mini-course 3: convergence analysis of neural …optimization i in practice, sgd always nds good...

Mini-Course 3:Convergence Analysis of Neural Network

Yang Yuan

Computer Science DepartmentCornell University

Deep learning is powerful

What is neural network?

one

laye

rA simplified view.Missing: Convolution/BatchNorm.· · · · · · · · ·

· · · · · · · · ·

← Input x = (1, 2,−4)>

← Weight W =

1 0 0 10 1 0 10 0 1 1

← W>x = (1, 2,−4,−1)>

← ReLU(W>x) = (1, 2, 0, 0)>

Three basic types of theory questions

I RepresentationI Can we express any functions with neural networks?

I Yes, very expressive: [Hornik et al., 1989, Cybenko, 1992,Barron, 1993, Eldan and Shamir, 2015,Safran and Shamir, 2016, Lee et al., 2017]

I OptimizationI Efficient methods for finding good parameters (i.e.,

representations)?

I GeneralizationI Training data used for optimization step.I Does it generalize to unseen data (test data)?

I Little is know. Flat minima? [Shirish Keskar et al., 2016,Hochreiter and Schmidhuber, 1995, Chaudhari et al., 2016,Zhang et al., 2016]

I In practice: neural network is doing great in ALL THREE!I What about in theory?


I RepresentationI Can we express any functions with neural networks?I Yes, very expressive: [Hornik et al., 1989, Cybenko, 1992,

Barron, 1993, Eldan and Shamir, 2015,Safran and Shamir, 2016, Lee et al., 2017]


representations)?

I GeneralizationI Training data used for optimization step.I Does it generalize to unseen data (test data)?

I Little is know. Flat minima? [Shirish Keskar et al., 2016,Hochreiter and Schmidhuber, 1995, Chaudhari et al., 2016,Zhang et al., 2016]



I RepresentationI Can we express any functions with neural networks?I Yes, very expressive: [Hornik et al., 1989, Cybenko, 1992,

Barron, 1993, Eldan and Shamir, 2015,Safran and Shamir, 2016, Lee et al., 2017]


representations)?

I GeneralizationI Training data used for optimization step.I Does it generalize to unseen data (test data)?I Little is know. Flat minima? [Shirish Keskar et al., 2016,

Hochreiter and Schmidhuber, 1995, Chaudhari et al., 2016,Zhang et al., 2016]


Optimization

I In practice, SGD always finds good local minima.I SGD: stochastic gradient descentI xt+1 = xt − ηgt , E [gt ] = ∇f (xt)

I Some results are negative, saying optimization for neuralnetworks is in general hard.

I [Sıma, 2002, Livni et al., 2014, Shamir, 2016].

I Or positive but with special algorithms (tensor decomposition,half space intersection, etc.)

I [Janzamin et al., 2015, Zhang et al., 2015,Sedghi and Anandkumar, 2015, Goel et al., 2016],

I With strong assumptions on the model (weights are complexnumbers, learning polynomials only, weights are iid random)

I [Andoni et al., 2014, Arora et al., 2014]

Recent work: independent activations

Independent activation assumption

I The outputs of ReLU units are independent of the input x,and independent of each other. [Choromanska et al., 2015,Kawaguchi, 2016, Brutzkus and Globerson, 2017].

← Input x = (1, 2,−4)>

← Weight W =

1 0 0 10 1 0 10 0 1 1

← W>x = (1, 2,−4,−1)>

← ReLU(W>x) = (1, 2, 0, 0)>

Independent!

Independent!

CNN Model in [Brutzkus and Globerson, 2017]

Recent work: guarantees of other algorithm

I Tensor Decomposition+ Gradient Descent converges toground truth for one hidden layer network.[Zhong et al., 2017]

I Kernel methods could learn deep neural network witheigenvalue decay assumption. [Goel and Klivans, 2017]

Recent work: deep linear models

Ignore the activation functions.

I [Saxe et al., 2013, Kawaguchi, 2016, Hardt and Ma, 2016]

I Only learn a linear function.I Proof idea (for deep linear residual network)

I Loss: ‖(I + W`) · · · (I + W1)x − (Ax + b)‖2

I Compute ∂f∂Wi

.I Get lower bound for the gradient norm:‖∇f (W)‖2

F ≥ C (f (W)− f ∗)I Show C ≥ 0.I So f (W) = f ∗.




I Only learn a linear function.

I Proof idea (for deep linear residual network)

I Loss: ‖(I + W`) · · · (I + W1)x − (Ax + b)‖2

I Compute ∂f∂Wi







I Loss: ‖(I + W`) · · · (I + W1)x − (Ax + b)‖2

I Compute ∂f∂Wi







I Loss: ‖(I + W`) · · · (I + W1)x − (Ax + b)‖2

I Compute ∂f∂Wi

.

I Get lower bound for the gradient norm:‖∇f (W)‖2






I Loss: ‖(I + W`) · · · (I + W1)x − (Ax + b)‖2

I Compute ∂f∂Wi


F ≥ C (f (W)− f ∗)

I Show C ≥ 0.I So f (W) = f ∗.





I Loss: ‖(I + W`) · · · (I + W1)x − (Ax + b)‖2

I Compute ∂f∂Wi


F ≥ C (f (W)− f ∗)I Show C ≥ 0.

I So f (W) = f ∗.





I Loss: ‖(I + W`) · · · (I + W1)x − (Ax + b)‖2

I Compute ∂f∂Wi



Today’s paper: convergence analysis for two layer withReLU [Li and Yuan, 2017]

Below: our thought process, might be messy.

Our starting model

I A two layer model, not deep.

f (x ,W) = ‖ReLU(W>x)‖1

I Assume there exists a teacher network producing thelabels.

f (x ,W∗) = ‖ReLU(W∗>x)‖1

I Square loss.

L(W) = Ex [(f (x ,W)− f (x ,W∗))2]

I x ∼ N (0, I).I A common assumption [Choromanska et al., 2015,

Tian, 2016, Xie et al., 2017].

input x

W>x

ReLU(W>x)

Take sum

output

However..

[Tian, 2016] showed, even if

I W initialized symmetrically

I W∗ forms orthonormal basis.

I Gradient descent may stuck atsaddle points.

Complicated surface, hard to analyze.

From [Tian, 2016]

However.. [stuck]

[Tian, 2016] showed, even if

I W initialized symmetrically

I W∗ forms orthonormal basis.

I Gradient descent may stuck atsaddle points.

Complicated surface, hard to analyze.

From [Tian, 2016]

Residual network [He et al., 2016]

I state-of-the-art structure. (3300citations)

I Won a few competitions.

I Very powerful after stacking multipleblocks together.

I Easy to train.

A ResNet Block. From[He et al., 2016]

Residual network [He et al., 2016]

I state-of-the-art structure. (3300citations)

I Won a few competitions.

I Very powerful after stacking multipleblocks together.

I Easy to train.A ResNet Block. From[He et al., 2016]

Adding residual link?

I We modify the network:

f (x ,W) = ‖ReLU(W>x)‖1

to (adding identity)

f (x ,W) = ‖ReLU((I + W)>x)‖1

I Same for the ground truth f (x ,W∗).

I Essentially: move the weight by I.

input x

W>x

⊕ResidualLink +x

ReLU((I + W)>x)

Take sum

output

Adding residual link?

I We modify the network:

f (x ,W) = ‖ReLU(W>x)‖1

to (adding identity)

f (x ,W) = ‖ReLU((I + W)>x)‖1

I Same for the ground truth f (x ,W∗).

I Essentially: move the weight by I.input x

W>x

⊕ResidualLink +x

ReLU((I + W)>x)

Take sum

output

Ask Simulation: does SGD converge for this model?

Ask Simulation: does SGD converge for this model?

Simulation says: yes.

Illustration of our key observation

O

I

I + W∗

I + W

Easy for SGD

Unknown

Seems hard

Residual link

How to prove this?

How to prove this?

One-point convexity.

A function f (x) is called δ-one point strongly convex in domain Dwith respect to point x∗, if ∀x ∈ D,〈−∇f (x), x∗ − x〉 > δ‖x∗ − x‖2

2.

How to prove this?

One-point convexity.

A function f (x) is called δ-one point strongly convex in domain Dwith respect to point x∗, if ∀x ∈ D,〈−∇f (x), x∗ − x〉 > δ‖x∗ − x‖2

2.

I A weaker condition than convexity.

I if it’s one point convex, we get toW∗ closer after every step, as longas the step size is small.

0

50

100

150

200

0

50

100

150

200

−5

0

5

10

15

One-point convex: an illustration

W∗

W1

W5

Ask Simulation: is it one point convex?

Ask Simulation: is it one point convex?

Simulation says: yes.

Compute 〈−∇L(W),W∗ −W〉

−∇L(W)j

=d∑

i=1

[π2

(w∗i − wi ) +(π

2− θi∗,j

)(ei + w∗i )−

(π2− θi ,j

)(ei + wi )

+ (‖ei + w∗i ‖2 sin θi∗,j − ‖ei + wi‖2 sin θi ,j)ej + wj

]

Compute 〈−∇L(W),W∗ −W〉

−∇L(W)j

=d∑

i=1

[π2

(w∗i − wi ) +(π

2− θi∗,j

)(ei + w∗i )−

(π2− θi ,j

)(ei + wi )

+ (‖ei + w∗i ‖2 sin θi∗,j − ‖ei + wi‖2 sin θi ,j)ej + wj

]

I ei ,wi ,w∗i are column vectors of I,W,W∗.

I θi ,j∗ : angle between ei + wi and ej + w∗j (Hard)

I sin θi ,j (Hard)

Taylor expansion: tedious calculation

What is θi ,j∗? It is the angle between wi + ei and w∗j + ej . We have

cos(θi ,j∗)

=〈wi + ei ,w

∗j + ej〉

‖wi + ei‖‖w∗j + ej‖=〈wi ,w

∗j 〉+ wi ,j + w∗j ,i

‖wi + ei‖‖w∗j + ej‖≈ (1− wi ,i )(1− w∗j ,j)(〈wi ,w

∗j 〉+ wi ,j + w∗j ,i )

≈ (1− wi ,i − w∗j ,j)(〈wi ,w∗j 〉+ wi ,j + w∗j ,i )

≈ 〈wi ,w∗j 〉+ wi ,j + w∗j ,i − wi ,iwi ,j − wi ,iw

∗j ,i − w∗j ,jwi ,j − w∗j ,jw

∗j ,i

Since we know arccos(x) ≈ π/2− x , we have

θi ,j∗ ≈π

2−〈wi ,w

∗j 〉+ wi ,j + w∗j ,i

‖wi + ei‖‖w∗j + ej‖

However direct Taylor expansion is too loose

I In order to show 〈−∇L(W),W∗ −W〉 ≥ 0, we need toassume γ , max‖W‖2, ‖W∗‖2 ≤ O( 1

d )

I Super local region.

However direct Taylor expansion is too loose [stuck]

I In order to show 〈−∇L(W),W∗ −W〉 ≥ 0, we need toassume γ , max‖W‖2, ‖W∗‖2 ≤ O( 1

d )

I Super local region.

Ask Simulation: what is the largest γ to satisfy one pointconvexity?

Ask Simulation: what is the largest γ to satisfy one pointconvexity?

Simulation says: Ω(1).

O

I

I + W∗

I + W

Easy for SGD

Unknown

Seems hard

Residual link

Taylor expansion: tedious calculation

What is θi ,j∗? It is the angle between wi + ei and w∗j + ej . We have

cos(θi ,j∗)

=〈wi + ei ,w

∗j + ej〉

‖wi + ei‖‖w∗j + ej‖=〈wi ,w

∗j 〉+ wi ,j + w∗j ,i

‖wi + ei‖‖w∗j + ej‖≈ (1− wi ,i )(1− w∗j ,j)(〈wi ,w

∗j 〉+ wi ,j + w∗j ,i )

≈ (1− wi ,i − w∗j ,j)(〈wi ,w∗j 〉+ wi ,j + w∗j ,i )

≈ 〈wi ,w∗j 〉+ wi ,j + w∗j ,i − wi ,iwi ,j − wi ,iw

∗j ,i − w∗j ,jwi ,j − w∗j ,jw

∗j ,i

Since we know arccos(x) ≈ π/2− x , we have

θi ,j∗ ≈π

2−〈wi ,w

∗j 〉+ wi ,j + w∗j ,i

‖wi + ei‖‖w∗j + ej‖

Use geometry to get tighter bounds!

I Denote ei + w∗i as−→OC , ei + wi as

−→OD, ei + w∗i as

−→OA, ei + wi as

−→OB. Thus, ‖w∗i − wi‖2 = ‖

−→DC‖2.

I Draw−→HB ‖

−→CD, so ‖

−→OH‖2 ≥ ‖

−→OB‖2 = ‖

−→OA‖2.

I Since 4CDO ∼ 4HBO, we have

‖−→CD‖2

‖−→HB‖2

=‖−→OD‖2

‖−→OB‖2

= ‖−→OD‖2 ≥ 1− γ

I So ‖−→CD‖2 ≥ (1− γ)‖

−→HB‖2.

O

A B

CD

E

FGH

OC : ei + w∗iOA : ei + w∗i

OD : ei + wi

OB : ei + wi

Central Lemma: geometric lemma

I 4ABO is a isosceles triangle, so ‖−→AG‖2 = ‖

−→GB‖2.

I ‖−→HB‖2 ≥ ‖

−→AB‖2 = 2‖

−→GB‖2.

I ‖−→GB‖2 ≤ ‖

−→HB‖2

2 ≤ ‖−→CD‖2

2(1−γ) .

I ‖−→GB‖2 ≤ ‖

−→CD‖2

2(1−γ) .

O

A B

CD

E

FGH


OD : ei + wi

OB : ei + wi

‖−→CD‖2 ≥ (1− γ)‖

−→HB‖2


I 4ABE ∼ 4BGO

I

‖−→AE‖2

‖−→AB‖2

=‖−→OG‖2

‖−→OB‖2

=

√1− ‖

−→GB‖2

2

1

I

‖−→AE‖2

‖−→AB‖2

≥

√1−

‖−→CD‖2

2

4(1− γ)2≥

√1−

(γ

1− γ

)2

O

A B

CD

E

FGH


OD : ei + wi

OB : ei + wi

‖−→GB‖2 ≤ ‖

−→CD‖2

2(1−γ)

‖wi − w∗i ‖2 ≤ 2γ


I ‖−→CD‖2 ≥ ‖

−→CF‖2

I 4CFO ∼ 4AEO

I ‖W∗‖2 ≤ γI

‖−→CD‖2

‖−→AE‖2

≥ ‖−→CF‖2

‖−→AE‖2

=‖−→OC‖2

‖−→OA‖2

= ‖ei + w∗i ‖2 ≥ 1− γ

I ‖−→CD‖2 ≥ (1− γ)‖

−→AE‖2 ≥

(1− γ)

√1−

(γ

1−γ

)2‖−→AB‖2

I ‖−→AB‖2 ≤ ‖

−→CD‖2√1−2γ

=‖w∗i −wi‖2√

1−2γ

O

A B

CD

E

FGH


OD : ei + wi

OB : ei + wi

‖−→AE‖2

‖−→AB‖2

≥√

1−(

γ1−γ

)2


I ‖−→CD‖2 ≥ ‖

−→CF‖2

I 4CFO ∼ 4AEO

I ‖W∗‖2 ≤ γ

I

‖−→CD‖2

‖−→AE‖2

≥ ‖−→CF‖2

‖−→AE‖2

=‖−→OC‖2

‖−→OA‖2

= ‖ei + w∗i ‖2 ≥ 1− γ

I ‖−→CD‖2 ≥ (1− γ)‖

−→AE‖2 ≥

(1− γ)

√1−

(γ

1−γ

)2‖−→AB‖2

I ‖−→AB‖2 ≤ ‖

−→CD‖2√1−2γ

=‖w∗i −wi‖2√

1−2γ

O

A B

CD

E

FGH


OD : ei + wi

OB : ei + wi

‖−→AE‖2

‖−→AB‖2

≥√

1−(

γ1−γ

)2


I ‖−→CD‖2 ≥ ‖

−→CF‖2

I 4CFO ∼ 4AEO

I ‖W∗‖2 ≤ γI

‖−→CD‖2

‖−→AE‖2

≥ ‖−→CF‖2

‖−→AE‖2

=‖−→OC‖2

‖−→OA‖2

= ‖ei + w∗i ‖2 ≥ 1− γ

I ‖−→CD‖2 ≥ (1− γ)‖

−→AE‖2 ≥

(1− γ)

√1−

(γ

1−γ

)2‖−→AB‖2

I ‖−→AB‖2 ≤ ‖

−→CD‖2√1−2γ

=‖w∗i −wi‖2√

1−2γ

O

A B

CD

E

FGH


OD : ei + wi

OB : ei + wi

‖−→AE‖2

‖−→AB‖2

≥√

1−(

γ1−γ

)2


I |〈ei + w∗i − ei + wi , ei + wi 〉| = ‖−→BE‖2

I 4ABE ∼ 4GBO

I

‖−→BE‖2

‖−→AB‖2

=‖−→GB‖2

‖−→BO‖2

=‖−→AB‖2

2

O

A B

CD

E

FGH


OD : ei + wi

OB : ei + wi

‖wi − w∗i ‖2 ≤ 2γ

‖−→AB‖2 ≤

‖w∗i −wi‖2√1−2γ


I |〈ei + w∗i − ei + wi , ei + wi 〉| = ‖−→BE‖2

I 4ABE ∼ 4GBO

I

‖−→BE‖2

‖−→AB‖2

=‖−→GB‖2

‖−→BO‖2

=‖−→AB‖2

2

|〈ei + w∗i − ei + wi , ei + wi 〉| =‖−→AB‖2

2

2≤‖w∗i − wi‖2

2

2(1− 2γ)

This is a very tight bound O(γ2).

O

A B

CD

E

FGH


OD : ei + wi

OB : ei + wi

‖wi − w∗i ‖2 ≤ 2γ

‖−→AB‖2 ≤

‖w∗i −wi‖2√1−2γ

Is it enough..?

With tight bounds, we get, if ‖W0‖2, ‖W∗‖2 ≤ γ = Ω(1):

〈−∇L(W),W∗ −W〉 >(

0.084− (1 + γ)g

2(1− 2γ)

)‖W∗ −W‖2

F

I g is a potential function:

g ,d∑

i=1

(‖ei + w∗i ‖2 − ‖ei + wi‖2)

I d is input dimension.

I How does g affect OPC?

Ask Simulation: what if g is large?

Ask Simulation: what if g is large?

Simulation:

Ask Simulation: what if g is large? [stuck]

Simulation:

Ask Simulation: will g always decrease with SGD?

Ask Simulation: will g always decrease with SGD?

Simulation: Yes (for lots of instances)

g controls the dynamics!

PI PII

The actual dynamics

W1

W∗

W6

W10

Phase I: W1 →W6, W may go to the wrong direction. Phase II:W6 →W10, W gets closer to W∗ in every step by one point convexity.

Two phase framework

I Phase II g decreases to a small value.I Technique: analyze the dynamics of SGD

I Phase III One point convex regionI Get closer to W∗ after every stepI g is always smallI Technique: compute inner product

Phase I: g keeps decreasing


I g ,∑d

i=1(‖ei + w∗i ‖2 − ‖ei + wi‖2).


I g ,∑d

i=1(‖ei + w∗i ‖2 − ‖ei + wi‖2).

What is ∆g here?

ei + wi

O

ei ∆wi

∆‖ei + wi‖2


I g ,∑d

i=1(‖ei + w∗i ‖2 − ‖ei + wi‖2).

What is ∆g here?

∆g ≈∑i

〈−∆wi , ei + wi 〉 = 〈η∇L(W), I + W〉

≈〈η∇L(W), I〉 = ηTr(∇L(W))

ei + wi

O

ei ∆wi

∆‖ei + wi‖2


I g ,∑d

i=1(‖ei + w∗i ‖2 − ‖ei + wi‖2).

What is ∆g here?

∆g ≈∑i

〈−∆wi , ei + wi 〉 = 〈η∇L(W), I + W〉

≈〈η∇L(W), I〉 = ηTr(∇L(W))

What is Tr(∇L(W))?

Tr(∇L(W)) ≈ O(Tr(W∗ −W)) + O(Tr((W∗ −W)uu>)) + dg ei + wi

O

ei ∆wi

∆‖ei + wi‖2


Observation:

Tr(W∗ −W) =d∑i

(1 + w∗i,i − 1− wi,i ) ≈d∑i

(‖ei + w∗i ‖2 − ‖ei + wi‖2) = g


Observation:

Tr(W∗ −W) =d∑i

(1 + w∗i,i − 1− wi,i ) ≈d∑i

(‖ei + w∗i ‖2 − ‖ei + wi‖2) = g

But Tr((W∗ −W)uu>) is hard to bound. Therefore, we considerthe joint updating rule of s , (W∗ −W)u and g .(Below is for illustration only)

‖st+1‖2 ≈ 0.9‖st‖2 + 10η|gt ||gt+1| ≈ 0.9|gt |+ 10η‖st‖2

When 10η < 0.05, ‖st+1‖2 + |gt+1| ≤ 0.95(‖st‖2 + |gt |). So |gt |will be very small.

Main result

Main Theorem (informal).

If input is from Gaussian distribution, ‖W0‖2, ‖W∗‖2 ≤ γ(constant), step size is small, SGD with mini batch and initial pointW0 will reach W∗ after polynomial number of steps, in two phases.

Main result

Main Theorem (informal).

If input is from Gaussian distribution, ‖W0‖2, ‖W∗‖2 ≤ γ(constant), step size is small, SGD with mini batch and initial pointW0 will reach W∗ after polynomial number of steps, in two phases.

Matches with standard O(1/√d) initialization schemes. (d is

input dimension.)

Open questions

I Multiple layers?

I Other input distributions?

I Convolutional networks?

I Residual link that skips two layers?

I Identify different potential functions for other non-convexproblems?

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mini-course 3: convergence analysis of neural …optimization i in practice, sgd always nds good...

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