G RADIENT DESCENT ALIGNS THE LAYERS OF DEEP LINEAR NETWORKS
BSTRACT
This paper establishes risk convergence and asymptotic weight matrix alignment — a form of implicit regularization — of gradient ﬂow and gradient descent when applied to deep linear networks on linearly separable data.
In more detail, for gradient ﬂow applied to strictly decreasing loss functions (with similar results for gradient descent with particular decreasing step sizes): (i) the risk converges to 0 ; (ii) the normalized i th weight matrix asymptotically equals its rank- 1 approximation u i v (cid:62) i ; (iii) these rank- 1 matrices are aligned across layers, meaning | v (cid:62) i +1 u i | → 1 .
In the case of the logistic loss (binary cross entropy), more can be said: the linear function induced by the network — the product of its weight matrices — converges to the same direction as the maximum margin solution.
This last property was identiﬁed in prior work, but only under assumptions on gradient descent which here are implied by the alignment phenomenon.
I NTRODUCTION
Efforts to explain the effectiveness of gradient descent in deep learning have uncovered an exciting possibility: it not only ﬁnds solutions with low error but also biases the search for low complexity solutions which generalize well (Zhang et al., 2017; Bartlett et al., 2017; Soudry et al., 2017;
Gunasekar et al., 2018).
This paper analyzes the implicit regularization of gradient descent and gradient ﬂow on deep linear networks and linearly separable data.
For strictly decreasing losses, the optimum is at inﬁnity and we establish various alignment phenomena :
• For each weight matrix W i , the corresponding normalized weight matrix W i / (cid:107) W
i (cid:107) F asymptotically equals its rank- 1 approximation u i v (cid:62) i , where the Frobenius norm (cid:107) W i (cid:107) F satisﬁes (cid:107) W i (cid:107) F → ∞ .
In other words, (cid:107) W i (cid:107) 2 / (cid:107) W i (cid:107) F → 1 , and asymptotically only the rankapproximation of W i contributes to the ﬁnal predictor, a form of implicit regularization.
• Adjacent rank- 1 weight matrix approximations are aligned: | v (cid:62) i +1 u
i | → 1 . •
For the logistic loss, the ﬁrst right singular vector v 1 of W 1 is aligned with the data, meaning v 1 converges to the unique maximum margin predictor ¯ u deﬁned by the data.
Moreover, the linear predictor induced by the network, w prod := W L · · · W 1 , is also aligned with the data, meaning w prod / (cid:107) w prod (cid:107) → ¯ u .
Simultaneously, this work proves that the risk is globally optimized: it asymptotes to 0.
Alignment and risk convergence are proved simultaneously; the phenomena are coupled within the proofs.
Since the layers align, they can be viewed as a minimum norm solution : they do not “waste norm” on components which are killed off when the layers are multiplied together.
Said another way, given data (( x i , y i ))
ni =1 , the normalized matrices ( W 1 / (cid:107) W 1 (cid:107) F , . . .
, W L / (cid:107) W L (cid:107) F ) asymptotically solve a maximum margin problem which demands all weight matrices be small, not merely their product:
[Equation]
1 layer 4 layers (a) Margin maximization.
risk ||
W
|| 2 /
|| W
||
F || W 2 || 2 /
|| W
|| F || W
|| 2 /
|| W
|| F || W
|| 2 /
|| W
|| F
[Figure]
The paper is organized as follows.
This introduction continues with related work, notation, and assumptions in Sections 1.1 and 1.2.
The analysis of gradient ﬂow is in Section 2, and gradient descent is analyzed in Section 3.
The paper closes with future directions in Section 4; a particular highlight is a preliminary experiment on CIFAR-10 which establishes empirically that a form of the alignment phenomenon occurs on the standard nonlinear network AlexNet.
1. R ELATED WORK
On the implicit regularization of gradient descent, Soudry et al.
(2017) show that for linear predictors and linearly separable data, the gradient descent iterates converge to the same direction as the maximum margin solution.
Ji & Telgarsky (2018) further characterize such an implicit bias for general nonseparable data.
Gunasekar et al.
(2018) consider gradient descent on fully connected linear networks and linear convolutional networks.
In particular, for the exponential loss, assuming the risk is minimized to 0 and the gradients converge in direction, they show that the whole network converges in direction to the maximum margin solution.
These two assumptions are on the gradient descent process itself, and speciﬁcally the second one might be hard to interpret and justify.
Compared with Gunasekar et al.
, this paper proves that the risk converges to 0 and the weight matrices align; moreover the proof here proves the properties simultaneously, rather than assuming one and deriving the other.
Lastly, Arora et al.
(2018) show for deep linear networks (and later Du et al.
(2018) for ReLU networks) that gradient ﬂow does not change the difference between squared
Frobenius norms of any two layers.
We use a few of these tools in our proofs; please see Sectionsand 3 for details.
For a smooth (nonconvex) function, Lee et al.
(2016) show that any strict saddle can be avoided almost surely with small step sizes.
If there are only countably many saddle points and they are all strict, and if gradient descent iterates converge, then this implies (almost surely) they converge to a local minimum.
In the present work, since there is no ﬁnite local minimum, gradient descent will go to inﬁnity and never converge, and thus these results of Lee et al.
(2016) do not show that the risk converges to 0 .
There has been a rich literature on linear networks.
Saxe et al.
(2013) analyze the learning dynamics of deep linear networks, showing that they exhibit some learning patterns similar to nonlinear networks, such as a long plateau followed by a rapid risk drop.
Arora et al.
(2018) show that depth can help accelerate optimization.
On the landscape properties of deep linear networks, Lu & Kawaguchi (2017); Laurent & von Brecht (2017) show that under various structural assumptions, all local optima are global.
Zhou & Liang (2018) give a necessary and sufﬁcient characterization of critical points for deep linear networks.
1 layer 3 layers (a) Overall margin maximization.
2 principal component 1 of input 1.
p r i n c i pa l c o m ponen t 2 o f i npu
t rsv 1 of layer(b)
Margin maximization at layer 1.
3 principal component 1 of input 1.
p r i n c i pa l c o m ponen t 2 o f i npu
t rsv 1 of layer(c
[Figure]
maximization at layer 2.
3 principal component 1 of input 1.
p r i n c i pa l c o m ponen t 2 o f i npu
t rsv 1 of layer(
[Figure]
1. N OTATION , SETTING , AND ASSUMPTIONS
Consider a data set { ( x i , y i ) } ni =1 , where x
i ∈ R d , (cid:107) x
i (cid:107) ≤ 1 , and y i ∈ {− 1 , +1 } .
The data set is assumed to be linearly separable, i.e., there exists a unit vector u which correctly classiﬁes every data point: for any 1 ≤ i ≤ n , y i (cid:104) u, x i (cid:105) > 0 .
Furthermore, let γ :=
max (cid:107) u (cid:107) =
1 min 1 ≤ i ≤ n y i (cid:104) u, x i (cid:105) >denote the maximum margin, and ¯ u := arg max (cid:107) u (cid:107) =
1 min 1 ≤ i ≤ n y i (cid:104) u, x i (cid:105) denote the maximum margin solution (the solution to the hard-margin SVM).
A linear network of depth L is parameterized by weight matrices W L , . .
, W 1 , where
W k ∈ R
d
k × d k − 1 , d 0 = d , and d L = 1 .
Let W =
( W L , . . .
, W 1 ) denote all parameters of the network.
The (empirical) risk induced by the network is given by
[Equation]
, where w prod :=
( W L · · · W 1 ) (cid:62) , and z i :=
y i x i .
The loss (cid:96) is assumed to be continuously differentiable, unbounded, and strictly decreasing to 0 .
Examples include the exponential loss (cid:96) exp ( x ) =
e − x and the logistic loss (cid:96) log ( x ) =
ln (cid:0) 1 + e − x
[Equation]
Assumption 1.
(cid:96) (cid:48) < 0 is continuous, lim x →−∞ (cid:96) ( x ) = ∞ and lim x →∞ (cid:96) ( x ) = 0 .
This paper considers gradient ﬂow and gradient descent, where gradient ﬂow (cid:8) W ( t ) (cid:12)(cid:12) t ≥ 0 , t ∈ R (cid:9) can be interpreted as gradient descent with inﬁnitesimal step sizes.
It starts from some W (0) at t = 0 , and proceeds as
[Equation]
By contrast, gradient descent (cid:8) W ( t ) (cid:12)(cid:12) t ≥ 0 , t ∈ Z (cid:9) is a discrete-time process given by
[Equation]
where η t is the step size at time t .
We assume that the initialization of the network is not a critical point and induces a risk no larger than the risk of the trivial linear predictor 0 .
Assumption 2.
The initialization W (0) satisﬁes ∇R (cid:0) W (0) (cid:1) (cid:54) = 0 and R (cid:0) W (0) (cid:1) ≤ R (0) =
(cid:96) (0) .
It is natural to require that the initialization is not a critical point, since otherwise gradient ﬂow/descent will never make a progress.
The requirement R (cid:0) W (0) (cid:1) ≤ R (0) can be easily satisﬁed, for example, by making W 1 (0)
= 0 and W L (0) · · · W 2 (0) (cid:54) = 0 .
On the other hand, if
R (cid:0) W (0) (cid:1) > R (0) , gradient ﬂow/descent may never minimize the risk to 0 .
Proofs of those claims are given in Appendix A.
R ESULTS FOR GRADIENT FLOW
In this section, we consider gradient ﬂow.
Although impractical when compared with gradient descent, gradient ﬂow can simplify the analysis and highlight proof ideas.
For convenience, we usually use W , W k , and w prod , but they all change with (the continuous time) t .
Only proof sketches are given here; detailed proofs are deferred to Appendix B.
2. R ISK CONVERGENCE
One key property of gradient ﬂow is that it never increases the risk:
[Equation]
We now state the main result: under Assumptions 1 and 2, gradient ﬂow minimizes the risk, W k and w prod all go to inﬁnity, and the alignment phenomenon occurs.
Theorem 1.
Under Assumptions 1 and 2, gradient ﬂow iterates satisfy the following properties:
• lim t →∞ R ( W ) = 0 .
• For any 1 ≤ k ≤ L , lim t →∞ (cid:107) W k (cid:107) F = ∞ .
• For any 1 ≤ k ≤ L , letting ( u k , v k ) denote the ﬁrst left and right singular vectors of W k ,
[Equation]
Moreover, for any 1 ≤ k < L , lim t →∞ (cid:12)(cid:12)
(cid:104)
v k +1 , u k (cid:105) (cid:12)(cid:12) = 1 .
As a result,
[Equation]
and thus lim t →∞ (cid:107)
[Equation]
Theorem 1 is proved using two lemmas, which may be of independent interest.
To show the ideas, let us ﬁrst introduce a little more notation.
Recall that R ( W ) denotes the empirical risk induced by the deep linear network W .
Abusing the notation a little, for any linear predictor w ∈ R d , we also use R ( w ) to denote the risk induced by w .
With this notation, R ( W )
=
R ( w prod ) , while
[Equation]
is in R d and different from ∇R ( W ) , which has (cid:80)
Lk
=
d k d k − 1 entries, as given below:
[Equation]
Furthermore, for any R > 0 , let
[Equation]
The ﬁrst lemma shows that for any R > 0 , the time spent by gradient ﬂow in B ( R ) is ﬁnite.
Lemma 1.
Under Assumption 1 and 2, for any R > 0 , there exists a constant (cid:15) ( R ) > 0 , such that for any t ≥ 1 and any W ∈ B ( R ) , (cid:107) ∂ R /∂W 1 (cid:107) F ≥ (cid:15) ( R ) .
As a result, gradient ﬂow spends a ﬁnite amount of time in B ( R ) for any R > 0 , and max 1 ≤ k ≤ L (cid:107) W k (cid:107) F is unbounded.
Here is the proof sketch.
If (cid:107) W k (cid:107) F are bounded, then (cid:107)∇R ( w prod ) (cid:107) will be lower bounded by a positive constant, therefore if (cid:107) ∂ R /∂W 1 (cid:107) F = (cid:107) W L · · · W 2 (cid:107)(cid:107)∇R ( w prod ) (cid:107) can be arbitrarily small, then (cid:107) W L · · · W 2 (cid:107) and (cid:107) w prod (cid:107) can also be arbitrarily small, and thus R ( W ) can be arbitrarily close to R (0) .
This cannot happen after t = 1 , otherwise it will contradict Assumption 2 and eq.
To proceed, we need the following properties of linear networks from prior work (Arora et al., 2018;
Du et al., 2018).
For any time t ≥ 0 and any 1 ≤ k < L ,
[Equation]
To see this, just notice that
[Equation]
Taking the trace on both sides of eq.
(2), we have
[Equation]
In other words, the difference between the squares of Frobenius norms of any two layers remains a constant.
Together with Lemma 1, it implies that all (cid:107) W k (cid:107) F are unbounded.
However, even if (cid:107) W k (cid:107) F are large, it does not follow necessarily that (cid:107) w prod (cid:107) is also large.
Lemma 2 shows that this is indeed true: for gradient ﬂow, as (cid:107) W k (cid:107) F get larger, adjacent layers also get more aligned to each other, which ensures that their product also has a large norm.
For 1 ≤ k ≤ L , let σ k , u k , and v k denote the ﬁrst singular value (the 2 -norm), the ﬁrst left singular vector, and the ﬁrst right singular vector of W k , respectively.
Furthermore, deﬁne
[Equation]
which depends only on the initialization.
If for any 1 ≤ k < L , W k (0) W (cid:62) k (0) =
W (cid:62) k +1 (0) W k +1 (0) , then D = 0 .
Lemma 2.
The gradient ﬂow iterates satisfy the following properties:
• For any 1 ≤ k ≤ L , (cid:107) W k (cid:107) 2 F − (cid:107) W k (cid:107) 22 ≤ D .
• For any 1 ≤ k < L , (cid:104) v k +1 , u k (cid:105) 2 ≥ 1 − (cid:0) D + (cid:107) W k +1 (0) (cid:107)
+ (cid:107) W k
(0) (cid:107) 22 (cid:1) / (cid:107) W k +1 (cid:107) 22 .
• Suppose max 1 ≤ k ≤ L (cid:107) W k (cid:107) F → ∞ , then (cid:12)(cid:12)(cid:12)(cid:10) w prod / (cid:81)
Lk =1 (cid:107) W k (cid:107) F , v 1 (cid:11)(cid:12)(cid:12)(cid:12)
→ 1 .
The proof is based on eq.
(2) and eq.
If W k (0) W (cid:62) k (0) =
W (cid:62) k +1 (0) W k +1 (0) , then eq.
(2) gives that W k +1 and W k have the same singular values, and
W k +1 ’s right singular vectors and W k ’s left singular vectors are the same.
If it is true for any two adjacent layers, since W L is a row vector, all layers have rank 1 .
With general initialization, we have similar results when (cid:107) W k (cid:107) F is large enough so that the initialization is negligible.
Careful calculations give the exact results in Lemma 2.
An interesting point is that the implicit regularization result in Lemma 2 helps establish risk convergence in Theorem 1.
Speciﬁcally, by Lemma 2, if all layers have large norms, (cid:107) W L · · · W 2 (cid:107) will be large.
If the risk is not minimized to 0 , (cid:107)∇R ( w prod ) (cid:107) will be lower bounded by a positive constant, and thus (cid:107) ∂ R /∂W 1 (cid:107) F = (cid:107) W L · · · W 2 (cid:107)(cid:107)∇R ( w prod ) (cid:107) will be large.
Invoking eq.
(1), Lemma 1 and eq.
(3) gives a contradiction.
Since the risk has no ﬁnite optimum, (cid:107) W k (cid:107) F → ∞ .
2. C ONVERGENCE TO THE MAXIMUM MARGIN SOLUTION
Here we focus on the exponential loss (cid:96) exp ( x ) =
e − x and the logistic loss (cid:96) log ( x ) =
ln(1 + e − x ) .
In addition to risk convergence, these two losses also enable gradient descent to ﬁnd the maximum margin solution.
To get such a strong convergence, we need one more assumption on the data set.
Recall that γ = max (cid:107) u (cid:107) =1 min 1 ≤ i ≤ n (cid:104) u, z i (cid:105) > 0 denotes the maximum margin, and ¯ u denotes the unique maximum margin predictor which attains this margin γ .
Those data points z i for which (cid:104) ¯ u, z i (cid:105)
=
γ are called support vectors.
Assumption 3.
The support vectors span the whole space R d .
Assumption 3 appears in prior work Soudry et al.
(2017), and can be satisﬁed in many cases: for example, it is almost surely true if the number of support vectors is larger than or equal to d and the data set is sampled from some density w.r.t.
the Lebesgue measure.
It can also be relaxed to the situation that the support vectors and the whole data set span the same space; in this case ∇R ( w prod ) will never leave this space, and we can always restrict our attention to this space.
With Assumption 3, we can state the main theorem.
Theorem 2.
Under Assumptions 2 and 3, for almost all data and for losses (cid:96) exp and (cid:96) log , then lim t →∞ (cid:12)(cid:12) (cid:104) v 1 , ¯ u (cid:105) (cid:12)(cid:12) = 1 , where v 1 is the ﬁrst right singular vector of W 1 .
As a result,
[Equation]
Before summarizing the proof, we can simplify both theorems into the following minimum norm property mentioned in the introduction.
Corollary 1.
Under Assumptions 2 and 3, for almost all data and for losses (cid:96) exp and (cid:96) log , min i
y
i
[Equation]
Theorem 2 relies on two structural lemmas.
The ﬁrst one is based on a similar almost-all argument due to Soudry et al.
(2017, Lemma 12).
Let S ⊂ { 1 , . . . , n } denote the set of indices of support vectors.
Lemma 3.
Under Assumption 3, if the data set is sampled from some density w.r.t.
the Lebesgue measure, then with probability 1 ,
[Equation]
Let ¯ u ⊥ denote the orthogonal complement of span(¯ u ) , and let Π ⊥ denote the projection onto ¯ u ⊥ .
We prove that if (cid:107) Π ⊥ w (cid:107) is large enough, gradient ﬂow starting from w will tend to decrease (cid:107) Π ⊥ w (cid:107) .
Lemma 4.
Under Assumption 3, for almost all data, (cid:96) exp and (cid:96) log , and any w ∈ R d , if (cid:104) w, ¯ u (cid:105)
≥and (cid:107) Π
⊥ w (cid:107) is larger than 1+ln( n ) / α for (cid:96) exp or 2 n / eα for (cid:96) log , then (cid:104)
Π ⊥ w, ∇R ( w ) (cid:105) ≥ 0 .
With Lemma 3 and Lemma 4 in hand, we can prove Theorem 2.
Let Π ⊥ W 1 denote the projection of rows of W 1 onto ¯ u ⊥ .
Notice that
[Equation]
and
[Equation]
If (cid:107) Π ⊥ W 1 (cid:107) F is large compared with (cid:107) W 1 (cid:107) F , since layers become aligned, (cid:107)
Π ⊥ w prod (cid:107) will also be large, and then Lemma 4 implies that (cid:107) Π ⊥ W 1 (cid:107) F will not increase.
At the same time, (cid:107) W 1 (cid:107) F → ∞ , and thus for large enough t , (cid:107) Π ⊥ W 1 (cid:107) F must be very small compared with (cid:107) W 1 (cid:107) F .
Many details need to be handled to make this intuition exact; the proof is given in Appendix B.
R ESULTS FOR GRADIENT DESCENT
One key property of gradient ﬂow which is used in the previous proofs is that it never increases the risk, which is not necessarily true for gradient descent.
However, for smooth losses (i.e, with
Lipschitz continuous derivatives), we can design some decaying step sizes, with which gradient descent never increases the risk, and basically the same results hold as in the gradient ﬂow case.
Deferred proofs are given in Appendix C.
We make the following additional assumption on the loss, which is satisﬁed by the logistic loss (cid:96) log .
Assumption 4.
(cid:96) (cid:48) is β -Lipschitz (i.e, (cid:96) is β -smooth), and | (cid:96) (cid:48) | ≤ G (i.e., (cid:96) is G -Lipschitz).
Under Assumption 4, the risk is also a smooth function of W , if all layers are bounded.
Lemma 5.
Suppose (cid:96) is β -smooth.
If R ≥ 1 , then β ( R ) = 2 L 2 R 2 L − 2 ( β + G ) , and R ( W ) is a β ( R ) -smooth function on the set B ( R )
= (cid:8) W (cid:12)(cid:12) (cid:107) W k (cid:107) F ≤ R, 1 ≤ k ≤ L (cid:9) .
Smoothness ensures that for any W, V ∈ B ( R ) , R ( W ) − R ( V ) ≤ (cid:104)∇R ( V ) , W − V (cid:105) +
β
( R ) (cid:107) W − V (cid:107) 2 / 2 (see Bubeck et al.
Lemma 3.4).
In particular, if we choose some R and set a constant step size η t = 1 /β
( R ) , then as long as W ( t + 1) and W ( t ) are both in B ( R ) ,
R (cid:0) W ( t + 1) (cid:1) − R (cid:0) W ( t ) (cid:1) ≤ (cid:68) ∇R (cid:0) W ( t ) (cid:1) , − η t ∇R (cid:0) W ( t ) (cid:1)(cid:69) + β ( R ) η 2 t
[Equation]
In other words, the risk does not increase at this step.
However, similar to gradient ﬂow, the gradient descent iterate will eventually escape B ( R ) , which may increase the risk.
Lemma 6.
Under Assumption 1, 2 and 4, suppose gradient descent is run with a constant step size 1 /β ( R ) .
Then there exists a time t
when W ( t ) (cid:54)∈ B ( R ) , in other words, max 1 ≤ k ≤ L (cid:107) W k
( t ) (cid:107) F > R .
Fortunately, this issue can be handled by adaptively increasing R and correspondingly decreasing the step sizes, formalized in the following assumption.
Assumption 5.
The step size η t = min { 1 /β
( R t ) , 1 } , where R t satisﬁes
W ( t ) ∈ B ( R t − 1) , and if
W ( t + 1) ∈ B ( R t − 1) , R t +1 =
R t .
Assumption 5 can be satisﬁed by a line search, which ensures that the gradient descent update is not too aggressive and the boundary R is increased properly.
With the additional Assumptions 4 and 5, exactly the same theorems can be proved for gradient descent.
We restate them brieﬂy here.
Theorem 3.
Under Assumption 1, 2, 4, and 5, gradient descent satisﬁes
• lim t →∞ R (cid:0) W ( t ) (cid:1) = 0 .
• For any 1 ≤ k ≤ L , lim t →∞ (cid:107) W k ( t ) (cid:107) F = ∞ .
• lim t →∞ (cid:12)(cid:12) (cid:12) (cid:10) w prod ( t ) / (cid:81)
Lk =1 (cid:107) W k ( t ) (cid:107) F , v 1 ( t ) (cid:11)(cid:12)(cid:12) (cid:12) = 1 , where v 1 ( t ) is the ﬁrst right singular vector of W 1 ( t ) .
Theorem 4.
Under Assumptions 2, 3 and 5, for the logistic loss (cid:96) log and almost all data, lim
[Equation]
v [Equation] t
[Equation]
, ¯ u
[Equation]
= 1 , and lim t
[Equation]
Corollary 2.
Under Assumptions 2, 3 and 5, for the logistic loss (cid:96) log and almost all data, min i
[Equation]
Proofs of Theorem 3 and 4 are given in Appendix C, and are basically the same as the gradient ﬂow proofs.
The key difference is that an error of (cid:80) ∞ t =
η 2 t (cid:107)∇R ( W ( t ))
(cid:107) 2 will be introduced in many parts of the proof.
However, it is bounded in light of eq.
[Equation]
[Table]
Since all weight matrices go to inﬁnity, such a bounded error does not matter asymptotically, and thus proofs still go through.
risk || W 3 || 2 /
|| W
|| F || W
|| 2 /
|| W
|| F || W
|| 2 /
|| W
|| F (a) Default initialization.
risk || W 3 || 2 /
|| W
|| F || W
|| 2 /
|| W
|| F || W
|| 2 /
|| W
|| F
[Figure]
S UMMARY AND FUTURE DIRECTIONS
This paper rigorously proves that, for deep linear networks on linearly separable data, gradient ﬂow and gradient descent minimize the risk to 0 , align adjacent weight matrices, and align the ﬁrst right singular vector of the ﬁrst layer to the maximum margin solution determined by the data.
There are many potential future directions; a few are as follows.
Convergence rates and practical step sizes.
This paper only proves asymptotic convergence, moreover for adaptive step sizes depending on the current weight matrix norms.
A reﬁned analysis with rates for practical step sizes (e.g., constant step sizes) would allow the algorithm to be compared to other methods which also globally optimize this objective, would suggest ways to improve step sizes and initialization, and ideally even exhibit a sensitivity to the network architecture and suggest how it could be improved.
Nonseparable data and nonlinear networks.
Real-world data is generally not linearly separable, but nonlinear deep networks can reliably decrease the risk to 0, even with random labels (Zhang et al., 2017).
This seems to suggest that a nonlinear notion of separability is at play; is there some way to adapt the present analysis?
The present analysis is crucially tied to the alignment of weight matrices: alignment and risk are analyzed simultaneously.
Motivated by this, consider a preliminary experiment, presented in Figure 3, where stochastic gradient descent was used to minimize the risk of a standard AlexNet on CIFAR-
(Krizhevsky et al., 2012; Krizhevsky & Hinton, 2009).
Even though there are ReLUs, max-pooling layers, and convolutional layers, the alignment phenomenon is occurring in a reduced form on the dense layers (the last three layers of the network).
Speciﬁcally, despite these weight matrices having shape (1024 , 4096) , (4096 , 4096) , and (4096 , 10) the key alignment ratios (cid:107) W i (cid:107) 2 / (cid:107) W
i
(cid:107) F are much larger than their respective lower bounds (1024 − 1 / 2 , 4096 − 1 / 2 , 10 − 1 / 2 ) .
Two initializations were tried: default PyTorch initialization, and a Gaussian initialization forcing all initial Frobenius norms to be just 4 , which is suggested by the norm preservation property in the analysis and removes noise in the weights.
A CKNOWLEDGEMENTS
The authors are grateful for support from the NSF under grant IIS-1750051.
This grant allowed them to focus on research, and when combined with an NVIDIA GPU grant, led to the creation of their beloved GPU machine D UTCH C RUNCH .
A R EGARDING A SSUMPTION
Suppose W 1 (0) =
while W L (0) · · · W 2 (0) (cid:54) = 0 .
First of all, W L (0) · · · W 1 (0) = 0 and thus
R (cid:0) W (0) (cid:1) =
R (0) .
Moreover,
[Equation]
n n
[Equation] [Equation]
≤
[Equation]
(cid:48)
[Equation] [Equation]
< 0 , which implies ∇R (cid:0) w prod (0) (cid:1) (cid:54) = 0 and ∂ R /∂W
= (cid:0) W L (0) · · · W 2 (0) (cid:1) (cid:62) ∇R (cid:0) w prod (0) (cid:1) (cid:62) (cid:54) = 0 .
On the other hand, if R (cid:0) W (0) (cid:1) > R (0) , gradient ﬂow/descent may never minimize the risk to 0 .
For example, suppose the network has two layers, and both the input and output have dimension 1 ; the network just computes the dot product of two vectors w 1 and w 2 .
Consider minimizing R ( w 1 , w 2 )
=
exp (cid:0) −(cid:104)
w 1 , w 2 (cid:105) (cid:1) .
If w 1 (0) = − w 2 (0) (cid:54) = 0 , then R (cid:0) w 1 (0) , w 2 (0) (cid:1) =
exp (cid:0) (cid:107) w 1 (cid:107) 2 (cid:1) > exp(0) .
It is easy to verify that for any t , w 1 ( t ) = − w 2 ( t ) , and R (cid:0) w 1 ( t ) , w 2 ( t ) (cid:1) ≥ exp(0) > 0 .
B O MITTED PROOFS FROM S ECTION
Proof of Lemma 1.
Fix an arbitrary R > 0 .
If the claim is not true, then for any (cid:15) > 0 , there exists some t ≥ 1 such that (cid:107) W k (cid:107) F ≤ R for all k while (cid:13)(cid:13)
∂ R /∂W 1 (cid:13)(cid:13) 2 F ≤ (cid:15) 2 , which means (cid:13)(cid:13)(cid:13)(cid:13) ∂
R ∂W 1 (cid:13)(cid:13)(cid:13)(cid:13)
F =
[Equation]
W
[Equation]
( w
[Equation]
(cid:62) (cid:13)(cid:13)(cid:13)
[Equation]
( w
[Equation]
) (cid:13)(cid:13) 2 ≤ (cid:15) 2 .
Since (cid:107) w prod (cid:107) ≤ R L , we have (cid:104)∇R ( w prod ) , ¯ u (cid:105) = 1 n n (cid:88) i = (cid:96) (cid:48) (cid:0) (cid:104) w prod , z i (cid:105) (cid:1) (cid:104)
z i , ¯ u (cid:105) ≤ 1 n n (cid:88) i = (cid:96) (cid:48) (cid:0) (cid:104) w prod , z i (cid:105) (cid:1) γ ≤ − Mγ, where − M = max − R L ≤ x ≤ R L (cid:96) (cid:48)
( x ) .
Since (cid:96) (cid:48) is continuous and the domain is bounded, the maximum is attained and negative, and thus M > 0 .
Therefore (cid:13)(cid:13) ∇R ( w prod ) (cid:13)(cid:13) ≥ Mγ , and thus (cid:107) W L · · · W 2 (cid:107) ≤ (cid:15)/Mγ .
Since (cid:107) W 1 (cid:107) F ≤ R , we further have (cid:107) w prod (cid:107) ≤ (cid:15)R/Mγ .
In other words, after t = 1 , (cid:13)(cid:13) w prod (cid:13)(cid:13) may be arbitrarily small, which implies R (cid:0) w prod (cid:1) can be arbitrarily close to
R (0) .
On the other hand, by Assumption 2, d R ( W ) / d t = −(cid:107)∇R ( W ) (cid:107) 2 < 0 at t = 0 .
This implies that
R (cid:0) W (1) (cid:1) <
R (cid:0) W (0) (cid:1) , and for any t ≥ 1 , R (cid:0) W ( t ) (cid:1) ≤ R (cid:0) W (1) (cid:1) <
R (cid:0) W (0) (cid:1) ≤ R (0) , which is a contradiction.
Since the risk is always positive, we have
[Equation]
F d t ≥ (cid:90) ∞ t = (cid:13)(cid:13)(cid:13)(cid:13) ∂
R ∂W 1 (cid:13)(cid:13)(cid:13)(cid:13)
F 1 (cid:20) max 1 ≤ k ≤ L (cid:107) W k (cid:107) F ≤ R (cid:21) d t ≥ (cid:90) ∞ t = (cid:13)(cid:13) (cid:13) (cid:13) ∂ R ∂W(cid:13)(cid:13)
(cid:13) (cid:13)
F 1 (cid:20) max 1 ≤ k ≤ L (cid:107) W k (cid:107) F ≤ R (cid:21) d t ≥ (cid:15) ( R ) 2 (cid:90) ∞ t = 1 (cid:20) max 1 ≤ k ≤ L (cid:107) W k (cid:107) F ≤ R (cid:21) d t, which implies gradient ﬂow only spends a ﬁnite amount of time in (cid:8) W (cid:12)(cid:12) max 1 ≤ k ≤ L (cid:107) W k (cid:107) F ≤ R (cid:9) .
This directly implies that max 1 ≤ k ≤ L (cid:107) W k (cid:107) F is unbounded.
Proof of Lemma 2.
The ﬁrst claim is true for k
=
L since W L is a row vector.
For any 1 ≤ k < L , recall that Arora et al.
Du et al.
(2018) give the following relation:
[Equation]
Let A k,k
+
=
W k (0) W (cid:62) k
(0) − W (cid:62) k +1 (0) W k +1 (0) .
By eq.
(5) and the deﬁnition of singular vectors and singular values, we have
[Equation]
Moreover, by taking the trace on both sides of eq.
(5), we have
[Equation]
Summing eq.
(6) and eq.
(7) from k to L − 1 , we get
[Equation]
Next we prove that singular vectors get aligned.
Consider u (cid:62) k W (cid:62)
k +1 W k +1 u k .
On one hand, similar to eq.
[Equation]
we can get that
[Equation] [Equation]
On the other hand
[Equation]
it follows from the deﬁnition of singular vectors and eq.
(8) that
[Equation]
Combining eq.
(9) and eq.
(10), we get
[Equation]
Similar to eq.
(9), we can get
[Equation]
Therefore
[Equation]
Combining eq.
(11) and eq.
(12), we ﬁnally get
[Equation]
Regarding the last claim, ﬁrst recall that since the difference between the squares of Frobenius norms of any two layers is a constant, max 1 ≤ k ≤ L (cid:107) W k (cid:107) F → ∞ implies (cid:107) W k (cid:107) F → ∞ for any k .
We further have the following.
• Since (cid:107) W k (cid:107) 2 F − (cid:107) W k (cid:107) 2 2 ≤ D , (cid:107) W k (cid:107) 2 → ∞ for any k , and W k / (cid:107)
W k (cid:107) F → u k v (cid:62) k .
• Since (cid:107) W k (cid:107) 2 → ∞ , |(cid:104) u k , v k +
(cid:105)| → 1 .
As a result,
[Equation]
w
[Equation] [Equation]
W k (cid:107) F
[Equation] [Equation]
[Table]
Proof of Theorem 1.
Suppose for some (cid:15) > 0 , R ( W ) ≥ (cid:15) for any t .
Then there exists some 1 ≤ j ≤ n such that (cid:96) (cid:0) (cid:104) w prod , z j (cid:105) (cid:1) ≥ (cid:15) , and thus (cid:104) w prod , z j (cid:105) ≤ (cid:96) − 1 ( (cid:15) ) .
On the other hand, since R ( W ) ≤ R (0) =
(cid:96) (0) , (cid:96) (cid:0) (cid:104) w prod , z j (cid:105) (cid:1) ≤ n(cid:96) (0) , and thus (cid:104) w prod , z j (cid:105) ≥ (cid:96) − 1 (cid:0) n(cid:96) (0) (cid:1) .
Let − M = max (cid:96) − 1 ( n(cid:96) (0) )
≤ x ≤ (cid:96) − 1 ( (cid:15)/n ) (cid:96) (cid:48) ( x ) < 0 , we have for any t , (cid:104)∇R ( w prod ) , ¯ u (cid:105) = 1 n n (cid:88) i = (cid:96) (cid:48)
(cid:0) (cid:104) w prod , z i (cid:105) (cid:1) (cid:104)
z i , ¯ u (cid:105) ≤n n (cid:88) i
=
(cid:96) (cid:48) (cid:0) (cid:104) w prod , z i (cid:105) (cid:1) γ ≤ 1 n(cid:96) (cid:48) (cid:0) (cid:104) w prod , z j (cid:105) (cid:1) γ ≤ − Mγ n < 0 , and thus (cid:107)∇R ( w prod ) (cid:107) ≥ Mγ/n .
Similar to the proof of Lemma 2, we can show that if (cid:107) W k (cid:107) F → ∞ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:42) ( W L · · · W 2 ) (cid:62) (cid:107) W k (cid:107) F · · · (cid:107) W 2 (cid:107) F , v 2 (cid:43)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) → 1 .
In other words, there exists some C > 0 , such that when min 1 ≤ k ≤ L (cid:107) W k (cid:107) F > C , (cid:107) W L · · · W 2 (cid:107) ≥ (cid:107) W k (cid:107) F · · · (cid:107) W 2 (cid:107) F / 2 > C L / 2 .
Lemma 1 shows that gradient ﬂow spends a ﬁnite amount of time in (cid:8) W (cid:12)(cid:12) max 1 ≤ k ≤ L (cid:107) W k (cid:107) F ≤ R (cid:9) for any R > 0 .
Since the difference between the squares of Frobenius norms of any two layers is a constant, gradient ﬂow also spends a ﬁnite amount of time in (cid:8) W (cid:12)(cid:12) min 1 ≤ k ≤ L (cid:107) W k (cid:107) F ≤ C (cid:9) .
Now we have
[Equation]
which is a contradiction.
Therefore R ( (cid:15) )
→ 0 .
This further implies (cid:107) W k (cid:107) F → ∞ , since R ( W ) has no ﬁnite optimum.
Finally, invoking Lemma 2 proves the ﬁnal claim of Theorem 1.
Proof of Lemma 3.
Soudry et al.
Lemma 12 proves that, with probability 1 , there are at most d support vectors, and moreover, the i -th support vector z i has a positive dual variable α i , such that (cid:80) i ∈
S
α
i z i
= ¯ u .
Suppose there exists some ξ ⊥ ¯ u , such that max i ∈ S (cid:104) ξ, z i (cid:105) ≤ 0 .
Since
[Equation]
we actually have (cid:104) ξ, z i (cid:105) =
for all i ∈ S .
This is impossible under Assumption 3, since the support vectors span the whole space.
Proof of Lemma 4.
For the sake of presentation, we leave out the subscript in z i and denote a data point by z generally.
For any data point z and predictor w , let z ⊥ and w ⊥ denote their projection onto ¯ u ⊥ .
Let z (cid:48) ∈ arg max i ∈ S (cid:104)− w ⊥ , z (cid:105) , and thus (cid:104)− w ⊥ , z (cid:48) (cid:105) ≥ α (cid:107) w ⊥ (cid:107) .
For (cid:96) exp , we have
[Equation]
The ﬁrst part can be lower bounded as below (recall that (cid:104)− w ⊥ ,
z (cid:48)⊥
[Equation]
= (cid:104)− w ⊥
[Equation]
z
[Equation]
≥ α
[Equation]
⊥
[Equation] [Equation]
To bound the second part, ﬁrst notice that since we assume (cid:104) w, ¯ u (cid:105) ≥ 0 , for any z ,
[Equation]
The reason is that every data point has margin at least γ , and thus z − γ ¯ u − z ⊥ =
c ¯ u for some c ≥ 0 .
Using eq.
(15), we can bound the second part of eq.
[Equation] [Equation]
On the third line eq.
(15) is applied.
The fourth line applies the property that f ( x ) = − xe − x ≥ − 1 /e when x ≥ 0 .
Combining eq.
(13), eq.
(14) and eq.
(16), we get
[Equation]
As long as (cid:107) w ⊥ (cid:107) ≥ (1 + ln( n )) /α , (cid:104) w ⊥ , ∇R ( w ) (cid:105) ≥ 0 .
For (cid:96) log , similar to eq.
(13), we have
[Equation] [Equation]
The second part of eq.
(17) can be bounded again by eq.
To bound the ﬁrst part of eq.
(17), ﬁrst notice that (recall (cid:104) w, ¯ u (cid:105) ≥ 0 )
[Equation]
Using eq.
(18), and recall that (cid:104)− w ⊥ , z (cid:48)⊥ (cid:105) =
(cid:104)− w ⊥ , z (cid:48) (cid:105) ≥ α (cid:107) w ⊥ (cid:107) ≥ 0 , we can bound the ﬁrst part of eq.
(17) as below.
[Equation] [Equation]
Combining eq.
(17), eq.
(19) and eq.
(16), we get
[Equation]
As long as
[Equation] [Equation]
Proof of Theorem 2.
Recall that
[Equation]
and thus
[Equation]
Let Π ¯
u
denote the projection onto span(¯ u ) , and let Π ⊥ denote the projection onto ¯ u ⊥ .
Also let Π ¯ u W 1 and Π ⊥ W 1 denote the projection of rows of W 1 onto span(¯ u ) and ¯ u ⊥ , respectively.
Notice that
[Equation]
We further have
[Equation]
Let W 1 = u 1 σ 1 v (cid:62) 1 + S .
We have (cid:107) S (cid:107) 2 ≤ σ 1 , 2 ≤ (cid:113) σ 21 , 2 ≤ (cid:112) (cid:107) W 1 (cid:107) 2 F −
[Equation]
W 1 (cid:107) 22 ≤ √ D , where σ 1 , 2 is the second singular value of W 1 and D is the constant introduced in Lemma 2.
Then
[Equation]
and
[Equation]
It follows that
[Equation]
Fix an arbitrary (cid:15) > 0 .
By Theorem 1, we can ﬁnd some t 0 large enough such that for any t ≥ t 0 :
[Equation]
K
[Equation]
, where K is the threshold given in Lemma 4, i.e., 1+ln( n ) / α for (cid:96) exp , 2 n / eα for (cid:96) log .
R ( W ) ≤ (cid:96) (0) /n , which implies (cid:104) w prod , z i (cid:105) ≥ 0 for all 1 ≤ i ≤ n .
By Lemma 3, there always exists a support vector z for which (cid:104)
Π ⊥ w prod , z (cid:105) ≤ 0 , and therefore (cid:104) w prod , ¯ u (cid:105) ≥ 0 .
Suppose for some t ≥ t 0 , (cid:107) Π ⊥ W 1 (cid:107) F / (cid:107) W 1 (cid:107) F ≥ (cid:15) .
By eq.
(22) and bullet 1 above, (cid:107) Π ⊥ v 1 (cid:107) ≥ 2 (cid:15)/ 3 .
Bullet 2 above then gives (cid:107)
Π ⊥ w prod / (cid:107) W L (cid:107) F ···(cid:107) W 1 (cid:107) F (cid:107) ≥ (cid:15)/ 3 , which together with bulletabove implies (cid:107)
Π ⊥ w prod (cid:107) ≥ K .
Since also (cid:104) w prod , ¯ u (cid:105) ≥ 0 , we can apply Lemma 4 and get that (cid:104) Π ⊥ w prod , ∇R ( w prod ) (cid:105) ≥ 0 .
In light of eq.
(21), d (cid:107) Π ⊥ W 1 (cid:107) 2 F / d t ≤ 0 .
On the other hand, since after t ≥ t 0 , (cid:104) w prod , z i (cid:105) ≥ 0 , we have d (cid:107) W 1 (cid:107) 2 F / d t ≥ 0 by eq.
Therefore (cid:107) Π ⊥ W 1 (cid:107) F / (cid:107) W 1 (cid:107) F will not increase, and since (cid:107) W 1 (cid:107) F → ∞ , it will eventually drop below (cid:15) , and will never exceed (cid:15) again.
Therefore, lim
[Equation]
(cid:107)
[Equation]
≤
[Equation]
Since (cid:15) is arbitrary, we have
[Equation]
and thus lim t →∞ (cid:12)(cid:12) (cid:104) v 1 , ¯ u (cid:105) (cid:12)(cid:12)
= 1 .
An application of Theorem 1 gives the other part of Theorem
[Equation]
Proof of Corollary 1.
By Theorem 2,
[Equation]
Next, pick arbitrary unit vectors a i ∈ R d i
[Equation]
and note
[Equation]
On the other hand
[Equation]
if matrices ( A L , . . . , A 1 ) are feasible
[Equation]
then
[Equation]
whereby
[Equation]
C O MITTED PROOFS FROM S ECTION
Proof of Lemma 5.
Given W, V ∈ B ( R ) , we need to show that (cid:107)∇R ( W ) −∇R
( V ) (cid:107) ≤ β
( R ) (cid:107) W −
V (cid:107) for some β ( R )
[Equation]
Consider k = 1 ﬁrst.
Let w
=
( W L · · · W 1 ) (cid:62)
[Equation]
and v = ( V L · · · V 1 ) (cid:62) .
Since | (cid:96) (cid:48) | ≤ G , (cid:107)∇R ( w ) (cid:107) , (cid:107)∇R ( v ) (cid:107) ≤ G .
We have
[Equation]
Proceeding in this way, we can get
[Equation] [Equation]
Since (cid:107) z i (cid:107) ≤ 1 , (cid:96) (cid:48) is β -Lipschitz, we have
[Equation]
25) where the last inequality follows from a similar one-by-one replacement procedure as in eq.
Combining eq.
(24) and eq.
(25), we get for R ≥ 1 ,
[Equation]
R
[Equation]
The same procedure can be done for other layers, and together
[Equation]
Proof of Lemma 6.
Recall that if W ( t ) , W ( t + 1) ∈ B ( R ) and η t = 1 /β ( R ) ,
[Equation]
Suppose W ( t ) ∈
B
( R ) for all t .
By Assumption 2 and eq.
[Equation]
[Equation]
(cid:1)(cid:13)(cid:13)(cid:13) 2 <
[Equation]
W (0) (cid:1) .
By eq.
(26), gradient descent never increases the risk, and thus for all t ≥ 1 , R (cid:0) W ( t ) (cid:1) ≤
R (cid:0) W (1) (cid:1) <
R (cid:0) W (0) (cid:1) .
In exactly the same way as in the proof of Lemma 1, one can show that there exists some constant (cid:15) ( R ) > 0 , so that (cid:107) ∂ R /∂W 1 ( t ) (cid:107) F ≥ (cid:15) ( R ) for all t .
Invoking eq.
(26) again, we will get
[Equation]
which is a contradiction.
Therefore W ( t ) must go out of B ( R ) at some time.
Next we prove Theorem 3 and 4.
The proofs depend on several lemmas which are similar to the gradient ﬂow ones.
The following Lemma 7 is similar to Lemma 1.
Lemma 7.
Under Assumption 1, 2, 4, and 5, gradient descent ensures that
• max 1 ≤ k ≤ L (cid:107) W k
( t ) (cid:107) F is unbounded.
• (cid:80) ∞
t =0 η t = ∞ .
• For any R > 0 , (cid:80) t : W ( t )
∈ B ( R ) η t < ∞ .
Proof.
By Assumption 5, we always have that W ( t ) ∈ B ( R t ) .
Since β ( R t ) =
2 L 2 R 2 L − 2 t ( β + G ) ≥
R L − 1 t
G , we have for any 1 ≤ k ≤ L ,
[Equation] [Equation]
Moreover, Lemma 6 shows that R t → ∞ .
Since R t +1 =
R t as long as W ( t + 1) ∈ B ( R t − 1) , max 1 ≤ k ≤ L (cid:107) W k
( t ) (cid:107) F is unbounded.
It then follows that for any t , by Cauchy-Schwarz,
[Equation]
since by eq.
[Equation]
we have (cid:80) ∞ t =
η t = ∞ .
Since under Assumptions 4 and 5 gradient descent never increases the risk, it can be shown in exactly the same as in the proof of Lemma 1 that, for W ( t ) ∈ B ( R ) , (cid:107) ∂ R /∂W 1 ( t ) (cid:107) F ≥ (cid:15) ( R ) for some constant (cid:15) ( R ) > 0 .
Invoking eq.
(26) again, we get that (cid:80) t : W ( t )
∈
B ( R ) η t < ∞ .
The next lemma is an analogy to Lemma 2.
Lemma 8.
Under Assumption 1 and 4, the gradient descent iterates satisfy the following properties:
• For any 1 ≤ k ≤ L , (cid:107) W k (cid:107) 2 F − (cid:107) W k (cid:107) 22 ≤ D + 2 R (cid:0) W (0) (cid:1) .
• For any 1 ≤ k < L , (cid:104) v k +1 , u k (cid:105) 2 ≥ 1 − D +3 R ( W (0) )
+ (cid:107) W k +
0) (cid:107)
+ (cid:107) W k
(0) (cid:107) 22 / (cid:107) W k +1 (cid:107) 22 .
• Suppose max 1 ≤ k ≤ L (cid:107) W k (cid:107) F → ∞ , then (cid:12)(cid:12)(cid:12)(cid:10) w prod / (cid:81)
Lk =1 (cid:107) W k (cid:107) F , v 1 (cid:11)(cid:12)(cid:12)(cid:12) → 1 .
Proof.
Recall that for any W ,
[Equation]
For gradient descent iterates, summing eq.
(28) from 0 to t − 1 , we get
[Equation]
t
[Equation]
η
[Equation]
(cid:18)
∂ R ∂W k (
[Equation]
(cid:19)
[Equation] [Equation]
For any 1 ≤ k ≤ L and any t , let
P k ( t ) =
t − 1 (cid:88) τ
= η 2 τ (cid:18) ∂
R ∂W k ( τ ) (cid:19) (cid:18)
∂ R ∂W k ( τ ) (cid:19) (cid:62) , and
Q k ( t ) =
t − 1 (cid:88) τ
= η 2 τ (cid:18) ∂
R ∂W k ( τ ) (cid:19) (cid:62) (cid:18)
∂ R ∂W k ( τ ) (cid:19) .
We have (cid:107) P k ( t ) (cid:107)
= (cid:107) Q k ( t ) (cid:107) 2 ≤ tr (cid:0) Q k ( t ) (cid:1) =
tr (cid:0) P k ( t ) (cid:1) .
Moreover, invoking eq.
L (cid:88) k
= tr (cid:0) P k ( t ) (cid:1) =
L (cid:88) k
= t − 1 (cid:88) τ
= η 2 τ (cid:13)(cid:13)(cid:13)(cid:13)
∂
R ∂W k ( τ ) (cid:13)(cid:13)(cid:13)(cid:13)
F = t − 1 (cid:88) τ
= η 2 τ (cid:13)(cid:13)(cid:13) ∇R (cid:0) W ( τ ) (cid:1)(cid:13)(cid:13)(cid:13)≤ t − 1 (cid:88) τ
= η τ (cid:13) (cid:13)(cid:13) ∇R (cid:0) W ( τ ) (cid:1) (cid:13) (cid:13)(cid:13)≤ 2 R (cid:0) W (0) (cid:1)
[Equation]
Still let σ k ( t ) , u k ( t ) and v k ( t ) denote the ﬁrst singular value, left singular vector and right singular vector of W k ( t ) .
We can then proceed basically in the same way as in the proof of Lemma 2.
For example, eq.
(6) becomes
[Equation]
31) while eq.
(7) becomes
[Equation]
Summing eq.
(31) and eq.
(32) from k to L −
[Equation]
and invoke eq.
(30), we get
[Equation]
To prove singular vectors get aligned, we can still proceed in nearly the same way as in the proof of
Lemma 2.
eq.
(9) becomes
[Equation]
while eq.
(10) becomes
[Equation]
Combining eq.
(33) and eq.
[Equation]
Similar to eq.
[Equation]
we can get
[Equation]
and thus eq.
(12) becomes
[Equation]
Combining eq.
(35) and eq.
(36), we get
[Equation]
The ﬁnal claim of Lemma 8 can be proved in exactly the same way as Lemma 2.
Proof of Theorem 3.
Summing eq.
(32), we know that for any two different layers j > k ,
[Equation]
Recall eq.
(30), we know that
[Equation]
In other words, the difference between the squares of Frobenius norms of any two layers is still bounded.
The proof then goes in the same way as the proof of Theorem 1.
Suppose the risk is always above (cid:15) > 0 .
Then there exists some c ( (cid:15) ) > 0 such that (cid:107)∇R ( w prod ) (cid:107) ≥ c ( (cid:15) ) .
By Lemma 8, there exists some C such that if min 1 ≤ k ≤ L (cid:107) W k
( t ) (cid:107) F > C , (cid:107) W L ( t ) · · · W 2 ( t ) (cid:107) ≥ C L / 2 .
By eq.
(37) and
Lemma 7, (cid:80) t : (cid:107) W k
( t ) (cid:107) F ≤ C for some k η t is ﬁnite.
On the other hand, by Lemma 7, (cid:80) ∞ i =
η t = ∞ , and thus (cid:80) t : (cid:107) W k
( t ) (cid:107) F >C for all k η t = ∞ .
Therefore we have, by invoking eq.
[Equation]
which is a contradiction.
Therefore R (cid:0) W ( t ) (cid:1) → 0 , and since it has no ﬁnite optimum, (cid:107) W k (cid:107) F → ∞ .
The other results follow from Lemma 7.
Proof of Theorem 4.
Recall that
[Equation]
and thus
[Equation]
If (cid:104) w prod , z i (cid:105) ≥ 0 for all i , then (cid:107) W 1 ( t + 1) (cid:107) F ≥ (cid:107) W 1 ( t ) (cid:107) F .
Also recall that Π ⊥ W 1 ( t ) denote the projection of rows of W 1 ( t ) onto ¯ u ⊥ , the orthogonal complement of span(¯ u ) .
We have
[Equation] [Equation]
Invoking eq.
(26) again gives
[Equation]
The proof then goes in almost the same way as the proof of Theorem 2.
For any (cid:15) > 0 , we can ﬁnd some large enough time t 0 , such that for any t ≥ t 0 ,
(cid:107) Π ⊥ W 1 ( t ) (cid:107) F / (cid:107) W 1 ( t ) (cid:107) F ≥ (cid:15) implies that (cid:68) Π ⊥ w prod ( t ) , ∇R (cid:0) w prod ( t ) (cid:1)(cid:69) ≥ 0 . 2.
(cid:104) w prod ( t ) , z i (cid:105) ≥ 0 for all i , and thus (cid:107) W 1 ( t + 1) (cid:107) F ≥ (cid:107) W 1 ( t ) (cid:107) F . 3.
(cid:107) W 1 ( t ) (cid:107) F ≥ 1+ √ 2 R ( W (0)) / (cid:15) .
Suppose at some time t 1 ≥ t 0 , (cid:107) Π ⊥ W 1 ( t 1 ) (cid:107) F / (cid:107) W 1 ( t 1 ) (cid:107) F ≥ (cid:15) .
As long as this still holds, in light of bullet (1) above, eq.
(38) and eq.
(39), (cid:107) Π ⊥ W 1 (cid:107) 2 F will increase by at most 2 R (cid:0) W ( t 1 ) (cid:1) ≤ 2 R (cid:0) W (0) (cid:1) .
On the other hand, (cid:107) W 1 (cid:107) F → ∞ , and thus there exists some t 2 > t 1 such that (cid:107) Π ⊥ W 1 ( t 2 ) (cid:107) F / (cid:107) W 1 ( t 2 ) (cid:107) F < (cid:15) .
Let t 3 denote the smallest time after t 2 such that (cid:107) Π ⊥ W 1 ( t 3 ) (cid:107) F / (cid:107) W 1 ( t 3 ) (cid:107) F ≥ (cid:15)
(if it exists).
Recall that (cid:107) W 1 ( t + 1) (cid:107) F ≤ (cid:107) W 1 ( t ) (cid:107) F + 1 for any t ≥ 0 , and (cid:107) W 1 ( t + 1) (cid:107) F ≥ (cid:107) W 1 ( t ) (cid:107) F for any t ≥ t 0 , we have
[Equation]
After t 3 , (cid:107) Π ⊥ W 1 (cid:107) 2 F will increase by at most 2 R (cid:0) W (0) (cid:1) , and thus (cid:107) Π ⊥ W 1 (cid:107) F will increase by at most (cid:113) 2 R (cid:0) W (0) (cid:1) .
Therefore, for any t 4 ≥ t 3 , as long as (cid:107) Π ⊥ W 1 ( t 4 ) (cid:107) F / (cid:107) W 1 ( t 4 ) (cid:107) F ≥ (cid:15) , we have
[Equation]
+ (cid:107) W 1 ( t 3 − 1) (cid:107) F + (cid:113) 2 R (cid:0) W (0) (cid:1) (cid:107) W 1 ( t 3 ) (cid:107) F ≤ 2 (cid:15), since (cid:107) W 1 ( t ) (cid:107) F ≥ 1+ √ 2 R ( W (0)) / (cid:15) after t 0 .
In other words,
[Equation]
Since (cid:15) is arbitrary, we have
[Equation]
, and thus lim t →∞ (cid:12)(cid:12) (cid:104) v 1 , ¯ u (cid:105) (cid:12)(cid:12)
= 1 .
Proof of Corollary 2.
The proof is analogous to that of Corollary 1, except using Theorem 4 in place of Theorem 2.