1 A Statistical Mechanical Analysis of Online Learning: Seiji MIYOSHI Kobe City College of Technology

2 Background (1) Batch Learning –Examples are used repeatedly –Correct answers for all examples –Long time –Large memory Online Learning –Examples used once are discarded –Cannot give correct answers for all examples –Large memory isn't necessary –Time variant teacher

3 A Statistical Mechanical Analysis of Online Learning: Can Student be more Clever than Teacher ? Seiji MIYOSHI Kobe City College of Technology Jan. 2006

4 Moving Teacher Student True Teacher A Jan. 2006

5 A Statistical Mechanical Analysis of Online Learning: Seiji MIYOSHI Kobe City College of Technology Many Teachers or Few Teachers ?

6 True teacher Student Ensemble teachers

7 P U R P O S E To analyze generalization performance of a model composed of a student, a true teacher and K teachers (ensemble teachers) who exist around the true teacher To discuss the relationship between the number, the diversity of ensemble teachers and the generalization error

8 M O D E L (1/4) True teacher Student J learns B 1,B 2, ・・・ in turn. J can not learn A directly. A, B 1,B 2, ・・・,J are linear perceptrons with noises. Ensemble teachers

9 Simple Perceptron Output Inputs Connection weights +1

10 Output Inputs Connection weights Simple Perceptron Linear Perceptron

11 M O D E L (2/4) Linear Perceptrons with Noises

12 M O D E L (3/4) Inputs: Initial value of student: True teacher: Ensemble teachers: N→∞ (Thermodynamic limit) Order parameters –Length of student –Direction cosines

13 True teacher Student Ensemble teachers

14 fkmfkm Student learns K ensemble teachers in turn. M O D E L (4/4) Gradient method Squared errors

15 GENERALIZATION ERROR A goal of statistical learning theory is to obtain generalization error theoretically. Generalization error = mean of errors over the distribution of new input

16 Simultaneous differential equations in deterministic forms, which describe dynamical behaviors of order parameters

17 Analytical solutions of order parameters

18 GENERALIZATION ERROR A goal of statistical learning theory is to obtain generalization error theoretically. Generalization error = mean of errors over the distribution of new input

19 Dynamical behaviors of generalization error, R J and l （ η=0.3, K=3, R B =0.7, σ A 2 =0.0, σ B 2 =0.1, σ J 2 =0.2 ） Student Ensemble teachers J

20 Analytical solutions of order parameters

21 Steady state analysis （ t → ∞ ） ・ If η ＜０ or η ＞ ２ ・ If ０＜ η ＜２ Generalization error and length of student diverge. If η ＜１, the more teachers exist or the richer the diversity of teachers is, the cleverer the student can become. If η ＞１, the fewer teachers exist or the poorer the diversity of teachers is, the cleverer the student can become.

22 Steady value of generalization error, R J and l （ K=3, R B =0.7, σ A 2 =0.0, σ B 2 =0.1, σ J 2 =0.2 ） J

23 Steady value of generalization error, R J and l （ q=0.49, R B =0.7, σ A 2 =0.0, σ B 2 =0.1, σ J 2 =0.2 ） J

24 CONCLUSIONS We have analyzed the generalization performance of a student in a model composed of linear perceptrons: a true teacher, K teachers, and the student. Calculating the generalization error of the student analytically using statistical mechanics in the framework of on-line learning, we have proven that when the learning rate satisfies η 1, the properties are completely reversed. If the diversity of the K teachers is rich enough, the direction cosine between the true teacher and the student becomes unity in the limit of η→0 and K→∞.