G LOBAL OPTIMALITY CONDITIONS FOR DEEP NEURAL NETWORKS
A BSTRACT
We study the error landscape of deep linear and nonlinear neural networks with the squared error loss.
Minimizing the loss of a deep linear neural network is a nonconvex problem, and despite recent progress, our understanding of this loss surface is still incomplete.
For deep linear networks, we present necessary and sufﬁcient conditions for a critical point of the risk function to be a global minimum.
Surprisingly, our conditions provide an efﬁciently checkable test for global optimality, while such tests are typically intractable in nonconvex optimization.
We further extend these results to deep nonlinear neural networks and prove similar sufﬁcient conditions for global optimality, albeit in a more limited function space setting.
I NTRODUCTION
Since the advent of AlexNet (Krizhevsky et al., 2012), deep neural networks have surged in popularity, and have redeﬁned the state-of-the-art across many application areas of machine learning and artiﬁcial intelligence, such as computer vision, speech recognition, and natural language processing.
However, a concrete theoretical understanding of why deep neural networks work well in practice remains elusive.
From the perspective of optimization, a signiﬁcant barrier is imposed by the nonconvexity of training neural networks.
Moreover, it was proved by Blum & Rivest (1988) that training even a 3-node neural network to global optimality is NP-Hard in the worst case, so there is little hope that neural networks have properties that make global optimization tractable.
Despite the difﬁculties of optimizing weights in neural networks, the empirical successes suggest that the local minima of their loss surfaces could be close to global minima; and several papers have recently appeared in the literature attempting to provide a theoretical justiﬁcation for the success of these models.
For example, by relating neural networks to spherical spin-glass models from statistical physics, Choromanska et al.
(2015) provided some empirical evidence that size of neural networks makes the performance of local minima close to that of global minima.
Another line of results (Yu & Chen, 1995; Soudry & Carmon, 2016; Xie et al., 2016; Nguyen & Hein, 2017) provides conditions under which a critical point of the empirical risk is a global minimum.
Such results roughly involve proving that if full rank conditions of certain matrices (as well as some additional technical conditions) are satisﬁed, derivative of the risk being zero implies loss being zero.
However, these results are obtained under restrictive assumptions; for example, Nguyen &
Hein (2017) require the width of one of the hidden layers to be as large as the number of training examples.
Soudry & Carmon (2016) and Xie et al.
(2016) require the product of widths of two adjacent layers to be at least as large as the number of training examples, meaning that the number of parameters in the model must grow rapidly as we have more training data available.
Another recent paper (Haeffele & Vidal, 2017) provides a sufﬁcient condition for global optimality when the neural network is composed of subnetworks with identical architectures connected in parallel and a regularizer is designed to control the number of parallel architectures.
Towards obtaining a more precise characterization of the loss-surfaces, a valuable conceptual simpliﬁcation of deep nonlinear networks is deep linear neural networks, in which all activation functions are linear and the output of the entire network is a chained product of weight matrices with the input vector.
Although at ﬁrst sight a deep linear model may appear overly simplistic, even its opti- mization is nonconvex, and only recently theoretical results on this problem have started emerging.
Interestingly, already in 1989, Baldi & Hornik (1989) showed that some shallow linear neural networks have no local minima.
More recently, Kawaguchi (2016) extended this result to deep linear networks and proved that any local minimum is also global while any other critical point is a saddle point.
Subsequently, Lu & Kawaguchi (2017) provided a simpler proof that any local minimum is also global, with fewer assumptions than (Kawaguchi, 2016).
Motivated by the success of deep residual networks (He et al., 2016a;b), Hardt & Ma (2017) investigated loss surfaces of deep linear residual networks and showed every critical point is a global minimum in a near-identity region; subsequently, Bartlett et al.
(2017) extended this result to a nonlinear function space setting.
1. O UR CONTRIBUTIONS
Inspired by this recent line of work, we study deep linear and nonlinear networks, in settings either similar to or more general than existing work.
We summarize our main contributions below.
• We provide both necessary and sufﬁcient conditions for a critical point of the empirical risk to be a global minimum.
Speciﬁcally, Theorem 2.1 shows that if the hidden layers are wide enough, then a critical point of the risk function is a global minimum if and only if the product of all parameter matrices is full-rank.
In Theorem 2.2, we consider the case where some hidden layers have smaller width than both the input and output layers, and again provide necessary and sufﬁcient conditions for global optimality. In comparison, Kawaguchi (2016) only proves that every critical point of the risk is either a global minimum or a saddle; it is an “existence” result without any computational implication.
In contrast, we present efﬁciently checkable conditions for distinguishing the two different types of critical points; we can even use these conditions while running optimization to test whether the critical points we encounter are saddle points or not, if desired. It is also worth noting that such tests are intractable for general nonconvex optimization (Murty & Kabadi, 1987).
• Under the same assumption as (Hardt & Ma, 2017) on the data distribution, namely, a linear model with Gaussian noise, we can modify Theorem 2.1 to handle the population risk.
As a corollary, we not only recover Theorem 2.2 in (Hardt & Ma, 2017), but also extend it to a strictly larger set, while removing their assumption that the true underlying linear model has a positive determinant.
• Motivated by (Bartlett et al., 2017), we extend our results on deep linear networks to obtain sufﬁcient conditions for global optimality in deep nonlinear networks, although only via a function space view; these are presented in Theorems 4.1 and 4.2.
G LOBAL OPTIMALITY CONDITIONS FOR DEEP LINEAR NEURAL NETWORKS
In this section, we describe the problem formulation and notations for deep linear neural networks, state main results (Theorems 2.1 and 2.2), and explain their implication.
2. P ROBLEM FORMULATION AND NOTATION
Suppose we have m input-output pairs, where the inputs are of dimension d x and outputs of dimension d y .
Let X ∈ R
d
x × m be the data matrix and
Y ∈ R d y × m be the output matrix.
Suppose we have H hidden layers in the network, each having width d 1 , . . .
, d H .
For notational simplicity we let d 0 = d x and d H +1 =
d y .
The weights between adjacent layers can be represented as matrices W k ∈ R d k × d k − 1 , for k = 1 , . . .
, H + 1 , and the output of the network can be written as the product of weight matrices W H
+1 , . .
, W 1 and data matrix X : W H +1 W H · · · W 1 X .
We consider minimizing the summation of squared error loss over all data points (i.e. empirical risk), minimize
[Equation]
where W is a shorthand notation for the tuple ( W 1 , .
, W H +1 ) .
Assumptions.
We assume that d x ≤ m and d y ≤ m , and that XX T and Y X T have full ranks.
These assumptions are common when we consider supervised learning problems with deep neural networks (e.g. Kawaguchi (2016)).
We also assume that the singular values of Y X T ( XX T ) − 1 X are all distinct, which is made for notational simplicity and can be relaxed without too much difﬁculty.
Notation.
Given a matrix A , let σ max ( A ) and σ min ( A ) denote the largest and smallest singular values of A, respectively.
Let row( A ) , col( A ) , null( A ) , rank( A ) , and (cid:107) A (cid:107) F be respectively the row space, column space, null space, rank, and Frobenius norm of matrix A .
Given a subspace V of R n , we denote V ⊥ as its orthogonal complement.
Given a set V , let V c denote the complement of V .
Let us denote k :=
min i ∈{ 0 ,...,H +1 } d i , and deﬁne p ∈ argmin i ∈{ 0 ,...,H +1 } d i .
That is, p is any layer with the smallest width, and k = d p is the width of that layer.
Here, p might not be unique, but our results hold for any layer p with smallest width.
Notice also that the product W H +1 · · · Wcan have rank at most k .
Let Y X T ( XX T ) − 1 X
= U Σ V T be the singular value decomposition of Y X T ( XX T ) − 1 X ∈ R d y × d x .
Let ˆ U ∈ R d
y × k be a matrix consisting of the ﬁrst k columns of U .
N ECESSARY AND SUFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY
We now present two main theorems for deep linear neural networks.
The theorems describe two sets, one for the case k = min { d x , d y } and the other for k < min { d x , d y } , inside which every critical point of L ( W ) is a global minimum.
Moreover, the sets have another remarkable property that every critical point outside of these sets is a saddle point.
Previous works (Kawaguchi, 2016; Lu & Kawaguchi, 2017) showed that any critical point is either a global minimum or a saddle point, without providing any condition to distinguish between the two; here, we take a step further and partition the domain of L ( W ) into two sets clearly delineating one set which only contains global minima and the other set with only saddle points.
Theorem 2.1.
If k = min { d x , d y } , deﬁne the following set
[Equation]
Then, every critical point of L ( W ) in V 1 is a global minimum.
Moreover, every critical point of
L ( W ) in V c 1 is a saddle point.
Theorem 2.2.
If k < min { d x , d y } , deﬁne the following set
[Equation]
Then, every critical point of L ( W ) in V 2 is a global minimum.
Moreover, every critical point of
L ( W ) in V c 2 is a saddle point.
Theorems 2.1 and 2.2 provide necessary and sufﬁcient conditions for a critical point of L ( W ) to be globally optimal.
From an algorithmic perspective, they provide easily checkable conditions, which we can use to determine if the critical point the algorithm encountered is a global optimum or not.
Given that L ( W ) is nonconvex, it is interesting to have such efﬁcient tests for global optimality, which is not possible in general (Murty & Kabadi, 1987).
In Hardt & Ma (2017), the authors consider minimizing population risk of linear residual networks: minimize
[Equation]
where d x = d 1 = · · · = d H =
d y = d .
They assume that x is drawn from a zero-mean distribution with a ﬁxed covariance matrix, and y =
Rx + ξ where ξ is iid standard Gaussian noise and R is the true underlying matrix with det( R ) > 0 .
With these assumptions they prove that whenever σ max ( W i ) < 1 for all i , any critical point is a global minimum (Hardt & Ma, 2017, Theorem 2.2).
Under the same assumptions on data distribution, we can slightly modify Theorem 2.1 to derive a population risk counterpart, and in fact notice that the result proved in Hardt & Ma (2017) is a corollary of this modiﬁcation because having σ max ( W i ) <
1 for all i is a sufﬁcient condition for ( I + W H +1 ) · · · ( I + W 1 ) having full rank.
Moreover, notice that we can remove the assumption det( R ) > 0 which was required by Hardt & Ma (2017).
We state this special case as a corollary:
Corollary 2.3 (Theorem 2.2 of Hardt & Ma (2017)) .
Under assumptions on data distribution as described above, any critical point of 12 E x,y (cid:104) (cid:107) ( I + W H +1 ) · · · ( I + W 1 ) x − y (cid:107) 2F (cid:105) is a global minimum if σ max ( W i ) < 1 for all i .
We also note in passing that the classical problem of matrix factorization is a special case of deep linear neural networks, so our theorems can also be directly applied to such a setting.
Remarks.
The previous result (Kawaguchi, 2016) assumed d y ≤ d x and showed that: 1) every local minimum is a global minimum, and 2) any other critical point is a saddle point.
A subsequent paper by Lu & Kawaguchi (2017) proved 1) without the assumption d y ≤ d x , but as far as we know there is no result showing 2) in the case of d y > d x .
We provide the proof for this case in Lemma B.1.
In fact, we propose an alternative proof technique for handling degenerate critical points, which is much simpler than the technique presented by Kawaguchi (2016).
A NALYSIS OF DEEP LINEAR NETWORKS
In this section, we provide proofs for Theorems 2.1 and 2.2.
3. S OLUTIONS OF THE RELAXED PROBLEM
We ﬁrst analyze the globally optimal solution of a “relaxation” of L ( W ) , which turns out to be very useful while proving Theorems 2.1 and 2.2.
Consider the relaxed risk function
[Equation]
where R ∈ R d y × d x and rank( R ) ≤ k .
For any W , the product W H +1 W H · · · W 1 has rank at most k and setting R to be this product gives the same loss values: L 0 ( W H +1 W H · · · W 1 ) =
L ( W ) .
Therefore, L 0 is a relaxation of L and
[Equation]
This means that if there exists W such that L ( W ) =
inf R :rank( R ) ≤ k L 0 ( R ) , then W is a global minimum of the function L .
This observation is very important in proofs; we will show that inside certain sets, any critical point W of L ( W ) must satisfy R ∗ =
W H +1 · · · W 1 , where R ∗ is a global optimum of L 0 ( R ) .
This proves that L ( W ) =
L 0 ( R ∗ ) =
inf R :rank( R ) ≤ k L 0 ( R ) , thus showing that
W is a global minimum of L .
By restating this observation as an optimization problem, the solution of problem in (1) is bounded below by the minimum value of the following: minimize
[Equation]
In case where k = min { d x , d y } , (2) is actually an unconstrained optimization problem.
Note that
L 0 is a convex function of R , so any critical point is a global minimum.
By differentiating and setting the derivative to zero, we can easily get the unique globally optimal solution
[Equation]
In case of k < min { d x , d y } , the problem becomes non-convex because of the rank constraint, but its exact solution can still be computed easily.
We present the solution of this case as a proposition and defer the proof to Appendix C due to its technicalities.
Proposition 3.1.
Suppose k < min { d x , d y } .
Then the optimal solution to (2) is
[Equation]
which is the orthogonal projection of Y X T ( XX T ) − 1 onto the column space of ˆ U .
3. P ARTIAL DERIVATIVES OF L ( W )
By simple matrix calculus, we can calculate the derivatives of L ( W ) with respect to W i , for i = 1 , . . .
, H + 1 .
We present the result as the following lemma, and defer the details to Appendix C.
Lemma 3.2.
The partial derivative of L ( W ) with respect to W i is given as
[Equation]
for i = 1 , . . .
, H + 1 .
This result will be used throughout the proof of Theorems 2.1 and 2.2.
For clarity in notation, note that when i = 1 , W T 1 · · · W T 0 is just an identity matrix in R d x × d x .
Similarly, when i =
H + 1 ,
W TH +2 · · · W TH +1 is an identity matrix in R d y × d y .
We also state an elementary lemma which proves useful in our proofs, whose proof we defer to
Appendix C.
Lemma 3.3.
For any A ∈ R m × n and B ∈ R n × l where m ≥ n ,
[Equation]
For any
A ∈ R m × n
and B ∈ R n × l where n ≤ l ,
[Equation]
3. P ROOF OF T HEOREM 2.
We prove Theorem 2.1, which addresses the case k = min { d x , d y } .
First, recall that the set deﬁned in Theorem 2.1 is
[Equation]
As seen in (3), the unique minimum point of L 0 has rank k .
So, no point W ∈ V c 1 can be a global minimum of L .
Therefore, by Kawaguchi (2016, Theorem 2.3.(iii)) and Lemma B.1, any critical point in V c 1 must be a saddle point.
For the rest of our proof, we need to consider two cases: d y ≤ d x and d x ≤ d y .
If d x = d y , both cases work.
The outline of the proof is as follows: we deﬁne a new set W (cid:15) , show that any critical point in the set W (cid:15) is a global minimum, and then show that every W ∈ V 1 is also in W (cid:15) for some (cid:15) > 0 .
This proves that any critical point of L ( W ) in V 1 is also a critical point in W (cid:15) for some (cid:15) > 0 , hence a global minimum.
The following proposition proves the ﬁrst step:
Proposition 3.4.
Assume that k = min { d x , d y } .
For any (cid:15) > 0 , deﬁne the following set:
[Equation]
Then any critical point of L ( W ) in W (cid:15) is a global minimum point.
Proof.
(If d y ≤ d x ) Consider (5) in the case of i = 1 .
We can observe that W T 2 · · · W TH +1 ∈ R
d 1 × d y and that d 1 ≥ d y .
Then by Lemma 3.3.1,
[Equation]
By the above inequality, any critical point in W satisﬁes
[Equation]
which means that W H +1 W H · · · W 1 = Y X T ( XX T ) − 1 .
The product is the unique globally optimal solution (3) of the relaxed problem in (2), so W is a global minimum point of L .
(If d x ≤ d y ) Consider (5) for i
= H + 1 .
We can observe that W T 1 · · · W TH ∈ R
d x × d H and that d x ≤ d H .
Then by Lemma 3.3.2,
[Equation]
and the rest of the proof ﬂows in a similar way as the previous case.
The next proposition proves the theorem:
Proposition 3.5.
For any point W ∈ V 1 , there exists an (cid:15) >
such that W ∈ W (cid:15) .
Proof.
Deﬁne a new set W , a “limit” version (as (cid:15) → 0 ) of W (cid:15) , as
W := (cid:26) { ( W 1 , . . .
, W H +1 ) :
rank( W H +1 · · · W 2 ) =
d y } , if d y ≤ d x , { ( W 1 , . .
, W H +1 ) :
rank( W H · · · W 1 ) =
d x } , if d x ≤ d y .
We show that V 1 ⊂ W by showing that W c ⊂ V c 1 .
Consider
W [Equation] [Equation] [Equation]
+ [Equation]
W [Equation] ) <
d y } , if d y ≤ d x , { ( W 1 , . .
, W H +1 ) :
rank( W H · · · W 1 ) <
d x } , if d x ≤ d y .
Then any W ∈ W c must have rank( W H +1 · · · W 1 ) < min { d x , d y } =
k , so W ∈ V c 1 .
Thus, any
W ∈ V 1 is also in W , so either rank( W H +1 · · · W 2 ) =
d y or rank( W H · · · W 1 ) =
d x , depending on the cases.
Then, we can set (cid:15) =
[Equation] min ( W H +1 · · · W 2 ) , if d y ≤ d x , σ min ( W H · · · W 1 ) , if d x ≤ d y .
We always have (cid:15) > 0 because the matrices are full rank, and we can see that W ∈ W (cid:15) .
3. P ROOF OF T HEOREM 2.
In this section we prove Theorem 2.2, which tackles the case k < min { d x , d y } .
Note that this assumption also implies that 1 ≤ p ≤ H .
As for the proof of Theorem 2.1, deﬁne
[Equation]
The globally optimal point of the relaxed problem (2) has rank k , as seen in (4).
Thus, any point outside of V 1 cannot be a global minimum.
Then, by Kawaguchi (2016, Theorem 2.3.(iii)) and
Lemma B.1, it follows that any critical point in V c 1 must be a saddle point.
The remaining proof considers points in V 1 .
For this section, let us introduce some additional notations to ease presentation.
Deﬁne
[Equation]
so that ∂L∂W i
=
A i EB i .
Notice that A H +1 and B 1 are identity matrices.
Now consider any tuple W ∈ V 1 .
Since the full product W H +1 · · · W 1 has rank k , any partial products A i and B i must have rank( A i ) ≥ k and rank( B i ) ≥ k , for all i .
Then, consider A p ∈ R k × d y and B p
+1 ∈ R d x × k .
Since rank( A p ) ≤ k and rank( B p +1 ) ≤ k , we can see that rank( A p ) =
rank( B p +1 )
=
k .
Also, notice that A
i
=
W i +
A
i +1 and B i +
=
B i
W i , so that rank( A 1 ) ≤ rank( A 2 ) ≤ · · · ≤ rank( A p ) and rank( B H +1 ) ≤ rank( B H ) ≤ · · · ≤ rank( B p +1 ) .
However, we have k ≤ rank( A 1 ) and k ≤ rank( B H +1 ) , so the ranks are all identically k .
Also, row( A 1 ) ⊂ row( A 2 ) ⊂ · · · ⊂ row( A p ) and col( B H +1 ) ⊂ col( B H ) ⊂ · · · ⊂ col( B p +1 ) , but it was just shown that the these spaces have the same dimensions, which equals k , meaning row( A 1 ) = row( A 2 ) = · · · = row( A p ) and col( B H +1 ) =
col( B H ) = · · · = col( B p +1 ) .
Using this observation, we can now state a proposition showing necessary and sufﬁcient conditions for a tuple W ∈ V 1 to be a critical point of L ( W ) .
Proposition 3.6.
A tuple W ∈ V 1 is a critical point of L if and only if A p E = 0 and EB p
+
= 0 .
Proof.
(If part)
A p E = 0 implies that col( E ) ⊂ row( A p ) ⊥ = · · · = row( A 1 ) ⊥ , so ∂L∂W
i =
A i EB i =
· B i = 0 , for i = 1 , . . . , p .
Similarly, EB p +1 = 0 implies row( E ) ⊂ col( B p +1 ) ⊥ = · · · = col( B H +1 ) ⊥ , so ∂L∂W
i =
A i EB i
=
A i · 0 = 0 for i
= p + 1 , . .
, H + 1 .
(Only if part)
We have ∂L∂W
i
=
A i EB i =
for all i .
This means that
[Equation]
for
[Equation]
, p row [Equation] [Equation]
[Equation] for i = p + 1 , . . .
, H + 1 .
Now recall that B 1 and A H +1 are identity matrices, so col( E ) ⊂ row( A p ) ⊥ and row( E ) ⊂ col( B p +1 ) ⊥ , which proves A p E = 0 and EB p +1 = 0 .
Now we present a proposition that speciﬁes the necessary and sufﬁcient condition in which a critical point of L ( W ) in V 1 is a global minimum.
Recall that when we take the SVD Y X T ( XX T ) − 1 X =
U Σ V T , ˆ U ∈ R d y × k is deﬁned to be a matrix consisting of the ﬁrst k columns of U .
Proposition 3.7.
A critical point W ∈ V 1 of L ( W ) is a global minimum point if and only if col( W H +1 · · · W p +1 )
= row( A p ) =
col( ˆ U ) .
Proof.
Since W is a critical point, by Proposition 3.6 we have A p E = 0 .
Also note from the deﬁnitions of A i ’s and B i ’s
that W H +1 · · · W 1 = A Tp B Tp +1 , so
[Equation]
Because rank( A p ) =
k , and A p A Tp ∈ R
k × k is invertible, so B p +1 is determined uniquely as
[Equation]
thus
[Equation]
Comparing this with (4), W is a global minimum solution if and only if
[Equation]
This equation holds if and only if A Tp ( A p A Tp ) − 1 A p
=
ˆ
U ˆ U T , meaning that they are projecting
Y X T ( XX T ) − 1 onto the same subspace.
The projection matrix A Tp ( A p A Tp ) − 1 A p is onto row( A p ) , while ˆ U ˆ U T is onto col( ˆ U ) .
From this, we conclude that W is a global minimum point if and only if row( A p ) =
col( ˆ U ) .
From Proposition 3.7, we can deﬁne the set V 2 that appeared in Theorem 2.2, and conclude that every critical point of L ( W ) in V 2 is a global minimum, and any other critical points are saddle points.
E XTENSION TO DEEP NONLINEAR NEURAL NETWORKS
In this section, we present some sufﬁcient conditions for global optimality for deep nonlinear neural networks via a function space view.
Given a smooth nonlinear function h ∗ that maps input to output, Bartlett et al.
(2017) described a method to decompose it into a number of smooth nonlinear functions h ∗ =
h H +1 ◦ · · · ◦ h 1 where h
i ’s are close to identity.
Using Fr ´ echet derivatives of the population risk with respect to each function
h i , they showed that when all h i ’s are close to identity, any critical point of the population risk is a global minimum.
One can see that these results are direct generalization of Theorems 2.1 and 2.2 of Hardt & Ma (2017) to nonlinear networks and utilize the classical “small gain” arguments often used in nonlinear analysis and control (Khalil, 1996; Zames, 1966).
Motivated by this result, we extended Theorem 2.1 to deep nonlinear neural networks and obtained sufﬁcient conditions for global optimality in function space.
4. P ROBLEM FORMULATION AND NOTATION
Suppose the data X ∈ R d x and its corresponding label Y ∈
R
d y are drawn from some distribution.
Notice that in this section, X and Y are random vectors instead of matrices.
We want to predict Y given X with a deep nonlinear neural network that has H hidden layers.
We express each layer of the network as functions h
i : R d i − 1 → R d i , so the entire network can be expressed as a composition of functions: h H +1 ◦ h H ◦ · · · ◦ h 1 .
Our goal is to obtain functions h 1 , . .
h H
+1 that minimize the population risk functional:
[Equation]
where h is a shorthand notation for ( h 1 , . .
h H +1 ) .
It is well-known that the minimizer of squared error risk is the conditional expectation of Y given X , which we will denote h ∗ ( x ) =
E [ Y | X = x ] .
With this, we can separate the risk functional into two terms
[Equation]
where the constant C denotes the variance that is independent of h 1 , . .
h H
+1 .
Note that if h H +1 ◦ · · · ◦ h 1 = h ∗ almost surely, the ﬁrst term in L ( h ) vanishes and the optimal value L ∗ of L ( h ) is C .
Assumptions.
Deﬁne the function spaces as the following:
[Equation]
where F i are deﬁned for all i = 1 , . .
, H + 1 .
Assume that h ∗ ∈ F , and that we are optimizing
L ( h ) with h 1 ∈ F 1 , . .
, h H +1 ∈ F H
+1 .
In other words, the functions in F , F 1 , . .
. , F H +1 are differentiable and show sublinear growth starting from 0.
Notice that h H +1 ◦ · · · ◦ h 1 ∈ F , because a composition of differentiable functions is also differentiable, and a composition of sublinear functions is also sublinear.
We also assume that d i ≥ min { d x , d y } for all i = 1 , . . .
, H + 1 , which is identical to the assumption k = min { d x , d y } in Theorem 2.1.
Notation.
To simplify multiple composition of functions, we denote h i : j = h
i ◦ h
i − 1 ◦· · ·◦ h
j +1 ◦ h j .
As in the matrix case, h 0:1 and h H +1:
H +2 mean identity maps in R d x and R d y , respectively.
Given a function f , let J [ f ]( x ) be the Jacobian matrix of function f evaluated at x .
Let D h
i
[ L ( h )] be the Fr´echet derivative of L ( h ) with respect to h i evaluated at h .
The Fr´echet derivative D h i
[ L ( h )] is a linear functional that maps a function (direction) η ∈ F i to a real number (directional derivative).
4. S UFFICIENT CONDITIONS FOR GLOBAL OPTIMALITY
Here, we present two theorems which give sufﬁcient conditions for a critical point ( D h i
[ L ( h )]
=for all i ) in the function space to be a global optimum.
The proofs are deferred to Appendix A.
Theorem 4.1.
Consider the case d x ≥ d y .
If there exists (cid:15) > 0 such that
J [ h H +1:2 ]( z ) ∈ R d y ×
d 1 has σ min ( J [ h H +1:2 ]( z )) ≥ (cid:15) for all z ∈ R d 1 , 2.
h H +1:2 ( z ) is twice-differentiable,
then any critical point of L ( h ) , in terms of D h
[ L ( h )] , . . .
, D h H +
[ L ( h )] , is a global minimum.
Theorem 4.2.
Consider the case d x ≤ d y .
Assume that there exists some j ∈ { 1 , . . .
, H + 1 } such
that d x = d j − 1 and d y ≤ d j .
If there exist (cid:15) 1 , (cid:15) 2 > 0 such that
h j − 1:1 : R d x → R d j − 1 = R d x is invertible, 2.
h j − 1:1 satisﬁes (cid:107) h
j − 1:1 ( u ) (cid:107) 2 ≥ (cid:15)
(cid:107) u (cid:107) 2 for all u ∈ R d x , 3.
J [ h H +1: j +1 ]( z ) ∈ R d
y
×
d j has σ min ( J [ h H
+1: j +1 ]( z )) ≥ (cid:15) 2 for all z ∈ R d j , 4.
h H
+1: j +1 ( z ) is twice-differentiable,
then any critical point of L ( h ) , in terms of D h
[ L ( h )] , . . .
, D h H +
[ L ( h )] , is a global minimum.
Note that these theorems give sufﬁcient conditions, whereas Theorems 2.1 and 2.2 provide necessary and sufﬁcient conditions.
So, if the sets we are describing in Theorems 4.1 and 4.2 do not contain any critical point, the claims would be vacuous.
We ensure that there are critical points in the sets, by presenting the following proposition, whose proof is also deferred to Appendix A.
Proposition 4.3.
For each of Theorems 4.1 and 4.2, there exists at least one global minimum solution of L ( h ) satisfying the conditions of the theorem.
Discussion and Future work.
Theorems 4.1 and 4.2 state that in certain sets of ( h 1 , . .
h H +1 ) , any critical point in function space a global minimum.
However, this does not imply that any critical point for a ﬁxed sigmoid or arctan network is a global minimum.
As noted in (Bartlett et al., 2017), there is a downhill direction in function space at any suboptimal point, but this direction might be orthogonal to the function space represented by a ﬁxed network, and may hence result in local minima in the parameter space of the ﬁxed architecture.
Understanding the connection between the function space and parameter space of commonly used architectures is an open direction for future research, and we believe that these results can be good initial steps from the theoretical point of view.
For example, we can see that one of the sufﬁcient conditions for global optimality is the Jacobian matrix being full rank.
Given that a nonlinear function can locally be linearly approximated using Jacobians, this connection is already interesting. An extension of the function space viewpoint to cover different architectures or design new architectures (that have “better” properties when viewed via the function space view) should also be possible and worth studying.
This research project was supported in parts by DARPA DSO’s Fundamental Limits of Learning program.
A A NALYSIS OF DEEP NONLINEAR NETWORKS A. N OTATION
In this section, we introduce additional notation that is used in the proofs.
To emphasize that the Fr´echet derivative D h i [ L ( h )] is a linear functional that outputs a real number, we will write
D h
i
[ L ( h )]
( η ) in an inner-product form (cid:104) D
h
i
[ L ( h )] , η (cid:105) .
This notation also helps avoiding confusion coming from multiple parentheses and square brackets.
There are many different kinds of norms that appear in the proofs.
Given a ﬁnite-dimensional real vector v , (cid:107) v (cid:107) 2 denotes its (cid:96) 2 norm.
For a matrix A , its operator norm is deﬁned as (cid:107) A (cid:107) op =
sup x (cid:107) Ax (cid:107) 2 (cid:107) x (cid:107) 2 .
Let h ∈ F .
Then deﬁne a “generalized” induced norm for nonlinear functions with sublinear growth: (cid:107) h (cid:107) nl = sup x (cid:107) h ( x ) (cid:107) 2 (cid:107) x (cid:107) 2 , where the subscript nl is used to emphasize that this norm is for nonlinear functions.
The norm (cid:107)·(cid:107) nl is deﬁned in the same way for F i ’s. Now, given a linear functional G that maps a function f ∈ F i to a real number (cid:104) G, f (cid:105) , deﬁne the operator norm (cid:107) G (cid:107) op = sup f ∈F i (cid:104) G,f (cid:105) (cid:107) f (cid:107) nl .
A. F R ´ ECHET D ERIVATIVES
By deﬁnition of Fr ´ echet derivatives, we have
[Equation]
where η ∈ F i is the direction of perturbation and (cid:104) D
h i
[ L ( h )] , η (cid:105) is the directional derivative along that direction η .
From the deﬁnition of L ( h ) ,
[Equation]
Therefore,
[Equation]
i [Equation] [Equation] This equation (6) will be used in the proof of Theorems 4.1 and 4.2.
A. P ROOF OF T HEOREM 4.
From (6), consider D h 1 [ L ( h )] .
For any η ∈ F 1 ,
[Equation]
Let A ( X ) =
J [ h H +1:2 ]( h 1 ( X )) .
Since A ( X ) has full row rank by assumption, A ( X ) A ( X ) T is invertible.
Then deﬁne a particular direction
[Equation]
so that
[Equation]
It remains to check if ˜ η ∈ F 1 .
It is easily checked that ˜ η (0) = 0 because h H +1:1 (0) − h ∗ (0) = 0 .
Since J [ h H +1:2 ] is differentiable by assumption and h 1 ∈ F 1 , A ( X ) is differentiable and A ( X ) T , ( A ( X )
A ( X ) T ) − 1 are differentiable functions.
Also, h H +1:1 − h ∗ ∈ F , so we can conclude that ˜ η is differentiable.
Moreover, if we decompose A ( X ) with SVD, A ( X ) = U Σ V T , Σ is of the form Σ =
[Σ 1 0] and
[Equation]
from which we can see that
[Equation]
by our assumption.
Note that, for any X ∈ R d x ,
[Equation]
Since this holds for any X
[Equation]
we have
[Equation]
which ensures that ˜ η ∈ F 1 .
Finally,
[Equation]
which yields
[Equation]
From this we can see that if we have a critical point of L ( h ) , then (cid:107) D h 1 [ L ( h )] (cid:107) op = 0 implies
L ( h ) =
L ∗ , which means that the critical point is a global minimum of L ( h ) .
A. P ROOF OF T HEOREM 4.
Recall that by assumption we have j ∈ { 1 , . . .
, H + 1 } such
that d x = d j − 1 and d y ≤ d j .
Consider
D h
j
[ L ( h )] , then for any η ∈ F j , (cid:10) [Equation] [ L
[Equation]
h [Equation] ] , η (cid:11) =
E (cid:2) ( h H +1:1 ( X ) − h [Equation]
( X )) T J [ h H +1: j +1 ]( h j :1 ( X ))
η ( h j − 1:1 ( X )) (cid:3) .
As done in the previous theorem, for any w ∈ R d j − 1 , let A ( w ) =
J [ h H +1: j +1 ]( h j ( w )) .
Since A ( w ) has full row rank by assumption, A ( w ) A ( w ) T is invertible.
Then deﬁne
[Equation]
, so that
[Equation]
We need to check if ˜ η ∈ F j .
It is easily checked that ˜ η (0) = 0 .
Since J [ h H +1: j +1 ] is differentiable by assumption and h j ∈ F j , A ( w ) is differentiable, and so are A ( w ) T and ( A ( w ) A ( w ) T ) − 1 .
The inverse function of a differentiable and invertible function is also differentiable, so ( h H +1:1 − h ∗ ) ◦ h − 1 j − 1:1 is differentiable.
Hence, we can conclude that ˜ η is differentiable.
As seen in the previous section,
[Equation]
By the assumption that h j − 1:1 is invertible and (cid:107) h
j − 1:1 ( u ) (cid:107) 2 ≥ (cid:15)
(cid:107) u (cid:107) 2 , (cid:107) v (cid:107) 2 ≥ (cid:15) 1 (cid:13)(cid:13)
h − 1 j − 1:1 ( v ) (cid:13)(cid:13) 2 ,
[Table]
[Equation]
Finally,
[Equation]
which yields
[Equation]
A. P ROOF OF P ROPOSITION 4.
(Theorem 4.1) By assumption, we have d 1 ≥ d y .
Set h 1 ( x ) =
( h ∗ ( x ) , 0 , . . . , 0) where for every x ∈ R d x , the ﬁrst d y components of h 1 ( x ) are identical to h ∗ ( x ) , and all other components are zero.
For the rest of h i ’s, deﬁne h i : R d i − 1 → R d i to be h
i ( w ) = (cid:26) ( w 1 , . . . , w d i ) , if d i ≤ d i − 1 , ( w 1 , . . . , w d i − 1 , 0 , . . . , 0) , if d i > d i − 1 , (7) for all w ∈ R d i − 1 .
Since d i ≥ d y for all i , we can check that h H +1 ◦ · · · ◦ h 1 =
h ∗ , and h i ∈
F i for all i .
Moreover, for all z ∈ R d 1 , J [ h H +1:2 ]( z ) is all 0 except 1’s in diagonal entries, so σ min ( J [ h H +1:2 ]( z )) ≥ 1 and h H +1:
( z ) is twice-differentiable.
(Theorem 4.2)
It is given that we have j ∈ { 1 , . . .
, H + 1 } such
that d x = d j − 1 and d y ≤ d j .
Set h j ( x ) =
( h ∗ ( x ) , 0 , . . . , 0) , where the ﬁrst d y components are h ∗ ( x ) and the rest are zero.
All the rest of h i are set as in (7).
Then, it can be easily checked that h i ∈ F i for all i and all the conditions of the theorem are satisﬁed.
B D EFERRED L EMMA
Lemma B.1.
Suppose we are given a data matrix
X ∈ R d x × m and an output matrix Y ∈ R d y × m , where d x < d y .
Assume XX T and Y X T have full ranks.
Consider minimizing the empirical squared error risk:
[Equation]
where
W k ∈ R
d
k × d k − 1 , k
= 1 , . .
, H + 1 are weight matrices of the linear neural network, and d 0 = d x and d H +1 =
d y for simplicity in notation.
Also let W denote the tuple ( W 1 , . . .
, W H +1 ) .
Then, any critical point of L ( W ) that is not a local minimum is a saddle point.
Proof.
For this lemma, we separate the proof into two cases: W H · · · W 1 (cid:54) = 0 and W H · · · W 1 = 0 .
The crux of the proof is to show that any critical point cannot be a local maximum.
Then, any critical point is either a local minimum or a saddle point, so the conclusion of this lemma follows.
In case of W H · · · W 1 (cid:54) = 0 , we use some of the results in Kawaguchi (2016) and examine the
Hessian of L ( W ) with respect to vec( W T H +1 ) , where vec( A ) denotes vectorization of matrix A .
Let D vec( W TH +1 )
L ( W ) be the partial derivative of L ( W ) with respect to vec( W TH +1 ) in numerator layout.
It was shown by Kawaguchi (2016, Lemma 4.3) that the Hessian matrix
[Equation]
where ⊗ denotes the Kronecker product of two matrices.
Notice that H ( W ) is positive semideﬁnite.
Since XX T is full rank, whenever W H · · · W 1 (cid:54) =
there exists a strictly positive eigenvalue in
H ( W ) , which means that there exists an increasing direction.
So W cannot be a local maximum.
The case where W H · · · W 1 = 0 requires a bit more careful treatment.
Note that this case corresponds to where we have degenerate critical points, which are in many cases much harder to handle.
For any arbitrary (cid:15) > 0 , we describe a procedure that perturbs the matrices W 1 , . .
, W H +1 by perturbations sampled from Frobenius norm balls of radius (cid:15) centered at 0 , which we will denote as
B i ( (cid:15) ) , i = 1 , . . .
, H +1 .
Let U ( B i ( (cid:15) )) be the uniform distribution over the ball B i ( (cid:15) ) .
The algorithm goes as the following:
For i ∈ { 1 , . . .
, H + 1 } 1.1.
Sample ∆ i ∼ U
( B i ( (cid:15) )) , and deﬁne V i
= W i + ∆ i .
If W H +1 · · · W i +1 V i · · · V
(cid:54) = 0 , stop and return i ∗
= i .
First, recall that the set of rank-deﬁcient matrices have Lebesgue measure zero, so for any sample ∆ i ∼ U
( B i ( (cid:15) )) , V i
= W i + ∆ i has full rank with probability 1.
If we proceed the for loop until i = H +1 , we have a full-rank V H +1 · · · V 1 with probability 1, which means that the algorithm must return i ∗ ∈ { 1 , . .
, H + 1 } with probability 1.
Notice that before and after the i ∗ -th iteration, we have
[Equation]
This means that if we deﬁne ˆ∆ =
W H +1 · · · W i ∗ +1 ∆ i ∗ V i ∗ − 1 · · · V 1 , then ˆ∆ (cid:54) = 0 . Also, notice that
[Equation]
Now, deﬁne two points
[Equation]
and notice that they are all in the neighborhood of W , that is, the Cartesian product of (cid:15) -radius balls centered at W 1 , . .
W H
+1 .
Moreover, we have
[Equation]
from which we can see that at least one of L ( W ) <
L ( U (1) ) or L ( W ) <
L ( U (2) ) must hold.
This shows that for any (cid:15) > 0 , there is a point U in (cid:15) -neighborhood of W with a strictly greater function value
L ( U ) .
This proves that W cannot be a local maximum.
C D EFERRED P ROOFS C. P ROOF OF P ROPOSITION 3.
In case of k < min { d x , d y } , we can decompose the loss function in the following way:
[Equation]
Let us take a close look into the last term in the RHS.
Note that Y X T ( XX T ) − 1 X is the orthogonal projection of Y onto row( X ) , so each row of Y X T ( XX T ) − 1 X − Y must be in null( X ) .
Also, ( RX − Y X T ( XX T ) − 1 X ) T = X T ( R T − ( XX T ) − 1 XY T ) .
It is X T right-multiplied with some matrix, so its columns must lie in col( X T )
= row( X ) .
By the fact that null( X ) ⊥ = row
( X ) ,
[Equation]
thus
[Equation]
holds.
Now, (2) becomes a problem of minimizing (cid:13)(cid:13) RX − Y X T ( XX T ) − 1 X (cid:13)(cid:13) 2 F subject to the rank constraint rank( R ) ≤ k .
The optimal solution for this is obtained when RX is the k -rank approximation of Y X T ( XX T ) − 1 X .
Then, k -rank approximation of Y X T ( XX T ) − 1 X can be expressed as ˆ U
ˆ U T Y X T ( XX T ) − 1 X , where ˆ U is unique due to our assumption that all singular values are distinct.
Therefore,
R ∗ = ˆ U
ˆ U T Y X T ( XX T ) −is the unique global minimum solution of (2) when k < min { d x , d y } .
C. P ROOF OF L EMMA 3.
[Equation]
From this, we can conclude that ∂L
[Equation]
Y )
X T W T 1 · · · W Ti − 1 .
C. P ROOF OF L EMMA 3.
Since A T A (cid:23) σ 2min ( A ) I , B T A T AB (cid:23) σ 2min ( A ) B T B .
Then (cid:107) AB (cid:107) 2F = tr( B T A T AB ) ≥ σ 2min ( A ) tr
( B T B ) =
σ 2min ( A ) (cid:107) B (cid:107) 2F . 2.
Since BB T (cid:23) σ 2min ( B ) I ,
ABB T
A T (cid:23) σ 2min ( B ) AA T .
Then (cid:107) AB (cid:107) 2F = tr( B T A T AB ) = tr( ABB T A T ) ≥ σ 2min ( B ) tr
( AA T ) =
σ 2min ( B ) (cid:107) A (cid:107) 2F .