Large-Scale Convex Optimization over Matrices for Multi-task Learning

for Multi-task Learning

Paul Tseng

Mathematics, University of Washington Seattle

Optima, Univ. Illinois, Urbana-Champaign March 26, 2009

Joint work with Ting Kei Pong (UW) and Jieping Ye (ASU)

Prologue

This story began with an innocent looking email..

A Question..

On Thu, 18 Sep 2008, Jieping Ye wrote:

Dr. Tseng,

I recently came across your interesting work on the block

coordinate descent method for non-differentiable optimization.

I wonder whether the convergence result will apply for the

matrix case where each block is a positive definite matrix, i.e., min f(X, Y, Z), where X, Y, Z are positive definite matrices.

I will appreciate it if you can provide some relevant references on this if any. Thanks!

Best, Jieping

The Problem

Q0,Wmin f (Q,W) := tr(W^TQ⁻¹W) + tr(Q) + kAW − Bk²_F

Q ∈ <^n×n, W ∈ <^n×m (A ∈ <^p×n and B ∈ <^p×m are given), kBk_F = (P

i,j B_ij² )^1/2

The Problem

Q0,Wmin f (Q,W) := tr(W^TQ⁻¹W) + tr(Q) + kAW − Bk²_F

Q ∈ <^n×n, W ∈ <^n×m (A ∈ <^p×n and B ∈ <^p×m are given), kBk_F = (P

i,j B_ij² )^1/2

Note

• f (Q,W) is diff., convex in Q for each W, convex in W for each Q.

• The min is finite, but may not be attained (e.g., when B = 0).

• If min is attained, it’s attained at a critical pt, i.e., ∇f (Q,W) = 0.

• m = # tasks. Q⁻¹ = covar. matrix.

On Wed, 24 Sep 2008, Jieping Ye wrote:

Dear Paul, ...

The multi-task learning problem comes from our biological application: Drosophila gene expression pattern analysis (funded by NSF and NIH).

...

Thanks, Jieping

First Try

∇f (Q,W) = −Q⁻¹W W^TQ⁻¹ + I, 2Q⁻¹W + 2A^T(AW − B)
So ∇f (Q,W) = 0 implies

(W^TQ⁻¹)^TW^TQ⁻¹ = I, Q⁻¹W + MW = A^TB where M := A^TA.

First Try

∇f (Q,W) = −Q⁻¹W W^TQ⁻¹ + I, 2Q⁻¹W + 2A^T(AW − B)
So ∇f (Q,W) = 0 implies

(W^TQ⁻¹)^TW^TQ⁻¹ = I, Q⁻¹W + MW = A^TB where M := A^TA.

So rank(W) = n, rank(A^TB) = n, ... , and

(I + MQ)(I + QM ) = A^TBB^TA

First Try

∇f (Q,W) = −Q⁻¹W W^TQ⁻¹ + I, 2Q⁻¹W + 2A^T(AW − B)
So ∇f (Q,W) = 0 implies

(W^TQ⁻¹)^TW^TQ⁻¹ = I, Q⁻¹W + MW = A^TB where M := A^TA.

So rank(W) = n, rank(A^TB) = n, ... , and

(I + MQ)(I + QM ) = A^TBB^TA

Prop. 1

Q = (M⁻¹A^TBB^TAM⁻¹)¹² − M⁻¹, W = (M + Q⁻¹)⁻¹A^TB

But..

Date: Sat, 25 Oct 2008 16:48:20 -0700 Dear Paul,

Thanks for the writeup. Very interesting.

...

Unfortunately, M is commonly not positive definite in our applications.

...

Thanks, Jieping

Second Try

Suppose M = A^TA is singular, so r := rank(A) < n.

Use SVD or QR decomp. of A:

A = R h

A 0e i S^T

with A ∈ <e ^p×r, R^TR = I and S^TS = I. Let B := Re ^TB.

Prop. 2

Q0,Wmin f (Q,W) = min

Q0,fe W

f (˜ Q,e Wf), where

f (˜ Q,e Wf) := tr(fW^TQe⁻¹Wf) + tr(Q) +e

AeWf − eB

2 F

.

Then recover Q,W from Q,e Wf.

Moreover, f˜ has a stationary pt iff

( fM⁻¹Ae^TB eeB^TA feM⁻¹)¹²  fM⁻¹

where M := ef A^TA.e

Done?

But..

Date: Thu, 30 Oct 2008 10:44:56 -0700 Dear Tseng,

Thanks. I like the derivation.

It seems the condition in Eq. (4) is the key.

We need to somehow relax this condition.

Will perturbation solve this problem?

Best, Jieping

Third Try

Assume w.l.o.g. rankA = n. Let

h(Q) := inf

W f (Q,W)

= inf

W tr(W^TQ⁻¹W) + tr(Q) + kAW − Bk²_F

= tr(Q) + tr(E^TE(Q + C)⁻¹) + const.

with C := M⁻¹  0 and E := B^TAC. (M = A^TA)

Third Try

Assume w.l.o.g. rankA = n. Let

h(Q) := inf

W f (Q,W)

= inf

W tr(W^TQ⁻¹W) + tr(Q) + kAW − Bk²_F

= tr(Q) + tr(E^TE(Q + C)⁻¹) + const.

with C := M⁻¹  0 and E := B^TAC. (M = A^TA)

Then h(Q) is cont. over Q  0 (!) so

minQ0 h(Q) = min

Q0,W f (Q,W).

Moreover, (Q,W) 7→ W^TQ⁻¹W is operator-convex, so f is convex, and hence h is convex.

Prop. 3

∇h(Q) = I − (Q + C)⁻¹E^TE(Q + C)⁻¹

is Lipschitz cont. over Q  0.

Prop. 3

∇h(Q) = I − (Q + C)⁻¹E^TE(Q + C)⁻¹

is Lipschitz cont. over Q  0.

Moreover, using Schur complement, min_Q0 h(Q) reduces to an SDP:

min tr(Q) + tr(U) s.t. Q  0,

 Q 0 0 U

 +

 C E^T

E 0



 0

Recall C  0 is n × n and E is m × n. This SDP is solvable by existing IP solvers (SeDuMi, SDPT³, CSDP, ..) for around m + n ≤ 500.

But..

Date: Mon, 1 Dec 2008 09:33:13 -0700 ...

In our application, n is around 1000-2000 and m is around 50-100.

...

It contains 1000-3000 rows depending on the feature extraction scheme. In general, X is dense. However, one of our recent feature extraction schemes produces sparse X. By the way, the columns of X correpspond

to biological images.

Best, Jieping

For m = 100, n = 2000, (Q,W) comprises 2201000 variables. A ∈ <^p×n may be dense.

6. .

_

Fourth Try

Lesson from my graduate student days:

“When stuck, look at the dual”

Consider the dual problem

maxΛ0 min

Q0,U L(Q,U,Λ), with Lagrangian (hW, Zi = tr(W^TZ))

L(Q,U,Λ) := hI,Qi + hI,Ui −

 Λ₁ Λ^T₂ Λ₂ Λ₃

 ,

 Q 0 0 U

 +

 C E^T

E 0



= hI − Λ₁,Qi + hI − Λ₃,Ui − hΛ₁, Ci − 2hΛ₂, Ei

Consider the dual problem

maxΛ0 min

Q0,U L(Q,U,Λ), with Lagrangian (hW, Zi = tr(W^TZ))

L(Q,U,Λ) := hI,Qi + hI,Ui −

 Λ₁ Λ^T₂ Λ₂ Λ₃

 ,

 Q 0 0 U

 +

 C E^T

E 0



= hI − Λ₁,Qi + hI − Λ₃,Ui − hΛ₁, Ci − 2hΛ₂, Ei

For dual feas., need I  Λ₁, I = Λ₃, Λ₁  Λ^T₂ Λ₂. Dual problem reduces to

min

IΛ₁Λ^T

2 Λ₂

hC, Λ₁i + 2hE,Λ₂i

Since C  0, minimum w.r.t. Λ₁ is attained at Λ₁ = Λ^T₂ Λ₂.

The dual problem reduces to (recall Λ₂ ∈ <^m×n)

min

IΛ^T₂ Λ₂

d₂(Λ₂) := 1

2hC,Λ^T₂ Λ₂i + hE, Λ₂i.

The dual problem reduces to (recall Λ₂ ∈ <^m×n)

min

IΛ^T₂ Λ₂

d₂(Λ₂) := 1

2hC,Λ^T₂ Λ₂i + hE, Λ₂i.

• No duality gap since the primal problem has interior soln.

• Recovers Q as Lagrange multiplier assoc. with I  Λ^T₂ Λ₂.

• ∇d₂(Λ₂) = Λ₂C + E is Lipschitz cont. with constant L = λ_max(C).

The dual problem reduces to (recall Λ₂ ∈ <^m×n)

min

IΛ^T₂ Λ₂

d₂(Λ₂) := 1

2hC,Λ^T₂ Λ₂i + hE, Λ₂i.

• No duality gap since the primal problem has interior soln.

• Recovers Q as Lagrange multiplier assoc. with I  Λ^T₂ Λ₂.

• ∇d₂(Λ₂) = Λ₂C + E is Lipschitz cont. with constant L = λ_max(C).

What about the constraint I  Λ^T₂ Λ₂?

Prop. 4

D 0 

S^T,

Proj(Λ₂) := arg min

IΨ^T₂ Ψ₂

kΛ₂ − Ψ₂k²_F = R

min {D, I} 0  S^T

Solving the reduced dual

Coded 3 methods in Matlab: Frank-Wolfe, grad.-proj. with LS Goldstein, Levitin, Polyak, and accel. grad.-proj. ^Nesterov.

Accel. grad.-proj. seems most efficient.

0. Choose I  Λ^T₂ Λ₂. Set Λ^prev₂ = Λ₂, θ^prev = θ = 1. Fix L = λ_max(C). Go to 1.

1. Set

Λ^ext₂ = Λ₂ +

 θ

θ^prev − θ



Λ₂ − Λ^prev₂  . Update Λ^prev₂ ← Λ₂, θ^prev ← θ, and

Λ₂ ← Proj



Λ^ext₂ − 1

L∇d₂(Λ^ext₂ )



θ ←

√θ⁴ + 4θ² − θ²

2 .

If relative duality gap ≤ tol, stop. Else to to 1.

Test Results (Preliminary)

Tested on random data: A ∼ U [0, 1]^p×n and B ∼ U [0, 1]^p×m. tol = .001 n = 2000 m = 100 p= 1000 tol= 0.001

reduce A to have full column rank:

done reducing A, time: 38.9895

done computing C and E, time: 4.05682

termination due to negligible change in U = 3.4469e-11 iter= 10 dobj= -96.7469 dual feas= 8.88178e-15

pobj= -96.7469 primal feas= 1.43293e-15 accel. grad-proj: iter= 10 total_time= 67.9021

fmin = 193.494 fval = 193.494

n = 2000 m = 100 p= 3000 tol= 0.001 done computing C and E, time: 31.5357

termination due to negligible change in U = 7.32917e-11 iter= 10 dobj= -137.14 dual feas= 6.21725e-15

pobj= -137.14 primal feas= 3.06165e-15 accel. grad-proj: iter= 10 total_time= 190.652 fmin = 8632.32 fval = 8632.32

However

Maybe finally..

Date: Sun, 11 Jan 2009 11:09:15 -0700 Dear Paul,

Sorry for the delay.

Some preliminary results prepared by my student are

attached. Overall, it performs well, especially when the number of labels is large. We will conduct more extensive experimental studies and keep you updated.

Best, Jieping

In preliminary result on Drosophila gene expression pattern annotation, a group of images are associated with variable number of terms using a controlled vocabulary.

k-means clustering and feature extractions are used to obtain a global

histogram counting the number of features closest to the visual words in the codebook obtained from the clustering algorithm (with 3000 clusters), etc.

Hard (soft) assignment: a feature assigned to one (multiple) word.

• n = 3000 (#clusters)

• 10 ≤ m ≤ 60 (#terms/tasks)

• 2200 ≤ p ≤ 2800 (#samples).

m MTL_hard MTL_soft SVM_hard SVM_soft PMK_star PMK_clique 10 77.22±0.63 78.86±0.58 74.89±0.68 78.51±0.60 71.80±0.81 71.98±0.87 20 78.95±0.82 80.90±1.02 76.38±0.84 78.94±0.86 72.01±1.01 71.70±1.20 10 52.57±1.19 54.89±1.24 52.25±0.98 55.64±0.69 46.20±1.18 47.06±1.16 20 33.15±1.37 37.25±1.25 35.62±0.99 39.18±1.18 28.21±1.00 28.11±1.09 10 59.92±1.04 60.84±0.99 55.74±1.02 59.27±0.80 53.25±1.15 53.36±1.20 20 55.33±0.88 56.79±0.72 51.70±1.17 54.25±0.93 49.59±1.24 48.14±1.34

Table 1: Performance (top: AUC, middle: macro F1, bottom: micro F1) of MTL, SVM, PMK on data sets in stage range 9-10 (m = 10, 20 and p = 919, 1015).

m MTL_hard MTL_soft SVM_hard SVM_soft PMK_star PMK_clique 10 84.06±0.53 86.18±0.45 83.05±0.54 84.84±0.57 78.68±0.58 78.52±0.55 30 81.83±0.46 83.85±0.36 79.18±0.51 81.31±0.48 71.85±0.61 71.13±0.53 50 80.56±0.53 82.87±0.53 76.19±0.72 78.75±0.68 69.66±0.81 68.80±0.68 10 60.30±0.92 64.00±0.85 60.37±0.88 62.61±0.82 54.61±0.68 55.19±0.62 30 35.20±0.85 39.15±0.83 35.32±0.75 37.38±0.95 22.30±0.70 24.85±0.63 50 23.07±0.86 26.67±1.05 23.46±0.60 26.26±0.65 14.07±0.48 15.04±0.46 10 66.89±0.79 68.92±0.68 65.67±0.60 66.73±0.74 62.06±0.54 61.84±0.51 30 55.66±0.64 56.70±0.68 48.87±0.85 51.52±0.96 47.08±0.81 44.81±0.66 50 52.92±0.78 54.54±0.70 47.18±0.84 47.97±0.90 44.25±0.65 42.49±0.70

Table 2: Performance (top: AUC, middle: macro F1, bottom: micro F1) of MTL, SVM, PMK on data sets in stage range 11-12 (10 ≤ m ≤ 50, 1622 ≤ p ≤ 2070).

m MTL_hard MTL_soft SVM_hard SVM_soft PMK_star PMK_clique 10 87.38±0.36 89.43±0.31 86.66±0.35 88.42±0.35 82.07±0.41 82.53±0.62 30 82.76±0.36 85.86±0.34 81.13±0.46 83.45±0.38 73.34±0.46 73.73±0.52 60 80.17±0.40 83.32±0.45 77.18±0.46 79.75±0.47 67.15±0.57 67.11±0.64 10 64.43±0.77 67.42±0.78 62.97±0.68 66.38±0.71 57.37±0.91 58.42±0.94 30 42.48±0.87 47.39±0.91 41.92±0.76 45.07±0.68 29.62±0.67 31.04±0.82 60 24.78±0.67 29.84±0.62 25.49±0.55 28.72±0.57 15.65±0.46 16.13±0.48 10 67.85±0.60 70.50±0.58 66.67±0.45 68.79±0.60 60.98±0.74 61.87±0.77 30 53.74±0.45 57.04±0.69 48.11±0.90 51.19±0.83 43.50±0.70 44.14±0.78 60 48.79±0.60 51.35±0.58 42.84±0.76 44.48±0.84 37.28±0.81 38.29±0.78

Table 3: Performance (top: AUC, middle: macro F1, bottom: micro F1) of MTL, SVM, PMK on data sets in stage range 13-16 (10 ≤ m ≤ 60, 2228 ≤ p ≤ 2754).

Conclusions & Extensions

1. A seemingly nasty problem arising from application is tamed by a mix of convex/matrix analysis, and modern algorithms.

Conclusions & Extensions

1. A seemingly nasty problem arising from application is tamed by a mix of convex/matrix analysis, and modern algorithms.

2. Extension to related convex optimization problems in learning?

Conclusions & Extensions

1. A seemingly nasty problem arising from application is tamed by a mix of convex/matrix analysis, and modern algorithms.

2. Extension to related convex optimization problems in learning?

3. Better algorithms to handle the case of p ≈ n?

Conclusions & Extensions

1. A seemingly nasty problem arising from application is tamed by a mix of convex/matrix analysis, and modern algorithms.

2. Extension to related convex optimization problems in learning?

3. Better algorithms to handle the case of p ≈ n?

The End?

Nooo..

On Sun, 22 Mar 2009, Jieping Ye wrote:

Dear Paul,

Thanks for the updated version.

...

There are two tex files: introduction.tex and experiment.tex and two bib files.

...

Thanks, Jieping

The introduction shows the original problem is a reformulation of

minW kWk_∗ + kAW − Bk²_F

with kWk_∗ = P

i σ_i(W) (“trace/nuclear-norm”).

This can be solved by accel. gradient method (one SVD per iter.) too.

Which is faster? Will see..