related to AdaGrad, as both employ the square of the gradient to scale the update coefficients on a per-coordinate basis. However, RMSProp overcomes AdaGrad's tendency for radically diminishing learning rates by using a leaky (exponentially weighted) average of squared gradients rather than a cumulative sum.
- RMSProp also shares the leaky averaging mechanism with the momentum method, but applies it differently: whereas momentum uses leaky averaging to smooth the gradient direction, RMSProp uses the technique to adjust the coefficient-wise preconditioner that rescales the learning rate independently for each parameter.
- Because RMSProp does not automatically schedule the learning rate (unlike AdaGrad, whose learning rate decays implicitly through accumulation), the learning rate must be explicitly scheduled by the practitioner in practice.
- The decay coefficient $$\gamma$$ governs how long the gradient history is retained when adjusting the per-coordinate scale: a larger $$\gamma$$ produces a longer memory, while a smaller $$\gamma$$ makes the algorithm more responsive to recent gradients.

RMSprop (Deep Learning Optimization Algorithm)

```python
class RMSprop():
  
    # Object of this classes goes together with a layer parameters
    def __init__(self, input_dims, nodes):
  
      self.learning_rate = 0.01
      self.beta = 0.9

      # G matrx for parameters
      self.G_weights = np.zeros((input_dims, nodes))

      # G matrix for biases
      self.G_biases = np.zeros(nodes)


    # Function gets gradient of the weigths and biases
    # Returns the update that we need to substract from
    # the current weights and biases
    def get_steps(self, grad_weights, grad_biases):

      eps = 1e-8
      
      # updating G matrixes
      self.G_weights = self.beta * self.G_weights + (1 - self.beta) * np.power(grad_weights, 2)
      self.G_biases = self.beta * self.G_biases + (1 - self.beta) * np.power(grad_biases, 2)

      weights_step = np.multiply((self.learning_rate / np.sqrt(self.G_weights + eps)), grad_weights)
      biases_step = np.multiply((self.learning_rate / np.sqrt(self.G_biases + eps)), grad_biases)

      return weights_step, biases_step
```

RMSprop (Deep Learning Optimization Algorithm) Python implementation

- Adam is fast, but tends to overfit
- SGD is slow but gives great results
- RMSProp sometimes works best
- SWA can easily improve quality
- AdaTune magically improves the learning rate

Adam vs. SGD vs. RMSProp vs. SWA vs. AdaTune

RMSprop (Deep Learning Optimization Algorithm) Pseudocode

Adam stands for adaptive moment estimation.
It combines gradient descent with momentum, and RMSProp. It brings the benefits from both sides - adaptive learning rate and faster convergence with momentum.

Adam (Deep Learning Optimization Algorithm)

To visualize RMSProp's convergence behavior, the algorithm is applied to the two-dimensional quadratic function $$f(\mathbf{x}) = 0.1 x_1^2 + 2 x_2^2$$ with a learning rate of $$\eta = 0.4$$ and decay parameter $$\gamma = 0.9$$. The coordinate-wise implementation computes gradients $$g_1 = 0.2 x_1$$ and $$g_2 = 4 x_2$$, updates the leaky averages of squared gradients as $$s_i = \gamma s_i + (1 - \gamma) g_i^2$$, and adjusts each coordinate by $$x_i \leftarrow x_i - \frac{\eta}{\sqrt{s_i + \epsilon}} g_i$$. After $$20$$ epochs, the variables converge near the origin ($$x_1 \approx -0.0106$$, $$x_2 \approx 0$$). Unlike AdaGrad, which stalls in later iterations because the learning rate decreases too quickly, RMSProp maintains effective progress throughout training because $$\eta$$ is controlled independently from the state variable rescaling.

python
def rmsprop_2d(x1, x2, s1, s2):
    g1, g2, eps = 0.2 * x1, 4 * x2, 1e-6
    s1 = gamma * s1 + (1 - gamma) * g1 ** 2
    s2 = gamma * s2 + (1 - gamma) * g2 ** 2
    x1 -= eta / math.sqrt(s1 + eps) * g1
    x2 -= eta / math.sqrt(s2 + eps) * g2
    return x1, x2, s1, s2

eta, gamma = 0.4, 0.9


RMSProp Optimization Trajectory in 2D

A from-scratch implementation of the RMSProp optimizer for deep networks requires maintaining an auxiliary state variable for each parameter tensor, initialized to zeros with the same shape. During each update step, the state is updated as a leaky average of squared gradients: $$\mathbf{s} \leftarrow \gamma \mathbf{s} + (1 - \gamma) \mathbf{g}^2$$, where $$\gamma$$ is the decay factor and $$\mathbf{g}$$ is the current gradient. The parameter is then decremented by the learning rate times the gradient divided by the square root of the state plus a numerical stability constant ($$\epsilon = 10^{-6}$$). Finally, the parameter gradients are zeroed out.

In PyTorch, this can be implemented as follows:

python
def init_rmsprop_states(feature_dim):
    s_w = torch.zeros((feature_dim, 1))
    s_b = torch.zeros(1)
    return (s_w, s_b)

def rmsprop(params, states, hyperparams):
    gamma, eps = hyperparams['gamma'], 1e-6
    for p, s in zip(params, states):
        with torch.no_grad():
            s[:] = gamma * s + (1 - gamma) * torch.square(p.grad)
            p[:] -= hyperparams['lr'] * p.grad / torch.sqrt(s + eps)
        p.grad.data.zero_()


RMSProp Optimizer From-Scratch Implementation

When the weighting term $$\gamma$$ is set to $$0.9$$, the state variable $$\mathbf{s}$$ in RMSProp effectively aggregates information over the past $$\frac{1}{1 - \gamma} = \frac{1}{1 - 0.9} = 10$$ observations of the squared gradient. This applies the general exponentially weighted average principle—where a decay factor $$\beta$$ yields an effective window of $$\frac{1}{1 - \beta}$$—specifically to RMSProp's squared gradient state variable. The quantity $$\frac{1}{1 - \gamma}$$ defines the effective observation window: a larger $$\gamma$$ produces a longer memory (smoother average), while a smaller $$\gamma$$ makes the algorithm more responsive to recent gradients.

Effective Observation Window of RMSProp

The RMSProp update rule maintains a state variable $$G^{t}$$ that tracks the exponentially weighted average of squared gradients, and uses it to adaptively scale the learning rate:

$$G^{t} = \beta G^{t-1} + (1 - \beta) (
abla J(W^{t}))^2$$

$$W^{t} = W^{t-1} - \frac{\alpha}{\sqrt{G^{t} + \epsilon}} 
abla J(W^{t})$$

The same principle applies to the bias parameters.

- $$G^{t}$$: helper matrix for the algorithm
- $$\beta$$: the decay factor controlling how quickly the running average forgets old observations (typically around $$0.9$$)
- $$W^{t}$$: the model parameters
- $$\alpha$$: the initial learning rate (typically around $$0.01$$)
- $$\epsilon$$: a small constant to prevent division by zero (typically around $$10^{-6}$$ or $$10^{-8}$$)

RMSprop (Deep Learning Optimization Algorithm) Mathematical Implementations

Adadelta is an optimization algorithm that has no explicit learning rate parameter. Instead, it uses the rate of change in the parameters themselves to dynamically adapt the learning rate. To accomplish this, the algorithm utilizes two specific state variables: $$\mathbf{s}_t$$ to track a leaky average of the second moment of the gradient, and $$\Delta\mathbf{x}_t$$ to track a leaky average of the second moment of the model's parameter changes. The algorithm retains standard naming conventions for these variables to maintain consistency with similar optimization methods like momentum, AdaGrad, and RMSProp.

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