LFD Book Forum  

Go Back   LFD Book Forum > Book Feedback - Learning From Data > Chapter 4 - Overfitting

Thread Tools Display Modes
Prev Previous Post   Next Post Next
Old 05-10-2013, 05:21 AM
Elroch Elroch is offline
Invited Guest
Join Date: Mar 2013
Posts: 143
Default Zeroth regularization coefficient

There is clearly a lot of scope for varying the form of cost function, as demonstrated by Tikhonov's work (which I have only a passing familiarity with so far. It is neat to note that this is the same guy who has separation axioms named after him in topology ).

But here the question I have is a much simpler one. Elsewhere I have seen a similar one parameter regularization to the one we have used for linear and polynomial hypotheses, with the single exception that the zeroth term is omitted, so only higher order parameters are penalised. The question is whether this is an improvement in some quantifiable sense. The mathematics is only marginally less simple, with the term \lambda I being replaced by the same matrix with the top left term set to zero.

As the simplest example, if our hypothesis set consists just of all constant functions, a single regularization parameter has the effect of replacing the equation:

\theta_0 = \overline { \{y^{(i)}\}_{i\leq N}}

by the equation

\theta_0 = \overline {\left\{{y^{(i)} \over{1 + \lambda}}\right\}_{i\leq N}}

This is systematically underestimating the absolute value of the mean in an interesting but not easily justified way.

The question is whether when the function is more complex, the zeroth regularisation coefficient makes any more sense.

Eg, suppose the hypothesis set is all quadratics without a linear term, i.e. H(\theta_0,\theta_2) = \theta_0 + \theta_2 x^2

Does it make sense to penalise just the \theta_2 term here? [If \theta_0 is penalised, a constant function will again be inaccurately modelled]
Reply With Quote

Thread Tools
Display Modes

Posting Rules
You may not post new threads
You may not post replies
You may not post attachments
You may not edit your posts

BB code is On
Smilies are On
[IMG] code is On
HTML code is Off

Forum Jump

All times are GMT -7. The time now is 03:57 PM.

Powered by vBulletin® Version 3.8.3
Copyright ©2000 - 2019, Jelsoft Enterprises Ltd.
The contents of this forum are to be used ONLY by readers of the Learning From Data book by Yaser S. Abu-Mostafa, Malik Magdon-Ismail, and Hsuan-Tien Lin, and participants in the Learning From Data MOOC by Yaser S. Abu-Mostafa. No part of these contents is to be communicated or made accessible to ANY other person or entity.