# 11.8. RMSProp¶ Open the notebook in Colab Open the notebook in Colab Open the notebook in Colab

One of the key issues in Section 11.7 is that the learning rate decreases at a predefined schedule of effectively $$\mathcal{O}(t^{-\frac{1}{2}})$$. While this is generally appropriate for convex problems, it might not be ideal for nonconvex ones, such as those encountered in deep learning. Yet, the coordinate-wise adaptivity of Adagrad is highly desirable as a preconditioner.

[Tieleman & Hinton, 2012] proposed the RMSProp algorithm as a simple fix to decouple rate scheduling from coordinate-adaptive learning rates. The issue is that Adagrad accumulates the squares of the gradient $$\mathbf{g}_t$$ into a state vector $$\mathbf{s}_t = \mathbf{s}_{t-1} + \mathbf{g}_t^2$$. As a result $$\mathbf{s}_t$$ keeps on growing without bound due to the lack of normalization, essentially linearly as the algorithm converges.

One way of fixing this problem would be to use $$\mathbf{s}_t / t$$. For reasonable distributions of $$\mathbf{g}_t$$ this will converge. Unfortunately it might take a very long time until the limit behavior starts to matter since the procedure remembers the full trajectory of values. An alternative is to use a leaky average in the same way we used in the momentum method, i.e., $$\mathbf{s}_t \leftarrow \gamma \mathbf{s}_{t-1} + (1-\gamma) \mathbf{g}_t^2$$ for some parameter $$\gamma > 0$$. Keeping all other parts unchanged yields RMSProp.

## 11.8.1. The Algorithm¶

Let us write out the equations in detail.

(11.8.1)\begin{split}\begin{aligned} \mathbf{s}_t & \leftarrow \gamma \mathbf{s}_{t-1} + (1 - \gamma) \mathbf{g}_t^2, \\ \mathbf{x}_t & \leftarrow \mathbf{x}_{t-1} - \frac{\eta}{\sqrt{\mathbf{s}_t + \epsilon}} \odot \mathbf{g}_t. \end{aligned}\end{split}

The constant $$\epsilon > 0$$ is typically set to $$10^{-6}$$ to ensure that we do not suffer from division by zero or overly large step sizes. Given this expansion we are now free to control the learning rate $$\eta$$ independently of the scaling that is applied on a per-coordinate basis. In terms of leaky averages we can apply the same reasoning as previously applied in the case of the momentum method. Expanding the definition of $$\mathbf{s}_t$$ yields

(11.8.2)\begin{split}\begin{aligned} \mathbf{s}_t & = (1 - \gamma) \mathbf{g}_t^2 + \gamma \mathbf{s}_{t-1} \\ & = (1 - \gamma) \left(\mathbf{g}_t^2 + \gamma \mathbf{g}_{t-1}^2 + \gamma^2 \mathbf{g}_{t-2} + \ldots, \right). \end{aligned}\end{split}

As before in Section 11.6 we use $$1 + \gamma + \gamma^2 + \ldots, = \frac{1}{1-\gamma}$$. Hence the sum of weights is normalized to $$1$$ with a half-life time of an observation of $$\gamma^{-1}$$. Let us visualize the weights for the past 40 time steps for various choices of $$\gamma$$.

%matplotlib inline
from d2l import mxnet as d2l
import math
from mxnet import np, npx

npx.set_np()

d2l.set_figsize()
gammas = [0.95, 0.9, 0.8, 0.7]
for gamma in gammas:
x = d2l.numpy(np.arange(40))
d2l.plt.plot(x, (1-gamma) * gamma ** x, label=f'gamma = {gamma:.2f}')
d2l.plt.xlabel('time'); from d2l import torch as d2l
import torch
import math

d2l.set_figsize()
gammas = [0.95, 0.9, 0.8, 0.7]
for gamma in gammas:
x = d2l.numpy(torch.arange(40))
d2l.plt.plot(x, (1-gamma) * gamma ** x, label=f'gamma = {gamma:.2f}')
d2l.plt.xlabel('time'); from d2l import tensorflow as d2l
import tensorflow as tf
import math

d2l.set_figsize()
gammas = [0.95, 0.9, 0.8, 0.7]
for gamma in gammas:
x = d2l.numpy(tf.range(40))
d2l.plt.plot(x, (1-gamma) * gamma ** x, label=f'gamma = {gamma:.2f}')
d2l.plt.xlabel('time'); ## 11.8.2. Implementation from Scratch¶

As before we use the quadratic function $$f(\mathbf{x})=0.1x_1^2+2x_2^2$$ to observe the trajectory of RMSProp. Recall that in Section 11.7, when we used Adagrad with a learning rate of 0.4, the variables moved only very slowly in the later stages of the algorithm since the learning rate decreased too quickly. Since $$\eta$$ is controlled separately this does not happen with RMSProp.

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

def f_2d(x1, x2):
return 0.1 * x1 ** 2 + 2 * x2 ** 2

eta, gamma = 0.4, 0.9
d2l.show_trace_2d(f_2d, d2l.train_2d(rmsprop_2d)) 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

def f_2d(x1, x2):
return 0.1 * x1 ** 2 + 2 * x2 ** 2

eta, gamma = 0.4, 0.9
d2l.show_trace_2d(f_2d, d2l.train_2d(rmsprop_2d)) 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

def f_2d(x1, x2):
return 0.1 * x1 ** 2 + 2 * x2 ** 2

eta, gamma = 0.4, 0.9
d2l.show_trace_2d(f_2d, d2l.train_2d(rmsprop_2d)) Next, we implement RMSProp to be used in a deep network. This is equally straightforward.

def init_rmsprop_states(feature_dim):
s_w = np.zeros((feature_dim, 1))
s_b = np.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):
s[:] = gamma * s + (1 - gamma) * np.square(p.grad)
p[:] -= hyperparams['lr'] * p.grad / np.sqrt(s + eps)

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):
s[:] = gamma * s + (1 - gamma) * torch.square(p.grad)
p[:] -= hyperparams['lr'] * p.grad / torch.sqrt(s + eps)

def init_rmsprop_states(feature_dim):
s_w = tf.Variable(tf.zeros((feature_dim, 1)))
s_b = tf.Variable(tf.zeros(1))
return (s_w, s_b)

gamma, eps = hyperparams['gamma'], 1e-6
for p, s, g in zip(params, states, grads):
s[:].assign(gamma * s + (1 - gamma) * tf.math.square(g))
p[:].assign(p - hyperparams['lr'] * g / tf.math.sqrt(s + eps))


We set the initial learning rate to 0.01 and the weighting term $$\gamma$$ to 0.9. That is, $$\mathbf{s}$$ aggregates on average over the past $$1/(1-\gamma) = 10$$ observations of the square gradient.

data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(rmsprop, init_rmsprop_states(feature_dim),
{'lr': 0.01, 'gamma': 0.9}, data_iter, feature_dim);

loss: 0.245, 0.078 sec/epoch data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(rmsprop, init_rmsprop_states(feature_dim),
{'lr': 0.01, 'gamma': 0.9}, data_iter, feature_dim);

loss: 0.245, 0.014 sec/epoch data_iter, feature_dim = d2l.get_data_ch11(batch_size=10)
d2l.train_ch11(rmsprop, init_rmsprop_states(feature_dim),
{'lr': 0.01, 'gamma': 0.9}, data_iter, feature_dim);

loss: 0.244, 0.125 sec/epoch ## 11.8.3. Concise Implementation¶

Since RMSProp is a rather popular algorithm it is also available in the Trainer instance. All we need to do is instantiate it using an algorithm named rmsprop, assigning $$\gamma$$ to the parameter gamma1.

d2l.train_concise_ch11('rmsprop', {'learning_rate': 0.01, 'gamma1': 0.9},
data_iter)

loss: 0.243, 0.039 sec/epoch trainer = torch.optim.RMSprop
d2l.train_concise_ch11(trainer, {'lr': 0.01, 'alpha': 0.9},
data_iter)

loss: 0.249, 0.011 sec/epoch trainer = tf.keras.optimizers.RMSprop
d2l.train_concise_ch11(trainer, {'learning_rate': 0.01, 'rho': 0.9},
data_iter)

loss: 0.250, 0.125 sec/epoch ## 11.8.4. Summary¶

• RMSProp is very similar to Adagrad insofar as both use the square of the gradient to scale coefficients.

• RMSProp shares with momentum the leaky averaging. However, RMSProp uses the technique to adjust the coefficient-wise preconditioner.

• The learning rate needs to be scheduled by the experimenter in practice.

• The coefficient $$\gamma$$ determines how long the history is when adjusting the per-coordinate scale.

## 11.8.5. Exercises¶

1. What happens experimentally if we set $$\gamma = 1$$? Why?

2. Rotate the optimization problem to minimize $$f(\mathbf{x}) = 0.1 (x_1 + x_2)^2 + 2 (x_1 - x_2)^2$$. What happens to the convergence?

3. Try out what happens to RMSProp on a real machine learning problem, such as training on Fashion-MNIST. Experiment with different choices for adjusting the learning rate.

4. Would you want to adjust $$\gamma$$ as optimization progresses? How sensitive is RMSProp to this?