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Coordinate Descent slide
Coordinate descent
Geoff Gordon Ryan Tibshirani
Optimization 10-725 / 36-725
1
Adding to the toolbox, with stats and ML in mind
We’ve seen several general and useful minimization tools
? First-order methods
? Newton’s method
? Dual methods
? Interior-point methods
These are some of the core methods in optimization, and they are
the main objects of study in this field
In statistics and machine learning, there are a few other techniques
that have received a lot of attention; these are not studied as much
by those purely in optimization
They don’t apply as broadly as above methods, but are interesting
and useful when they do apply ... our focus for the next 2 lectures
2
Coordinate-wise minimization
We’ve seen (and will continue to see) some pretty sophisticated
methods. Today, we’ll see an extremely simple technique that is
surprisingly efficient and scalable
Focus is on coordinate-wise minimization
Q: Given convex, differentiable f : Rn → R, if we are at a point x
such that f(x) is minimized along each coordinate axis, have we
found a global minimizer?
I.e., does f(x+ d · ei) ≥ f(x) for all d, i ? f(x) = minz f(z)?
(Here ei = (0, . . . , 1, . . . 0) ∈ Rn, the ith standard basis vector)
3
x1 x2
f
A: Yes! Proof:
?f(x) =
(
?f
?x1
(x), . . .
?f
?xn
(x)
)
= 0
Q: Same question, but for f convex (not differentiable) ... ?
4
x1
x
2
f
x1
x
2
?4 ?2 0 2 4
?
4
?
2
0
2
4
●
A: No! Look at the above counterexample
Q: Same question again, but now f(x) = g(x) +
∑n
i=1 hi(xi), with
g convex, differentiable and each hi convex ... ? (Non-smooth part
here called separable)
5
x1
x
2
f
x1
x
2
?4 ?2 0 2 4
?
4
?
2
0
2
4
●
A: Yes! Proof: for any y,
f(y)? f(x) ≥ ?g(x)T (y ? x) +
n∑
i=1
[hi(yi)? hi(xi)]
=
n∑
i=1
[?ig(x)(yi ? xi) + hi(yi)? hi(xi)]︸ ︷︷ ︸
≥0
≥ 0
6
Coordinate descent
This suggests that for f(x) = g(x) +
∑n
i=1 hi(xi) (with g convex,
differentiable and each hi convex) we can use coordinate descent
to find a minimizer: start with some initial guess x(0), and repeat
for k = 1, 2, 3, .
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