A C ONVERGENCE A NALYSIS OF G RADIENT D ESCENT FOR D EEP L INEAR N EURAL N ETWORKS
A BSTRACT
We analyze speed of convergence to global optimum for gradient descent training a deep linear neural network (parameterized as x (cid:55)→ W N W N − 1 · · · W 1 x ) by minimizing the (cid:96) 2 loss over whitened data.
Convergence at a linear rate is guaranteed when the following hold: (i) dimensions of hidden layers are at least the minimum of the input and output dimensions; (ii) weight matrices at initialization are approximately balanced; and (iii) the initial loss is smaller than the loss of any rank-deﬁcient solution.
The assumptions on initialization (conditions (ii) and (iii) ) are necessary, in the sense that violating any one of them may lead to convergence failure.
Moreover, in the important case of output dimension 1 , i.e. scalar regression, they are met, and thus convergence to global optimum holds, with constant probability under a random initialization scheme.
Our results signiﬁcantly extend previous analyses, e.g. , of deep linear residual networks (Bartlett et al., 2018).
I NTRODUCTION
Deep learning builds upon the mysterious ability of gradient-based optimization methods to solve related non-convex problems.
Immense efforts are underway to mathematically analyze this phenomenon.
The prominent landscape approach focuses on special properties of critical points ( i.e. points where the gradient of the objective function vanishes) that will imply convergence to global optimum.
Several papers ( e.g. Ge et al.
; Lee et al.
(2016)) have shown that (given certain smoothness properties) it sufﬁces for critical points to meet the following two conditions: (i) no poor local minima — every local minimum is close in its objective value to a global minimum; and (ii) strict saddle property — every critical point that is not a local minimum has at least one negative eigenvalue to its Hessian.
While condition (i) does not always hold ( cf.
Safran and
Shamir (2018)), it has been established for various simple settings ( e.g. Soudry and Carmon (2016);
Kawaguchi (2016)).
Condition (ii) on the other hand seems less plausible, and is in fact provably false for models with three or more layers ( cf.
Kawaguchi (2016)), i.e. for deep networks.
It has only been established for problems involving shallow (two layer) models, e.g. matrix factorization (Ge et al.
; Du et al. (2018a)).
The landscape approach as currently construed thus suffers from inherent limitations in proving convergence to global minimum for deep networks.
A potential path to circumvent this obstacle lies in realizing that landscape properties matter only in the vicinity of trajectories that can be taken by the optimizer, which may be a negligible portion of the overall parameter space.
Several papers ( e.g. Saxe et al.
; Arora et al.
(2018)) have taken this trajectory-based approach, primarily in the context of linear neural networks — fully-connected neural networks with linear activation.
Linear networks are trivial from a representational perspective, but not so in terms of optimization — they lead to non-convex training problems with multiple minima and saddle points.
Through a mix of theory and experiments, Arora et al.
(2018) argued that such non-convexities may in fact be beneﬁcial for gradient descent, in the sense that sometimes, adding (redundant) linear layers to a classic linear prediction model can accelerate the optimization.
This phenomenon challenges the holistic landscape view, by which convex problems are always preferable to non-convex ones.
Even in the linear network setting, a rigorous proof of efﬁcient convergence to global minimum has proved elusive.
One recent progress is the analysis of Bartlett et al.
(2018) for linear residual networks — a particular subclass of linear neural networks in which the input, output and all hidden dimensions are equal, and all layers are initialized to be the identity matrix ( cf.
Hardt and Ma (2016)).
Through a trajectory-based analysis of gradient descent minimizing (cid:96) 2 loss over a whitened dataset (see Section 2), Bartlett et al.
(2018) show that convergence to global minimum at a linear rate — loss is less than (cid:15) > 0 after O (log 1 (cid:15) ) iterations — takes place if one of the following holds: (i) the objective value at initialization is sufﬁciently close to a global minimum; or (ii) a global minimum is attained when the product of all layers is positive deﬁnite.
The current paper carries out a trajectory-based analysis of gradient descent for general deep linear neural networks, covering the residual setting of Bartlett et al.
(2018), as well as many more settings that better match practical deep learning.
Our analysis draws upon the trajectory characterization of Arora et al.
(2018) for gradient ﬂow (inﬁnitesimally small learning rate), together with signiﬁcant new ideas necessitated due to discrete updates.
Ultimately, we show that when minimizing (cid:96) 2 loss of a deep linear network over a whitened dataset, gradient descent converges to the global minimum, at a linear rate, provided that the following conditions hold: (i) the dimensions of hidden layers are greater than or equal to the minimum between those of the input and output; (ii) layers are initialized to be approximately balanced (see Deﬁnition 1) — this is met under commonplace near-zero, as well as residual (identity) initializations; and (iii) the initial loss is smaller than any loss obtainable with rank deﬁciencies — this condition will hold with probability close to 0 .
5 if the output dimension is(scalar regression) and standard (random) near-zero initialization is employed.
Our result applies to networks with arbitrary depth and input/output dimensions, as well as any conﬁguration of hidden layer widths that does not force rank deﬁciency ( i.e. that meets condition (i) ).
The assumptions on initialization (conditions (ii) and (iii) ) are necessary, in the sense that violating any one of them may lead to convergence failure.
Moreover, in the case of scalar regression, they are met with constant probability under a random initialization scheme.
We are not aware of any similarly general analysis for efﬁcient convergence of gradient descent to global minimum in deep learning.
The remainder of the paper is organized as follows.
In Section 2 we present the problem of gradient descent training a deep linear neural network by minimizing the (cid:96) 2 loss over a whitened dataset.
Section 3 formally states our assumptions, and presents our convergence analysis.
Key ideas brought forth by our analysis are demonstrated empirically in Section 4.
Section 5 gives a review of relevant literature, including a detailed comparison of our results against those of Bartlett et al.
Finally, Section 6 concludes.
G RADIENT D ESCENT FOR D EEP L INEAR N EURAL N ETWORKS
We denote by (cid:107) v (cid:107) the Euclidean norm of a vector v , and by (cid:107) A (cid:107) F the Frobenius norm of a matrix A .
We are given a training set { ( x ( i ) , y ( i ) ) } mi =1 ⊂ R d x × R d y , and would like to learn a hypothesis (predictor) from a parametric family H :
= { h θ : R d x → R d y | θ ∈ Θ } by minimizing the (cid:96) 2 loss:
[Equation]
[Equation] [Equation] When the parametric family in question is the class of linear predictors, i.e. H = { x (cid:55)→ W x | W ∈ R d y × d x } , the training loss may be written as L ( W ) =m (cid:107) WX − Y (cid:107) 2 F , where X ∈ R d x × m and Y ∈ R d y × m are matrices whose columns hold instances and labels respectively.
Suppose now that the dataset is whitened , i.e. has been transformed such that the empirical (uncentered) covariance matrix for instances — Λ xx :
= 1 m XX (cid:62) ∈ R d x × d x — is equal to identity.
Standard calculations (see Appendix A) show that in this case:
[Equation]
where Λ yx := 1 m Y X (cid:62) ∈ R d y × d x is the empirical (uncentered) cross-covariance matrix between instances and labels, and c is a constant (that does not depend on W ).
Denoting Φ := Λ yx for brevity, we have that for linear models, minimizing (cid:96) 2 loss over whitened data is equivalent to minimizing the squared Frobenius distance from a target matrix Φ :
[Equation]
Our interest in this work lies on linear neural networks — fully-connected neural networks with linear activation.
A depth- N ( N ∈ N ) linear neural network with hidden widths d 1 , . . .
, d N − 1 ∈ N corresponds to the parametric family of hypotheses H :=
{ x (cid:55)→ W N W N − 1 · · · W 1 x | W j ∈ R d j × d j − 1 , j = 1 , . .
. , N } , where d 0 :=
d x , d N := d y .
Similarly to the case of a (directly parameterized) linear predictor (Equation (2)), with a linear neural network, minimizing (cid:96) 2 loss over whitened data can be cast as squared Frobenius approximation of a target matrix Φ :
[Equation]
Note that the notation L N ( · ) is consistent with that of Equation (2), as a network with depth N =precisely reduces to a (directly parameterized) linear model.
We focus on studying the process of training a deep linear neural network by gradient descent , i.e. of tackling the optimization problem in Equation (3) by iteratively applying the following updates:
[Equation]
where η > 0 is a conﬁgurable learning rate.
In the case of depth N = 1 , the training problem in
Equation (3) is smooth and strongly convex, thus it is known ( cf.
Boyd and Vandenberghe (2004))
that with proper choice of η , gradient descent converges to global minimum at a linear rate.
In contrast, for any depth greater than 1 , Equation (3) comprises a fundamentally non-convex program, and the convergence properties of gradient descent are highly non-trivial.
Apart from the case N =(shallow network), one cannot hope to prove convergence via landscape arguments, as the strict saddle property is provably violated (see Section 1).
We will see in Section 3 that a direct analysis of the trajectories taken by gradient descent can succeed in this arena, providing a guarantee for linear rate convergence to global minimum.
We close this section by introducing additional notation that will be used in our analysis.
For an arbitrary matrix A , we denote by σ max ( A ) and σ min ( A ) its largest and smallest (respectively) singular values.
For d ∈ N , we use I d to signify the identity matrix in R d × d .
Given weights W 1 , . . .
, W N of a linear neural network, we let W 1: N be the direct parameterization of the end-to-end linear mapping realized by the network, i.e. W 1: N := W N W N − 1 · · · W 1 .
Note that
L N ( W 1 , . . . , W N )
=
L 1 ( W 1: N ) , meaning the loss associated with a depth- N network is equal to the loss of the corresponding end-to-end linear model.
In the context of gradient descent, we will oftentimes use (cid:96) ( t ) as shorthand for the loss at iteration t :
[Equation]
C ONVERGENCE A NALYSIS
In this section we establish convergence of gradient descent for deep linear neural networks (Equations (4) and (3)) by directly analyzing the trajectories taken by the algorithm.
We begin in Subsection 3.1 with a presentation of two concepts central to our analysis: approximate balancedness and deﬁciency margin .
These facilitate our main convergence theorem, delivered in Subsection 3.2.
We conclude in Subsection 3.3 by deriving a convergence guarantee that holds with constant probability over a random initialization.
3. A PPROXIMATE B ALANCEDNESS AND D EFICIENCY M ARGIN
In our context, the notion of approximate balancedness is formally deﬁned as follows:
Deﬁnition 1.
For δ ≥ 0 , we say that the matrices W j ∈ R d j × d j − 1 , j =1 , . . .
, N , are δ - balanced if:
[Equation]
Note that in the case of 0 -balancedness, i.e. W (cid:62)
j +1 W j +
=
W j W (cid:62) j , ∀ j ∈ { 1 , . . .
, N − 1 } , all matrices W j share the same set of non-zero singular values.
Moreover, as shown in the proof of
Theorem 1 in Arora et al.
(2018), this set is obtained by taking the N -th root of each non-zero singular value in the end-to-end matrix W 1: N .
We will establish approximate versions of these facts for δ -balancedness with δ > 0 , and admit their usage by showing that if the weights of a linear neural network are initialized to be approximately balanced, they will remain that way throughout the iterations of gradient descent.
The condition of approximate balancedness at initialization is trivially met in the special case of linear residual networks ( d 0 = · · · = d N = d and W 1 (0) = · · · = W N (0) =
I d ).
Moreover, as Claim 2 in Appendix B shows, for a given δ > 0 , the customary initialization via random Gaussian distribution with mean zero leads to approximate balancedness with high probability if the standard deviation is sufﬁciently small.
The second concept we introduce — deﬁciency margin — refers to how far a ball around the target is from containing rank-deﬁcient ( i.e. low rank) matrices.
Deﬁnition 2.
Given a target matrix Φ ∈ R d N × d 0 and a constant c > 0 , we say that a matrix W ∈ R d N × d 0 has deﬁciency margin c with respect to Φ if:
[Equation]
The term “deﬁciency margin” alludes to the fact that if Equation (6) holds, every matrix W (cid:48) whose distance from Φ is no greater than that of W , has singular values c -bounded away from zero:
Claim 1.
Suppose W has deﬁciency margin c with respect to Φ .
Then, any matrix W (cid:48)
(of same size as Φ and W ) for which (cid:107) W (cid:48) − Φ (cid:107) F ≤ (cid:107) W − Φ (cid:107) F satisﬁes σ min ( W (cid:48) ) ≥ c .
Proof.
Our proof relies on the inequality σ min ( A + B ) ≥ σ min ( A ) − σ max ( B ) — see Appendix D.1.
We will show that if the weights W 1 , . . .
, W N are initialized such that (they are approximately balanced and) the end-to-end matrix W 1: N has deﬁciency margin c > 0 with respect to the target Φ , convergence of gradient descent to global minimum is guaranteed.
Moreover, the convergence will outpace a particular rate that gets faster when c grows larger.
This suggests that from a theoretical perspective, it is advantageous to initialize a linear neural network such that the end-to-end matrix has a large deﬁciency margin with respect to the target.
Claim 3 in Appendix B provides information on how likely deﬁciency margins are in the case of a single output model (scalar regression) subject to customary zero-centered Gaussian initialization.
It shows in particular that if the standard deviation of the initialization is sufﬁciently small, the probability of a deﬁciency margin being met is close to 0 .
5 ; on the other hand, for this deﬁciency margin to have considerable magnitude, a non-negligible standard deviation is required.
Taking into account the need for both approximate balancedness and deﬁciency margin at initialization, we observe a delicate trade-off under the common setting of Gaussian perturbations around zero: if the standard deviation is small, it is likely that weights be highly balanced and a deﬁciency margin be met; however overly small standard deviation will render high magnitude for the deﬁciency margin improbable, and therefore fast convergence is less likely to happen; on the opposite end, large standard deviation jeopardizes both balancedness and deﬁciency margin, putting the entire convergence at risk.
This trade-off is reminiscent of empirical phenomena in deep learning, by fact a implies that all critical points in the respective of smaller loss value) are minima.
This is far from sufﬁcient for proving convergence, as sublevel sets are unbounded, and the loss landscape over them is non- and non-smooth we show to which small initialization can bring forth efﬁcient convergence, while if exceedingly small, rate of convergence may plummet (“vanishing gradient problem”), and if made large, divergence becomes inevitable (“exploding gradient problem”).
The common resolution of residual connections (He et al., 2016) is analogous in our context to linear residual networks, which ensure perfect balancedness, and allow large deﬁciency margin if the target is not too far from identity.
3. M AIN T HEOREM
Using approximate balancedness (Deﬁnition 1) and deﬁciency margin (Deﬁnition 2), we present our main theorem — a guarantee for linear convergence to global minimum:
Theorem 1.
Assume that gradient descent is initialized such that the end-to-end matrix W 1: N (0) has deﬁciency margin c > 0 with respect to the target Φ , and the weights W 1 (0) , . .
. , W N (0) are δ -balanced with δ =
c 2 (cid:14)(cid:0) 256 · N 3 · (cid:107) Φ (cid:107)
( N
[Equation]
1) /N
[Equation]
(cid:1) .
Suppose also that the learning rate η meets:
[Equation] [Equation]
Then, for any (cid:15) > 0 and:
[Equation]
the loss at iteration T of gradient descent — (cid:96) ( T ) — is no greater than (cid:15) .
3.2. O N THE A SSUMPTIONS M ADE
The assumptions made in Theorem 1 — approximate balancedness and deﬁciency margin at initialization — are both necessary, in the sense that violating any one of them may lead to convergence failure.
We demonstrate this in Appendix C.
In the special case of linear residual networks (uniform dimensions and identity initialization), a sufﬁcient condition for the assumptions to be met is that the target matrix have (Frobenius) distance less than 0 .
5 from identity.
This strengthens one of the central results in Bartlett et al.
(see Section 5).
For a setting of random near-zero initialization, we present in Subsection 3.3 a scheme that, when the output dimension is 1 (scalar regression), ensures assumptions are satisﬁed (and therefore gradient descent efﬁciently converges to global minimum) with constant probability.
It is an open problem to fully analyze gradient descent under the common initialization scheme of zero-centered Gaussian perturbations applied to each layer independently.
We treat this scenario in Appendix B, providing quantitative results concerning the likelihood of each assumption (approximate balancedness or deﬁciency margin) being met individually.
However the question of how likely it is that both assumptions be met simultaneously, and how that depends on the standard deviation of the Gaussian, is left for future work.
An additional point to make is that Theorem 1 poses a structural limitation on the linear neural network.
Namely, it requires the dimension of each hidden layer ( d i , i = 1 , . . .
, N − 1 ) to be greater than or equal to the minimum between those of the input ( d 0 ) and output ( d N ).
Indeed, in order for the initial end-to-end matrix W 1: N (0) to have deﬁciency margin c > 0 , it must (by Claim 1) have full rank, and this is only possible if there is no intermediate dimension
d i smaller than min { d 0 , d N } .
We make no other assumptions on network architecture (depth, input/output/hidden dimensions).
3.2. P ROOF
The cornerstone upon which Theorem 1 rests is the following lemma, showing non-trivial descent whenever σ min ( W 1: N ) is bounded away from zero:
Lemma 1.
Under the conditions of Theorem 1, we have that for every t = 0 , 1 , 2 , . . . :
[Equation]
Proof of Lemma 1 (in idealized setting; for complete proof see Appendix D.2).
We prove the lemma here for the idealized setting of perfect initial balancedness ( δ = 0 ):
[Equation]
and inﬁnitesimally small learning rate ( η → 0 + ) — gradient ﬂow :
[Equation]
where τ is a continuous time index, and dot symbol (in ˙ W j ( τ ) ) signiﬁes derivative with respect to time.
The complete proof, for the realistic case of approximate balancedness and discrete updates ( δ, η > 0 ), is similar but much more involved, and appears in Appendix D.2.
Recall that (cid:96) ( t ) — the objective value at iteration t of gradient descent — is equal to L 1 ( W 1: N ( t ))
(see Equation (5)).
Accordingly, for the idealized setting in consideration, we would like to show:
[Equation]
We will see that a stronger version of Equation (10) holds, namely, one without the 1 / 2 factor (which only appears due to discretization).
By (Theorem 1 and Claim 1 in) Arora et al.
(2018), the weights W 1 ( τ ) , . . .
, W N ( τ ) remain balanced throughout the entire optimization, and that implies the end-to-end matrix W 1: N ( τ ) moves according to the following differential equation:
[Equation]
where vec ( A ) , for an arbitrary matrix A , stands for vectorization in column-ﬁrst order, and
P W 1: N ( τ ) is a positive semideﬁnite matrix whose eigenvalues are all greater than or equal to σ min ( W 1: N ( τ ))
( N − 1) /N .
Taking the derivative of L 1 ( W 1: N ( τ )) with respect to time, we obtain the sought-after Equation (10) (with no 1 / 2 factor):
[Equation]
The ﬁrst transition here (equality) is an application of the chain rule; the second (equality) plugs in Equation (11); the third (inequality) results from the fact that the eigenvalues of the symmetric matrix P W 1: N ( τ ) are no smaller than σ min ( W 1: N ( τ )) 2( N − 1) /N
(recall that (cid:107)·(cid:107) stands for Euclidean norm); and the last (equality) is trivial — (cid:107) A (cid:107) F = (cid:107) vec ( A ) (cid:107) for any matrix A .
With Lemma 1 established, the proof of Theorem 1 readily follows:
Proof of Theorem 1.
By the deﬁnition of L 1 ( · ) (Equation (2)), for any W ∈ R d N × d 0 :
[Equation]
Plugging this into Equation (9) while recalling that (cid:96) ( t ) =
L 1 ( W 1: N ( t ))
(Equation (5)), we have (by Lemma 1) that for every t = 0 , 1 , 2 , . . . :
[Equation]
− 1) N (cid:17) .
Since the coefﬁcients 1 − η · σ min ( W 1: N ( t )) 2( N [Equation] [Equation] N are necessarily non-negative (otherwise would contradict non-negativity of L 1 ( · ) ), we may unroll the inequalities, obtaining:
[Equation]
Now, this in particular means that for every t (cid:48)
= 0 , 1 , 2 , . .
[Equation]
Deﬁciency margin c of W 1: N (0) along with Claim 1 thus imply σ min (cid:0) W 1: N ( t (cid:48) ) (cid:1) ≥ c , which when inserted back into Equation (12) yields, for every t = 1 , 2 , 3 , . . . :
[Equation]
η
[Equation]
is obviously non-negative, and it is also no greater than 1 (otherwise would contradict non-negativity of L 1 ( · ) ).
We may therefore incorporate the inequality 1 − η · c 2( N − 1) /N
≤ exp (cid:0) − η · c 2( N − 1) /N (cid:1) into Equation (13):
[Equation]
from which it follows that L 1 ( W 1: N ( t )) ≤ (cid:15)
if: t ≥
[Equation]
· c 2( N [Equation] 1) /N · log
[Equation]
W
[Equation]
N
[Equation]
(cid:19) .
Recalling again that (cid:96) ( t )
=
L 1 ( W 1: N ( t ))
(Equation (5)), we conclude the proof.
3. B ALANCED I NITIALIZATION
We deﬁne the following procedure, balanced initialization , which assigns weights randomly while ensuring perfect balancedness:
Procedure 1 (Balanced initialization) .
Given d 0 , d 1 , . . .
, d N ∈ N such that min { d 1 , . . .
, d N − 1 } ≥ min { d 0 , d N } and a distribution D over d N × d 0 matrices, a balanced initialization of W j ∈ R d j × d j − 1 , j =1 , . . .
, N , assigns these weights as follows:
(i) Sample A ∈ R d N × d 0 according to D .
(ii) Take singular value decomposition
A = U Σ V (cid:62) ,
where U ∈ R
d N × min { d 0 ,d N } , V ∈ R d 0 × min { d 0 ,d N } have orthonormal columns, and Σ ∈ R min { d 0 ,d N }× min { d 0 ,d N } is diagonal and holds the singular values of A .
(iii) Set W N (cid:39) U Σ 1 /N , W N − 1 (cid:39) Σ 1 /N , . .
, W 2 (cid:39) Σ 1 /N , W 1 (cid:39) Σ
1 /N V (cid:62) , where the symbol “ (cid:39) ” stands for equality up to zero-valued padding.
The concept of balanced initialization, together with Theorem 1, leads to a guarantee for linear convergence (applicable to output dimension 1 — scalar regression) that holds with constant probability over the randomness in initialization:
Theorem 2.
For any constant 0 < p < 1 / 2 , there are constants d (cid:48) 0 , a > 0 8 such that the following holds.
Assume d N = 1 , d 0 ≥ d (cid:48) 0 , and that the weights W 1 (0) , . .
. , W N (0) are subject to balanced initialization (Procedure 1) such that the entries in W 1: N (0) are independent zero-centered Gaussian perturbations with standard deviation s ≤ (cid:107) Φ (cid:107) 2 / (cid:112) ad 20 .
Suppose also that we run gradient (i) here ( (cid:48) 20 ) serves solely for simpliﬁcation of expressions in the theorem.
descent with learning rate η ≤ ( s 2 d 0 ) 4 − 2 /N
(cid:14)(cid:0) 10 5 N 3 (cid:107) Φ (cid:107) 10 − 6 /N(cid:1) .
Then, with probability at least p over the random initialization, we have that for every (cid:15) >
and:
[Equation]
the loss at iteration T of gradient descent — (cid:96) ( T ) — is no greater than (cid:15) .
Proof.
See Appendix D.3.
E XPERIMENTS
Balanced initialization (Procedure 1) possesses theoretical advantages compared with the customary layer-wise independent scheme — it allowed us to derive a convergence guarantee that holds with constant probability over the randomness of initialization (Theorem 2).
In this section we present empirical evidence suggesting that initializing with balancedness may be beneﬁcial in practice as well.
For conciseness, some of the details behind our implementation are deferred to Appendix E.
We began by experimenting in the setting covered by our analysis — linear neural networks trained via gradient descent minimization of (cid:96) 2 loss over whitened data.
The dataset chosen for the experiment was UCI Machine Learning Repository’s “Gas Sensor Array Drift at Different Concentrations” (Vergara et al., 2012; Rodriguez-Lujan et al., 2014).
Speciﬁcally, we used the dataset’s “Ethanol” problem — a scalar regression task with 2565 examples, each comprising 128 features (one of the largest numeric regression tasks in the repository).
Starting with the customary initialization of layer-wise independent random Gaussian perturbations centered at zero, we trained a three layer network ( N = 3 ) with hidden widths ( d 1 , d 2 ) set to 32 , and measured the time (number of iterations) it takes to converge (reach training loss within (cid:15)
=
10 − 5 from optimum) under different choices of standard deviation for the initialization.
To account for the possibility of different standard deviations requiring different learning rates (values for η ), we applied, for each standard deviation independently, a grid search over learning rates, and recorded the one that led to fastest convergence.
The result of this test is presented in Figure 1(a).
As can be seen, there is a range of standard deviations that leads to fast convergence (a few hundred iterations or less), below and above which optimization decelerates by orders of magnitude.
This accords with our discussion at the end of Subsection 3.3, by which overly small initialization ensures approximate balancedness (small δ ; see Deﬁnition 1) but diminishes deﬁciency margin (small c ; see Deﬁnition 2) — “vanishing gradient problem” — whereas large initialization hinders both approximate balancedness and deﬁciency margin — “exploding gradient problem”.
In that regard, as a sanity test for the validity of our analysis, in a case where approximate balancedness is met at initialization (small standard deviation), we measured its persistence throughout optimization.
As Figure 1(c) shows, our theoretical ﬁndings manifest themselves here — trajectories of gradient descent indeed preserve weight balancedness.
In addition to a three layer network, we also evaluated a deeper, eight layer model (with hidden widths identical to the former — N = 8 , d 1 = · · · = d 7 = 32 ).
In particular, using the same experimental protocol as above, we measured convergence time under different choices of standard deviation for the initialization.
Figure 1(a) displays the result of this test alongside that of the three layer model.
As the ﬁgure shows, transitioning from three layers to eight aggravated the instability with respect to initialization — there is now a narrow band of standard deviations that lead to convergence in reasonable time, and outside of this band convergence is extremely slow, to the point where it does not take place within the duration we allowed ( 10 6 iterations).
From the perspective of our analysis, a possible explanation for the aggravation is as follows: under layer-wise independent initialization, the magnitude of the end-to-end matrix W 1: N depends on the standard deviation in a manner that is exponential in depth, thus for large depths the range of standard deviations that lead to moderately sized W 1: N (as required for a deﬁciency margin) is limited, and within this range, there may not be many standard deviations small enough to ensure approximate balancedness.
The procedure of balanced initialization (Procedure 1) circumvents these difﬁculties — it assigns W 1: N directly (no exponential dependence on depth), and distributes its content between the individual weights W 1 , . . .
, W N in a perfectly balanced fashion.
Rerunning the experiment of Figure 1(a) with this initialization replacing the customary layer-wise scheme (using same experimental protocol), we obtained the results shown in Figure 1(b) — both the original three layer network, and the deeper eight layer model, converged quickly under virtually all standard deviations tried.
[Figure]
[Figure] [Figure]
[Figure] [Figure]
As a ﬁnal experiment, we evaluated the effect of balanced initialization in a setting that involves non-linear activation, softmax-cross-entropy loss and stochastic optimization (factors not accounted for by our analysis).
For this purpose, we turned to the MNIST tutorial built into TensorFlow (Abadi et al., 2016), 9 which comprises a fully-connected neural network with two hidden layers (widthfollowed by 32 ) and ReLU activation (Nair and Hinton, 2010), trained through stochastic gradient descent (over softmax-cross-entropy loss) with batch size 100 , initialized via customary layer-wise independent Gaussian perturbations centered at zero.
While keeping the learning rate at its default value 0 .
01 , we varied the standard deviation of initialization, and for each value measured the training loss after 10 epochs.
10 We then replaced the original (layer-wise independent) initialization with a balanced initialization based on Gaussian perturbations centered at zero (latter was implemented per Procedure 1, disregarding non-linear activation), and repeated the process.
The results of this experiment are shown in Figure 1(d).
Although our theoretical analysis does not cover non-linear activation, softmax-cross-entropy loss or stochasticity in optimization, its conclusion of balanced initialization leading to improved (faster and more stable) convergence carried over to such setting.
R ELATED W ORK
Theoretical study of gradient-based optimization in deep learning is a highly active area of research.
As discussed in Section 1, a popular approach is to show that the objective landscape admits the properties of no poor local minima and strict saddle, which, by Ge et al.
; Lee et al.
Panageas and Piliouras (2017), ensure convergence to global minimum.
Many works, both classic ( e.g. Baldi and Hornik (1989)) and recent ( e.g. Choromanska et al.
; Kawaguchi (2016); Hardt and Ma (2016); Soudry and Carmon (2016); Haeffele and Vidal (2017); Nguyen and Hein (2017);
Safran and Shamir (2018); Nguyen and Hein (2018); Laurent and Brecht (2018)), have focused on the validity of these properties in different deep learning settings.
Nonetheless, to our knowledge,
examples/tutorials/mnist
As opposed to the dataset used in our experiments with linear networks, measuring the training loss with MNIST is non-trivial computationally (involves passing through 60 K examples).
Therefore, rather than continuously polling training loss until it reaches a certain threshold, in this experiment we chose to evaluate speed of convergence by measuring the training loss once after a predetermined number of iterations.
the success of landscape-driven analyses in formally proving convergence to global minimum for a gradient-based algorithm, has thus far been limited to shallow (two layer) models only ( e.g. Ge et al.
(2016); Du and Lee (2018); Du et al.
(2018a)).
An alternative to the landscape approach is a direct analysis of the trajectories taken by the optimizer.
Various papers ( e.g. Brutzkus and Globerson (2017); Li and Yuan (2017); Zhong et al. (2017); Tian (2017); Brutzkus et al.
; Li et al.
Du et al.
(2018c;b);
Liao et al.
(2018)) have recently adopted this strategy, but their analyses only apply to shallow models.
In the context of linear neural networks, deep (three or more layer) models have also been treated — cf.
Saxe et al.
(2014) and Arora et al.
(2018), from which we draw certain technical ideas for proving Lemma 1.
However these treatments all apply to gradient ﬂow (gradient descent with inﬁnitesimally small learning rate), and thus do not formally address the question of computational efﬁciency.
To our knowledge, Bartlett et al.
(2018) is the only existing work rigorously proving convergence to global minimum for a conventional gradient-based algorithm training a deep model.
This work is similar to ours in the sense that it also treats linear neural networks trained via minimization of (cid:96) 2 loss over whitened data, and proves linear convergence (to global minimum) for gradient descent.
It is more limited in that it only covers the subclass of linear residual networks, i.e. the speciﬁc setting of uniform width across all layers ( d 0 = · · · = d N ) along with identity initialization.
We on the other hand allow the input, output and hidden dimensions to take on any conﬁguration that avoids “bottlenecks” ( i.e. admits min { d 1 , . . .
, d N − 1 } ≥ min { d 0 , d N } ), and from initialization require only approximate balancedness (Deﬁnition 1), supporting many options beyond identity.
In terms of the target matrix Φ , Bartlett et al.
(2018) treats two separate scenarios: 11 (i) Φ is symmetric and positive deﬁnite; and (ii) Φ is within distance 1 / 10 e from identity.
12 Our analysis does not fully account for scenario (i) , which seems to be somewhat of a singularity, where all layers are equal to each other throughout optimization (see proof of Theorem 2 in Bartlett et al.
We do however provide a strict generalization of scenario (ii) — our assumption of deﬁciency margin (Deﬁnition 2), in the setting of linear residual networks, is met if the distance between target and identity is less than 0 .
C ONCLUSION
For deep linear neural networks, we have rigorously proven convergence of gradient descent to global minima, at a linear rate, provided that the initial weight matrices are approximately balanced and the initial end-to-end matrix has positive deﬁciency margin.
The result applies to networks with arbitrary depth, and any conﬁguration of input/output/hidden dimensions that supports full rank, i.e. in which no hidden layer has dimension smaller than both the input and output.
Our assumptions on initialization — approximate balancedness and deﬁciency margin — are both necessary, in the sense that violating any one of them may lead to convergence failure, as we demonstrated explicitly.
Moreover, for networks with output dimension 1 (scalar regression), we have shown that a balanced initialization, i.e. a random choice of the end-to-end matrix followed by a balanced partition across all layers, leads assumptions to be met, and thus convergence to take place, with constant probability.
Rigorously proving efﬁcient convergence with signiﬁcant probability under customary layer-wise independent initialization remains an open problem.
The recent work of Shamir (2018) suggests that this may not be possible, as at least in some settings, the number of iterations required for convergence is exponential in depth with overwhelming probability.
This negative result, a theoretical manifestation of the “vanishing gradient problem”, is circumvented by balanced initialization.
Through simple experiments we have shown that the latter can lead to favorable convergence in deep learning practice, as it does in theory.
Further investigation of balanced initialization, including development of variants for convolutional layers, is regarded as a promising direction for future research.
The analysis in this paper uncovers special properties of the optimization landscape in the vicinity of gradient descent trajectories.
We expect similar ideas to prove useful in further study of gradient descent on non-convex objectives, including training losses of deep non-linear neural networks. A CKNOWLEDGMENTS
This work is supported by NSF, ONR, Simons Foundation, Schmidt Foundation, Mozilla Research,
Amazon Research, DARPA and SRC.
Nadav Cohen is a member of the Zuckerman Israeli Postdoctoral Scholars Program, and is supported by Schmidt Foundation
A (cid:96) 2 L OSS OVER W HITENED D ATA
Recall the (cid:96) 2 loss of a linear predictor W ∈ R d y × d x as deﬁned in Section 2:
[Equation]
where X ∈ R d x × m and Y ∈ R d y × m .
Deﬁne Λ xx :
= 1 m XX (cid:62) ∈ R d x × d x , Λ yy := 1 m Y Y (cid:62) ∈
R d y × d y and Λ yx :=
1 m Y X
(cid:62) ∈ R d y × d x .
Using the relation (cid:107) A (cid:107) 2 F [Equation]
Tr( AA (cid:62) ) , we have:
[Equation]
By deﬁnition, when data is whitened, Λ xx is equal to identity, yielding:
[Equation]
where c := − 12 Tr(Λ yx Λ (cid:62) yx ) + 12 Tr(Λ yy ) does not depend on W .
Hence we arrive at Equation (1).
B A PPROXIMATE B ALANCEDNESS AND D EFICIENCY M ARGIN U NDER C USTOMARY I NITIALIZATION
Two assumptions concerning initialization facilitate our main convergence result (Theorem 1): (i) the initial weights W 1 (0) , . . .
, W N (0) are approximately balanced (see Deﬁnition 1); and (ii) the initial end-to-end matrix W 1: N (0) has positive deﬁciency margin with respect to the target
Φ (see
Deﬁnition 2).
The current appendix studies the likelihood of these assumptions being met under customary initialization of random (layer-wise independent)
Gaussian perturbations centered at zero.
For approximate balancedness we have the following claim, which shows that it becomes more and more likely the smaller the standard deviation of initialization is:
Claim 2.
Assume all entries in the matrices W j ∈ R d j × d j − 1 , j = 1 , . .
, N , are drawn independently at random from a Gaussian distribution with mean zero and standard deviation s > 0 .
Then, for any δ > 0 , the probability of W 1 , . . .
, W N being δ -balanced is at least max { 0 , 1 − 10 δ − 2 Ns 4 d 3 max } , where d max := max { d 0 , . . .
, d N } .
Proof.
See Appendix D.4.
In terms of deﬁciency margin, the claim below treats the case of a single output model (scalar regression), and shows that if the standard deviation of initialization is sufﬁciently small, with probability close to 0 . 5 , a deﬁciency margin will be met.
However, for this deﬁciency margin to meet a chosen threshold c , the standard deviation need be sufﬁciently large.
Claim 3.
There is a constant C 1 > 0 such that the following holds.
Consider the case where d N = 1 , d 0 ≥ 20 , 13 and suppose all entries in the matrices W j ∈ R d j × d j − 1 , j = 1 , . .
, N , are drawn independently at random from a Gaussian distribution with mean zero, whose standard deviation s > 0 is small with respect to the target, i.e. s ≤ (cid:107) Φ (cid:107) 1 /NF
[Equation]
(10 5 d 30 d 1 · · · d N − 1 C 1 ) 1 / (2 N ) .
Then, for any c with 0 < c ≤ (cid:107) Φ (cid:107) F (cid:14)(cid:0) 10 5 d 30 C 1 ( C 1 N ) 2 N (cid:1) , the probability of the end-to-end matrix W 1: N having deﬁciency margin c with respect to Φ is at least 0 .
if:
[Equation]
Proof.
See Appendix D.5.
C C ONVERGENCE F AILURES
In this appendix we show that the assumptions on initialization facilitating our main convergence result (Theorem 1) — approximate balancedness and deﬁciency margin — are both necessary, by demonstrating cases where violating each of them leads to convergence failure.
This accords with widely observed empirical phenomena, by which successful optimization in deep learning crucially depends on careful initialization ( cf.
Sutskever et al.
Claim 4 below shows 16 that if one omits from Theorem 1 the assumption of approximate balancedness at initialization, no choice of learning rate can guarantee convergence:
Claim 4.
Assume gradient descent with some learning rate η > 0 is a applied to a network whose depth N is even, and whose input, output and hidden dimensions d 0 , . . .
, d N are all equal to some d ∈ N .
Then, there exist target matrices Φ such that the following holds.
For any c with 0 < c < σ min (Φ) , there are initializations for which the end-to-end matrix W 1: N (0) has deﬁciency margin c with respect to Φ , and yet convergence will fail — objective will never go beneath a positive constant.
Proof.
See Appendix D.6.
In terms of deﬁciency margin, we provide (by adapting Theorem 4 in Bartlett et al.
a different, somewhat stronger result — there exist settings where initialization violates the assumption of deﬁciency margin, and despite being perfectly balanced, leads to convergence failure, for any choice of learning rate:
Claim 5. Consider a network whose depth N is even, and whose input, output and hidden dimensions d 0 , . . .
, d N are all equal to some d ∈ N .
Then, there exist target matrices Φ for which there are non-stationary initializations W 1 (0) , . . .
, W N (0) that are 0 -balanced, and yet lead gradient descent, under any learning rate, to fail — objective will never go beneath a positive constant.
Proof.
See Appendix D.7.
D D EFERRED P ROOFS
We introduce some additional notation here in addition to the notation speciﬁed in Section 2.
We use (cid:107) A (cid:107) σ to denote the spectral norm (largest singular value) of a matrix A , and sometimes (cid:107) v (cid:107) 2 as an alternative to (cid:107) v (cid:107) — the Euclidean norm of a vector v .
Recall that for a matrix A , vec ( A ) is its vectorization in column-ﬁrst order.
We let F ( · ) denote the cumulative distribution function of the standard normal distribution, i.e. F ( x ) =
(cid:82) x
−∞ 1 √ 2 π e − 12 u 2 du ( x ∈ R ).
To simplify the presentation we will oftentimes use W as an alternative (shortened) notation for W 1: N — the end-to-end matrix of a linear neural network.
We will also use L ( · ) as shorthand for L 1 ( · ) — the loss associated with a (directly parameterized) linear model, i.e. L ( W ) :
= 12 (cid:107) W − Φ (cid:107) 2 F .
Therefore, in the context of gradient descent training a linear neural network, the following expressions all represent the loss at iteration t :
[Equation]
Also, for weights W j ∈ R d j × d j − 1 , j = 1 , . .
, N of a linear neural network, we generalize the notation W 1: N , and deﬁne W j : j (cid:48) :
= W j (cid:48)
W j (cid:48) − 1 · · · W j for every 1 ≤ j ≤ j (cid:48) ≤ N .
Note that
W (cid:62) j : j (cid:48)
=
W (cid:62) j W (cid:62)
j +1 · · · W (cid:62) j (cid:48) .
Then, by a simple gradient calculation, the gradient descent updates (4) can be written as
[Equation]
where we deﬁne W 1:0 ( t ) :
= I d 0 and W N +1: N ( t ) :=
I d N for completeness.
Finally, recall the standard deﬁnition of the tensor product of two matrices (also known as the
Kronecker product): for matrices A ∈ R m A × n A ,
B ∈ R m B × n B , their tensor product
A ⊗ B ∈ R m
A m B × n A n B is deﬁned as
[Equation]
where a i,j is the element in the i -th row and j -th column of A .
D. P ROOF OF C LAIM
Proof.
Recall that for any matrices A and B of compatible sizes σ min ( A + B ) ≥ σ min ( A ) − σ max ( B ) , and that the Frobenius norm of a matrix is always lower bounded by its largest singular value (Horn and Johnson (1990)).
Using these facts, we have:
[Equation]
D. P ROOF OF L EMMA
To prove Lemma 1, we will in fact prove a stronger result, Lemma 2 below, which states that for each iteration t , in addition to (9) being satisﬁed, certain other properties are also satisﬁed, namely: (i) the weight matrices W 1 ( t ) , . .
, W N ( t ) are 2 δ -balanced, and (ii) W 1 ( t ) , . .
. , W N ( t ) have bounded spectral norms.
Lemma 2.
Suppose the conditions of Theorem 1 are satisﬁed.
Then for all
t ∈ N ∪ { 0 } , ( A ( t ) )
For 1 ≤ j ≤ N − 1 , (cid:107) W (cid:62)
j +1 ( t ) W j +1 ( t ) − W j ( t ) W (cid:62) j ( t ) (cid:107) F ≤ 2 δ .
( A (cid:48) ( t ) )
If t ≥ 1 , then for 1 ≤ j ≤ N − 1 ,
[Equation]
W (cid:62) j +1 ( t )
[Equation]
+
[Equation]
t )
[Equation]
t )
[Equation]
t )
[Equation]
F
[Equation]
( B ( t ) )
If t = 0 , then (cid:96) ( t ) ≤ 12 (cid:107) Φ (cid:107) 2 F .
If t ≥ 1 , then
[Equation]
( t ) ≤ (cid:96) ( t − 1) − η 2 σ min ( W ( t − 1)) 2( N − 1)
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table] [Equation]
( C ( t ) )
For 1 ≤ j ≤ N , (cid:107) W j ( t
[Table]
First we observe that Lemma 1 is an immediate consequence of Lemma 2.
Proof of Lemma 1.
Notice that condition B ( t ) of Lemma 2 for each t ≥ 1 immediately establishes the conclusion of Lemma 1 at time step t − 1 .
D.2. P RELIMINARY LEMMAS
We next prove some preliminary lemmas which will aid us in the proof of Lemma 2.
The ﬁrst is a matrix inequality that follows from Lidskii’s theorem.
For a matrix A , let Sing( A ) denote the rectangular diagonal matrix of the same size, whose diagonal elements are the singular values of A arranged in non-increasing order (starting from the (1 , 1) position).
Lemma 3 (Bhatia (1997), Exercise IV.3.5) .
For any two matrices A, B of the same size, (cid:107) Sing( A ) −
Sing( B ) (cid:107) σ ≤ (cid:107) A − B (cid:107) σ and (cid:107) Sing( A )
− Sing( B ) (cid:107) F ≤ (cid:107)
A − B (cid:107) F .
Using Lemma 3, we get:
Lemma 4.
Suppose D 1 , D 2 ∈ R d × d are non-negative diagonal matrices with non-increasing values along the diagonal and O ∈ R
d × d is an orthogonal matrix.
Suppose that (cid:107) D 1 − OD 2 O (cid:62) (cid:107) F ≤ (cid:15) , for some (cid:15) > 0 .
Then:
(cid:107) D 1 − OD 1 O (cid:62) (cid:107)
F ≤ 2 (cid:15) .
(cid:107) D 1 − D 2 (cid:107) F ≤ (cid:15) .
Proof.
Since D 1 and OD 2 O T are both symmetric positive semi-deﬁnite matrices, their singular values are equal to their eigenvalues.
Moreover, the singular values of D 1 are simply its diagonal elements and the singular values of OD 2 O T are simply the diagonal elements of D 2 .
Thus by
Lemma 3 we get that (cid:107) D 1 − D 2 (cid:107) F ≤ (cid:107) D 1 − OD 2 O T (cid:107) F ≤ (cid:15) .
Since the Frobenius norm is unitarily invariant, (cid:107) D 1 − D 2 (cid:107) F = (cid:107) OD 1 O T − OD 2 O T (cid:107) F , and by the triangle inequality it follows that [Equation] T
[Equation]
F
[Equation]
Lemma 5 below states that if W 1 , . .
, W N are approximately balanced matrices, i.e. W (cid:62)
j +1 W j +1 −
W j W (cid:62) j has small Frobenius norm for 1 ≤ j ≤ N − 1 , then we can bound the Frobenius distance between W (cid:62) 1: j W 1: j and ( W (cid:62) 1 W 1 ) j (as well as between W j : N W (cid:62) j : N and ( W N W (cid:62) N ) N − j +1 ).
Lemma 5.
Suppose that d N ≤ d N − 1 , d 0 ≤ d 1 , and that for some ν > 0 , M > 0 , the matrices
W j ∈ R d j × d j − 1 , 1 ≤ j ≤ N satisfy, for 1 ≤ j ≤ N − 1 ,
[Equation]
and for 1 ≤ j ≤ N , (cid:107) W j (cid:107) σ ≤ M .
Then, for 1 ≤ j ≤ N , (cid:107)
[Equation]
16) and
[Equation]
Moreover, if σ min denotes the minimum singular value of W 1: N , σ 1 ,min denotes the minimum singular value of W 1 and σ N,min denotes the minimum singular value of W N , then
[Equation]
1 ,min :
[Equation]
[Equation]
Proof.
For 1 ≤ j ≤ N , let us write the singular value decomposition of W j as W j = U j Σ j V (cid:62) j , where U j ∈ R d j × d j and V j ∈ R d j − 1 ×
d j − 1 are orthogonal matrices and Σ j ∈ R d j × d j − 1 is diagonal.
We may assume without loss of generality that the singular values of W j are non-increasing along the diagonal of Σ j .
Then we can write (15) as
[Equation]
Since the Frobenius norm is invariant to orthogonal transformations, we get that
[Equation]
By Lemma 4, we have that (cid:107) Σ (cid:62) j +1 Σ
j +1 − Σ j Σ (cid:62) j (cid:107) F ≤ ν and (cid:107)
[Equation]
U
[Equation]
V j
[Equation]
1 (cid:107) F ≤ 2 ν .
We may rewrite the latter of these two inequalities as
[Equation]
Note that
[Equation]
For matrices A, B , we have that (cid:107) AB (cid:107) F ≤ (cid:107) A (cid:107) σ · (cid:107) B (cid:107) F .
Therefore, for j + 1 ≤ i ≤ N , we have that
[Table]
F (cid:1) .
Next, we have that
[Equation] [Equation]
We now argue that (cid:107) (Σ i − 1 Σ (cid:62) i − 1 ) k − (Σ (cid:62) i Σ i ) k (cid:107) F ≤ ν · kM 2( k − 1) .
Note that (cid:107) Σ
i − 1 Σ (cid:62) i − 1 − Σ (cid:62) i Σ i (cid:107) F ≤ ν , verifying the case k = 1 .
To see the general case, since square diagonal matrices commute, we have that
[Equation]
= (Σ i − 1 Σ (cid:62) i − 1 ) (cid:96) (Σ (cid:62) i Σ i ) k − 1 − (cid:96) (cid:33)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F ≤ ν · k − [Equation] (cid:96) =
[Equation]
(cid:107) [Equation] i − 1 (cid:107)
[Equation]
(cid:96)σ · (cid:107)
[Equation]
i (cid:107)
[Equation]
( k − (cid:96) −
[Equation]
( k [Equation]
[Equation]
It then follows that
[Equation]
By the triangle inequality
[Equation]
we then have that
[Equation]
By an identical argument (formally, by replacing W j with W (cid:62) N − j +1 ), we get that
[Equation]
(19) and (20) verify (17) and (16), respectively, so it only remains to verify (18).
Letting j = 1 in (19), we get
[Equation]
Let us write the eigendecomposition of W 1: N W (cid:62) 1: N with an orthogonal eigenbasis as W 1: N W (cid:62) 1: N =
U Σ U (cid:62) , where Σ is diagonal with its (non-negative) elements arranged in non-increasing order and
U is orthogonal.
We can write the left hand side of (21) as (cid:107) U Σ U (cid:62) − U N (Σ N Σ (cid:62) N ) N U (cid:62) N (cid:107) F = (cid:107) Σ − U
[Equation]
U N
[Equation]
N U (cid:62)
[Equation]
U
[Equation]
By Lemma 4, we have that
[Equation]
Recall that W ∈ R d N × d 0 .
Suppose ﬁrst that d N ≤ d 0 .
Let σ min denote the minimum singular value of W 1: N (so that σ 2 min is the element in the ( d N , d N ) position of Σ ∈ R d N × d N ), and σ N,min denote the minimum singular value (i.e. diagonal element) of Σ N , which lies in the ( d N , d N ) position of Σ N .
(Note that the ( d N , d N ) position of Σ N ∈ R d N × d N − 1 exists since d N − 1 ≥ d N by assumption.)
Then
[Equation]
By an identical argument using (20), we get that, in the case that d 0 ≤ d N , if σ 1 ,min denotes the minimum singular value of Σ 1 , then
[Equation]
(Notice that we have used the fact that the nonzero eigenvalues of W 1: N W (cid:62) 1: N are the same as the nonzero eigenvalues of W (cid:62) 1: N W 1: N .)
This completes the proof of (18).
Using Lemma 5, we next show in Lemma
that if W 1 , . . . , W N are approximately balanced, then an upper bound on (cid:107) W N · · · W 1 (cid:107) σ implies an upper bound on (cid:107) W j (cid:107) σ for 1 ≤ j ≤ N .
Lemma 6.
Suppose ν, C are real numbers satisfying C > 0 and 0 < ν ≤ C 2 /N 30 N 2 .
Moreover suppose that the matrices W 1 , . . .
, W N satisfy the following:
For 1 ≤ j ≤ N − 1 , (cid:107) W (cid:62) j +1 W j +1 − W j W (cid:62) j (cid:107) F ≤ ν . 2.
(cid:107) W N · · · W 1 (cid:107) σ ≤ C .
Then for 1 ≤ j ≤ N , (cid:107) W j (cid:107) σ ≤ C 1 /N · 2 1 / (2 N ) .
Proof.
For 1 ≤ j ≤ N , let us write the singular value decomposition of W j as W j = U j Σ j V (cid:62) j , where the singular values of W j are decreasing along the main diagonal of Σ j .
By Lemma 4, we have that for 1 ≤ j ≤ N − 1 , (cid:107) Σ (cid:62) j +1 Σ
j +1 − Σ j Σ (cid:62) j (cid:107) F ≤ ν , which implies that (cid:12)(cid:12) (cid:107) Σ (cid:62) j +1 Σ
j +1 (cid:107) σ − (cid:107) Σ j Σ (cid:62) j
(cid:107) σ (cid:12)(cid:12) ≤ ν .
Write M = max 1 ≤ j ≤ N (cid:107) W j (cid:107) σ = max 1 ≤ j ≤ N (cid:107) Σ j (cid:107) σ .
By the above we have that (cid:107) Σ j Σ (cid:62) j
(cid:107) σ ≥
M 2 − Nν for 1 ≤ j ≤ N .
Let the singular value decomposition of W 1: N be denoted by W 1: N = U Σ V (cid:62) , so that (cid:107) Σ (cid:107) σ ≤ C .
Then by (17) of Lemma 5 and Lemma 4 (see also (22), where the same argument was used), we have that
[Equation]
Then
[Equation]
Now recall that ν is chosen
so that ν ≤ C 2 /N 30 · N 2 .
Suppose for the purpose of contradiction that there is some j such that (cid:107) W j W
[Equation]
j
[Equation]
> 2 1 /N C 2 /N .
Then it must be the case that
[Equation]
where we have used that
[Equation]
for all N ≥ 2 , which follows by considering the Laurent series exp(1 /z )
= (cid:80) ∞
i =1 1 i !
z
i , which converges in | z | > 0 for z ∈ C .
We now rewrite inequality (24) as
[Equation]
Next, using (25) and (1 + 1 /x ) x ≤ e for all x > 0 ,
[Equation]
Since (cid:107) (Σ N Σ (cid:62) N ) N (cid:107) σ = (cid:107) Σ N Σ (cid:62) N (cid:107) Nσ , we get by combining (23) and (26)
that
[Equation]
/N · C 2 /N , and since 1 − e/ 20 > 1 / (5 / 4) , it follows that (cid:107) Σ N Σ (cid:62) N (cid:107) σ < (5 / 4) 1 /N C 2 /N , which contradicts (24).
It follows that for all 1 ≤ j ≤ N , (cid:107) W j W (cid:62) j (cid:107) σ ≤ 2 1 /N C 2 /N .
The conclusion of the lemma then follows from the fact that (cid:107) W j W (cid:62) j (cid:107) σ = (cid:107) W j (cid:107) 2 σ .
D.2. S INGLE -S TEP D ESCENT
Lemma 7 below states that if certain conditions on W 1 ( t ) , . . . , W N ( t ) are met, the sought-after descent — Equation (9) — will take place at iteration t .
We will later show (by induction) that the required conditions indeed hold for every t , thus the descent persists throughout optimization.
The proof of Lemma 7 is essentially a discrete, single-step analogue of the continuous proof for
Lemma 1 (covering the case of gradient ﬂow) given in Section 3.
Lemma 7.
Assume the conditions of Theorem 1.
Moreover, suppose that for some t , the matrices
W 1 ( t ) , . . .
, W N ( t ) and the end-to-end matrix W ( t ) :
= W 1: N ( t ) satisfy the following properties:
(cid:107) W j ( t ) (cid:107) σ ≤ (4 (cid:107) Φ (cid:107) F ) 1 /N for 1 ≤ j ≤ N . 2.
(cid:107) W ( t ) − Φ (cid:107) σ ≤ (cid:107) Φ (cid:107) F . 3.
(cid:107) W (cid:62)
j +1 ( t ) W j +1 ( t ) − W j ( t ) W (cid:62) j
( t ) (cid:107) F ≤ 2 δ for 1 ≤ j ≤ N − 1 . 4.
σ min := σ min ( W ( t )
) ≥ c .
Then, after applying a gradient descent update (4) we have that
[Equation]
F
[Equation]
Proof.
For simplicity write M = (4 (cid:107) Φ (cid:107) F ) 1 /N and B = (cid:107) Φ (cid:107) F .
We ﬁrst claim that η ≤ min
[Equation]
Since c ≤ σ min , for (27) to hold it sufﬁces to have η ≤ min
[Equation]
As the minimum singular value of Φ must be at least c , we must have c ≤ (cid:107) Φ (cid:107) σ .
Since then c (cid:107) Φ
[Equation]
≤
[Equation]
1 , it holds that
[Equation]
meaning that it sufﬁces to have
[Equation]
which is guaranteed by (7).
Next, we claim that
[Equation]
The second inequality above is trivial, and for the ﬁrst to hold, since c ≤ (cid:107) Φ (cid:107) F , it sufﬁces to take
[Equation]
which is guaranteed by the deﬁnition of δ in Theorem 1.
Next we continue with the rest of the proof.
It follows from (14) that
W
[Equation]
(29) where ( (cid:63) ) denotes higher order terms in η .
We now bound the Frobenius norm of ( (cid:63) ) .
To do this, note that since L ( W ) = 12 (cid:107) W − Φ (cid:107) 2 F , dLdW ( W ( t ))
=
W ( t ) − Φ .
Then (cid:107) ( (cid:63) ) (cid:107) F ≤
N (cid:88) k
= η k · M k ( N − [Equation] N − k · (cid:13)(cid:13)(cid:13)(cid:13) dL
[Equation]
( t )
[Equation]
(cid:13)(cid:13)(cid:13)(cid:13)
F
[Equation]
(cid:13)(cid:13)(cid:13)(cid:13)
dL
[Equation]
( W ( t )) (cid:13)(cid:13)(cid:13)(cid:13) k −σ · (cid:18) N k (cid:19) ≤ ηM 2 N − 2 N (cid:13)(cid:13)(cid:13)(cid:13) dL dW ( W ( t )) (cid:13)(cid:13)(cid:13)(cid:13)
F
N (cid:88) k
= (cid:0) ηM
[Equation]
[Equation]
where the last inequality uses ηM N − 2 BN ≤ 1 / 2 , which is a consequence of (27).
Next, by Lemma 5 with ν = 2 δ ,
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
Next, by standard properties of tensor product, we have that
[Equation]
Let us write eigenvalue decompositions W (cid:62)
1 W 1 = UDU (cid:62) , W N W (cid:62) N = V EV (cid:62) .
Then
N (cid:88)
j = (cid:0) ( W (cid:62) 1 W 1 )
j − 1 ⊗ ( W N W (cid:62) N ) N − j (cid:1) =
N (cid:88)
j = (cid:0) UD j − 1 U (cid:62) ⊗ V E
N − j V (cid:62) (cid:1) =
( U ⊗ V )  
N (cid:88) j =
D j − 1 ⊗ E N −
j  
( U ⊗ V ) (cid:62) =
O Λ O (cid:62) , with O =
U ⊗ V , and Λ = (cid:80) Nj =1 D j − 1 ⊗ E
N − j .
As W 1 ∈ R d 1 × d 0 , and W N ∈ R d N × d N − 1 , then
D ∈ R d 0 × d 0 , E ∈ R d N × d N , so Λ ∈ R d 0 d N × d 0 d N .
Moreover note that Λ (cid:23) D 0 ⊗ E N − 1 + D N − 1 ⊗
E
= I d 0 ⊗ E N − 1 + D N − 1 ⊗ I d N .
If λ D denotes the minimum diagonal element of D and λ E denotes the minimum diagonal element of E , then the minimum diagonal element of Λ is therefore at least λ N − 1 D + λ N − 1 E .
But, it follows from Lemma 5 (with ν = 2 δ )
that
[Equation]
where the second inequality follows from (28).
Hence the minimum diagonal element of Λ is at least ( σ 2 min / (4 / 3))
( N − 1) /N ≥ σ 2( N − 1) /N
min / (4 / 3) .
It follows as a result of the above inequalities that if we write E ( t ) = vec ( W ( t +1))
− vec ( W ( t ))
+ η ( O Λ O (cid:62) ) vec (cid:0) dLdW ( W ( t ))
(cid:1) , then (cid:107) E [Equation] t ) [Equation]
=
[Equation]
≤
[Equation]
[Table]
[Equation]
Then we have
[Equation]
where the ﬁrst inequality follows since L ( W )
= 12 (cid:107) W − Φ (cid:107) 2 F is 1 -smooth as a function of W .
Next, by (29) and (30),
[Equation]
2 F ≤ 2 η 2 · (cid:18) NM 2( N − 1) · (cid:13)(cid:13)(cid:13)(cid:13) dL dW ( W ( t ))
(cid:13)(cid:13)(cid:13)(cid:13)
F (cid:19)+ 2 η 2 · (2 ηM 3 N − 4 N 2 B ) 2 · (cid:13)(cid:13)(cid:13)(cid:13) dL dW ( W ( t ))
(cid:13)(cid:13)(cid:13)(cid:13)
F = 2 η 2 (cid:13)(cid:13)(cid:13)(cid:13)
[Equation]
( W ( t )) (cid:13)(cid:13)(cid:13)(cid:13)
F
[Equation]
(cid:16) N
[Equation]
( N
[Equation] [Equation]
N
[Equation]
) (cid:17) .
Thus
L ( W ( t + 1))
− L ( W ( t )) ≤ η · (cid:13)(cid:13)(cid:13)(cid:13) dL dW ( W ( t ))
(cid:13)(cid:13)(cid:13)(cid:13)
F · (cid:32) − σ 2( N − 1) /N
min 4 /+ 2 ηM 3 N − 4 N 2 B + 2 δN 3 M 2( N − 2) + η · ( N 2 M 4( N − 1) + 4 η 2 M 6 N − 8 N 4 B 2 ) (cid:17) .
By (27, 28), which bound η, 2 δ , respectively, we have that
L ( W ( t + 1))
− L ( W ( t )) ≤ η · (cid:13)(cid:13)(cid:13)(cid:13) dL dW ( W ( t ))
(cid:13)(cid:13)(cid:13)(cid:13)
F · (cid:32) − σ 2( N − 1) /N
min 4 /+ σ 2( N − 1) /N
min + σ 2( N − 1) /N
min + σ 2( N − 1) /N min + σ 2( N − 1) /N
min (cid:33) =
− 1 2 σ 2( N − 1) /N
min
[Equation]
(cid:13)(cid:13)(cid:13)(cid:13)
[Equation]
D.2. P ROOF OF L EMMA
Proof of Lemma 2.
We use induction on t , beginning with the base case t = 0 .
Since the weights
W 1 (0) , . . .
, W N (0) are δ -balanced, we get that A (0) holds automatically.
To establish B (0) , note that since W 1: N (0) has deﬁciency margin c > 0 with respect to Φ , we must have (cid:107) W 1: N (0) − Φ (cid:107) F ≤ σ min (Φ) ≤ (cid:107) Φ (cid:107) F , meaning that L 1 ( W 1: N (0)) ≤ 12 (cid:107) Φ (cid:107) 2 F .
Finally, by B (0) , which gives (cid:107) W (0) − Φ (cid:107) F ≤ (cid:107) Φ (cid:107) F , we have that (cid:107)
[Equation]
[Equation]
σ
[Equation]
[Equation] [Equation] To show that the above implies C (0) , we use condition A (0) and Lemma 6 with C = 2 (cid:107) Φ (cid:107) F and ν = 2 δ .
By the deﬁnition of δ in Theorem 1 and since c ≤ (cid:107) Φ (cid:107) F , we have that
[Equation]
as required by Lemma 6.
As A (0) and (33) verify the preconditions 1.
and 2., respectively, of
Lemma 6, it follows that for 1 ≤ j ≤ N , (cid:107) W j ( t ) (cid:107) σ ≤ (2 (cid:107) Φ (cid:107) F ) 1 /N · 2 1 / (2 N ) <
(4 (cid:107) Φ (cid:107) F ) 1 /N , verifying C (0) and completing the proof of the base case.
The proof of Lemma 2 follows directly from the following inductive claims.
A ( t ) , B ( t ) , C ( t ) ⇒ B ( t + 1) .
To prove this, we use Lemma 7.
We verify ﬁrst that the preconditions hold.
First, C ( t ) immediately gives condition 1. of Lemma 7.
By B ( t ) , we have that (cid:107) W ( t ) − Φ (cid:107) σ ≤ (cid:107) W ( t ) − Φ (cid:107) F ≤ (cid:107) Φ (cid:107) F , giving condition 2. of Lemma 7.
A ( t ) immediately gives condition 3.
of Lemma 7.
Finally, by B ( t ) , we have that L N ( W 1 ( t ) , . . .
, W N ( t ))
≤ L N ( W 1 (0) , . .
W N (0)) , so σ min ( W 1: N ( t )) ≥ c by Claim 1.
This veriﬁes condition 4. of Lemma 7.
Then
Lemma 7 gives that
L N ( W 1 ( t + 1) , . . .
, W N ( t + 1)) ≤ L N ( W 1 ( t )
W N ( t ))
−
12 σ min ( W ( t )) 2( N − 1) /N
η (cid:13)(cid:13) dLdW ( W ( t ))
(cid:13)(cid:13) 2 F , establishing B ( t + 1) .
A (
0) , A (cid:48) (1) , . . .
, A (cid:48) ( t ) , A ( t ) , B (0) , . . .
, B ( t ) , C ( t ) ⇒
A ( t + 1) , A (cid:48)
( t + 1) .
To prove this, note that for 1 ≤ j ≤ N − 1 ,
[Equation]
By B (0) , . . .
, B ( t ) , (cid:107) W 1: N ( t ) − Φ (cid:107) F ≤ (cid:107) Φ (cid:107) F .
By the triangle inequality it then follows that (cid:107) W 1: N ( t ) (cid:107) σ ≤ 2 (cid:107) Φ (cid:107) F .
Also A ( t ) gives that for 1 ≤ j ≤ N − 1 , (cid:107) W j ( t ) W (cid:62) j ( t ) −
W (cid:62) j +1 ( t ) W j +1 ( t ) (cid:107) F ≤ 2 δ .
By Lemma 6 with C = 2 (cid:107) Φ (cid:107) F , ν = 2 δ (so that (34) is satisﬁed),
[Table]
[Equation]
In the ﬁrst inequality above, we have also used the fact that for matrices A, B such that AB is deﬁned, (cid:107) AB (cid:107) F ≤ (cid:107) A (cid:107) σ (cid:107) B (cid:107) F . (35) gives us A (cid:48)
( t + 1) .
We next establish A ( t + 1) .
By B ( i ) for 0 ≤ i ≤ t , we have that (cid:13)(cid:13) dLdW ( W ( i ))
(cid:13)(cid:13) F = (cid:107) W − Φ (cid:107) F ≤ (cid:107) Φ (cid:107) F .
Using A (cid:48)
( i ) for 0 ≤ i ≤ t and summing over i gives
[Equation]
[Equation]
Next, by B (0) , . . .
, B ( t ) , we have that L ( W ( i ))
≤ L ( W (0)) for i ≤ t .
Since W (0) has deﬁciency margin of c and by Claim 1, it then follows that σ min ( W ( i )) ≥ c for all i ≤ t .
Therefore, by summing B (0) , . . .
, B ( t ) ,
[Equation]
(cid:13)(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)(cid:13)
[Equation] F ≤ [Equation] t (cid:88) i = σ min
[Equation]
( i ) [Equation] ( N − 1) /N (cid:13)(cid:13)(cid:13)(cid:13)
[Equation]
(cid:13)(cid:13)(cid:13)(cid:13)
F [Equation] [Equation] L (
[Equation]
(0)) ≤
[Equation]
(cid:107) Φ (cid:107) 2 F . Therefore, 4 (2 (cid:107) Φ (cid:107) F ) 2( N − 1) /N
η 2 t (cid:88) i = (cid:13)(cid:13)(cid:13)(cid:13) dL dW
[Equation]
( i ) (cid:13)(cid:13)(cid:13)(cid:13)
F ≤ 16 (cid:107) Φ (cid:107)
N − 1) /N
F η (cid:107) Φ (cid:107) 2 F c 2( N − 1) /N
≤ 16 (cid:107) Φ (cid:107) 2( N − 1) /N
F · 3 · 2 11 · N 3 · c (4 N − 2) /N (cid:107) Φ (cid:107) (6 N − 4) /N
F · (cid:107) Φ (cid:107) 2 F c 2( N − 1) /N
(37) ≤ c256 N 3 (cid:107) Φ (cid:107) 2( N − 1) /N
F = δ, where (37) follows from the deﬁnition of η in (7), and the last equality follows from deﬁnition of δ in Theorem 1.
By (36), it follows that
[Equation]
verifying A ( t + 1) .
A ( t ) , B ( t ) ⇒ C ( t ) .
We apply Lemma 6 with ν = 2 δ and C = 2 (cid:107) Φ (cid:107) F .
First, the triangle inequality and B ( t ) give
[Equation]
verifying precondition 2. of Lemma 6.
A ( t ) veriﬁes condition 1. of Lemma 6, so for
[Equation]
giving C
[Equation]
t
[Equation]
The proof of Lemma 2 then follows by induction on t .
D. P ROOF OF T HEOREM
Theorem 2 is proven by combining Lemma 8 below, which implies that the balanced initialization is likely to lead to an end-to-end matrix W 1: N (0) with sufﬁciently large deﬁciency margin, with
Theorem 1, which establishes convergence.
Lemma 8.
Let d ∈ N , d ≥ 20 ; b 2 > b 1 ≥ 1 be real numbers (possibly depending on d ); and
Φ ∈ R
d be a vector.
Suppose that µ is a rotation-invariant distribution 19 over R d with a well-deﬁned density, such that, for some 0 < (cid:15) < 1 ,
[Equation]
Then, with probability at least (1 − (cid:15) ) · 3 − 4 F (2 / √ b 1 ), V will have deﬁciency margin (cid:107) Φ (cid:107) 2 / ( b 2 d ) with respect to Φ .
The proof of Lemma 8 is postponed to Appendix D.5, where Lemma 8 will be restated as Lemma 16.
One additional technique is used in the proof of Theorem 2, which leads to an improvement in the guaranteed convergence rate.
Because the deﬁciency margin of W 1: N (0) is very small, namely
O ( (cid:107) Φ (cid:107) 2 /d 0 ) (which is necessary for the theorem to maintain constant probability), at the beginning of optimization, (cid:96) ( t ) will decrease very slowly.
However, after a certain amount of time, the deﬁciency margin of W 1: N ( t ) will increase to a constant, at which point the decrease of (cid:96) ( t ) will be much faster.
To capture this acceleration, we apply Theorem 1 a second time, using the larger deﬁciency margin at the new “initialization.”
From a geometric perspective, we note that the matrices W 1 (0) , . . .
, W N (0) are very close to 0, and the point at which W j (0) = 0 for all j is a saddle.
Thus, the increase in (cid:96) ( t ) − (cid:96) ( t +1) over time captures the fact that the iterates ( W 1 ( t ) , . . .
, W N ( t )) escape a saddle point.
Proof of Theorem 2.
Choose some a ≥ 2 , to be speciﬁed later.
By assumption, all entries of the end-to-end matrix at time 0, W 1: N (0) , are distributed as independent Gaussians of mean 0 and standard deviation s ≤ (cid:107) Φ (cid:107) 2 / (cid:112) ad 20 .
We will apply Lemma 8 to the vector W 1: N (0)
∈ R d 0 .
Since its distribution is obviously rotation-invariant, in remains to show that the distribution of the norm (cid:107) W 1: N (0) (cid:107) 2 is not too spread out.
The following lemma — a direct consequence of the Chernoff bound applied to the χ 2 distribution with d 0 degrees of freedom — will give us the desired result:
Lemma 9 (Laurent and Massart (2000), Lemma 1) .
Suppose that d ∈ N and V ∈ R
d
is a vector whose entries are i.i.d.
Gaussians with mean 0 and standard deviation s .
Then, for any k > 0 ,
[Equation]
By Lemma 9 with k = d 0 / 16 , we have that
[Equation]
We next use Lemma 8, with b 1 = (cid:107) Φ (cid:107) 22 / (2 s 2 d 20 ) , b 2 = 2 (cid:107) Φ (cid:107) 22 / ( s 2 d 20 ) ; note that since a ≥ 2 , b 1 ≥ 1 , as required by the lemma.
Lemma 8 then implies that with probability at least
[Equation]
W 1: N (0) will have deﬁciency margin s 2 d 0 / 2 (cid:107) Φ (cid:107) 2 with respect to Φ .
By the deﬁnition of balanced initialization (Procedure 1) W 1 (0) , . . .
, W N (0) are 0 -balanced.
Since 2 4 · 6144 < 10 5 , our assumption on η gives
[Equation]
so that Equation (7) holds with c = s 2 d 0 2 (cid:107) Φ (cid:107) 2 .
The conditions of Theorem 1 thus hold with probability at least that given in Equation (38).
In such a constant probability event, by Theorem 1 (and the fact that a positive deﬁciency margin implies L 1 ( W 1: N (0)) ≤ 12 (cid:107) Φ (cid:107) 22 ), if we choose
[Equation]
[Equation] then L 1 ( W 1: N ( t 0 )) ≤ 1 8 (cid:107) Φ (cid:107) 2 2 , meaning that (cid:107) W 1: N ( t 0 ) − Φ (cid:107) 2 ≤ 1 2 (cid:107) Φ (cid:107)
= (cid:107) Φ (cid:107) 2 − 1 2 σ min (Φ) .
Moreover, by condition A ( t 0 ) of Lemma 2 and the deﬁnition of δ in Theorem 1, we have, for 1 ≤ j ≤ N − 1 ,
[Equation]
≤ [Equation] [Equation] We now apply Theorem 1 again, verifying its conditions again, this time with the initialization ( W 1 ( t 0 ) , . .
W N ( t 0 )) .
First note that the end-to-end matrix W 1: N ( t 0 ) has deﬁciency margin c
= (cid:107) Φ (cid:107) 2 / 2 as shown above.
The learning rate η , by Equation (39), satisﬁes Equation (7) with c = (cid:107) Φ (cid:107) 2 / 2 .
Finally, since
[Equation]
for d 0 ≥ 2 , by Equation (41), the matrices W 1 ( t 0 ) , . . .
, W N ( t 0 ) are δ -balanced with δ = (cid:107) [Equation] N
[Equation]
256 N. Iteration t 0 thus satisﬁes the conditions of Theorem 1 with deﬁciency margin (cid:107) Φ (cid:107) 2 / 2 , meaning that for
[Equation]
[Equation] we will have (cid:96) ( T ) ≤ (cid:15) .
Therefore, by Equations (40) and (42), to ensure that (cid:96) ( T ) ≤ (cid:15) , we may take
T ≥ 4 η −(cid:32) ln(4) (cid:18) (cid:107) Φ (cid:107)s 2 d(cid:19)
2 − 2 /N
+ (cid:107) Φ (cid:107) 2 /N − 2ln
( (cid:107) Φ (cid:107) 22 / (8 (cid:15) )) (cid:33) .
Recall that this entire analysis holds only with the probability given in Equation (38).
As lim d →∞ (1 − 2 exp( − d/ 16))
= 1 and lim a →∞ (3 − 4 F (2 (cid:112) 2 /a )) / 2 = 1 / 2 , for any 0 < p < 1 / 2 , there exist a, d (cid:48) 0 > 0 such that for d 0 ≥ d (cid:48) 0 , the probability given in Equation (38) is at least p .
This completes the proof.
In the context of the above proof, we remark that the expressions 1 − 2 exp( − d 0 / 16) and (3 − 4 F (2 (cid:112) 2 /a )) / 2 converge to their limits of 1 and 1 / 2 , respectively, as d 0 , a → ∞ quite quickly.
For instance, to obtain a probability of greater than 0 .
25 of the initialization conditions being met, we may take d 0 ≥ 100 , a ≥ 100 .
D. P ROOF OF C LAIM
We ﬁrst consider the probability of δ -balancedness holding between any two layers:
Lemma 10.
Suppose a, b, d ∈ N and A ∈ R a × d , B ∈ R d × b are matrices whose entries are distributed as i.i.d.
Gaussians with mean 0 and standard deviation s .
Then for k ≥ 1 ,
[Equation]
Proof.
Note that for 1 ≤ i, j ≤ d , let X ij be the random variable ( A T A − BB T ) ij , so that
[Equation]
If i (cid:54) =
j , then
[Equation]
We next note that for a normal random variable Y of variance s 2 and mean 0, E [ Y 4 ] = 3 s 4 .
Then if i = j ,
E [
[Equation] [Equation]
Thus
[Equation]
Then (43) follows from Markov’s inequality.
Now the proof of Claim 2 follows from a simple union bound:
Proof of Claim 2.
By (43) of Lemma 10, for each 1 ≤ j ≤ N − 1 , k ≥ 1 ,
[Equation]
By the union bound,
[Equation]
and the claim follows with δ =
ks 2 (cid:112) 10 d 3 max .
D
P ROOF OF C LAIM
We begin by introducing some notation.
Given d ∈ N and r > 0 , we let B d ( r ) denote the open ball of radius r centered at the origin in R d .
For an open subset U ⊂ R d , let ∂U :
=
¯ U \
U be its boundary, where ¯ U denotes the closure of U .
For the special case of U = B d ( r ) , we will denote by S d ( r ) the boundary of such a ball, i.e. the sphere of radius r centered at the origin in R d .
Let
S d := S d (1) and B d := B d (1) .
There is a well-deﬁned uniform (Haar) measure on S d ( r ) for all d, r , which we denote by σ d,r ; we assume σ d,r is normalized so that σ d,r ( S d ( r ))
= 1 .
Finally, since in the context of this claim we have d N = 1 , we allow ourselves to regard the end-to-end matrix
W 1: N ∈ R 1 × d 0 as both a matrix and a vector.
To establish Claim 3, we will use the following low-degree anti-concentration result of Carbery and
Wright (2001) (see also Lovett (2010); Meka et al.
Lemma 11 (Carbery and Wright (2001)) .
There is an absolute constant C 0 such that the following holds.
Suppose that h is a multilinear polynomial of K variables X 1 , . . .
, X K and of degree N .
Suppose that X 1 , . . .
, X K are i.i.d.
Gaussian.
Then, for any (cid:15) > 0 :
[Equation]
The below lemma characterizes the norm of the end-to-end matrix W 1: N following zero-centered
Gaussian initialization:
Lemma 12.
For any constant 0 < C 2 < 1 , there is an absolute constant C 1 > 0 such that the following holds.
Let N, d 0 , . . .
, d N − 1 ∈ N .
Set d N = 1 .
Suppose that for 1 ≤ j ≤ N ,
W j ∈ R d
j × d j − 1 are matrices whose entries are i.i.d.
Gaussians of standard deviation s and mean 0.
Then
[Equation]
Proof.
Let f ( W 1 , . .
. , W N )
= (cid:107) W 1: N (cid:107) 22 , so that f is a polynomial of degree 2 N in the entries of
W 1 , . .
. , W N .
Notice that f [Equation]
[Equation] W N
[Equation]
( W 1 ) i 1 ,i  .
For 1 ≤ i 0 ≤ d 0 , set g i
[Equation] [Equation]
i 1 = · · · d N
[Equation]
(cid:88) i N −
[Equation]
( W N ) 1 ,i N − 1 ( W N − 1 ) i N − 1 ,i N − 2 · · · ( W 1 ) i 1 ,i 0 , so that f = (cid:80) d 0 i
=1 g 2 i 0 .
Since each g i 0 is a multilinear polynomial in W 1 , . .
, W N , we have that E [ g i
( W 1 , . . .
, W N )]
= 0 for all 1 ≤ i 0 ≤ d 0 .
Also
Var[ g i 0 ( W 1 , . .
. , W N )]
=
E [ g
[Equation]
0 ( W 1 , . . .
, W N ) 2 ] = d
[Equation]
(cid:88)
[Equation]
1 = · · · d N
[Equation]
(cid:88)
i N − 1 = E
[Equation]
W [Equation] [Equation] N
[Equation]
( W
[Equation]
[Equation]
−
[Equation]
i N − 2 · · · ( W [Equation] [Equation] = d
[Equation]
d
[Equation]
[Equation]
N .
It then follows by Markov’s inequality that for any k ≥ 1 , P [ g 2 i 0 ≥
ks 2 N d 1 · · · d N − 1 ] ≤ 1 /k .
For any constant B 1 (whose exact value will be speciﬁed below), it follows that
[Equation]
Next, by Lemma 11, there is an absolute constant C 0 > 0 such that for any (cid:15) > 0 , and any 1 ≤ i 0 ≤ d 0 ,
[Equation]
Since f 2 ≥ g 2 i 0 for each i 0 , it follows that
[Equation]
Next, given 0 < C 2 < 1 , choose (cid:15) =
(1 − C 2 ) / (2 C 0 N ) , and B 1 = 2 / (1 − C 2 ) .
Then by (44) and (45) and a union bound, we have that
[Equation]
The result of the lemma then follows by taking C 1 = max (cid:110) 2 1 − C 2 , 2 C 0 1 − C 2 (cid:111) .
Lemma 13.
Let N, d 0 , . . .
, d N − 1 ∈ N , and set d N = 1 .
Suppose W j ∈ R d
j × d j − 1 for 1 ≤ j ≤ N , are matrices whose entries are i.i.d.
Gaussians with mean 0 and standard deviation s .
Then, the distribution of W 1: N is rotation-invariant.
Proof.
First we remark that for any orthogonal matrix O ∈ R d 0 × d 0 , the distribution of W 1 is the same as that of W 1 O .
To see this, let us denote the rows of W 1 by ( W 1 ) 1 , . .
, ( W 1 ) d 1 , and the columns of O by O 1 , . . .
, O d 0 .
Then the ( i 1 , i 0 ) entry of W 1 O , for 1 ≤ i 1 ≤ d 1 , 1 ≤ i 0 ≤ d 0 is (cid:104)
( W 1 ) i 1 , O i 0 (cid:105) , which is a Gaussian with mean 0 and standard deviation s , since (cid:107) O i 0 (cid:107) 2 = 1 .
Since (cid:104) O i 0 , O i (cid:48)
(cid:105)
=
for i 0 (cid:54) =
i (cid:48) 0 , the covariance between any two distinct entries of W 1 O is 0.
Therefore, the entries of W 1 O are independent Gaussians with mean 0 and standard deviation s , just as are the entries of W 1 .
But now for any matrix O ∈ R d 0 × d 0 , the distribution of W 1: N O is the distribution of
W N W N − 1 · · · W 2 ( W 1 O ) , which is the same as the distribution of W N W N − 1 · · · W 2 W 1 = W 1: N , since W 1 , W 2 , . .
, W N are all independent.
For a dimension d ∈ N , radius r > 0 , and 0 < h < r , a ( d, r ) -hyperspherical cap of height h is a subset C ⊂ B d ( r ) of the form { x ∈ B d ( r ) : (cid:104) x, u (cid:105) ≥ r − h } , where u is any d -dimensional unit vector.
We deﬁne the area of a ( d, r ) -hyperspherical cap of height h — C — to be σ d,r ( ∂ C∩ S d ( r )) .
Lemma 14.
For d ≥ 20 , choose any 0 ≤ h ≤ 1 .
Then, the area of a ( d, 1) -hyperspherical cap of height h is at least
[Equation]
Proof.
In Chudnov (1986), it is shown that the area of a ( d, 1) -hyperspherical cap of height h is given by 1 −
[Equation]
, where
[Equation]
Next, by the inequality 1 − t 2 ≥ exp( − 2 t 2 ) for 0 ≤ t ≤ 1 / 2 ,
[Equation]
where the last inequality follows from the standard estimate F ( x ) ≥ 1 − exp( − x 2 / 2) for x ≥ 1 .
Also, since 1 − t 2 ≤ exp( − t 2 ) for all t ,
[Equation]
Therefore, for d ≥ 20 , by (46) and (47),
[Equation]
where the second inequality has used 1 / (1 − y ) ≤ 1 + 2 y for all 0 < y < 1 / 2 (and where y = 2 exp(( − ( d − 3) / 4)) < 2 exp( − 17 / 4) < 1 / 2 ), and the ﬁnal inequality uses 1 + 4 exp( − ( d − 3) / 4) ≤ √ 2 for d ≥ 20 .
The above chain of inequalities gives us the desired result.
Lemma 15.
Let d ∈ N , d ≥ 20 ; a ≥ 1 be a real number (possibly depending on d ); and Φ ∈ R d be some vector.
Set r = (cid:107) Φ (cid:107) 2 / √ ad , and suppose that V ∈ S d ( r ) is drawn according to the uniform measure.
Then, with probability at least 3 − 4 F (2 / √ a ), V will have deﬁciency margin (cid:107) Φ (cid:107) 2 / ( ad ) with respect to Φ .
Proof.
By rescaling, we may assume without loss of generality that (cid:107) Φ (cid:107)
= 1 , so that r = 1 / √ ad .
Let D denote the intersection of B d ( r ) with the open d -ball of radius 1 − 1 / ( ad ) centered at Φ .
Let
C ⊂ B d ( r ) denote the ( d, r ) -hyperspherical cap of height r · (cid:0) 1 − 2 / ( √ ad ) (cid:1) =
r − 2 / ( ad ) whose base is orthogonal to the line between 0 and Φ (see Figure 2).
Note that σ d,r ( ∂ D ∩ S d ( r )) , the Haar measure of the portion of ∂ D intersecting S d ( r ) , gives the probability that V belongs to the boundary of D .
By Lemma 14 above (along with rescaling arguments), since d ≥ 20 , σ d,r ( ∂ C ∩ S d ( r )) ≥ 12 · (3 − 4 F (2 / √ a )) , and therefore V ∈ ∂ C with at least this probability.
We next claim that C ⊆ D .
To see this, ﬁrst let T ⊂ R d denote the ( d − 1) -sphere of radius 1 − 1 / ( ad ) centered at Φ (see Figure 2).
Let P be the intersection of T with the line from 0 to Φ , and Q denote the intersection of this line with the unique hyperplane of codimension 1 containing
T ∩ ∂B d ( r ) — we denote this hyperplane by H .
If we can show that (cid:107) P − Q (cid:107) 2 ≤ 1 / ( ad ) , then it follows that C lies entirely on the other side of H as 0 , which will complete the proof that C ⊆ D .
The calculation of (cid:107) P − Q (cid:107) 2 is simply an application of the law of cosines: letting θ be the angle determining the intersection of ∂B d ( r ) and T (see Figure 2), note that
[Equation]
as desired.
[Figure]
[Equation]
[Figure]
Using that C ⊆ D , we continue with the proof.
Notice the fact that C ⊆ D is equivalent to ∂ C ∩
S d ( r ) ⊆
∂ D ∩ S d ( r ) , by the structure of C and D .
Since the probability that V lands in ∂ C is at least 3 − 4 F (2 / √ a ), this lower bound applies to V landing in ∂ D as well.
Since all V ∈ ∂ D have distance at most 1 − 1 / ( ad ) from Φ , and since σ min (Φ) = (cid:107) Φ (cid:107) 2 = 1 , it follows that for any
V ∈ ∂ D , (cid:107) V − Φ (cid:107) 2 ≤ σ min (Φ) − 1 / ( ad ) .
Therefore, with probability of at least 3 − 4 F (2 / √ a ), V has deﬁciency margin (cid:107) Φ (cid:107) 2 / ( ad ) with respect to Φ .
Lemma 16 (Lemma 8 restated) .
Let d ∈ N , d ≥ 20 ; b 2 > b 1 ≥ 1 be real numbers (possibly depending on d ); and
Φ ∈ R
d be a vector.
Suppose that µ is a rotation-invariant distribution over R d with a well-deﬁned density, such that, for some 0 < (cid:15) < 1 ,
[Equation]
Then, with probability at least (1 − (cid:15) ) · 3 − 4 F (2 / √ b 1 ), V will have deﬁciency margin (cid:107) Φ (cid:107) 2 / ( b 2 d ) with respect to Φ .
Proof.
By rescaling we may assume that (cid:107) Φ (cid:107) 2 = 1 without loss of generality.
Then the deﬁciency margin of V is equal to 1 − (cid:107) V − Φ (cid:107) 2 .
µ has a well-deﬁned density, so we can set ˆ µ to be the probability density function of (cid:107) V (cid:107) 2 .
Since µ is rotation-invariant, we can integrate over spherical coordinates, giving
[Equation]
where the ﬁrst inequlaity used Lemma 15 and the fact that the distribution of V conditioned on (cid:107) V (cid:107) 2 = r is uniform on S d ( r ) .
Now we are ready to prove Claim 3:
Proof of Claim 3.
We let W ∈ R 1 × d 0 (cid:39) R d 0 denote the random vector W 1: N ; also let µ denote the distribution of W , so that by Lemma 13, µ is rotation-invariant.
Let C 1 be the constant from
Lemma 12 for C 2 = 999 / 1000 .
For some a ≥ 10 5 , the standard deviation of the entries of each W j is given by
[Equation]
Then by Lemma 12,
[Equation]
Then Lemma 16, with d = d 0 , b 1 = a and b 2 = ad 20 C 1 · ( C 1 N ) 2 N , implies that with probability at least 9991000 · 3 − 4 F (2 / √ a ), W has deﬁciency margin (cid:107) Φ (cid:107) 2 / ( ad 30 C 2 N +
N 2 N ) with respect to Φ .
But a ≥ 10 5 implies that this probability is at least 0 .
49 , and from (48),
[Equation]
Next recall the assumption in the hypothesis that s ≥ C 1 N ( c · (cid:107) Φ (cid:107)
[Equation]
[Equation] [Equation] N − 1 )) 1 / 2 N .
Then the deﬁciency margin in (49) is at least
[Equation]
completing the proof.
D. P ROOF OF C LAIM
Proof.
The target matrices Φ that will be used to prove the claim satisfy σ min (Φ)
= 1 .
We may assume without loss of generality that c ≥ 3 / 4 , the reason being that if a matrix has deﬁciency margin c with respect to Φ and c (cid:48) <
c , it certainly has deﬁciency margin c (cid:48) with respect to Φ .
We ﬁrst consider the case d = 1 , so that the target and all matrices are simply real numbers; we will make a slight abuse of notation in identifying 1 × 1 matrices with their unique entries.
We set Φ = 1 .
For all choices of η , we will set the initializations W 1 (0) , . . .
, W N (0) so that W 1: N (0) = c .
Then
[Equation]
so the initial end-to-end matrix W 1: N (0)
∈ R 1 × 1 has deﬁciency margin c .
Now ﬁx η .
Choose
A ∈ R with
[Equation]
We will set:
[Equation]
so that W 1: N (0) =
c .
Then since L N ( W 1 , . . . , W N ) =
[Equation]
1 − W N · ·
[Equation]
W 1 ) 2 , the gradient descent updates are given by
[Equation]
where we view W 1 ( t ) , . .
, W N ( t ) as real numbers.
This gives
[Equation]
Since 3 / 4 ≤ c < 1 and − η ( c − 1) c ( N − 1) /N
A ≥ 0 , we have that A/ 2 ≤ 3 A/ 4 ≤ W j (1) for 1 ≤ j ≤ N/ 2 .
Next, since 1 − c 1 − c 1 /N ≤ N for 0 ≤ c < 1 , we have that A 2 ≥ ηN ≥ η (1 − c ) 1 − c 1 /N , which implies that A 2 ≥ c 1 /N
A 2 + η (1 − c ) , or c 1 /N
A
+ η (1 − c )
A ≤ A .
Thus W j (1) ≤ A for N/ 2 < j ≤ N .
Similarly, using the same bound 3 / 4 ≤ c < 1 and the fact that η (1 − c ) c ( N − 1) /N
A ≥ 2 we get 316 ηA ≤ W j (1) ≤ ηA for N/ 2 < j ≤ N .
In particular, for all 1 ≤ j ≤ N , we have that min { η, 1 } A ≤ W j ( [Equation] ≤ max
[Equation]
A .
We prove the following lemma by induction:
Lemma 17.
For each t ≥ 1 , the real numbers W 1 ( t ) , . . .
, W N ( t ) all have the same sign and this sign alternates for each integer t .
Moreover, there are real numbers 2 ≤ B ( t ) <
C ( t ) for t ≥such that for 1 ≤ j ≤ N , B ( t ) ≤ | W j ( t ) | ≤ C ( t ) and ηB ( t ) 2 N − 1 ≥ 20 C ( t ) .
Proof.
First we claim that we may take B (1)
=
min { η, 1 }
A and C (1) = max { η, 1 } A .
We have shown above that B (1) ≤ W j (1) ≤ C (1) for all j .
Next we establish that ηB (1) 2 N − 1 ≥ 20 C (1) .
If η ≤ 1 , then
[Equation] [Equation]
, where the inequality follows from A ≥ (cid:16) 20 · 10 2 N − 1 η 2 N (cid:17) 1 / (2 N − 2) by deﬁnition of A .
If η ≥ 1 , then ηB (1) 2 N − 1 = η ( A/ 10) 2 N − 1 ≥ 20 ηA = 20 C (1) , where the inequality follows from A ≥ 2000 ≥ (cid:0) 20 · 10 2 N − 1 (cid:1) 1 / (2 N − 2) by deﬁnition of A .
Now, suppose the statement of Lemma 17 holds for some t .
Suppose ﬁrst that W j ( t ) are all positive for 1 ≤ j ≤ N .
Then for all j , as B ( t ) ≥ 2 , and ηB ( t ) 2 N − 1 ≥ 20 C ( t ) ,
[Equation]
which establishes that W j ( t + 1) is negative for all j .
Moreover,
[Equation]
Now set B ( t + 1) = 9 C ( t ) and C ( t + 1) =
ηC ( t ) 2 N − 1 .
Since N ≥ 2 , we have that
[Equation]
The case that all W j ( t ) are negative for 1 ≤ j ≤ N is nearly identical, with the same values for
B ( t + 1) , C ( t + 1) in terms of B ( t ) , C ( t ) , except all W j ( t + 1) will be positive.
This establishes the inductive step and completes the proof of Lemma 17.
By Lemma 17, we have that for all t ≥ 1 , L N ( W 1 ( t ) , . . .
, W N ( t ))
=
12 ( W 1: N ( t ) − 1) 2 ≥ 12 (2 N − 1) 2 > 0 , thus completing the proof of Claim 4 for the case where all dimensions are equal to 1.
For the general case where d 0 = d 1 = · · · = d N = d for some d ≥ 1 , we set Φ = I d , and given c, η , we set W j (0) to be the d × d diagonal matrix where all diagonal entries except the ﬁrst one are equal to 1, and where the ﬁrst diagonal entry is given by Equation (51), where A is given by Equation (50).
It is easily veriﬁed that all entries of W j ( t ) , 1 ≤ j ≤ N , except for the ﬁrst diagonal element of each matrix, will remain constant for all t ≥ 0 , and that the ﬁrst diagonal elements evolve exactly as in the 1-dimensional case presented above.
Therefore the loss in the d -dimensional case is equal to the loss in the 1-dimensional case, which is always greater than some positive constant.
We remark that the proof of Claim 4 establishes that the loss (cid:96) ( t ) :
=
L N ( W 1 ( t ) , . . .
, W N ( t )) grows at least exponentially in t for the chosen initialization.
Such behavior, in which gradients and weights explode, indeed takes place in deep learning practice if initialization is not chosen with care.
D. P ROOF OF C LAIM
Proof.
We will show that a target matrix Φ ∈ R d × d which is symmetric with at least one negative eigenvalue, along with identity initialization ( W j (0) =
I d , ∀ j ∈ { 1 , . . .
, N } ), satisfy the conditions of the claim.
First, note that non-stationarity of initialization is met, as for any 1 ≤ j ≤ N ,
[Equation]
( W 1: N (0) − Φ) W 1: j − 1 (0)
=
I d − Φ (cid:54) = 0 , where the last inequality follows since Φ has a negative eigenvalue.
To analyze gradient descent we use the following result, which was established in Bartlett et al.
Lemma 18 (Bartlett et al.
(2018), Lemma 6) .
If W 1 (0) , . .
. , W N (0) are all initialized to identity, Φ is symmetric, Φ = UDU (cid:62) is a diagonalization of Φ , and gradient descent is performed with any learning rate, then for each t ≥ 0 there is a diagonal matrix ˆ D ( t ) such that W j ( t )
=
U ˆ D ( t ) U (cid:62) for each 1 ≤ j ≤ N .
By Lemma 18, for any choice of learning rate η , the end-to-end matrix at time t is given by
W 1: N ( t ) =
U ˆ D ( t ) N U (cid:62) .
As long as some diagonal element of D is negative, say equal to − λ < 0 , then
[Equation]
E I MPLEMENTATION D ETAILS
Below we provide implementation details omitted from our experimental report (Section 4).
The platform used for running the experiments is PyTorch (Paszke et al., 2017).
For compliance with our analysis, we applied PCA whitening to the numeric regression dataset from UCI Machine
Learning Repository.
That is, all instances in the dataset were preprocessed by an afﬁne operator that ensured zero mean and identity covariance matrix.
Subsequently, we rescaled labels such that the uncentered cross-covariance matrix Λ yx (see Section 2) has unit Frobenius norm (this has no effect on optimization other than calibrating learning rate and standard deviation of initialization to their conventional ranges).
With the training objective taking the form of Equation (1), we then computed c — the global optimum — in accordance with the formula derived in Appendix A.
In our experiments with linear neural networks, balanced initialization was implemented with the assignment written in step (iii) of Procedure 1.
In the non-linear network experiment, we added, for each j ∈ { 1 , . . .
, N − 1 } , a random orthogonal matrix to the right of W j , and its transpose to the left of W j +1 — this assignment maintains the properties required from balanced initialization (see
Footnote 7).
During all experiments, whenever we applied grid search over learning rate, values between 10 − 4 and 1 (in regular logarithmic intervals) were tried.