I have a large set of values (let's say 1M entries) where I need to apply an exponential smoothing algorithm, but only incrementing one value at a time (all others decay to zero). The trivial implementation would be (pseudo-code):

function smooth_once(index, x):
    // Decay everything
    for s in s_vec:
        s = s * (1 - alpha)
    // Only increase one value, for the other x=0
    s_vec[index] += alpha * x

function get_s(index):
    return s_vec[index]

This is rather slow, as every element of s_vec need to be scanned at every iteration, whereas only one need to be incremented. Also, I only need to check few values from s_vec from time to time (much less frequently than the frequency at which smooth_once is called).

Note: the smoothing and read sequence is unpredictable, and is usually interleaved, for example:

smooth_once(6573, 1.23)
smooth_once(3345, 2.45)
smooth_once(6874, 1.10)

In order to speed up this, I was thinking of multiplying the x value by a factor (scale) which is multiplied every time, and dividing everything when the factor grows too large. Basically, this boils down to rescale everything by an exponentially-incrementing factor.

scale = 1
k = 1 / (1 - alpha)
function optimized_smooth(index, x):
    // Rescale everything
    scale = scale * k
    if scale > scale_max:
        // Switch back scale to 1
        for s in s_vec:
            s = s / scale
        scale = 1
    s_vec[index] = v + alpha * x * scale

function get_s(index):
    // Scale back for reading
    return s_vec[index] / scale

IMHO this would be much faster (assuming k is not too large), as you only need to scan s_vec when the scale overflow and you rescale the whole array.

  • What do you think of this method?
  • Is this already known and has a name?
  • Apart from loosing a few bits of precision on the values (depending on beta_max), do you see any drawbacks?

Addendum. A small demo program shows that both algorithms end up with the same result on a controlled sequence of smoothing indexes.

  • 1
    It is not clear to me which problem you trying to solve. s_vec is modified in a specific manner by smooth_once. It seems smooth_once is called in a specific context which you did not describe. Doesn't that context need the intermediate values of the whole s_vec, as it was calculated by smooth_once?
    – Doc Brown
    May 17, 2016 at 12:34
  • smooth_once is called very frequently with unpredictable index and x. s_vec is "large" (think 1M entries), but I need to check its values only infrequently. Iterating 1M entries for each call to smooth_once is a performance killer. May 17, 2016 at 12:39

1 Answer 1


After thinking twice about your solution, I am sure that your algorithm works well. IMHO it could be seen as a variant of lazy evaluation. Here is another variant.

Change smooth_once so it only puts the pairs (index,x) into a queue (lets call the arrays indexes and xs).

function smooth_once(index, x):

Here is the method to calculcate the smoothing for the whole queue at once:

function nth_smooth(indexes, xs):
    n = size(indexes)  // equal to size(xs)
    if n==0: 
    cn = (1-alpha) ** n // (1-alpha) to the power of n
    for s in s_vec:
        s = s * cn
    for i = 0 to n-1:
       s_vec[indexes[i]] += alpha * xs[i] *c
       c = c * (1-alpha)

And here is the method to retrieve the result:

function get_s(index):
    return s_vec[index]

I guess it is not too hard to understand why this works, it simply replaces the repeated multiplications of s_vec with 1-alpha by one multiplication with (1-alpha) ** n, and adds the "x*alpha" additions afterwards, taking the influence of 1-alpha to those pertubances into account.

However, if this will be faster or slower than your own proposal depends on the number of calls of smooth_once before the next get_s occurs, compared to the frequency of having the condition scale > scale_max fulfilled (in your variant).

  • With this solution you can't read values of s_vec anytime during the smoothing sequence. Since my read/write sequence is not predictable this can't work. I'll update my question to add this context. May 18, 2016 at 7:21
  • 1
    @LaurentGrégoire: see my edit
    – Doc Brown
    May 18, 2016 at 13:18
  • 1
    Your other solution is really nice. I think picking the best method will also depend on alpha: for small alpha (k~=1) the "dynamic rescale" would work best as down-scaling will not happen often; for large alpha the "smooth Q" will perform better. In my specific case alpha is very low and the smooth/get pattern is interleaved a lot. May 18, 2016 at 13:49

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