I was reading about the Adam optimizer for Deep Learning and came across the following sentence in the new book Deep Learning by Begnio, Goodfellow and Courtville:

Adam includes bias corrections to the estimates of both the first-order moments (the momentum term) and the (uncentered) second-order moments to account for their initialization at the origin.

it seems that the main reason to include these bias correction terms is that somehow it removes the bias of the initialization of $m_t = 0$ and $v_t = 0$.

  • I am not 100% sure what that means but it seems to me that it probably means that the 1st and 2nd moment start at zero and somehow starting it off at zero slants the values closer to zero in an unfair (or useful) way for training?
  • Though I would love to know what that means a bit more precisely and how that damages the learning. In particular, what advantages does un-biasing the optimizer have in terms of optimization?
  • How does this help training deep learning models?
  • Also, what does it mean when it's unbiased? I am familiar what unbiased standard deviation means but it's not clear to me what it means in this context.
  • Is bias correction really a big deal or is that something overhyped in the Adam optimizer paper?

Just so people know I've tried really hard to understand the original paper but I've gotten very little out of reading and re-reading the original paper. I assume some of these question might be answered there but I can't seem to parse the answers.

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    $\begingroup$ Link:arxiv.org/pdf/1412.6980v8.pdf 1st and 2nd moment gradient estimates are updated via moving average, and started off with both estimat4es being zero, hence those initial values for which the true value is not zero, would bias the results, because the initial estimates of zero only gradually die out. What I don't understand is why the gradient at initial point is not used for the initial values of these things, and then the first parameter update. Then there would be no contamination by the initial zero values, which has to be undone.So there'd be no need for the bias correction. $\endgroup$ – Mark L. Stone Aug 31 '16 at 21:45
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    $\begingroup$ So it appears that rather than having special code for the initial iteration, the authors have decided to do a mathematically equivalent thing by introducing what would be a bias, and then undoing it. This adds unnecessary, though fast, calculations on all iterations. Bt doing this, they have maintained a purity of code that looks the same at all iterations. I would have just started with the first gradient evaluation instead, and have the gradient moment update formula only start on the 2nd iteration. $\endgroup$ – Mark L. Stone Aug 31 '16 at 21:48
  • $\begingroup$ @MarkL.Stone the authors emphasize so much the bias correction that it seemed to me that was what was novel or important in their paper. So they could have just not "corrected the bias" and have the same algorithm? If that is true I fail to see why adam is such an important optimizer or what the big deal is. I always thought it was the bias correction. $\endgroup$ – Charlie Parker Aug 31 '16 at 22:18
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    $\begingroup$ They introduce a bias and then correct it out, for no good reason apparent to me. It's like multiplying by 2 (oh my, the result is biased), and then dividing by 2 to "correct" it. The whole thing with the bias introduction and removal seems like an unnecessary sideshow. Maybe the paper wasn't long enough without doing it, so they added this spiel to make it longer :) Adam may have its merits, but they would have been the same doing it the way I proposed. I'd love for the authors to come on here and explain it though. Maybe I'm missing some subtle point or misunderstanding something. $\endgroup$ – Mark L. Stone Aug 31 '16 at 22:42

The problem of NOT correcting the bias
According to the paper

In case of sparse gradients, for a reliable estimate of the second moment one needs to average over many gradients by chosing a small value of β2; however it is exactly this case of small β2 where a lack of initialisation bias correction would lead to initial steps that are much larger.

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Normally in practice $\beta_2$ is set much closer to 1 than $\beta_1$ (as suggested by the author $\beta_2=0.999$, $\beta_1=0.9$), so the update coefficients $1-\beta_2=0.001$ is much smaller than $1-\beta_1=0.1$.

In the first step of training $m_1=0.1g_t$, $v_1=0.001g_t^2$, the $m_1/(\sqrt{v_1}+\epsilon)$ term in the parameter update can be very large if we use the biased estimation directly.

On the other hand when using the bias-corrected estimation, $\hat{m_1}=g_1$ and $\hat{v_1}=g_1^2$, the $\hat{m_t}/(\sqrt{\hat{v_t}}+\epsilon)$ term becomes less sensitive to $\beta_1$ and $\beta_2$.

How the bias is corrected
The algorithm uses moving average to estimate the first and second moments. The biased estimation would be, we start at an arbitrary guess $m_0$, and update the estimation gradually by $m_t=\beta m_{t-1}+(1-\beta)g_t$. So it's obvious in the first few steps our moving average is heavily biased towards the initial $m_0$.

To correct this, we can remove the effect of the initial guess (bias) out of the moving average. For example at time 1, $m_1=\beta m_0+(1-\beta)g_t$, we take out the $\beta m_0$ term from $m_1$ and divide it by $(1-\beta)$, which yields $\hat{m_1}=(m_1- \beta m_0)/(1-\beta)$. When $m_0=0$, $\hat{m_t}=m_t/(1-\beta^t)$. The full proof is given in Section 3 of the paper.

As Mark L. Stone has well commented

It's like multiplying by 2 (oh my, the result is biased), and then dividing by 2 to "correct" it.

Somehow this is not exactly equivalent to

the gradient at initial point is used for the initial values of these things, and then the first parameter update

(of course it can be turned into the same form by changing the update rule (see the update of the answer), and I believe this line mainly aims at showing the unnecessity of introducing the bias, but perhaps it's worth noticing the difference)

For example, the corrected first moment at time 2

$$\hat{m_2}=\frac{\beta(1-\beta)g_1+(1-\beta)g_2}{1-\beta^2}=\frac{\beta g_1+g_2}{\beta+1}$$

If using $g_1$ as the initial value with the same update rule, $$m_2=\beta g_1+(1-\beta)g_2$$ which will bias towards $g_1$ instead in the first few steps.

Is bias correction really a big deal
Since it only actually affects the first few steps of training, it seems not a very big issue, in many popular frameworks (e.g. keras, caffe) only the biased estimation is implemented.

From my experience the biased estimation sometimes leads to undesirable situations where the loss won't go down (I haven't thoroughly tested that so I'm not exactly sure whether this is due to the biased estimation or something else), and a trick that I use is using a larger $\epsilon$ to moderate the initial step sizes.

If you unfold the recursive update rules, essentially $\hat{m}_t$ is a weighted average of the gradients,
$$\hat{m}_t=\frac{\beta^{t-1}g_1+\beta^{t-2}g_2+...+g_t}{\beta^{t-1}+\beta^{t-2}+...+1}$$ The denominator can be computed by the geometric sum formula, so it's equivalent to following update rule (which doesn't involve a bias term)

$m_1\leftarrow g_1$
while not converge do
$\qquad m_t\leftarrow \beta m_t + g_t$ (weighted sum)
$\qquad \hat{m}_t\leftarrow \dfrac{(1-\beta)m_t}{1-\beta^t}$ (weighted average)

Therefore it can be possibly done without introducing a bias term and correcting it. I think the paper put it in the bias-correction form for the convenience of comparing with other algorithms (e.g. RmsProp).

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    $\begingroup$ Do you agree with my 2nd comment on the question? To me, that's the bottom line. The thing about the multiplying and dividing by 2 was just supposed to be an "easier to understand" analogy, not the math used in the matter at hand. if there were other papers, which i haven't looked at, which introduced a bias by the same mechanism, which in the case of ADAM seems entirely avoidable, but didn't correct it, then that's just totally STUPID (unless somehow the bias helped the performance of the algorithm). $\endgroup$ – Mark L. Stone Sep 13 '16 at 15:55
  • $\begingroup$ @MarkL.Stone yes! actually i upvoted it, sorry about my English. and i think the algorithm that didn't correct the bias is the rmsprop, but unlike adam rmsprop works fine with the bias. $\endgroup$ – dontloo Sep 13 '16 at 16:48
  • $\begingroup$ @dontloo does your answer address Mark L. Stone's comment on why the bias correction seems superfluous? (That I think is quite important, maybe even more than paraphrasing what the original paper says). $\endgroup$ – Charlie Parker Sep 20 '16 at 1:20
  • $\begingroup$ @CharlieParker you mean why the bias correction is superfluous or why the author makes it look superfluous? $\endgroup$ – dontloo Sep 20 '16 at 2:09
  • $\begingroup$ @dontloo I don't think the authors make it superfluous. I thought they did indeed need it (given their specific conditions). However, I thought that it wasn't necessary given Mark's suggestion. I guess my question in the comments section right now is if they really needed the correction term. $\endgroup$ – Charlie Parker Sep 20 '16 at 16:19

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