Advanced Topics in Machine Learning
A. LAZARIC (INRIA-Lille)
DEI, Politecnico di Milano
SequeL – INRIA Lille
April 2-15, 2012
Before You Attended the Course
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 2/8
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 3/8
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 4/8
Before You Attended the Course
This machine learning stuff looks cool!
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 5/8
Before You Attended the Course
This machine learning stuff looks cool!
Maybe I can even find a way to make my computer to learn how to write
my PhD thesis...
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 5/8
During the Course
Theorem
Let Zn be a training set of n i.i.d. samples from a distribution P and H be a hypothesis space with VC(H) = d. If
b
ĥ(·; Zn ) = arg min R(h;
Zn )
h∈H
then
∗
h (·; P) = arg min R(h; P)
h∈H
s
∗
R(ĥ; P) ≤ R(h ; P) + O
d log n/δ n
with probability at least 1 − δ (w.r.t. the randomness in the training set).
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 6/8
During the Course
Theorem
Let Zn be a training set of n i.i.d. samples from a distribution P and H be a hypothesis space with VC(H) = d. If
b
ĥ(·; Zn ) = arg min R(h;
Zn )
h∈H
∗
h (·; P) = arg min R(h; P)
h∈H
s
then
∗
R(ĥ; P) ≤ R(h ; P) + O
d log n/δ n
with probability at least 1 − δ (w.r.t. the randomness in the training set).
Theorem
If D is a convex decision space and the loss function is bounded and convex in the first argument, then on any
sequence yn , EWA(η) satisfies
n
n
Rn = Ln (A; y ) − min Li,n (y ) ≤
i
log N
η
+
ηn
A. LAZARIC – Advanced Topics in Machine Learning
8
.
April 2-15, 2012 - 6/8
During the Course
Theorem
Let Zn be a training set of n i.i.d. samples from a distribution P and H be a hypothesis space with VC(H) = d. If
b
ĥ(·; Zn ) = arg min R(h;
Zn )
h∈H
∗
h (·; P) = arg min R(h; P)
h∈H
s
then
∗
R(ĥ; P) ≤ R(h ; P) + O
d log n/δ n
with probability at least 1 − δ (w.r.t. the randomness in the training set).
Theorem
If D is a convex decision space and the loss function is bounded and convex in the first argument, then on any
sequence yn , EWA(η) satisfies
n
n
Rn = Ln (A; y ) − min Li,n (y ) ≤
i
log N
η
+
ηn
8
.
Theorem
For any set of N arms with distributions bounded in [0, b], if δ = 1/t, then UCB(ρ) with ρ > 1, achieves a regret
"
!#
X 4b 2
3
1
Rn (A) ≤
ρ log(n) + ∆i
+
∆i
2
2(ρ − 1)
i6=i ∗
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 6/8
After You Attended the Course
This machine learning stuff looks cool!
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 7/8
After You Attended the Course
This machine learning stuff looks cool!
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 7/8
After You Attended the Course
This machine learning stuff looks awesome!
A. LAZARIC – Advanced Topics in Machine Learning
April 2-15, 2012 - 7/8
Thank you!!
Alessandro Lazaric
[email protected]
sequel.lille.inria.fr