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I've been implementing RMSprop following this helpful blog post. The post doesn't talk about weight decay, i.e. regularization. What I'm implementing is effectively a ridge penalty.

The RMSprop update is defined as $$ \theta_{t+1} = \theta_t - \frac{\eta}{\sqrt{E[g^2]_t+\epsilon}}g_t $$ where $\eta$ is the learning rate, $E[g^2]$ is the RMSprop running average of the past squared gradients, $\epsilon$ is the don't-divide-by-zero fudge factor, and $g_t$ is the gradient.

Now, for normal SGD with weight decay, I would have $$ \theta_{t+1} = \theta_t - \eta (g_t + 2\lambda\theta_t) $$ For RMSprop, I first did

$$ \theta_{t+1} = \theta_t - \frac{\eta}{\sqrt{E[g^2]_t+\epsilon}}(g_t+2\lambda\theta_t) $$ That didn't work very well. MSE at convergence was essentially insensitive to the penalty factor. Without a whole lot of theoretical justification, I tried $$ \theta_{t+1} = \theta_t - \frac{\eta}{\sqrt{E[g^2]_t+\epsilon}}g_t+2\eta\lambda\theta_t $$ ...which worked a lot better.

My questions:

  1. Why did this work better? I guess it is because you don't want to adaptively penalize, but you do want to adaptively change the learning rate. Adaptively penalizing would basically shrink the ridge penalty with the step size.
  2. Is there a better to regularize in the context of RMSprop?
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  • $\begingroup$ Any penalty-based regularization works by adding a term to the objective function. So why not just include your penalty term in the objective function? (Then the gradient, however it is calculated, will incorporate the regularization.) $\endgroup$ – GeoMatt22 Nov 2 '16 at 23:16
  • $\begingroup$ Well, the question is more about steps sizes in various dimensions, rather than what the gradient is. RMSprop is an approach to differing step sizes in different directions. The objective function has nothing to do with the step size. $\endgroup$ – generic_user Nov 2 '16 at 23:44
  • $\begingroup$ In an optimization sense, regularization via a penalty term changes the problem to be solved (i.e. the objective function). To my understanding SGD, including variants such as RMSprop, are (approximate) solvers which are applied to a given problem. To put it another way: a penalty term should be "felt" by any solver type you wish to try (e.g. SGD, SQP, DFO, etc.). Same goes for line search/step-size strategies. $\endgroup$ – GeoMatt22 Nov 3 '16 at 0:10
  • $\begingroup$ Another way to think of it: If your update gives $\theta_{t+1}=\theta_t$, then the gradient of your "effective objective function" is zero, i.e. a fixed point of your "gradient descent" defines a local optimum. So you can use this to infer what the "effective penalty term" (and relative weights, "ridge parameter") are that correspond to your different variants. $\endgroup$ – GeoMatt22 Nov 3 '16 at 0:24
  • $\begingroup$ I put some thoughts into an answer below. Your method seems to have some similarities to Levenberg-Marquardt, but with a "stochastic" flavor in the sense of using limited data (i.e. vs. "batch" mode). $\endgroup$ – GeoMatt22 Nov 3 '16 at 1:55
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For straight SGD, you have $$\theta_{t+1} = \theta_t - \eta g_t$$ and $\eta$ is the "learning rate" (a.k.a. step size).

In principle any gradient descent, including SGD, is trying to find a stationary point of the objective function $f[\theta]$, where $g=\partial_{\theta}f$ (and hopefully a local minimum, rather than a maximum or saddle). It is easy to see that $$g_t=0 \implies \theta_{t+1} = \theta_t$$ so the SGD update is consistent with this (as is the RMSprop update you cite).

When you give the regularized update equation $$\theta_{t+1} = \theta_t - \eta (g_t + 2\lambda\theta_t)$$ notice how $$\theta_{t+1} = \theta_t \implies g_t + 2\lambda\theta_t = 0$$ So the stationary point no longer corresponds to $g=0$, but rather corresponds to $$\hat{g} = g + 2\lambda\theta = \partial_{\theta}(f+\lambda\theta^2) = 0$$ i.e. the gradient of the regularized objective function, which includes a penalty term. (This also applies to the first "regularized RMSprop update" formula you give.)

Your "adjusted RMSprop" update equation $$\theta_{t+1} = \theta_t - \frac{\eta}{\sqrt{E[g^2]_t+\epsilon}}g_t+2\eta\lambda\theta_t$$ corresponds to a stationary point $$\theta_{t+1} = \theta_t \implies g_t - 2\left(\lambda\sqrt{E[g^2]_t+\epsilon}\right)\theta_t = 0$$ This shows that the update does not correspond to any consistent objective function. Rather, it corresponds to an "evolving objective function" where the effective regularization weight $\hat{\lambda}$ changes through time, and depends on the path the optimization takes, i.e. $E[g^2]$. (Note: It appears you have a sign change in the last formula ... did you mean to have a $-\lambda$ perhaps?)

Most "momentum" techniques will try to preserve the stationary points of the objective function (which may include penalty terms). For your question 2, I would say the standard approach is simply to add the penalty term to the objective function, so that it shows up in the gradient $g$ automatically (and then RMSprop, or whatever method, will incorporate it into $E[g^2]$).

For your question 1, I would say that you are changing the penalty, so it is definitely not standard penalty-term regularization (which would change the objective function). It actually appears more similar to the Levenberg-Marquardt algorithm for nonlinear least squares, in that the "regularization" goes to zero as $E[g^2]$ goes to zero. (However there, I believe the averaging would always be over "all the data", so not path dependent.)

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  • $\begingroup$ Thanks for the insight. +1. This actually helped me figure out a bug in my RMSProp implementation. I was accumulating the squared energy before regularization so of course the gradient updates didn't make sense. $\endgroup$ – rayryeng Feb 7 '17 at 21:55

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