summaries: In this work the authors prove several claims regarding the inductive bias of gradient descent and gradient flow trained on deep linear networks with linearly separable data. They show that asymptotically gradient descent minimizes the risk, each weight matrix converges to its rank one approximation and the top singular vectors of two adjacent weight matrices align. Furthermore, for the logistic and exponential loss the induced linear predictor converges to the max margin solution.  This paper studies the properties of applying gradient flow and gradient descent to deep linear networks on linearly separable data. For strictly decreasing loss like the logistic loss, this paper shows 1) the loss goes to 0, 2) for every layer the normalized weight matrix converges to a rank-1 matrix 3) these rank-1 matrices are aligned. For the logistic loss, this paper further shows the linear function is the maximum margin solution.
abstractText: This paper establishes risk convergence and asymptotic weight matrix alignment — a form of implicit regularization — of gradient flow and gradient descent when applied to deep linear networks on linearly separable data. In more detail, for gradient flow applied to strictly decreasing loss functions (with similar results for gradient descent with particular decreasing step sizes): (i) the risk converges to 0; (ii) the normalized ith weight matrix asymptotically equals its rank-1 approximation uiv i ; (iii) these rank-1 matrices are aligned across layers, meaning |v> i+1ui| → 1. In the case of the logistic loss (binary cross entropy), more can be said: the linear function induced by the network — the product of its weight matrices — converges to the same direction as the maximum margin solution. This last property was identified in prior work, but only under assumptions on gradient descent which here are implied by the alignment phenomenon.
authors.name: Ziwei Ji
authors.name: Matus Telgarsky
references.authors: Sanjeev Arora Nadav Cohen Elad Hazan
references.title: On the optimization of deep networks: Implicit acceleration by overparameterization
references.venue: arXiv preprint arXiv:1802.06509,
references.authors: Peter Bartlett Dylan Foster Matus Telgarsky
references.title: Spectrally-normalized margin bounds for neural networks
references.authors: Simon S Du Wei Hu Jason D Lee
references.title: Algorithmic regularization in learning deep homogeneous models: Layers are automatically balanced
references.venue: arXiv preprint arXiv:1806.00900,
references.authors: Suriya Gunasekar Jason Lee Daniel Soudry Nathan Srebro
references.title: Implicit bias of gradient descent on linear convolutional networks
references.venue: arXiv preprint arXiv:1806.00468,
references.authors: Ziwei Ji Matus Telgarsky
references.title: Risk and parameter convergence of logistic regression
references.venue: arXiv preprint arXiv:1803.07300,
references.authors: Alex Krizhevsky Geoffrey Hinton
references.title: Learning multiple layers of features from tiny images
references.venue: Technical report, Citeseer,
references.authors: Alex Krizhevsky Ilya Sutskever Geoffery Hinton
references.title: Imagenet classification with deep convolutional neural networks
references.venue: In NIPS,
references.authors: Thomas Laurent James von Brecht
references.title: Deep linear neural networks with arbitrary loss: All local minima are global
references.venue: arXiv preprint arXiv:1712.01473,
references.authors: Jason D Lee Max Simchowitz Michael I Jordan Benjamin Recht
references.title: Gradient descent converges to minimizers
references.venue: arXiv preprint arXiv:1602.04915,
references.authors: Haihao Lu Kenji Kawaguchi
references.title: Depth creates no bad local minima
references.venue: arXiv preprint arXiv:1702.08580,
references.authors: Andrew M Saxe James L McClelland Surya Ganguli
references.title: Exact solutions to the nonlinear dynamics of learning in deep linear neural networks
references.venue: arXiv preprint arXiv:1312.6120,
references.authors: Daniel Soudry Elad Hoffer Nathan Srebro
references.title: The implicit bias of gradient descent on separable data
references.venue: arXiv preprint arXiv:1710.10345v3,
references.authors: Chiyuan Zhang Samy Bengio Moritz Hardt Benjamin Recht Oriol Vinyals
references.title: Understanding deep learning requires rethinking generalization
references.authors: Yi Zhou Yingbin Liang
references.title: Critical points of linear neural networks: Analytical forms and landscape properties
references.authors: Du
references.title: Proof of Lemma 2. The first claim is true for k = L since WL is a row vector. For any 1 ≤ k < L, recall that Arora et al
references.authors: ‖Wk‖F
references.title: sinceR(W ) has no finite optimum. Finally, invoking Lemma 2 proves the final claim of Theorem 1. Proof of Lemma 3. Soudry et al. (2017) Lemma 12 proves that, with probability 1, there are at most d support vectors, and moreover, the i-th support vector zi has a positive dual variable αi
sections.heading: 1 INTRODUCTION
sections.text: Efforts to explain the effectiveness of gradient descent in deep learning have uncovered an exciting possibility: it not only finds solutions with low error, but also biases the search for low complexity solutions which generalize well (Zhang et al., 2017; Bartlett et al., 2017; Soudry et al., 2017; Gunasekar et al., 2018).
  This paper analyzes the implicit regularization of gradient descent and gradient flow on deep linear networks and linearly separable data. For strictly decreasing losses, the optimum is at infinity, and we establish various alignment phenomena:
  • For each weight matrix Wi, the corresponding normalized weight matrix Wi/‖Wi‖F asymptotically equals its rank-1 approximation uiv>i , where the Frobenius norm ‖Wi‖F satisfies ‖Wi‖F → ∞. In other words, ‖Wi‖2/‖Wi‖F → 1, and asymptotically only the rank-1 approximation of Wi contributes to the final predictor, a form of implicit regularization. • Adjacent rank-1 weight matrix approximations are aligned: |v>i+1ui| → 1. • For the logistic loss, the first right singular vector v1 ofW1 is aligned with the data, meaning v1 converges to the unique maximum margin predictor ū defined by the data. Moreover, the linear predictor induced by the network, wprod := WL · · ·W1, is also aligned with the data, meaning wprod/‖wprod‖→ ū.
  Simultaneously, this work proves that the risk is globally optimized: it asymptotes to 0. Alignment and risk convergence are proved simultaneously; the phenomena are coupled within the proofs.
  Since the layers align, they can be viewed as a minimum norm solution: they do not “waste norm” on components which are killed off when the layers are multiplied together. Said another way, given data ((xi, yi))ni=1, the normalized matrices (W1/‖W1‖F , . . . ,WL/‖WL‖F ) asymptotically solve a maximum margin problem which demands all weight matrices be small, not merely their product:
  max WL∈R1×dL−1 ‖WL‖F =1 · · · max W1∈Rd1×d0 ‖W1‖F =1 min i yi(WL · · ·W1)xi.
  The paper is organized as follows. This introduction continues with related work, notation, and assumptions in Sections 1.1 and 1.2. The analysis of gradient flow is in Section 2, and gradient descent is analyzed in Section 3. The paper closes with future directions in Section 4; a particular highlight is a preliminary experiment on CIFAR-10 which establishes empirically that a form of the alignment phenomenon occurs on the standard nonlinear network AlexNet.
sections.heading: 1.1 RELATED WORK
sections.text: On the implicit regularization of gradient descent, Soudry et al. (2017) show that for linear predictors and linearly separable data, the gradient descent iterates converge to the same direction as the maximum margin solution. Ji & Telgarsky (2018) further characterize such an implicit bias for general nonseparable data. Gunasekar et al. (2018) consider gradient descent on fully connected linear networks and linear convolutional networks. In particular, for the exponential loss, assuming the risk is minimized to 0 and the gradients converge in direction, they show that the whole network converges in direction to the maximum margin solution. These two assumptions are on the gradient descent process itself, and specifically the second one might be hard to interpret and justify. Compared with Gunasekar et al. (2018), this paper proves that the risk converges to 0 and the weight matrices align; moreover the proof here proves the properties simultaneously, rather than assuming one and deriving the other. Lastly, Arora et al. (2018) show for deep linear networks (and later Du et al. (2018) for ReLU networks) that gradient flow does not change the difference between squared Frobenius norms of any two layers. We use a few of these tools in our proofs; please see Sections 2 and 3 for details.
  For a smooth (nonconvex) function, Lee et al. (2016) show that any strict saddle can be avoided almost surely with small step sizes. If there are only countably many saddle points and they are all strict, and if gradient descent iterates converge, then this implies (almost surely) they converge to a local minimum. In the present work, since there is no finite local minimum, gradient descent will go to infinity and never converge, and thus these results of Lee et al. (2016) do not show that the risk converges to 0.
  There has been a rich literature on linear networks. Saxe et al. (2013) analyze the learning dynamics of deep linear networks, showing that they exhibit some learning patterns similar to nonlinear networks, such as a long plateau followed by a rapid risk drop. Arora et al. (2018) show that depth can help accelerate optimization. On the landscape properties of deep linear networks, Lu & Kawaguchi (2017); Laurent & von Brecht (2017) show that under various structural assumptions, all local optima are global. Zhou & Liang (2018) give a necessary and sufficient characterization of critical points for deep linear networks.
sections.heading: 1.2 NOTATION, SETTING, AND ASSUMPTIONS
sections.text: Consider a data set {(xi, yi)}ni=1, where xi ∈ Rd, ‖xi‖ ≤ 1, and yi ∈ {−1,+1}. The data set is assumed to be linearly separable, i.e., there exists a unit vector uwhich correctly classifies every data point: for any 1 ≤ i ≤ n, yi〈u, xi〉 > 0. Furthermore, let γ := max‖u‖=1 min1≤i≤n yi〈u, xi〉 > 0 denote the maximum margin, and ū := arg max‖u‖=1 min1≤i≤n yi〈u, xi〉 denote the maximum margin solution (the solution to the hard-margin SVM).
  A linear network of depth L is parameterized by weight matrices WL, . . . ,W1, where Wk ∈ Rdk×dk−1 , d0 = d, and dL = 1. Let W = (WL, . . . ,W1) denote all parameters of the network. The (empirical) risk induced by the network is given by
  R(W ) = R (WL, . . . ,W1) = 1
  n n∑ i=1 ` (yiWL · · ·W1xi) = 1 n n∑ i=1 ` ( 〈wprod, zi〉 ) ,
  where wprod := (WL · · ·W1)>, and zi := yixi. The loss ` is assumed to be continuously differentiable, unbounded, and strictly decreasing to 0. Examples include the exponential loss `exp(x) = e−x and the logistic loss `log(x) = ln ( 1 + e−x ) .
  Assumption 1. `′ < 0 is continuous, limx→−∞ `(x) =∞ and limx→∞ `(x) = 0.
  This paper considers gradient flow and gradient descent, where gradient flow { W (t) ∣∣t ≥ 0, t ∈ R} can be interpreted as gradient descent with infinitesimal step sizes. It starts from some W (0) at
  t = 0, and proceeds as dW (t)
  dt = −∇R
  ( W (t) ) .
  By contrast, gradient descent { W (t) ∣∣t ≥ 0, t ∈ Z} is a discrete-time process given by W (t+ 1) = W (t)− ηt∇R ( W (t) ) ,
  where ηt is the step size at time t.
  We assume that the initialization of the network is not a critical point and induces a risk no larger than the risk of the trivial linear predictor 0. Assumption 2. The initialization W (0) satisfies∇R ( W (0) ) 6= 0 andR ( W (0) ) ≤ R(0) = `(0).
  It is natural to require that the initialization is not a critical point, since otherwise gradient flow/descent will never make a progress. The requirement R ( W (0) ) ≤ R(0) can be easily satisfied, for example, by making W1(0) = 0 and WL(0) · · ·W2(0) 6= 0. On the other hand, if R ( W (0) ) > R(0), gradient flow/descent may never minimize the risk to 0. Proofs of those claims are given in Appendix A.
sections.heading: 2 RESULTS FOR GRADIENT FLOW
sections.text: In this section, we consider gradient flow. Although impractical when compared with gradient descent, gradient flow can simplify the analysis and highlight proof ideas. For convenience, we usually useW ,Wk, andwprod, but they all change with (the continuous time) t. Only proof sketches are given here; detailed proofs are deferred to Appendix B.
sections.heading: 2.1 RISK CONVERGENCE
sections.text: One key property of gradient flow is that it never increases the risk:
  dR(W ) dt =
  〈 ∇R(W ), dW
  dt
  〉 = −‖∇R(W )‖2 = − L∑ k=1 ∥∥∥∥ ∂R∂Wk ∥∥∥∥2 F ≤ 0. (1)
  We now state the main result: under Assumptions 1 and 2, gradient flow minimizes the risk, Wk and wprod all go to infinity, and the alignment phenomenon occurs. Theorem 1. Under Assumptions 1 and 2, gradient flow iterates satisfy the following properties:
  • limt→∞R(W ) = 0.
  • For any 1 ≤ k ≤ L, limt→∞ ‖Wk‖F =∞.
  • For any 1 ≤ k ≤ L, letting (uk, vk) denote the first left and right singular vectors of Wk,
  lim t→∞ ∥∥∥∥ Wk‖Wk‖F − ukv>k ∥∥∥∥ F = 0.
  Moreover, for any 1 ≤ k < L, limt→∞ ∣∣〈vk+1, uk〉∣∣ = 1. As a result,
  lim t→∞ ∣∣∣∣∣∣ 〈 wprod∏L k=1 ‖Wk‖F , v1 〉∣∣∣∣∣∣ = 1, and thus limt→∞ ‖wprod‖ =∞.
  Theorem 1 is proved using two lemmas, which may be of independent interest. To show the ideas, let us first introduce a little more notation. Recall that R(W ) denotes the empirical risk induced by the deep linear network W . Abusing the notation a little, for any linear predictor w ∈ Rd, we also useR(w) to denote the risk induced by w. With this notation,R(W ) = R(wprod), while
  ∇R(wprod) = 1
  n n∑ i=1 `′ ( 〈wprod, zi〉 ) zi = 1 n n∑ i=1 `′ (WL · · ·W1zi) zi
  is in Rd and different from∇R(W ), which has ∑L k=1 dkdk−1 entries, as given below:
  ∂R ∂Wk = W>k+1 · · ·W>L ∇R(wprod)>W>1 · · ·W>k−1.
  Furthermore, for any R > 0, let
  B(R) = { W ∣∣∣∣ max1≤k≤L ‖Wk‖F ≤ R } .
  The first lemma shows that for any R > 0, the time spent by gradient flow in B(R) is finite. Lemma 1. Under Assumption 1 and 2, for any R > 0, there exists a constant (R) > 0, such that for any t ≥ 1 and any W ∈ B(R), ‖∂R/∂W1‖F ≥ (R). As a result, gradient flow spends a finite amount of time in B(R) for any R > 0, and max1≤k≤L ‖Wk‖F is unbounded.
  Here is the proof sketch. If ‖Wk‖F are bounded, then ‖∇R(wprod)‖ will be lower bounded by a positive constant, therefore if ‖∂R/∂W1‖F = ‖WL · · ·W2‖‖∇R(wprod)‖ can be arbitrarily small, then ‖WL · · ·W2‖ and ‖wprod‖ can also be arbitrarily small, and thusR(W ) can be arbitrarily close toR(0). This cannot happen after t = 1, otherwise it will contradict Assumption 2 and eq. (1). To proceed, we need the following properties of linear networks from prior work (Arora et al., 2018; Du et al., 2018). For any time t ≥ 0 and any 1 ≤ k < L, W>k+1(t)Wk+1(t)−W>k+1(0)Wk+1(0) = Wk(t)W>k (t)−Wk(0)W>k (0). (2) To see this, just notice that
  W>k+1 ∂R
  ∂Wk+1 = W>k+1 · · ·W>L ∇R(wprod)>W>1 · · ·W>k = ∂R ∂Wk W>k .
  Taking the trace on both sides of eq. (2), we have∥∥Wk+1(t)∥∥2F −∥∥Wk+1(0)∥∥2F =∥∥Wk(t)∥∥2F −∥∥Wk(0)∥∥2F . (3) In other words, the difference between the squares of Frobenius norms of any two layers remains a constant. Together with Lemma 1, it implies that all ‖Wk‖F are unbounded. However, even if ‖Wk‖F are large, it does not follow necessarily that ‖wprod‖ is also large. Lemma 2 shows that this is indeed true: for gradient flow, as ‖Wk‖F get larger, adjacent layers also get more aligned to each other, which ensures that their product also has a large norm.
  For 1 ≤ k ≤ L, let σk, uk, and vk denote the first singular value (the 2-norm), the first left singular vector, and the first right singular vector of Wk, respectively. Furthermore, define
  D := ( max
  1≤k≤L ‖Wk(0)‖2F
  ) − ‖WL(0)‖2F + L−1∑ k=1 ∥∥∥Wk(0)W>k (0)−W>k+1(0)Wk+1(0)∥∥∥ 2 ,
  which depends only on the initialization. If for any 1 ≤ k < L,Wk(0)W>k (0) = W>k+1(0)Wk+1(0), then D = 0. Lemma 2. The gradient flow iterates satisfy the following properties:
  • For any 1 ≤ k ≤ L, ‖Wk‖2F − ‖Wk‖22 ≤ D. • For any 1 ≤ k < L, 〈vk+1, uk〉2 ≥ 1− ( D+‖Wk+1(0)‖22+‖Wk(0)‖ 2 2 ) /‖Wk+1‖22.
  • Suppose max1≤k≤L ‖Wk‖F →∞, then ∣∣∣〈wprod/∏Lk=1 ‖Wk‖F , v1〉∣∣∣→ 1.
  The proof is based on eq. (2) and eq. (3). If Wk(0)W>k (0) = W > k+1(0)Wk+1(0), then eq. (2) gives that Wk+1 and Wk have the same singular values, and Wk+1’s right singular vectors and Wk’s left singular vectors are the same. If it is true for any two adjacent layers, since WL is a row vector, all layers have rank 1. With general initialization, we have similar results when ‖Wk‖F is large enough so that the initialization is negligible. Careful calculations give the exact results in Lemma 2.
  An interesting point is that the implicit regularization result in Lemma 2 helps establish risk convergence in Theorem 1. Specifically, by Lemma 2, if all layers have large norms, ‖WL · · ·W2‖ will be large. If the risk is not minimized to 0, ‖∇R(wprod)‖ will be lower bounded by a positive constant, and thus ‖∂R/∂W1‖F = ‖WL · · ·W2‖‖∇R(wprod)‖ will be large. Invoking eq. (1), Lemma 1 and eq. (3) gives a contradiction. Since the risk has no finite optimum, ‖Wk‖F →∞.
sections.heading: 2.2 CONVERGENCE TO THE MAXIMUM MARGIN SOLUTION
sections.text: Here we focus on the exponential loss `exp(x) = e−x and the logistic loss `log(x) = ln(1 + e−x). In addition to risk convergence, these two losses also enable gradient descent to find the maximum margin solution.
  To get such a strong convergence, we need one more assumption on the data set. Recall that γ = max‖u‖=1 min1≤i≤n〈u, zi〉 > 0 denotes the maximum margin, and ū denotes the unique maximum margin predictor which attains this margin γ. Those data points zi for which 〈ū, zi〉 = γ are called support vectors. Assumption 3. The support vectors span the whole space Rd.
  Assumption 3 appears in prior work Soudry et al. (2017), and can be satisfied in many cases: for example, it is almost surely true if the number of support vectors is larger than or equal to d and the data set is sampled from some density w.r.t. the Lebesgue measure. It can also be relaxed to the situation that the support vectors and the whole data set span the same space; in this case∇R(wprod) will never leave this space, and we can always restrict our attention to this space.
  With Assumption 3, we can state the main theorem. Theorem 2. Under Assumptions 2 and 3, for almost all data and for losses `exp and `log, then limt→∞ ∣∣〈v1, ū〉∣∣ = 1, where v1 is the first right singular vector of W1. As a result, limt→∞ wprod/ ∏L k=1 ‖Wk‖F = ū.
  Before summarizing the proof, we can simplify both theorems into the following minimum norm property mentioned in the introduction. Corollary 1. Under Assumptions 2 and 3, for almost all data and for losses `exp and `log,
  min i yi ( WL ‖WL‖F · · · W1 ‖W1‖F ) xi −−−→ t→∞ max
  AL∈R1×dL−1 ‖AL‖F =1 · · · max A1∈Rd1×d0 ‖A1‖F =1 min i yi (AL · · ·A1)xi.
  Theorem 2 relies on two structural lemmas. The first one is based on a similar almost-all argument due to Soudry et al. (2017, Lemma 12). Let S ⊂ {1, . . . , n} denote the set of indices of support vectors. Lemma 3. Under Assumption 3, if the data set is sampled from some density w.r.t. the Lebesgue measure, then with probability 1,
  α := min |ξ|=1,ξ⊥ū max i∈S 〈ξ, zi〉 > 0.
  Let ū⊥ denote the orthogonal complement of span(ū), and let Π⊥ denote the projection onto ū⊥. We prove that if ‖Π⊥w‖ is large enough, gradient flow starting fromw will tend to decrease ‖Π⊥w‖. Lemma 4. Under Assumption 3, for almost all data, `exp and `log, and any w ∈ Rd, if 〈w, ū〉 ≥ 0 and ‖Π⊥w‖ is larger than 1+ln(n)/α for `exp or 2n/eα for `log, then 〈Π⊥w,∇R(w)〉 ≥ 0.
  With Lemma 3 and Lemma 4 in hand, we can prove Theorem 2. Let Π⊥W1 denote the projection of rows of W1 onto ū⊥. Notice that
  Π⊥wprod = ( WL . . .W2(Π⊥W1) )> and
  d‖Π⊥W1‖2F dt = −2〈Π⊥wprod,∇R(wprod)〉.
  If ‖Π⊥W1‖F is large compared with ‖W1‖F , since layers become aligned, ‖Π⊥wprod‖ will also be large, and then Lemma 4 implies that ‖Π⊥W1‖F will not increase. At the same time, ‖W1‖F →∞, and thus for large enough t, ‖Π⊥W1‖F must be very small compared with ‖W1‖F . Many details need to be handled to make this intuition exact; the proof is given in Appendix B.
sections.heading: 3 RESULTS FOR GRADIENT DESCENT
sections.text: One key property of gradient flow which is used in the previous proofs is that it never increases the risk, which is not necessarily true for gradient descent. However, for smooth losses (i.e, with
  Lipschitz continuous derivatives), we can design some decaying step sizes, with which gradient descent never increases the risk, and basically the same results hold as in the gradient flow case. Deferred proofs are given in Appendix C.
  We make the following additional assumption on the loss, which is satisfied by the logistic loss `log. Assumption 4. `′ is β-Lipschitz (i.e, ` is β-smooth), and |`′| ≤ G (i.e., ` is G-Lipschitz).
  Under Assumption 4, the risk is also a smooth function of W , if all layers are bounded. Lemma 5. Suppose ` is β-smooth. If R ≥ 1, then β(R) = 2L2R2L−2(β + G), and R(W ) is a β(R)-smooth function on the set B(R) = { W ∣∣‖Wk‖F ≤ R, 1 ≤ k ≤ L}.
  Smoothness ensures that for any W,V ∈ B(R), R(W ) − R(V ) ≤ 〈∇R(V ),W − V 〉 + β(R)‖W−V ‖2/2 (see Bubeck et al. (2015) Lemma 3.4). In particular, if we choose some R and set a constant step size ηt = 1/β(R), then as long as W (t+ 1) and W (t) are both in B(R),
  R ( W (t+ 1) ) −R ( W (t) ) ≤ 〈 ∇R ( W (t) ) ,−ηt∇R ( W (t) )〉 + β(R)η2t
  2 ∥∥∥∇R (W (t))∥∥∥2 = − 1
  2β(R) ∥∥∥∇R (W (t))∥∥∥2 = −ηt 2 ∥∥∥∇R (W (t))∥∥∥2 . (4) In other words, the risk does not increase at this step. However, similar to gradient flow, the gradient descent iterate will eventually escape B(R), which may increase the risk. Lemma 6. Under Assumption 1, 2 and 4, suppose gradient descent is run with a constant step size 1/β(R). Then there exists a time t whenW (t) 6∈ B(R), in other words, max1≤k≤L ‖Wk(t)‖F > R.
  Fortunately, this issue can be handled by adaptively increasing R and correspondingly decreasing the step sizes, formalized in the following assumption. Assumption 5. The step size ηt = min{1/β(Rt), 1}, where Rt satisfies W (t) ∈ B(Rt− 1), and if W (t+ 1) ∈ B(Rt − 1), Rt+1 = Rt.
  Assumption 5 can be satisfied by a line search, which ensures that the gradient descent update is not too aggressive and the boundary R is increased properly.
  With the additional Assumptions 4 and 5, exactly the same theorems can be proved for gradient descent. We restate them briefly here. Theorem 3. Under Assumption 1, 2, 4, and 5, gradient descent satisfies
  • limt→∞R ( W (t) ) = 0.
  • For any 1 ≤ k ≤ L, limt→∞ ‖Wk(t)‖F =∞. • limt→∞ ∣∣∣〈wprod(t)/∏Lk=1 ‖Wk(t)‖F , v1(t)〉∣∣∣ = 1, where v1(t) is the first right singular vector
  of W1(t). Theorem 4. Under Assumptions 2, 3 and 5, for the logistic loss `log and almost all data, limt→∞ ∣∣〈v1(t), ū〉∣∣ = 1, and limt→∞ wprod(t)/∏Lk=1 ‖Wk(t)‖F = ū. Corollary 2. Under Assumptions 2, 3 and 5, for the logistic loss `log and almost all data,
  min i yi ( WL ‖WL‖F · · · W1 ‖W1‖F ) xi −−−→ t→∞ max
  AL∈R1×dL−1 ‖AL‖F =1 · · · max A1∈Rd1×d0 ‖A1‖F =1 min i yi (AL · · ·A1)xi.
  Proofs of Theorem 3 and 4 are given in Appendix C, and are basically the same as the gradient flow proofs. The key difference is that an error of ∑∞ t=0 η 2 t ‖∇R(W (t))‖2 will be introduced in many parts of the proof. However, it is bounded in light of eq. (4): ∞∑ t=0 η2t ∥∥∥∇R (W (t))∥∥∥2 ≤ ∞∑ t=0 ηt
  ∥∥∥∇R (W (t))∥∥∥2 ≤ 2R (W (0)) . Since all weight matrices go to infinity, such a bounded error does not matter asymptotically, and thus proofs still go through.
sections.heading: 4 SUMMARY AND FUTURE DIRECTIONS
sections.text: This paper rigorously proves that, for deep linear networks on linearly separable data, gradient flow and gradient descent minimize the risk to 0, align adjacent weight matrices, and align the first right singular vector of the first layer to the maximum margin solution determined by the data. There are many potential future directions; a few are as follows.
  Convergence rates and practical step sizes. This paper only proves asymptotic convergence, moreover for adaptive step sizes depending on the current weight matrix norms. A refined analysis with rates for practical step sizes (e.g., constant step sizes) would allow the algorithm to be compared to other methods which also globally optimize this objective, would suggest ways to improve step sizes and initialization, and ideally even exhibit a sensitivity to the network architecture and suggest how it could be improved.
  Nonseparable data and nonlinear networks. Real-world data is generally not linearly separable, but nonlinear deep networks can reliably decrease the risk to 0, even with random labels (Zhang et al., 2017). This seems to suggest that a nonlinear notion of separability is at play; is there some way to adapt the present analysis?
  The present analysis is crucially tied to the alignment of weight matrices: alignment and risk are analyzed simultaneously. Motivated by this, consider a preliminary experiment, presented in Figure 3, where stochastic gradient descent was used to minimize the risk of a standard AlexNet on CIFAR-10 (Krizhevsky et al., 2012; Krizhevsky & Hinton, 2009).
  Even though there are ReLUs, max-pooling layers, and convolutional layers, the alignment phenomenon is occurring in a reduced form on the dense layers (the last three layers of the network). Specifically, despite these weight matrices having shape (1024, 4096), (4096, 4096), and (4096, 10) the key alignment ratios ‖Wi‖2/‖Wi‖F are much larger than their respective lower bounds (1024−1/2, 4096−1/2, 10−1/2). Two initializations were tried: default PyTorch initialization, and a Gaussian initialization forcing all initial Frobenius norms to be just 4, which is suggested by the norm preservation property in the analysis and removes noise in the weights.
sections.heading: ACKNOWLEDGEMENTS
sections.text: The authors are grateful for support from the NSF under grant IIS-1750051. This grant allowed them to focus on research, and when combined with an NVIDIA GPU grant, led to the creation of their beloved GPU machine DUTCHCRUNCH.
sections.heading: A REGARDING ASSUMPTION 2
sections.text: Suppose W1(0) = 0 while WL(0) · · ·W2(0) 6= 0. First of all, WL(0) · · ·W1(0) = 0 and thus R ( W (0) ) = R(0). Moreover,〈
  ∇R ( wprod(0) ) , ū 〉 = 1
  n n∑ i=1 `′(0)〈zi, ū〉 ≤ `′(0)γ < 0,
  which implies∇R ( wprod(0) ) 6= 0 and ∂R/∂W1 = ( WL(0) · · ·W2(0) )>∇R (wprod(0))> 6= 0. On the other hand, if R ( W (0) ) > R(0), gradient flow/descent may never minimize the risk to 0. For example, suppose the network has two layers, and both the input and output have dimension 1; the network just computes the dot product of two vectors w1 and w2. Consider minimizing R(w1, w2) = exp ( −〈w1, w2〉 ) . If w1(0) = −w2(0) 6= 0, then R ( w1(0), w2(0) ) =
  exp ( ‖w1‖2 ) > exp(0). It is easy to verify that for any t, w1(t) = −w2(t), andR ( w1(t), w2(t) ) ≥ exp(0) > 0.
sections.heading: B OMITTED PROOFS FROM SECTION 2
sections.text: Proof of Lemma 1. Fix an arbitrary R > 0. If the claim is not true, then for any > 0, there exists some t ≥ 1 such that ‖Wk‖F ≤ R for all k while ∥∥∂R/∂W1∥∥2F ≤ 2, which means∥∥∥∥ ∂R∂W1 ∥∥∥∥2 F = ∥∥∥W>2 · · ·W>L ∇R(wprod)>∥∥∥2 F =‖WL · · ·W2‖2
  ∥∥∇R(wprod)∥∥2 ≤ 2. Since ‖wprod‖ ≤ RL, we have
  〈∇R(wprod), ū〉 = 1
  n n∑ i=1 `′ ( 〈wprod, zi〉 ) 〈zi, ū〉 ≤ 1 n n∑ i=1 `′ ( 〈wprod, zi〉 ) γ ≤ −Mγ,
  where −M = max−RL≤x≤RL `′(x). Since `′ is continuous and the domain is bounded, the maximum is attained and negative, and thus M > 0. Therefore ∥∥∇R(wprod)∥∥ ≥ Mγ, and thus ‖WL · · ·W2‖ ≤ /Mγ. Since ‖W1‖F ≤ R, we further have ‖wprod‖ ≤ R/Mγ. In other words, after t = 1,
  ∥∥wprod∥∥ may be arbitrarily small, which implies R (wprod) can be arbitrarily close to R (0). On the other hand, by Assumption 2, dR(W )/ dt = −‖∇R(W )‖2 < 0 at t = 0. This implies that R ( W (1) ) < R ( W (0) ) , and for any t ≥ 1,R ( W (t) ) ≤ R ( W (1) ) < R ( W (0) ) ≤ R(0), which is a contradiction.
  Since the risk is always positive, we have
  R ( W (0) ) ≥ ∫ ∞ t=0 L∑ k=1 ∥∥∥∥ ∂R∂Wk ∥∥∥∥2 F dt
  ≥ ∫ ∞ t=0 ∥∥∥∥ ∂R∂W1 ∥∥∥∥2 F dt
  ≥ ∫ ∞ t=0 ∥∥∥∥ ∂R∂W1 ∥∥∥∥2 F 1 [ max 1≤k≤L ‖Wk‖F ≤ R ] dt
  ≥ ∫ ∞ t=1 ∥∥∥∥ ∂R∂W1 ∥∥∥∥2 F 1 [ max 1≤k≤L ‖Wk‖F ≤ R ] dt
  ≥ (R)2 ∫ ∞ t=1 1 [ max 1≤k≤L ‖Wk‖F ≤ R ] dt,
  which implies gradient flow only spends a finite amount of time in { W ∣∣max1≤k≤L ‖Wk‖F ≤ R}. This directly implies that max1≤k≤L ‖Wk‖F is unbounded.
  Proof of Lemma 2. The first claim is true for k = L since WL is a row vector. For any 1 ≤ k < L, recall that Arora et al. (2018); Du et al. (2018) give the following relation:
  W>k+1(t)Wk+1(t)−W>k+1(0)Wk+1(0) = Wk(t)W>k (t)−Wk(0)W>k (0). (5)
  Let Ak,k+1 = Wk(0)W>k (0)−W>k+1(0)Wk+1(0). By eq. (5) and the definition of singular vectors and singular values, we have
  σ2k ≥ v>k+1WkW>k vk+1 = v>k+1W > k+1Wk+1vk+1 + v > k+1Ak,k+1vk+1
  = σ2k+1 + v > k+1Ak,k+1vk+1 ≥ σ2k+1 − ‖Ak,k+1‖2. (6)
  Moreover, by taking the trace on both sides of eq. (5), we have ‖Wk‖2F = tr ( WkW > k ) = tr ( W>k+1Wk+1 ) + tr ( Wk(0)W > k (0) ) − tr ( W>k+1(0)Wk+1(0) ) = ‖Wk+1‖2F + ‖Wk(0)‖2F − ‖Wk+1(0)‖2F . (7)
  Summing eq. (6) and eq. (7) from k to L− 1, we get
  ‖Wk‖2F − ‖Wk‖22 ≤ ‖Wk(0)‖2F − ‖WL(0)‖2F + L−1∑ k′=k ‖Ak′,k′+1‖2 ≤ D. (8)
  Next we prove that singular vectors get aligned. Consider u>kW > k+1Wk+1uk. On one hand, similar to eq. (6), we can get that
  u>kW > k+1Wk+1uk = u > kWkW > k uk − u>kWk(0)W>k (0)uk + u>kW>k+1(0)Wk+1(0)uk
  ≥ u>kWkW>k uk − u>kWk(0)W>k (0)uk ≥ σ2k − ‖Wk(0)‖22. (9)
  On the other hand, it follows from the definition of singular vectors and eq. (8) that
  u>kW > k+1Wk+1uk = 〈uk, vk+1〉2σ2k+1 + u>k ( W>k+1Wk+1 − vk+1σ2k+1v>k+1 ) uk
  ≤ 〈uk, vk+1〉2σ2k+1 + ‖Wk+1‖2F − ‖Wk+1‖22 ≤ 〈uk, vk+1〉2σ2k+1 +D. (10)
  Combining eq. (9) and eq. (10), we get
  σ2k ≤ 〈uk, vk+1〉2σ2k+1 +D + ‖Wk(0)‖22. (11)
  Similar to eq. (9), we can get
  σ2k ≥ v>k+1WkW>k vk+1 ≥ σ2k+1 − ‖Wk+1(0)‖22.
  Therefore σ2k σ2k+1 ≥ 1− ‖Wk+1(0)‖ 2 2 σ2k+1 . (12)
  Combining eq. (11) and eq. (12), we finally get
  〈uk, vk+1〉2 ≥ 1− D + ‖Wk(0)‖22 + ‖Wk+1(0)‖22
  σ2k+1 .
  Regarding the last claim, first recall that since the difference between the squares of Frobenius norms of any two layers is a constant, max1≤k≤L ‖Wk‖F → ∞ implies ‖Wk‖F → ∞ for any k. We further have the following.
  • Since ‖Wk‖2F − ‖Wk‖22 ≤ D, ‖Wk‖2 →∞ for any k, and Wk/‖Wk‖F → ukv>k .
  • Since ‖Wk‖2 →∞, |〈uk, vk+1〉| → 1.
  As a result, ∣∣∣∣∣∣ 〈 wprod∏L k=1 ‖Wk‖F , v1 〉∣∣∣∣∣∣ = ∣∣∣∣∣∣ 〈 L∏ k=1 Wk ‖Wk‖F , v1 〉∣∣∣∣∣∣ → ∣∣∣∣∣∣ 〈 L∏ k=1 uiv > i , v1
  〉∣∣∣∣∣∣ → 1.
  Proof of Theorem 1. Suppose for some > 0, R (W ) ≥ for any t. Then there exists some 1 ≤ j ≤ n such that ` ( 〈wprod, zj〉 ) ≥ , and thus 〈wprod, zj〉 ≤ `−1( ). On the other hand,
  since R(W ) ≤ R(0) = `(0), ` ( 〈wprod, zj〉 ) ≤ n`(0), and thus 〈wprod, zj〉 ≥ `−1 ( n`(0) ) . Let −M = max`−1(n`(0))≤x≤`−1( /n) ` ′(x) < 0, we have for any t,
  〈∇R(wprod), ū〉 = 1
  n n∑ i=1 `′ ( 〈wprod, zi〉 ) 〈zi, ū〉
  ≤ 1 n n∑ i=1 `′ ( 〈wprod, zi〉 ) γ
  ≤ 1 n `′ ( 〈wprod, zj〉 ) γ ≤ −Mγ n < 0,
  and thus ‖∇R(wprod)‖ ≥Mγ/n. Similar to the proof of Lemma 2, we can show that if ‖Wk‖F →∞,∣∣∣∣∣∣ 〈 (WL · · ·W2)> ‖Wk‖F · · · ‖W2‖F , v2
  〉∣∣∣∣∣∣→ 1. In other words, there exists some C > 0, such that when min1≤k≤L ‖Wk‖F > C, ‖WL · · ·W2‖ ≥ ‖Wk‖F · · · ‖W2‖F /2 > CL/2.
  Lemma 1 shows that gradient flow spends a finite amount of time in { W ∣∣max1≤k≤L ‖Wk‖F ≤ R} for any R > 0. Since the difference between the squares of Frobenius norms of any two layers is a constant, gradient flow also spends a finite amount of time in { W ∣∣min1≤k≤L ‖Wk‖F ≤ C}. Now we have
  R ( W (0) ) ≥ ∫ ∞ t=0 L∑ k=1 ∥∥∥∥ ∂R∂Wk ∥∥∥∥2 F dt
  ≥ ∫ ∞ t=0 ∥∥∥∥ ∂R∂W1 ∥∥∥∥2 F dt
  = ∫ ∞ t=0 ‖WL · · ·W2‖2‖∇R(wprod)‖2 dt
  ≥ ∫ ∞ t=0 ‖WL · · ·W2‖2‖∇R(wprod)‖21 [ W ∣∣∣∣ min1≤k≤L ‖Wk‖F > C ] dt
  ≥ ( Mγ
  n
  )2( CL
  2 )2 ∫ ∞ t=0 1 [ W ∣∣∣∣ min1≤k≤L ‖Wk‖F > C ] dt
  =∞,
  which is a contradiction. ThereforeR( )→ 0. This further implies ‖Wk‖F →∞, sinceR(W ) has no finite optimum. Finally, invoking Lemma 2 proves the final claim of Theorem 1.
  Proof of Lemma 3. Soudry et al. (2017) Lemma 12 proves that, with probability 1, there are at most d support vectors, and moreover, the i-th support vector zi has a positive dual variable αi, such that∑ i∈S αizi = ū.
  Suppose there exists some ξ ⊥ ū, such that maxi∈S〈ξ, zi〉 ≤ 0. Since∑ i∈S αi〈ξ, zi〉 = 〈 ξ, ∑ i∈S αizi 〉 = 〈ξ, ū〉 = 0,
  we actually have 〈ξ, zi〉 = 0 for all i ∈ S. This is impossible under Assumption 3, since the support vectors span the whole space.
  Proof of Lemma 4. For the sake of presentation, we leave out the subscript in zi and denote a data point by z generally. For any data point z and predictor w, let z⊥ and w⊥ denote their projection onto ū⊥. Let z′ ∈ arg maxi∈S〈−w⊥, z〉, and thus 〈−w⊥, z′〉 ≥ α‖w⊥‖. For `exp, we have
  〈w⊥,∇R(w)〉 = ∑ z 1 n [ − exp ( −〈w, z〉 )] 〈w⊥, z⊥〉
  = ∑ z 1 n [ exp ( 〈−w, z〉 )] 〈−w⊥, z⊥〉
  ≥ 1 n
  exp ( 〈−w, z′〉 ) 〈−w⊥, z′⊥〉+ ∑ 〈z⊥,w⊥〉≥0 1 n exp ( 〈−w, z〉 ) 〈−w⊥, z⊥〉. (13)
  The first part can be lower bounded as below (recall that 〈−w⊥, z′⊥〉 = 〈−w⊥, z′〉 ≥ α‖w⊥‖)
  1 n exp
  ( 〈−w, z′〉 ) 〈−w⊥, z′⊥〉 = 1
  n exp
  ( 〈−w, γū〉 ) exp ( 〈−w⊥, z′⊥〉 ) 〈−w⊥, z′⊥〉
  ≥ 1 n
  exp ( −〈w, γū〉 ) exp ( α‖w⊥‖ ) α‖w⊥‖. (14)
  To bound the second part, first notice that since we assume 〈w, ū〉 ≥ 0, for any z,
  〈w, z − γū〉 = 〈w, z⊥〉+ 〈w, z − γū− z⊥〉 ≥ 〈w, z⊥〉 = 〈w⊥, z⊥〉. (15)
  The reason is that every data point has margin at least γ, and thus z−γū−z⊥ = cū for some c ≥ 0. Using eq. (15), we can bound the second part of eq. (13).∑
  〈z⊥,w⊥〉≥0
  1 n exp
  ( 〈−w, z〉 ) 〈−w⊥, z⊥〉
  = ∑
  〈z⊥,w⊥〉≥0
  1 n exp
  ( 〈−w, γū〉 ) exp ( 〈−w, z − γū〉 ) 〈−w⊥, z⊥〉
  ≥ ∑
  〈z⊥,w⊥〉≥0
  1 n exp
  ( 〈−w, γū〉 ) exp ( 〈−w⊥, z⊥ ) 〈−w⊥, z⊥〉
  ≥ ∑
  〈z⊥,w⊥〉≥0
  1 n exp
  ( 〈−w, γū〉 )( −1 e )
  ≥ exp ( 〈−w, γū〉 )( −1 e ) . (16)
  On the third line eq. (15) is applied. The fourth line applies the property that f(x) = −xe−x ≥ −1/e when x ≥ 0.
  Combining eq. (13), eq. (14) and eq. (16), we get
  〈w⊥,∇R(w)〉 ≥ exp ( 〈−w, γū〉 )( 1 n exp ( α‖w⊥‖ ) α‖w⊥‖ − 1 e ) .
sections.heading: As long as ‖w⊥‖ ≥ (1 + ln(n))/α, 〈w⊥,∇R(w)〉 ≥ 0.
sections.text: For `log, similar to eq. (13), we have
  〈w⊥,∇R(w)〉 ≥ 1
  n
  exp ( 〈−w, z′〉 ) 1 + exp ( 〈−w, z′〉 ) 〈−w⊥, z′⊥〉+ ∑ 〈z⊥,w⊥〉≥0 1 n exp ( 〈−w, z〉 ) 1 + exp ( 〈−w, z〉
  ) 〈−w⊥, z⊥〉 ≥ 1 n exp ( 〈−w, z′〉 ) 1 + exp ( 〈−w, z′〉 ) 〈−w⊥, z′⊥〉+ ∑ 〈z⊥,w⊥〉≥0 1 n exp ( 〈−w, z〉 ) 〈−w⊥, z⊥〉.
  (17)
  The second part of eq. (17) can be bounded again by eq. (16). To bound the first part of eq. (17), first notice that (recall 〈w, ū〉 ≥ 0)
  exp ( 〈−w, z′〉 ) = exp ( 〈−w, γū〉 ) exp ( 〈−w⊥, z′⊥〉 ) ≤ exp ( 〈−w⊥, z′⊥〉 ) . (18)
  Using eq. (18), and recall that 〈−w⊥, z′⊥〉 = 〈−w⊥, z′〉 ≥ α‖w⊥‖ ≥ 0, we can bound the first part of eq. (17) as below.
  1
  n
  exp ( 〈−w, z′〉 ) 1 + exp ( 〈−w, z′〉 ) 〈−w⊥, z′⊥〉 = 1n exp (〈−w, γū〉) exp ( 〈−w⊥, z′⊥〉 ) 1 + exp ( 〈−w, z′〉
  ) 〈−w⊥, z′⊥〉 ≥ 1 n exp ( 〈−w, γū〉 ) exp (〈−w⊥, z′⊥〉) 1 + exp ( 〈−w⊥, z′⊥〉
  ) 〈−w⊥, z′⊥〉 ≥ 1
  2n exp
  ( 〈−w, γū〉 ) 〈−w⊥, z′⊥〉
  ≥ 1 2n
  exp ( 〈−w, γū〉 ) α‖w⊥‖. (19)
  Combining eq. (17), eq. (19) and eq. (16), we get
  〈w⊥,∇R(w)〉 ≥ exp ( 〈−w, γū〉 )( 1 2n α‖w⊥‖ − 1 e ) .
sections.heading: As long as ‖w⊥‖ ≥ 2n/eα, 〈w⊥,∇R(w)〉 ≥ 0.
sections.text: Proof of Theorem 2. Recall that
  dW1 dt = − ∂R ∂W1 = −W>2 · · ·W>L ∇R(wprod)>,
  and thus d‖W1‖2F
  dt =
  〈 W1,
  dW1 dt
  〉 = −2〈wprod,∇R(wprod)〉. (20)
  Let Πū denote the projection onto span(ū), and let Π⊥ denote the projection onto ū⊥. Also let ΠūW1 and Π⊥W1 denote the projection of rows of W1 onto span(ū) and ū⊥, respectively. Notice that
  Πūwprod = ( WL · · ·W2(ΠūW1) )> , and Π⊥wprod = ( WL · · ·W2(Π⊥W1) )> .
  We further have d‖Π⊥W1‖2F
  dt = −2〈Π⊥wprod,∇R(wprod)〉. (21)
  Let W1 = u1σ1v>1 + S. We have ‖S‖2 ≤ σ1,2 ≤ √ σ21,2 ≤ √ ‖W1‖2F − ‖W1‖22 ≤ √ D, where σ1,2 is the second singular value of W1 and D is the constant introduced in Lemma 2. Then
  Π⊥W1 = u1σ1 (Π⊥v1) > + Π⊥S,
  and ‖Π⊥W1‖F ≤ ∥∥∥u1σ1 (Π⊥v1)>∥∥∥
  F + ‖Π⊥S‖F = σ1‖Π⊥v1‖+ ‖Π⊥S‖F ≤ σ1‖Π⊥v1‖+
  √ dD.
  It follows that
  ‖Π⊥v1‖ ≥ ‖Π⊥W1‖F σ1 − √ dD σ1 ≥ ‖Π⊥W1‖F ‖W1‖F − √ dD ‖W1‖2 . (22)
  Fix an arbitrary > 0. By Theorem 1, we can find some t0 large enough such that for any t ≥ t0:
  1. √ dD/‖W1‖2 ≤ /3.
  2. ‖wprod/‖WL‖F ···‖W1‖F − v1‖ ≤ /3, or ‖wprod/‖WL‖F ···‖W1‖F + v1‖ ≤ /3.
  3. ‖WL‖F · · · ‖W1‖F ≥ 3K/ , where K is the threshold given in Lemma 4, i.e., 1+ln(n)/α for `exp, 2n/eα for `log.
  4. R(W ) ≤ `(0)/n, which implies 〈wprod, zi〉 ≥ 0 for all 1 ≤ i ≤ n. By Lemma 3, there always exists a support vector z for which 〈Π⊥wprod, z〉 ≤ 0, and therefore 〈wprod, ū〉 ≥ 0.
  Suppose for some t ≥ t0, ‖Π⊥W1‖F /‖W1‖F ≥ . By eq. (22) and bullet 1 above, ‖Π⊥v1‖ ≥ 2 /3. Bullet 2 above then gives ‖Π⊥wprod/‖WL‖F ···‖W1‖F‖ ≥ /3, which together with bullet 3 above implies ‖Π⊥wprod‖ ≥ K. Since also 〈wprod, ū〉 ≥ 0, we can apply Lemma 4 and get that 〈Π⊥wprod,∇R(wprod)〉 ≥ 0. In light of eq. (21), d‖Π⊥W1‖2F / dt ≤ 0.
  On the other hand, since after t ≥ t0, 〈wprod, zi〉 ≥ 0, we have d‖W1‖2F / dt ≥ 0 by eq. (20). Therefore ‖Π⊥W1‖F /‖W1‖F will not increase, and since ‖W1‖F → ∞, it will eventually drop below , and will never exceed again. Therefore,
  lim sup t→∞ ‖Π⊥W1‖F ‖W1‖F ≤ .
  Since is arbitrary, we have
  lim sup t→∞ ‖Π⊥W1‖F ‖W1‖F = 0,
  and thus limt→∞ ∣∣〈v1, ū〉∣∣ = 1. An application of Theorem 1 gives the other part of Theorem 2.
  Proof of Corollary 1. By Theorem 2,
  min i yi ( WL ‖WL‖F · · · W1 ‖W1‖F ) xi −→ min i yiū >xi.
  Next, pick arbitrary unit vectors ai ∈ Rdi , and note
  max A1∈Rd1×d0 ‖A1‖F =1 · · · max AL∈R1×dL−1 ‖AL‖F =1 min i yi (AL · · ·A1)xi ≥ min i yi
  ( (1a>L−1)(aL−1a > L−2) · · · (a2a>1 )(a1ū>) ) xi
  = min i yiū >xi.
  On the other hand, if matrices (AL, . . . , A1) are feasible, then
  ‖AL · · ·A1‖2 ≤ ‖AL‖2 · · · ‖A2‖2‖A1‖F ≤ ‖AL‖F · · · ‖A1‖F ≤ 1, whereby
  min i yi(AL · · ·A1)xi ≤ ‖AL · · ·A1‖2 min i yi ( AL · · ·A1 ‖AL · · ·A1‖2 ) xi
  ≤ 1 · max ‖w‖2=1 min i yiw >xi = min i yiū >xi.
sections.heading: C OMITTED PROOFS FROM SECTION 3
sections.text: Proof of Lemma 5. GivenW,V ∈ B(R), we need to show that ‖∇R(W )−∇R(V )‖ ≤ β(R)‖W− V ‖ for some β(R). Consider k = 1 first. Let w = (WL · · ·W1)>, and v = (VL · · ·V1)>. Since |`′| ≤ G, ‖∇R(w)‖, ‖∇R(v)‖ ≤ G. We have∥∥∥∥ ∂R∂W1 − ∂R∂V1
  ∥∥∥∥ =∥∥∥W>2 · · ·W>L ∇R(w)− V >2 · · ·V >L ∇R(v)∥∥∥ ≤ ∥∥∥W>2 · · ·W>L ∇R(w)− V >2 W>3 · · ·W>L ∇R(w)∥∥∥ + ∥∥∥V >2 W>3 · · ·W>L ∇R(w)− V >2 · · ·V >L ∇R(v)∥∥∥
  ≤ RL−2G‖W2 − V2‖ + ∥∥∥V >2 W>3 · · ·W>L ∇R(w)− V >2 · · ·V >L ∇R(v)∥∥∥
  ≤ RL−2G‖W − V ‖ + ∥∥∥V >2 W>3 · · ·W>L ∇R(w)− V >2 · · ·V >L ∇R(v)∥∥∥ . (23)
  Proceeding in this way, we can get∥∥∥∥ ∂R∂W1 − ∂R∂V1 ∥∥∥∥ ≤ (L− 1)RL−2G‖W − V ‖+RL−1‖∇R(w)−∇R(v)‖. (24) Since ‖zi‖ ≤ 1, `′ is β-Lipschitz, we have ‖∇R(w)−∇R(v)‖ ≤ β‖w − v‖ ≤ βLRL−1‖W − V ‖, (25)
  where the last inequality follows from a similar one-by-one replacement procedure as in eq. (23). Combining eq. (24) and eq. (25), we get for R ≥ 1,∥∥∥∥ ∂R∂W1 − ∂R∂V1 ∥∥∥∥ ≤ ((L− 1)RL−2G+ βLR2L−2) ‖W − V ‖ ≤ 2LR2L−2(β +G)‖W − V ‖. The same procedure can be done for other layers, and together
  ‖∇R(W )−∇R(V )‖ ≤ 2L2R2L−2(β +G)‖W − V ‖.
  Proof of Lemma 6. Recall that if W (t),W (t+ 1) ∈ B(R) and ηt = 1/β(R),
  R ( W (t+ 1) ) −R ( W (t) ) ≤ 〈∇R ( W (t) ) ,−ηt∇R ( W (t) ) 〉+ β(R)η 2 t
  2 ∥∥∥∇R (W (t))∥∥∥2 = − 1
  2β(R) ∥∥∥∇R (W (t))∥∥∥2 = −ηt
  2 ∥∥∥∇R (W (t))∥∥∥2 . (26) Suppose W (t) ∈ B(R) for all t. By Assumption 2 and eq. (26),
  R ( W (1) ) ≤ R ( W (0) ) − 1
  2β(R) ∥∥∥∇R (W (0))∥∥∥2 < R (W (0)) . By eq. (26), gradient descent never increases the risk, and thus for all t ≥ 1, R ( W (t) ) ≤
  R ( W (1) ) < R ( W (0) ) . In exactly the same way as in the proof of Lemma 1, one can show that there exists some constant (R) > 0, so that ‖∂R/∂W1(t)‖F ≥ (R) for all t. Invoking eq. (26) again, we will get
  R ( W (0) ) ≥ ∞∑ t=0 1 2β(R) (R)2 =∞,
  which is a contradiction. Therefore W (t) must go out of B(R) at some time.
  Next we prove Theorem 3 and 4. The proofs depend on several lemmas which are similar to the gradient flow ones. The following Lemma 7 is similar to Lemma 1. Lemma 7. Under Assumption 1, 2, 4, and 5, gradient descent ensures that
  • max1≤k≤L ‖Wk(t)‖F is unbounded. • ∑∞ t=0 ηt =∞.
  • For any R > 0, ∑ t:W (t)∈B(R) ηt <∞.
  Proof. By Assumption 5, we always have thatW (t) ∈ B(Rt). Since β(Rt) = 2L2R2L−2t (β+G) ≥ RL−1t G, we have for any 1 ≤ k ≤ L,
  ‖Wk(t+ 1)‖F ≤ ‖Wk(t)‖F + ηt ∥∥∥∥ ∂R∂Wk(t) ∥∥∥∥ F
  ≤ ‖Wk(t)‖F + 1
  β(Rt) ∥∥∥∥ ∂R∂Wk(t) ∥∥∥∥ F
  ≤ ‖Wk(t)‖F + 1
  β(Rt) RL−1t G
  ≤ ‖Wk(t)‖F + 1. (27) Moreover, Lemma 6 shows that Rt → ∞. Since Rt+1 = Rt as long as W (t + 1) ∈ B(Rt − 1), max1≤k≤L ‖Wk(t)‖F is unbounded. It then follows that for any t, by Cauchy-Schwarz,t−1∑
  τ=0
  ητ t−1∑ τ=0 ητ ∥∥∥∇R (W (τ))∥∥∥2  ≥ t−1∑ τ=0 ητ ∥∥∥∇R (W (τ))∥∥∥ 2 →∞.
  since by eq. (26), t−1∑ τ=0 ητ ∥∥∥∇R (W (τ))∥∥∥2 ≤ 2R (W (0))− 2R (W (t)) ≤ 2R (W (0)) , we have ∑∞ t=0 ηt =∞.
  Since under Assumptions 4 and 5 gradient descent never increases the risk, it can be shown in exactly the same as in the proof of Lemma 1 that, for W (t) ∈ B(R), ‖∂R/∂W1(t)‖F ≥ (R) for some constant (R) > 0. Invoking eq. (26) again, we get that ∑ t:W (t)∈B(R) ηt <∞.
  The next lemma is an analogy to Lemma 2. Lemma 8. Under Assumption 1 and 4, the gradient descent iterates satisfy the following properties:
  • For any 1 ≤ k ≤ L, ‖Wk‖2F − ‖Wk‖22 ≤ D + 2R ( W (0) ) .
  • For any 1 ≤ k < L, 〈vk+1, uk〉2 ≥ 1− D+3R(W (0))+‖Wk+1(0)‖22+‖Wk(0)‖22/‖Wk+1‖22. • Suppose max1≤k≤L ‖Wk‖F →∞, then ∣∣∣〈wprod/∏Lk=1 ‖Wk‖F , v1〉∣∣∣→ 1.
  Proof. Recall that for any W ,
  W>k+1 ∂R
  ∂Wk+1 = W>k+1 · · ·W>L ∇R(wprod)>W>1 · · ·W>k = ∂R ∂Wk W>k . (28)
  For gradient descent iterates, summing eq. (28) from 0 to t− 1, we get
  W>k+1(t)Wk+1(t)−W>k+1(0)Wk+1(0) + t−1∑ τ=0 η2τ ( ∂R ∂Wk+1(τ) )>( ∂R ∂Wk+1(τ) )
  = Wk(t)W > k (t)−Wk(0)W>k (0) + t−1∑ τ=0 η2τ ( ∂R ∂Wk(τ) )( ∂R ∂Wk(τ) )> . (29)
  For any 1 ≤ k ≤ L and any t, let
  Pk(t) = t−1∑ τ=0 η2τ ( ∂R ∂Wk(τ) )( ∂R ∂Wk(τ) )> ,
  and
  Qk(t) = t−1∑ τ=0 η2τ ( ∂R ∂Wk(τ) )>( ∂R ∂Wk(τ) ) .
  We have ‖Pk(t)‖2 = ‖Qk(t)‖2 ≤ tr ( Qk(t) ) = tr ( Pk(t) ) . Moreover, invoking eq. (26),
  L∑ k=1 tr ( Pk(t) ) = L∑ k=1 t−1∑ τ=0 η2τ ∥∥∥∥ ∂R∂Wk(τ) ∥∥∥∥2 F
  = t−1∑ τ=0 η2τ ∥∥∥∇R (W (τ))∥∥∥2 ≤
  t−1∑ τ=0 ητ ∥∥∥∇R (W (τ))∥∥∥2 ≤ 2R ( W (0) ) − 2R ( W (t)
  ) ≤ 2R ( W (0) ) . (30)
  Still let σk(t), uk(t) and vk(t) denote the first singular value, left singular vector and right singular vector of Wk(t). We can then proceed basically in the same way as in the proof of Lemma 2. For example, eq. (6) becomes
  σ2k(t) ≥ σ2k+1(t)− ‖Ak,k+1(t)‖2 − ‖Pk(t)‖2 ≥ σ2k+1(t)− ‖Ak,k+1(t)‖2 − tr ( Pk(t) ) , (31)
  while eq. (7) becomes ‖Wk(t)‖2F = ‖Wk+1(t)‖2F + ‖Wk(0)‖2F − ‖Wk+1(0)‖2F − tr ( Pk(t) ) + tr ( Qk+1(t) ) . (32)
  Summing eq. (31) and eq. (32) from k to L− 1, and invoke eq. (30), we get
  ‖Wk(t)‖2F − ‖Wk(t)‖22 ≤ D − tr ( Pk(t) ) + tr ( QL(t) ) + L−1∑ k′=k tr ( Pk′(t) ) ≤ D + 2R ( W (0) ) .
  To prove singular vectors get aligned, we can still proceed in nearly the same way as in the proof of Lemma 2. eq. (9) becomes
  u>kW > k+1Wk+1uk ≥ σ2k − ‖Wk(0)‖22 − ‖Qk+1(t)‖2, (33)
  while eq. (10) becomes
  u>kW > k+1Wk+1uk ≤ 〈uk, vk+1〉2σ2k+1 +D + 2R
  ( W (0) ) . (34)
  Combining eq. (33) and eq. (34) σ2k ≤ 〈uk, vk+1〉2σ2k+1 +D + 2R ( W (0) ) + ‖Qk+1(t)‖2 + ‖Wk(0)‖22. (35)
  Similar to eq. (33), we can get
  σ2k ≥ v>k+1WkW>k vk+1 ≥ σ2k+1 − ‖Wk+1(0)‖22 − ‖Pk(t)‖2, and thus eq. (12) becomes
  σ2k σ2k+1 ≥ 1− ‖Wk+1(0)‖ 2 2 + ‖Pk(t)‖2 σ2k+1 . (36)
  Combining eq. (35) and eq. (36), we get
  〈uk, vk+1〉2 ≥ 1− D + ‖Wk(0)‖22 + ‖Wk+1(0)‖22 + 3R
  ( W (0) ) σ2k+1 .
  The final claim of Lemma 8 can be proved in exactly the same way as Lemma 2.
  Proof of Theorem 3. Summing eq. (32), we know that for any two different layers j > k, ‖Wk(t)‖2F − ‖Wj(t)‖2F = ‖Wk(0)‖2F − ‖Wj(0)‖2F − tr ( Pk(t) ) + tr ( Qj(t) ) . Recall eq. (30), we know that∣∣∣∣(‖Wk(t)‖2F − ‖Wj(t)‖2F)− (‖Wk(0)‖2F − ‖Wj(0)‖2F)∣∣∣∣ ≤ 2R (W (0)) . (37) In other words, the difference between the squares of Frobenius norms of any two layers is still bounded.
  The proof then goes in the same way as the proof of Theorem 1. Suppose the risk is always above > 0. Then there exists some c( ) > 0 such that ‖∇R(wprod)‖ ≥ c( ). By Lemma 8, there exists some C such that if min1≤k≤L ‖Wk(t)‖F > C, ‖WL(t) · · ·W2(t)‖ ≥ CL/2. By eq. (37) and Lemma 7, ∑ t:‖Wk(t)‖F≤C for some k ηt is finite. On the other hand, by Lemma 7, ∑∞ i=0 ηt =∞, and
  thus ∑ t:‖Wk(t)‖F>C for all k ηt =∞. Therefore we have, by invoking eq. (26),
  2R ( W (0) ) ≥ ∞∑ t=0 ηt ∥∥∥R (W (t))∥∥∥2 ≥ ∞∑ t=0 ηt ∥∥∥∥ ∂R∂W1(t) ∥∥∥∥2
  ≥ c( )C L
  2 ∑ t:‖Wk(t)‖F>C for all k ηt
  =∞, which is a contradiction. Therefore R ( W (t) ) → 0, and since it has no finite optimum, ‖Wk‖F → ∞. The other results follow from Lemma 7.
  Proof of Theorem 4. Recall that
  ∂R ∂W1 = W>2 · · ·W>L ∇R(wprod),
  and thus ‖W1(t+ 1)‖2F = ‖W1(t)‖2F − 2ηt 〈 W1(t),
  ∂R ∂W1(t)
  〉 + η2t ∥∥∥∥ ∂R∂W1(t) ∥∥∥∥2 F
  = ‖W1(t)‖2F − 2ηt 〈 wprod(t),∇R ( wprod(t) )〉 + η2t ∥∥∥∥ ∂R∂W1(t) ∥∥∥∥2 F .
sections.heading: If 〈wprod, zi〉 ≥ 0 for all i, then ‖W1(t+ 1)‖F ≥ ‖W1(t)‖F .
sections.text: Also recall that Π⊥W1(t) denote the projection of rows of W1(t) onto ū⊥, the orthogonal complement of span(ū). We have
  ‖Π⊥W1(t+ 1)‖2F ≤ ‖Π⊥W1(t)‖2F − 2ηt 〈 Π⊥W1(t), ∂R
  ∂W1(t)
  〉 + η2t ∥∥∥∥ ∂R∂W1(t) ∥∥∥∥2 F
  = ‖Π⊥W1(t)‖2F − 2ηt 〈 Π⊥wprod(t),∇R ( wprod(t) )〉 + η2t ∥∥∥∥ ∂R∂W1(t) ∥∥∥∥2 F .
  (38)
  Invoking eq. (26) again gives
  η2t ∥∥∥∥ ∂R∂W1(t) ∥∥∥∥2 F ≤ ηt ∥∥∥∇R (W (t))∥∥∥2 ≤ 2(R (W (t))−R (W (t+ 1))) . (39)
  The proof then goes in almost the same way as the proof of Theorem 2. For any > 0, we can find some large enough time t0, such that for any t ≥ t0,
  1. ‖Π⊥W1(t)‖F /‖W1(t)‖F ≥ implies that 〈 Π⊥wprod(t),∇R ( wprod(t) )〉 ≥ 0.
  2. 〈wprod(t), zi〉 ≥ 0 for all i, and thus ‖W1(t+ 1)‖F ≥ ‖W1(t)‖F . 3. ‖W1(t)‖F ≥ 1+ √ 2R(W (0))/ .
  Suppose at some time t1 ≥ t0, ‖Π⊥W1(t1)‖F /‖W1(t1)‖F ≥ . As long as this still holds, in light of bullet (1) above, eq. (38) and eq. (39), ‖Π⊥W1‖2F will increase by at most 2R ( W (t1) ) ≤
  2R ( W (0) ) . On the other hand, ‖W1‖F → ∞, and thus there exists some t2 > t1 such that ‖Π⊥W1(t2)‖F /‖W1(t2)‖F < . Let t3 denote the smallest time after t2 such that ‖Π⊥W1(t3)‖F /‖W1(t3)‖F ≥ (if it exists). Recall that ‖W1(t+ 1)‖F ≤ ‖W1(t)‖F + 1 for any t ≥ 0, and ‖W1(t+ 1)‖F ≥ ‖W1(t)‖F for any t ≥ t0, we have
  ‖Π⊥W1(t3)‖F ‖W1(t3)‖F ≤ ‖Π⊥W1(t3)‖F ‖W1(t3 − 1)‖F ≤ ‖Π⊥W1(t3 − 1)‖F + 1 ‖W1(t3 − 1)‖F < + 1 ‖W1(t3 − 1)‖F .
  After t3, ‖Π⊥W1‖2F will increase by at most 2R ( W (0) ) , and thus ‖Π⊥W1‖F will increase by at
  most √ 2R ( W (0) ) . Therefore, for any t4 ≥ t3, as long as ‖Π⊥W1(t4)‖F /‖W1(t4)‖F ≥ , we
  have
  ‖Π⊥W1(t4)‖F ‖W1(t4)‖F ≤ ‖Π⊥W1(t4)‖F ‖W1(t3)‖F
  ≤ ‖Π⊥W1(t3)‖F +
  √ 2R ( W (0) ) ‖W1(t3)‖F
  ≤ + 1 ‖W1(t3 − 1)‖F +
  √ 2R ( W (0) ) ‖W1(t3)‖F ≤ 2 ,
  since ‖W1(t)‖F ≥ 1+ √ 2R(W (0))/ after t0. In other words,
  lim sup t→∞ ‖Π⊥W1‖F ‖W1‖F ≤ 2 .
  Since is arbitrary, we have
  lim sup t→∞ ‖Π⊥W1‖F ‖W1‖F = 0,
  and thus limt→∞ ∣∣〈v1, ū〉∣∣ = 1.
  Proof of Corollary 2. The proof is analogous to that of Corollary 1, except using Theorem 4 in place of Theorem 2.
title: GRADIENT DESCENT ALIGNS THE LAYERS OF DEEP LINEAR NETWORKS