:label:sec_rmsprop
One of the key issues in :numref:sec_adagrad
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.
:cite: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 linarly 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.
Let us write out the equations in detail.
$$\begin{aligned} \mathbf{s}_t & \leftarrow \gamma \mathbf{s}_{t-1} + (1 - \gamma) \mathbf{g}_t^2, \\ \mathbf{x}_t & \leftarrow \mathbf{x}_{t-1} - \frac{\eta}{\sqrt{\mathbf{s}_t + \epsilon}} \odot \mathbf{g}_t. \end{aligned}$$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
$$ \begin{aligned} \mathbf{s}_t & = (1 - \gamma) \mathbf{g}_t^2 + \gamma \mathbf{s}_{t-1} \\ & = (1 - \gamma) \left(\mathbf{g}_t^2 + \gamma \mathbf{g}_{t-1}^2 + \gamma^2 \mathbf{g}_{t-2} + \ldots, \right). \end{aligned} $$As before in :numref:sec_momentum
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 timesteps for various choices of $\gamma$.
%load ../utils/djl-imports
%load ../utils/plot-utils
%load ../utils/Functions.java
%load ../utils/GradDescUtils.java
%load ../utils/Accumulator.java
%load ../utils/StopWatch.java
%load ../utils/Training.java
%load ../utils/TrainingChapter11.java
NDManager manager = NDManager.newBaseManager();
float[] gammas = new float[]{0.95f, 0.9f, 0.8f, 0.7f};
NDArray timesND = manager.arange(40f);
float[] times = timesND.toFloatArray();
display(GradDescUtils.plotGammas(times, gammas, 600, 400));
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 :numref:sec_adagrad
, 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.
float eta = 0.4f;
float gamma = 0.9f;
Function<Float[], Float[]> rmsProp2d = (state) -> {
Float x1 = state[0], x2 = state[1], s1 = state[2], s2 = state[3];
float g1 = 0.2f * x1;
float g2 = 4 * x2;
float eps = (float) 1e-6;
s1 = gamma * s1 + (1 - gamma) * g1 * g1;
s2 = gamma * s2 + (1 - gamma) * g2 * g2;
x1 -= eta / (float) Math.sqrt(s1 + eps) * g1;
x2 -= eta / (float) Math.sqrt(s2 + eps) * g2;
return new Float[]{x1, x2, s1, s2};
};
BiFunction<Float, Float, Float> f2d = (x1, x2) -> {
return 0.1f * x1 * x1 + 2 * x2 * x2;
};
GradDescUtils.showTrace2d(f2d, GradDescUtils.train2d(rmsProp2d, 20));
Next, we implement RMSProp to be used in a deep network. This is equally straightforward.
NDList initRmsPropStates(int featureDimension) {
NDManager manager = NDManager.newBaseManager();
NDArray sW = manager.zeros(new Shape(featureDimension, 1));
NDArray sB = manager.zeros(new Shape(1));
return new NDList(sW, sB);
}
public class Optimization {
public static void rmsProp(NDList params, NDList states, Map<String, Float> hyperparams) {
float gamma = hyperparams.get("gamma");
float eps = (float) 1e-6;
for (int i = 0; i < params.size(); i++) {
NDArray param = params.get(i);
NDArray state = states.get(i);
// Update parameter and state
// state = gamma * state + (1 - gamma) * param.gradient^(1/2)
state.muli(gamma).addi(param.getGradient().square().mul(1 - gamma));
// param -= lr * param.gradient / sqrt(s + eps)
param.subi(param.getGradient().mul(hyperparams.get("lr")).div(state.add(eps).sqrt()));
}
}
}
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.
AirfoilRandomAccess airfoil = TrainingChapter11.getDataCh11(10, 1500);
public TrainingChapter11.LossTime trainRmsProp(float lr, float gamma, int numEpochs)
throws IOException, TranslateException {
int featureDimension = airfoil.getColumnNames().size();
Map<String, Float> hyperparams = new HashMap<>();
hyperparams.put("lr", lr);
hyperparams.put("gamma", gamma);
return TrainingChapter11.trainCh11(Optimization::rmsProp,
initRmsPropStates(featureDimension),
hyperparams, airfoil,
featureDimension, numEpochs);
}
trainRmsProp(0.01f, 0.9f, 2);
Since RMSProp is a rather popular algorithm it is also available in Optimizer
. We create an instance of RmsProp
and set its learning rate and optional gamma1
parameter.
Tracker lrt = Tracker.fixed(0.01f);
Optimizer rmsProp = Optimizer.rmsprop().optLearningRateTracker(lrt).optRho(0.9f).build();
TrainingChapter11.trainConciseCh11(rmsProp, airfoil, 2);