Francesco Parrella's Online Support Vector Regression PDF

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It is useful to save kernel function values in this way for computational reasons. In the next paragraph will be shown how, combining the results found until now, we can arrive to the definition of KKT conditions. 3. t. t. 3. 16) Like exposed above, this is an optimization problem with a convex domain. 3. 3. 4. 23) θi ∈ [0, C] θi = +C The Online Support Vector Regression algorithm main purpose is to verify these conditions after that a new sample is added. 24) (θi = +C ∧ h(xi ) ≤ − )} Remaining Set R = {i| θi = 0 ∧ |h(xi )| ≤ } The goal is to find a way to add a new sample to one of the three sets maintaining KKT conditions consistent.

13):       0 1 ··· 1   ∆h(xn1 ) ∆Qn1 c        1 Qn1 s1 · · · Qn1 sls  .. 38)  =  ∆θc +  .   .. ..     .   . .   ∆h(xnln ) ∆Qnln c 1 Qnln s1 · · · Qnln sls Now, γ can be defined as:     γ=   0 1 ··· 1  ∆Qn1 c   1 Qn1 s1 · · · Qn1 sls ..   + .  .. ...   ... .  ∆Qnln c 1 Qnln s1 · · · Qnln sls And rewriting the formula with new notations:   ∆h(xn1 )   ..     = γ∆θc . 40) In this way, ∆θi and ∆b values can be updated by computing β, and ∆(xi ) values can be updated by computing γ.

14 47 Stabilization Unfortunately, this algorithm produces a big number of instability errors due to an enormous number of floating-point operations. So it often happens that a sample exit from its set and does no more verify KKT conditions. To avoid this problem, there has been ideated a way to correct these samples and move back to its original sets. The basic idea is to use the same principles applied in normal algorithm and extend them with new possible moves permitted only when a sample exit from its set.

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Online Support Vector Regression by Francesco Parrella

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