 LFD Book Forum bias-variance plot on p67

#1
 Steve_Y Junior Member Join Date: May 2017 Posts: 2 bias-variance plot on p67

Hi Prof. Abu-Mostafa,

As you suggested, I post below the question that I emailed you earlier, in case other people also have similar questions. However, I couldn't seem to insert/upload images properly here (it showed only a link), so I'll just do a text-only question.

Specifically, Im a little confused about the bias-variance plot at the bottom of page 67. In the plot, the bias appears to be a flat line, i.e. constant, independent of the sample (training set) size, N. I wondered if this is (approx.) true in general, so I did some experiments (simulations). What I found was that while this was indeed approximately true for the linear regression; it didnt appear so true when I used the 1-nearest-neighbor (1-NN) algorithm. (Similar to Example 2.8, I tried to learn a sinusoid.)

More specifically, for the linear regression, the averaged learned hypothesis, i.e. "g bar", stays almost unchanged when the size of the training set (N) increases from 4 to 10 in my simulation. Even for N=2, "g bar" doesnt deviate too much.

However, for the 1-Nearest-Neighbor (1-NN) algorithm, "g bar" changes considerably as N grows from 2 to 4, and to 10. This seems reasonable to me though, because as N increases, the distance between a test point (x) and its nearest neighbor decreases, with high probability. So its natural to expect "g bar" to converge to the sinusoid, and the bias to decrease as N increases.

Here's the simulated average (squared) bias when N was 2, 4, and 8:
OLS: 0.205, 0.199, 0.198
1NN: 0.184, 0.052, 0.013
where OLS stands for ordinary least squares linear regression.

Do these results and interpretations look correct to you? Or am I mistaken somewhere? Id greatly appreciate it, if youd clarify this a little bit more for me. Thanks a lot!

BTW, in my simulation, the training set of size N is sampled independently and uniformly on the [0,1] interval. I then averaged the learned hypotheses from 5000 training sets to obtain each "g bar".

Best regards,
Steve

 Thread Tools Show Printable Version Email this Page Display Modes Switch to Linear Mode Switch to Hybrid Mode Threaded Mode 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 Rules
 Forum Jump User Control Panel Private Messages Subscriptions Who's Online Search Forums Forums Home General     General Discussion of Machine Learning     Free Additional Material         Dynamic e-Chapters         Dynamic e-Appendices Course Discussions     Online LFD course         General comments on the course         Homework 1         Homework 2         Homework 3         Homework 4         Homework 5         Homework 6         Homework 7         Homework 8         The Final         Create New Homework Problems Book Feedback - Learning From Data     General comments on the book     Chapter 1 - The Learning Problem     Chapter 2 - Training versus Testing     Chapter 3 - The Linear Model     Chapter 4 - Overfitting     Chapter 5 - Three Learning Principles     e-Chapter 6 - Similarity Based Methods     e-Chapter 7 - Neural Networks     e-Chapter 8 - Support Vector Machines     e-Chapter 9 - Learning Aides     Appendix and Notation     e-Appendices

All times are GMT -7. The time now is 04:13 AM.