Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data

Introduction

Neural networks trained by gradient descent appear to generalize well in many settings, even when trained without explicit regularization. It is thus understood that the usage of gradient-based optimization imposes an implicit bias towards particular solutions which enjoy favorable properties. The nature of this implicit regularization effect—and its dependence on the structure of the training data, the architecture of the network, and the particular gradient-based optimization algorithm—is thus a central object of study in the theory of deep learning.

For gradient flow, we consider the standard leaky ReLU activation, $\phi(z)=\max(\gamma z,z)$. Our starting point in this setting is recent work by [LL20, JT20] that show that, provided the network interpolates the training data at some time, gradient flow on homogeneous networks, such as two-layer leaky ReLU networks, converges (in direction) to a network that satisfies the Karush–Kuhn–Tucker (KKT) conditions for the margin-maximization problem,

For gradient descent, we consider a smoothed approximation to the leaky ReLU activation, and consider training that starts from a random initialization with small initialization variance. Our result for gradient flow on the standard leaky ReLU activation suggests that gradient descent with small-enough step size should eventually produce a network for which $W^{(t)}$ has small rank. However, the asymptotic characterization of trained neural networks in terms of KKT points of a margin-maximization problem relies heavily upon the infinite-time limit. This leaves open what happens in finite time. Towards this end, we consider the stable rank of the weight matrix $W^{(t)}$ found by gradient descent at time $t$, defined as $\|W^{(t)}\|_{F}^{2}/\|W^{(t)}\|_{2}^{2}$, the square of the ratio of the Frobenius norm to the spectral norm of $W^{(t)}$. We show that after the first step of gradient descent, the stable rank of the weight matrix $W^{(t)}$ reduces from something that is of order $\min(m,d)$ to that which is at most an absolute constant, independent of $m$, $d$, or the number of samples. Further, throughout the training trajectory the stable rank of the network is never larger than some absolute constant.

We conclude by verifying our results with experiments. We first confirm our theoretical predictions for binary classification problems with high-dimensional data. We then consider the stable rank of two-layer networks trained by SGD for the CIFAR10 dataset, which is not high-dimensional. We notice that the scale of the initialization plays a crucial role in the stable rank of the weights found by gradient descent: with default TensorFlow initialization, the stable rank of a network with $m=512$ neurons never falls below 74, while with a smaller initialization variance, the stable rank quickly drops to 3.25, and only begins to increase above 10 when the network begins to overfit.

The literature on the implicit bias in neural networks has rapidly expanded in recent years, and cannot be reasonably surveyed here (see [Var22] for a survey). In what follows, we discuss results which apply to two-layer ReLU or leaky ReLU networks in classification settings.

By [LL20] and [JT20], homogeneous neural networks (and specifically two-layer leaky ReLU networks, which are the focus of this paper) trained with exponentially-tailed classification losses converge in direction to a KKT point of the maximum-margin problem. Our analysis of the implicit bias relies on this result. We note that the aforementioned KKT point may not be a global optimum (see a discussion in Section 3).

[CB20] studied the dynamics of gradient flow on infinite-width homogeneous two-layer networks with exponentially-tailed losses, and showed bias towards margin maximization w.r.t. a certain function norm known as the variation norm. [PL20] studied the implicit bias in two-layer ReLU networks trained on orthogonally separable data (i.e., where for every pair of labeled examples $(x_{i},y_{i}),(x_{j},y_{j})$ we have $x_{i}^{\top}x_{j}>0$ if $y_{i}=y_{j}$ and $x_{i}^{\top}x_{j}\leq 0$ otherwise). [SVL22] proved implicit bias towards minimizing the number of linear regions in univariate two-layer ReLU networks. Implicit bias in neural networks trained with nearly-orthogonal data was previously studied in [VYS22]. Their assumptions on the training data are similar to ours, but they consider ReLU networks and prove bias towards non-robust networks. Their results do not have any clear implications for our setting.

Implicit bias towards rank minimization was also studied in several other papers. [JT19, JT20] showed that in linear networks of output dimension $1$, gradient flow with exponentially-tailed losses converges to networks where the weight matrix of every layer is of rank $1$. [TVS22] showed that the bias towards margin maximization in homogeneous ReLU networks may induce a certain bias towards rank minimization in the weight matrices of sufficiently deep ReLU networks. Finally, implicit bias towards rank minimization was also studied in regression settings. See, e.g., [Aro+19, RC20, LLL21, TVS22].

Neural network optimization.

This work can be considered in the context of other work on developing optimization guarantees for neural networks trained by gradient descent. A line of work based on the neural tangent kernel approximation [JGH18] showed that global convergence of gradient descent is possible if the network is sufficiently wide and stays close to its random initialization [ALS19, Zou+19, Du+19, Aro+19a, SJL19, FCG19]. These results do not hold if the network has constant width or if the variance of the random initialization is small, both of which are permitted with our analysis.

A series of works have explored the training dynamics of gradient descent when the data is linearly separable (such as is the case when the input dimension is larger than the number of samples, as we consider here). [Bru+18] showed that in two-layer leaky ReLU networks, SGD on the hinge loss for linearly separable data converges to zero loss. [FCG21] showed that even when a constant fraction of the training labels are corrupted by an adversary, in two-layer leaky ReLU networks, SGD on the logistic loss produces neural networks that have generalization error close to the label noise rate. As we mentioned above, both [Lyu+21] and [SBG21] considered two-layer leaky ReLU networks trained by gradient-based methods on linearly separable datasets.

Training of neural networks for high-dimensional data.

The training dynamics of neural networks for high-dimensional data has been studied in a number of recent works. [Cao+22] studied two-layer convolutional networks trained on an image-patch data model and showed how a low signal-to-noise ratio can result in harmful overfitting, while a high signal-to-noise ratio allows for good generalization performance. [SBG22] considered a similar image-patch signal model and studied how data augmentation can improve generalization performance of two-layer convolutional networks. [FCB22] showed that two-layer fully connected networks trained on high-dimensional mixture model data can exhibit a ‘benign overfitting’ phenomenon. [FCB22a] studied the feature-learning process for two-layer ReLU networks trained on noisy 2-xor clustered data and showed that early-stopped networks can generalize well even in high-dimensional settings. [BPF22] studied the dynamics of gradient flow on the squared loss for two-layer ReLU networks with orthogonal inputs.

Preliminaries

Neural networks.

In this work we consider depth-$2$ neural networks, where the second layer is fixed and only the first layer is trained. Thus, a neural network with parameters $W$ is defined as

Gradient descent and gradient flow.

Gradient flow captures the behavior of gradient descent with an infinitesimally small step size. The trajectory $W(t)$ of gradient flow is defined such that starting from an initial point $W(0)$, the dynamics of $W(t)$ obeys the differential equation $\frac{dW(t)}{dt}=-\nabla_{W}\widehat{L}(W(t))$. When $\widehat{L}(W)$ is non-differentiable, the dynamics of gradient flow obeys the differential equation $\frac{dW(t)}{dt}\in-\partial^{\circ}\widehat{L}(W(t))$, where $\partial^{\circ}$ denotes the Clarke subdifferential, which is a generalization of the derivative for non-differentiable functions (see Appendix A for a formal definition).

Asymptotic Analysis of the Implicit Bias

In this section, we study the implicit bias of gradient flow in the limit $t\to\infty$. Our results build on a theorem by [LL20] and [JT20], which considers the implicit bias in homogeneous neural networks. Let $f(x;\theta)$ be a neural network parameterized by $\theta$, where we view $\theta$ as a vector. The network $f$ is homogeneous if there exists $L>0$ such that for every $\beta>0$ and $x,\theta$ we have $f(x;\beta\theta)=\beta^{L}f(x;\theta)$. We say that a trajectory $\theta(t)$ of gradient flow converges in direction to $\theta^{*}$ if $\lim_{t\to\infty}\frac{\theta(t)}{\left\|\theta(t)\right\|}=\frac{\theta^{*}}{\left\|\theta^{*}\right\|}$. Their theorem can be stated as follows.

Let $f$ be a homogeneous ReLU or leaky ReLU neural network parameterized by $\theta$. Consider minimizing either the exponential or the logistic loss over a binary classification dataset $\{(x_{i},y_{i})\}_{i=1}^{n}$ using gradient flow. Assume that there exists time $t_{0}$ such that $\widehat{L}(\theta(t_{0}))<\frac{\log(2)}{n}$. Then, gradient flow converges in direction to a first order stationary point (KKT point) of the following maximum-margin problem in parameter space:

Moreover, $\widehat{L}(\theta(t))\to 0$ and $\left\|\theta(t)\right\|\to\infty$ as $t\to\infty$.

Note that in leaky ReLU networks Problem (1) is non-smooth. Hence, the KKT conditions are defined using the Clarke subdifferential. See Appendix A for more details of the KKT conditions. The theorem implies that even though there might be many possible directions $\frac{W}{\left\|W\right\|_{F}}$ that classify the dataset correctly, gradient flow converges only to directions that are KKT points of Problem (1). We note that such a KKT point is not necessarily a global/local optimum (cf. [VSS21, Lyu+21]). Thus, under the theorem’s assumptions, gradient flow may not converge to an optimum of Problem (1), but it is guaranteed to converge to a KKT point.

We now state our main result for this section. For convenience, we will use different notations for positive neurons (i.e., where $a_{j}=1/\sqrt{m}$) and negative neurons (i.e., where $a_{j}=-1/\sqrt{m}$). Namely,

Note that $m=m_{1}+m_{2}$. We assume that $m_{1},m_{2}\geq 1$.

Let $f$ be the leaky ReLU network from \tagform@2 and let $W$ be a KKT point of Problem (1). Then, the following hold:

$v=\frac{1}{\sqrt{m}}\sum_{i\in I_{+}}\lambda_{i}x_{i}-\frac{\gamma}{\sqrt{m}}\sum_{i\in I_{-}}\lambda_{i}x_{i}$ and $u=\frac{1}{\sqrt{m}}\sum_{i\in I_{-}}\lambda_{i}x_{i}-\frac{\gamma}{\sqrt{m}}\sum_{i\in I_{+}}\lambda_{i}x_{i}$, where $\lambda_{i}\in\left(\frac{1}{2R_{\text{max}}^{2}},\frac{3}{2\gamma^{2}R_{\text{min}}^{2}}\right)$ for every $i\in I$. Furthermore, for all $i\in I$ we have $y_{i}v^{\top}x_{i}>0$ and $y_{i}u^{\top}x_{i}<0$.

$W$ is a global optimum of Problem (1). Moreover, this global optimum is unique.

The pair $v,u$ from item 2 is the global optimum of the following convex problem:

Note that by the above theorem, the KKT points possess very strong properties: the weight matrix is of rank at most $2$, there is margin maximization in parameter space, in function space the predictor has a linear decision boundary, there may not be margin maximization in predictor space, but the predictor maximizes the margin approximately within a factor of $\frac{2}{\kappa+\gamma}$. Note that if $\kappa=1$ (i.e., $m_{1}=m_{2}$) and $\gamma$ is roughly $1$, then we get margin maximization also in predictor space. We remark that variants of items 2, 5 and 6 were shown in [SBG21] under a different assumption called Neural Agreement Regime (as we discussed in the related work section).In fact, the main challenge in our proof is to show that a property similar to their assumption holds in every KKT point in our setting.

The proof of Theorem 3.2 is given in Appendix B. We now briefly discuss the proof idea. Since $W$ satisfies the KKT conditions of Problem (1), then there are $\lambda_{1},\ldots,\lambda_{n}$ such that for every $j\in[m]$ we have

where $\phi^{\prime}_{i,w_{j}}$ is a subgradient of $\phi$ at $w_{j}^{\top}x_{i}$. Also we have $\lambda_{i}\geq 0$ for all $i$, and $\lambda_{i}=0$ if $y_{i}f(x_{i};W)\neq 1$. We prove strictly positive upper and lower bounds for each of the $\lambda_{i}$’s. Since the $\lambda_{i}$’s are strictly positive, the KKT conditions show that the margin constraints are satisfied with equalities, i.e., part 1 of the theorem. By leveraging these bounds on the $\lambda_{i}$’s we also derive the remaining parts of the theorem.

The proof of Lemma 3.3 is provided in Appendix C.

By Theorem 3.2, if the data points are nearly orthogonal then every KKT point of Problem (1) satisfies items 1-7 there. It leaves open the question of whether gradient flow converges to a KKT point. By Theorem 3.1, in order to prove convergence to a KKT point, it suffices to show that there exists time $t_{0}$ where $\widehat{L}(W(t_{0}))<\frac{\log(2)}{n}$. In the following theorem we show that such $t_{0}$ exists, regardless of the initialization of gradient flow (the theorem holds both for the logistic and the exponential losses).

Consider gradient flow on a the network from \tagform@2 w.r.t. a dataset that satisfies the assumption from Theorem 3.2. Then, there exists a finite time $t_{0}$ such that for all $t\geq t_{0}$ we have $\widehat{L}(W(t))<\log(2)/n$.

We prove the theorem in Appendix D. Combining Theorems 3.1, 3.2 and 3.4, we get the following corollary:

Consider gradient flow on the network from \tagform@2 w.r.t. a dataset that satisfies the assumption from Theorem 3.2. Then, gradient flow converges to zero loss, and converges in direction to a weight matrix $W$ that satisfies items 1-7 from Theorem 3.2.

Non-Asymptotic Analysis of the Implicit Bias

We shall refer to functions satisfying the above properties as $\gamma$-leaky, $H$-smooth. Note that such functions are not necessarily homogeneous. Examples of such functions are any smoothed approximation to the leaky ReLU that is zero at the origin. One such example is: $\phi(z)=\gamma z+(1-\gamma)\log\left(\frac{1}{2}(1+\exp(z))\right)$, which is $\gamma$-leaky and $\nicefrac{{1}}{{4}}$-smooth (see Figure 3 in the appendix for a side-by-side plot of this activation with the standard leaky ReLU).

We next introduce the definition of stable rank [RV07].

With the above conditions in hand, we can state our main theorem for this section.

The empirical risk under the logistic loss is driven to zero:

The stable rank of the weights throughout the gradient descent trajectory satisfies,

We now make a few remarks on the above theorem. We note that the assumption on the training data is the same as in Theorem 3.2 up to constants (treating $\gamma$ as a constant), and is satisfied in many settings when $d\gg n^{2}$ (see Lemma 3.3).

For the first part of the theorem, we show that despite the non-convexity of the underlying optimization problem, gradient descent can efficiently minimize the training error, driving the empirical risk to zero.

The third part of the theorem is perhaps the most interesting one. In Theorem 3.2, we showed that for the standard leaky ReLU activation trained on nearly-orthogonal data with gradient flow, the asymptotic true rank of the network is at most 2. By contrast, Theorem 4.2 shows that the stable rank of neural networks with $\gamma$-leaky, $H$-smooth activations trained by gradient descent have a constant stable rank after the first step of gradient descent and the rank remains bounded by a constant throughout the trajectory. Note that at initialization, by standard concentration bounds for random matrices (see, e.g., [Ver10]), the stable rank satisfies

so that Theorem 4.2 implies that gradient descent drastically reduces the rank of the matrix after just one step.

Implications of the Implicit Bias and Empirical Observations

The results in the preceding sections show a remarkable simplicity bias of gradient-based optimization when training two-layer networks with leaky activations on sufficiently high-dimensional data. For gradient flow, regardless of the initialization, the learned network has a linear decision boundary, even when the labels $y$ are some nonlinear function of the input features and when the network has the capacity to approximate any continuous function. With our analysis of gradient descent, we showed that the bias towards producing low-complexity networks (as measured by the stable rank of the network) is something that occurs quickly following random initialization, provided the initialization scale is small enough.

The linear classifier $x\mapsto\operatorname*{sign}(\langle\mu,x\rangle)$ performs optimally for this distribution, and so the implicit bias of gradient descent towards low-rank classifiers (and of gradient flow towards linear decision boundaries) for high-dimensional data could in principle be helpful for allowing neural networks trained on such data to generalize well for this distribution. Indeed, as shown by [CL21], since $\|x_{i}\|^{2}\approx d+\|\mu\|^{2}$ while $|\langle x_{i},x_{j}\rangle|\approx\|\mu\|^{2}+\sqrt{d}$ for $i\neq j$, provided $\|\mu\|=\Theta(d^{\beta})$ and $d\gg n^{\frac{1}{1-2\beta}}\vee n^{2}$ for $\beta\in(0,1/2)$, the assumptions in Theorem 4.2 hold. Thus, gradient descent on two-layer networks with $\gamma$-leaky, $H$-smooth activations, the empirical risk is driven to zero and the stable rank of the network is constant after the first step of gradient descent. In this setting, [FCB22] recently showed that such networks also achieve minimax-optimal generalization error. This shows that the implicit bias towards classifiers with constant rank can be beneficial in distributional settings where linear classifiers can perform well.

On the other hand, the same implicit bias can be harmful if the training data come from a distribution that does not align with this bias. Consider the noisy 2-xor distribution $\mathcal{D}_{\mathsf{xor}}$ defined by $x=z+\xi$ where $z\sim\mathsf{Uniform}(\{\pm\mu_{1},\pm\mu_{2}\})$, where $\mu_{1},\mu_{2}$ are orthogonal with identical norms, $\xi\sim\mathsf{N}(0,I_{d})$, and $y=\operatorname*{sign}(|\langle\mu_{1},x\rangle|-|\langle\mu_{2},x\rangle|)$. Then every linear classifier achieves 50% test error on $\mathcal{D}_{\mathsf{xor}}$. Moreover, provided $\|\mu_{i}\|=\Theta(d^{\beta})$ for $\beta<1/2$, by the same reasoning in the preceding paragraph the assumptions needed for Theorem 3.2 are satisfied provided $d\gg n^{\frac{1}{1-2\beta}}\vee n^{2}$. In this setting, regardless of the initialization, by Theorem 3.2 the limit of gradient flow produces a neural network which has a linear decision boundary and thus achieves 50% test error.

Thus, the implicit bias can be beneficial in some settings and harmful in others. Theorem 4.2 and Lemma 3.3 suggest that the relationship between the input dimension and the number of samples, as well as the initialization variance, can influence how quickly gradient descent finds low-rank networks. In Figure 1 we examine these factors for two-layer nets trained on a Gaussian mixture model distribution (see Appendix F for experimental details). We see that the bias towards rank reduction increases as the dimension increases and the initialization scale decreases, as suggested by our theory. Moreover, it appears that the initialization scale is more influential for determining the rank reduction than training gradient descent for longer. In Appendix F we provide more detailed empirical investigations into this phenomenon.

In Figure 2, we investigate whether or not the initialization scale’s effect on the rank reduction of gradient descent occurs in settings not covered by our theory, namely in two-layer ReLU networks with bias terms trained by SGD on CIFAR-10. We consider two different initialization schemes: (1) Glorot uniform, the default TensorFlow initialization scheme with standard deviation of order $1/\sqrt{m+d}$, and (2) a uniform initialization scheme with $50\times$ smaller standard deviation than that of the Glorot uniform initialization. In the default initialization scheme, it appears that a reduction in the rank of the network only comes in the late stages of training, and the smallest stable rank achieved by the network within $10^{6}$ steps is 74.0. On the other hand, with the smaller initialization scheme, the rank reduction comes rapidly, and the smallest stable rank achieved by the network is 3.25. It is also interesting to note that in the small initialization setting, after gradient descent rapidly produces low-rank weights, the rank of the trained network begins to increase only when the gap between the train and test accuracy begin to diverge.

Conclusion

In this work, we characterized the implicit bias of gradient flow and gradient descent for two-layer leaky ReLU networks when trained on high-dimensional datasets. For both gradient flow and gradient descent, we proved convergence to near-zero training loss and that there is an implicit bias towards low-rank networks. For gradient flow, we showed a number of additional implicit biases: the weights are (unique) global maxima of the associated margin maximization problem, and the decision boundary of the learned network is linear. For gradient descent, we provided experimental evidence which suggests that small initialization variance is important for gradient descent’s ability to quickly produce low-rank networks.

Acknowledgements

We thank Matus Telgarsky for helpful discussions. This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing as a part of the Deep Learning Theory Summer Cluster. SF, GV, PB, and NS acknowledge the support of the NSF and the Simons Foundation for the Collaboration on the Theoretical Foundations of Deep Learning through awards DMS-2031883 and #814639.

Appendix A Preliminaries on the Clarke Subdifferential and the KKT Conditions

Below we review the definition of the KKT conditions for non-smooth optimization problems (cf. [LL20, Dut+13]).

Consider the following optimization problem

${\mathbf{0}}\in\partial^{\circ}f(x)+\sum_{n\in[N]}\lambda_{n}\partial^{\circ}g_{n}(x)$;

For all $n\in[N]$ we have $\lambda_{n}g_{n}(x)=0$.

Appendix B Proof of Theorem 3.2

We start with some notations. We denote $p=\max_{i\neq j}|\langle x_{i},x_{j}\rangle|$. Thus, our assumption on $n$ can be written as $n\leq\frac{\gamma^{3}}{3}\cdot\frac{R_{\text{min}}^{2}}{p}\cdot\frac{R_{\text{min}}^{2}}{R_{\text{max}}^{2}}$. Since $W$ satisfies the KKT conditions of Problem (1), then there are $\lambda_{1},\ldots,\lambda_{n}$ such that for every $j\in[m_{1}]$ we have

where $\phi^{\prime}_{i,v_{j}}$ is a subgradient of $\phi$ at $v_{j}^{\top}x_{i}$, i.e., if $v_{j}^{\top}x_{i}>0$ then $\phi^{\prime}_{i,v_{j}}=1$, if $v_{j}^{\top}x_{i}<0$ then $\phi^{\prime}_{i,v_{j}}=\gamma$ and otherwise $\phi^{\prime}_{i,v_{j}}$ is some value in $[\gamma,1]$. Also we have $\lambda_{i}\geq 0$ for all $i$, and $\lambda_{i}=0$ if $y_{i}f(x_{i};W)\neq 1$. Likewise, for all $j\in[m_{2}]$ we have

where $\phi^{\prime}_{i,u_{j}}$ is defined similarly to $\phi^{\prime}_{i,v_{j}}$. The proof of the theorem follows from the following lemmas.

For all $i\in I$ we have $\sum_{j\in[m_{1}]}\lambda_{i}\phi^{\prime}_{i,v_{j}}+\sum_{j\in[m_{2}]}\lambda_{i}\phi^{\prime}_{i,u_{j}}<\frac{3m}{2\gamma R_{\text{min}}^{2}}$. Furthermore, $\lambda_{i}<\frac{3}{2\gamma^{2}R_{\text{min}}^{2}}$ for all $i\in I$.

Let $\xi=\max_{q\in I}\left(\sum_{j\in[m_{1}]}\lambda_{q}\phi^{\prime}_{q,v_{j}}+\sum_{j\in[m_{2}]}\lambda_{q}\phi^{\prime}_{q,u_{j}}\right)$ and suppose that $\xi\geq\frac{3m}{2\gamma R_{\text{min}}^{2}}$. Let $r=\operatorname*{argmax}_{q\in I}\left(\sum_{j\in[m_{1}]}\lambda_{q}\phi^{\prime}_{q,v_{j}}+\sum_{j\in[m_{2}]}\lambda_{q}\phi^{\prime}_{q,u_{j}}\right)$. Since $\xi\geq\frac{3m}{2\gamma R_{\text{min}}^{2}}>0$ then $\lambda_{r}>0$, and hence by the KKT conditions we must have $y_{r}f(x_{r};W)=1$.

Case 1: Assume that $r\in I_{-}$. Using \tagform@6 and (7), we have

Using $|x_{q}^{\top}x_{r}|\leq p$ for $q\neq r$, the above is at most

By our assumption on $n$, we can bound the above expression by

Thus, we obtain $f(x_{r};W)<-1$ in contradiction to $y_{r}f(x_{r};W)=1$.

Case 2: Assume that $r\in I_{+}$. A similar calculation to the one given in case 1 (which we do not repeat for conciseness) implies that $f(x_{r};W)>1$, in contradiction to $y_{r}f(x_{r};W)=1$. It concludes the proof of $\xi<\frac{3m}{2\gamma R_{\text{min}}^{2}}$.

Finally, since $\xi<\frac{3m}{2\gamma R_{\text{min}}^{2}}$ and the derivative of $\phi$ is lower bounded by $\gamma$, then for all $i\in I$ we have

and hence $\lambda_{i}<\frac{3}{2\gamma^{2}R_{\text{min}}^{2}}$. ∎

For all $i\in I$ we have $\sum_{j\in[m_{1}]}\lambda_{i}\phi^{\prime}_{i,v_{j}}+\sum_{j\in[m_{2}]}\lambda_{i}\phi^{\prime}_{i,u_{j}}>\frac{m}{2R_{\text{max}}^{2}}$. Furthermore, $\lambda_{i}>\frac{1}{2R_{\text{max}}^{2}}$ for all $i\in I$.

Suppose that there is $i\in I$ such that $\sum_{j\in[m_{1}]}\lambda_{i}\phi^{\prime}_{i,v_{j}}+\sum_{j\in[m_{2}]}\lambda_{i}\phi^{\prime}_{i,u_{j}}\leq\frac{m}{2R_{\text{max}}^{2}}$. Using \tagform@6 and (7), we have

Using $|x_{q}^{\top}x_{i}|\leq p$ for $q\neq i$ and $x_{i}^{\top}x_{i}\leq R_{\text{max}}^{2}$, the above is at most

Combining the above with our assumption on $n$, we get

in contradiction to Lemma B.1. It concludes the proof of $\sum_{j\in[m_{1}]}\lambda_{i}\phi^{\prime}_{i,v_{j}}+\sum_{j\in[m_{2}]}\lambda_{i}\phi^{\prime}_{i,u_{j}}>\frac{m}{2R_{\text{max}}^{2}}$.

Finally, since $\sum_{j\in[m_{1}]}\lambda_{i}\phi^{\prime}_{i,v_{j}}+\sum_{j\in[m_{2}]}\lambda_{i}\phi^{\prime}_{i,u_{j}}>\frac{m}{2R_{\text{max}}^{2}}$ and the derivative of $\phi$ is upper bounded by $1$, then for all $i\in I$ we have

and hence $\lambda_{i}>\frac{1}{2R_{\text{max}}^{2}}$. ∎

For all $i\in I$ we have $y_{i}f(x_{i};W)=1$.

By Lemma B.2 we have $\lambda_{i}>0$ for all $i\in I$, and hence by the KKT conditions we must have $y_{i}f(x_{i};W)=1$. ∎

Moreover, for all $i\in I$ we have: $y_{i}v_{j}^{\top}x_{i}>0$ for every $j\in[m_{1}]$, and $y_{i}u_{j}^{\top}x_{i}<0$ for every $j\in[m_{2}]$.

Fix $j\in[m_{1}]$. By \tagform@6 for all $i\in I_{+}$ we have

By Lemma B.1 and Lemma B.2, and using $\phi^{\prime}_{q,v_{j}}\in[\gamma,1]$ for all $q\in I$, the above is larger than

Thus, $v_{j}^{\top}x_{i}>0$, which implies $\phi^{\prime}_{i,v_{j}}=1$.

By Lemma B.1 and Lemma B.2, and using $\phi^{\prime}_{q,v_{j}}\in[\gamma,1]$ for all $q\in I$, the above is smaller than

Thus, $v_{j}^{\top}x_{i}<0$, which implies $\phi^{\prime}_{i,v_{j}}=\gamma$.

Since the above expression holds for all $j\in[m_{1}]$ then we have $v_{1}=\ldots=v_{m_{1}}$.

By similar arguments (which we do not repeat for conciseness) we also get

and $y_{i}u_{j}^{\top}x_{i}<0$ for all $i\in I$ and $j\in[m_{2}]$. ∎

By the above lemma, we may denote $v:=v_{1}=\ldots=v_{m_{1}}$ and $u:=u_{1}=\ldots=u_{m_{2}}$, and denote $z:=\frac{m_{1}}{\sqrt{m}}v-\frac{m_{2}}{\sqrt{m}}u$.

The pair $v,u$ is a unique global optimum of the Problem (3).

First, we remark that a variant of the this lemma appears in [SBG21]. They proved the claim under an assumption called Neural Agreement Regime (NAR), and Lemma B.4 implies that this assumption holds in our setting.

Note that the objective in Problem (3) is strictly convex and the constraints are affine. Hence, its KKT conditions are sufficient for global optimality, and the global optimum is unique. It remains to show that $v,u$ satisfy the KKT conditions.

Firstly, note that $v,u$ satisfy the constraints. Indeed, by Lemma B.4, for every $i\in I_{+}$ we have $v^{\top}x_{i}>0$ and $u^{\top}x_{i}<0$. Combining it with Lemma B.3 we get

Similarly, for every $i\in I_{-}$ we have $v^{\top}x_{i}<0$ and $u^{\top}x_{i}>0$. Together with Lemma B.3 we get

Next, we need to show that there are $\mu_{1},\ldots,\mu_{n}\geq 0$ such that

By setting $\mu_{i}=\lambda_{i}$ for all $i\in I$, Lemma B.4 implies that the above equations hold.

Finally, we need to show that $\mu_{i}=0$ for all $i\in I$ where the corresponding constraint holds with a strict inequality. However, by \tagform@8 and (9) all constraints hold with an equality. ∎

The weight matrix $W$ is a unique global optimum of Problem (1).

Now, let $W^{*}$ be a global optimum of Problem (1). By [LL20], the KKT conditions of this problem are necessary for optimality, and hence they are satisfied by $W^{*}$. Therefore, we have $W^{*}=W$. Thus, $W$ is a unique global optimum. ∎

First, We remark that a variant of the this lemma appears in [SBG21]. They proved the claim under an assumption called Neural Agreement Regime (NAR), and Lemma B.4 implies that this assumption holds in our setting.

Case 1: If $v^{\top}x\geq 0$ and $u^{\top}x\geq 0$ then $f(x;W)=\frac{m_{1}}{\sqrt{m}}v^{\top}x-\frac{m_{2}}{\sqrt{m}}u^{\top}x=z^{\top}x$, and thus $\operatorname*{sign}\left(f(x;W)\right)=\operatorname*{sign}(z^{\top}x)$.

Case 2: If $v^{\top}x\geq 0$ and $u^{\top}x<0$ then $f(x;W)=\frac{m_{1}}{\sqrt{m}}v^{\top}x-\frac{m_{2}}{\sqrt{m}}\gamma u^{\top}x>0$ and $z^{\top}x=\frac{m_{1}}{\sqrt{m}}v^{\top}x-\frac{m_{2}}{\sqrt{m}}u^{\top}x>0$.

Case 3: If $v^{\top}x<0$ and $u^{\top}x\geq 0$ then $f(x;W)=\frac{m_{1}}{\sqrt{m}}\gamma v^{\top}x-\frac{m_{2}}{\sqrt{m}}u^{\top}x<0$ and $z^{\top}x=\frac{m_{1}}{\sqrt{m}}v^{\top}x-\frac{m_{2}}{\sqrt{m}}u^{\top}x<0$.

Case 4: If $v^{\top}x<0$ and $u^{\top}x<0$ then $f(x;W)=\frac{m_{1}}{\sqrt{m}}\gamma v^{\top}x-\frac{m_{2}}{\sqrt{m}}\gamma u^{\top}x=\gamma z^{\top}x$, and thus $\operatorname*{sign}\left(f(x;W)\right)=\operatorname*{sign}(z^{\top}x)$. ∎

where $\lambda_{i}>0$ for all $i$. Since $x_{1},x_{2},x_{3}$ are linearly independent, then given $v,u$ there is a unique choice of $\lambda_{1},\lambda_{2},\lambda_{3}$ that satisfy the above equations.

Since $v,u$ satisfy the KKT conditions of Problem (3), we can find $\lambda_{1},\lambda_{2},\lambda_{3}$ as follows. Let $\mu_{1},\mu_{2},\mu_{3}\geq 0$ be such that the KKT conditions of Problem (3) hold. From the stationarity condition we have

Since $x_{1},x_{2},x_{3}$ are linearly independent, combining the above with \tagform@10 and (11) implies $\mu_{i}=\lambda_{i}>0$ for all $i$. Therefore, all constraints in Problem (3) must hold with an equality. Namely, we have

Solving the above equations, we get $\lambda_{1}=\lambda_{2}=\frac{8}{4\epsilon+5}$, and $\lambda_{3}=\frac{8}{5}$.

Thus, a KKT point of Problem (1) must satisfy \tagform@10 and (11) with the above $\lambda_{i}$’s. Now, consider

We need to show that $z$ does not satisfy the KKT conditions of the problem

Using the above equations, it is easy to verify that $z^{\top}x_{2}\neq z^{\top}x_{3}$ for all $\epsilon>0$. ∎

By Lemma B.4, for all $i\in I_{+}$ we have $v^{\top}x_{i}>0$ and $u^{\top}x_{i}<0$. Hence

Likewise, by Lemma B.4, for all $i\in I_{-}$ we have $v^{\top}x_{i}<0$ and $u^{\top}x_{i}>0$. Hence

Thus, it remains to obtain an upper bound for $\left\|z\right\|$.

Note that $v^{*},u^{*}$ satisfy the constraints in Problem (3). Indeed, for $i\in I_{-}$ we have

where the last inequality is since $0\leq(1-\kappa)(1-\gamma)=1+\kappa\gamma-\kappa-\gamma$.

By Lemma B.5 the pair $v,u$ is a global optimum of Problem (3). Hence

which implies $\left\|z\right\|\leq\frac{2\left\|z^{*}\right\|}{\kappa+\gamma}$ as required. ∎

Appendix C Proof of Lemma 3.3

According to the distribution assumption in the lemma, we can write $x_{i}=\Sigma^{1/2}\bar{x}_{i}$ where $\bar{x}_{i}\sim\mathsf{N}(0,I_{d})$.The proof below holds more generally when $\bar{x}_{i}$ has independent subgaussian entries. By Hanson-Wright inequality [RV13, Theorem 2.1], we have for any $t\geq 0$,

Let $t=C\sqrt{\log n}$ for a sufficiently large constant $C>0$. Taking a union bound over all $i\in[n]$, we have that with probability at least $1-n^{-20}$, $\left\|x_{i}\right\|=\sqrt{d}\pm O(\sqrt{\log n})$ for all $i\in[n]$ simultaneously.

For $i\not=j$, we have $\langle x_{i},x_{j}\rangle|x_{j}\sim\mathsf{N}(0,x_{j}^{\top}\Sigma x_{j})$. Hence we can apply a standard tail bound to obtain

Because we have known that $x_{j}^{\top}\Sigma x_{j}=O(\left\|x_{j}\right\|^{2})=O(d+\log n)=O(d)$ with probability at least $1-n^{-20}$, we have

Then we can take $t=C\sqrt{d\log n}$ for a sufficiently large constant $C$ and apply a union bound over all $i,j$, which gives $|\langle x_{i},x_{j}\rangle|=O(\sqrt{d\log n})$ for all $i\not=j$ with probability at least

Appendix D Proof of Theorem 3.4

To prove Theorem 3.4, we need to show that for some $t_{0}>0$, $\widehat{L}(W(t))<\log 2/n$ for all $t\geq t_{0}$. To do so, we will first show a proxy PL inequality [FG21], and then use this to argue that the loss must eventually be smaller than $\log 2/n$.

We begin by showing that the vector $\widehat{\mu}:=\sum_{i=1}^{n}y_{i}x_{i}$ correctly classifies the training data with a positive margin. To see this, note that for any $k\in[n]$,

Inequality $(i)$ uses the theorem’s assumption that $3n\max_{i\neq j}|\langle x_{i},x_{j}\rangle|\leq\gamma^{3}\min_{i}\|x_{i}\|^{2}$. Inequality $(ii)$ uses that $\gamma\leq 1$. To show how large of a margin $\widehat{\mu}$ gets on the training data, we bound its norm. We have,

Denoting $R_{\text{min}}:=\min_{i}\|x_{i}\|$, $R_{\text{max}}=\max_{i}\|x_{i}\|$, and $R=R_{\text{max}}/R_{\text{min}}$, substituting the above display into \tagform@13 we get for any $k\in[n]$,

then since $g(z)>0$ the above allows for the following proxy-PL inequality,

Let us now calculate how long until we reach the point where $\widehat{G}(W(t))<\log 2/(3n)$. Define

Since $\widehat{L}(W(t))$ is decreasing, we thus have for all times $t\geq\tau$, we have $\widehat{L}(W(t))\leq\widehat{L}(W(\tau))<\log(2)/n$.

Appendix E Proof of Theorem 4.2

In this section, we provide a proof of Theorem 4.2. An overview of our proof is as follows.

In Section E.1 we provide basic concentration arguments about the random initialization.

In Section E.2 we show that the neural network output and the logistic loss objective function are smooth as a function of the parameters.

In Section E.4 we leverage the above structural result to provide a good upper bound on $\|W^{(t)}\|_{F}$.

In Section E.5 we provide a lower bound for $\|W^{(t)}\|_{2}$.

In Section E.6 we show that a proxy-PL inequality is satisfied.

We conclude the proof of Theorem 4.2 in Section E.7 by putting together the preceding items to bound the stable rank $\mathsf{StableRank}(W^{(t)})=\|W^{(t)}\|_{F}^{2}/\|W^{(t)}\|_{2}^{2}$ and to show that $\widehat{L}(W^{(t)})\to 0$.

Let us denote by $C_{R}:=10R^{2}/\gamma^{2}+10$, where $R=R_{\text{max}}/R_{\text{min}}$ and $R_{\text{max}}=\max_{i}\|x_{i}\|$, $R_{\text{min}}=\min_{i}\|x_{i}\|$. For a given probability threshold $\delta\in(0,1)$, we make the following assumptions moving forward:

Step-size $\alpha\leq\gamma^{2}\left(5nR_{\text{max}}^{2}R^{2}C_{R}\max(1,H)\right)^{-1}$, where $\phi$ is $H$-smooth and $\gamma$-leaky.

We shall also use the following notation to refer to the sigmoid losses that appear throughout the analysis of gradient descent training for the logistic loss,

With probability at least $1-\delta$ over the random initialization, the following holds. First, we have the following upper bounds for the spectral norm and per-neuron norms at initialization,

For the first part of the lemma, note that for fixed $j\in[m]$, there are i.i.d. $z_{i}\sim\mathsf{N}(0,1)$ such that

By concentration of the $\chi^{2}$ distribution [LM00, Lemma 1], for any $t>0$,

In particular, if we let $t=\log(4m/\delta)$, we have that with probability at least $1-\delta/4$, for all $j\in[m]$,

E.2 Smoothness of network output and loss

In this sub-section, we show that the network output and the logistic loss satisfy a number of smoothness properties, owing to the fact that $\phi$ is $H$-smooth (i.e., $\phi^{\prime\prime}$ exists and $|\phi^{\prime\prime}(z)|\leq H$).

We next show that the empirical risk is smooth, in the sense that the gradient norm is bounded by the loss itself and that the gradients are Lipschitz.

where $\widehat{G}(W)$ is defined in \tagform@16. Additionally,

E.3 Loss ratio bound

In this section, we prove a key structural result which we will refer to as a ‘loss ratio bound’.

Let $\phi$ be a $\gamma$-leaky, $H$-smooth activation. Define $R=\max_{i,j}\nicefrac{{\|x_{i}\|}}{{\|x_{j}\|}}$, and let us denote $C_{R}=10R^{2}\gamma^{-2}+10$. Suppose that for all $i\in[n]$, we have,

Then under Assumptions (A1) and (A2), we have with probability at least $1-\delta$,

Our proof largely follows that used by [FCB22], who showed a loss ratio bound for gradient descent-trained two-layer networks with $\gamma$-leaky, $H$-smooth activations when the data comes from a mixture of isotropic log-concave distributions. We generalize their proof technique to accommodate general training data for which the samples are nearly orthogonal in the sense that $\|x_{i}\|^{2}\gg n\max_{k\neq i}|\langle x_{i},x_{k}\rangle|$. Additionally, we provide a more general proof technique that illustrates how a loss ratio bound could hold for activations $\phi$ for which $\phi^{\prime}(z)$ is not bounded from below by an absolute constant (like the ReLU), as well as for training data which are not necessarily nearly-orthogonal. We begin by describing two conditions which form the basis of this more general proof technique. The first condition concerns near-orthogonality of the gradients of the network, rather than the samples as in the assumption for Theorem 4.2.

We say that near-orthogonality of gradients holds at time $t$ if, for a some absolute constant $C^{\prime}>1$, for any $i\in[n]$,

Note that for linear classifiers—i.e., $m=1$ with $\phi(z)=z$—near-orthogonality of gradients is equivalent to near-orthogonality of samples, since in this setting $\nabla f(x_{i};W)=x_{i}$. It is clear that this is a more general condition than near-orthogonality of samples.

The next condition we call gradient persistence, which roughly states that the gradients of the network with respect to a sample has large norm whenever that sample has large norm.

We say that gradient persistence holds at time $t$ if there is a constant $c>0$ such that for all $i\in[n]$,

Gradient persistence essentially states that there is no possibility of a ‘vanishing gradient’ problem.

Next, we show that Lipschitz activation functions that are also ‘leaky’ in the sense that $\phi^{\prime}(z)\geq\gamma>0$ everywhere, allow for both gradient persistence and, when the samples are nearly-orthogonal, near-orthogonality of gradients.

Suppose $\phi$ is such that $\phi^{\prime}(z)\in[\gamma,1]$ for all $z$ for some absolute constant $\gamma>0$. Suppose that for some $C>\gamma^{-2}$, for all $i\in[n]$ we have,

Then for all times $t\geq 0$, the gradients are nearly-orthogonal (Condition E.5) with $C^{\prime}=C\gamma^{2}$ and gradient persistence (Condition E.6) holds for $c=\gamma^{2}$.

Since $\phi^{\prime}(z)\in[\gamma,1]$ for all $z$, we therefore see that gradient persistence holds with $c=\gamma^{2}$:

Similarly, we see that the gradients are nearly-orthogonal, since

where $(i)$ uses that $\phi$ is 1-Lipschitz and $(ii)$ uses the assumption on the near-orthogonality of the samples. ∎

We can now begin to prove Lemma E.4. We remind the reader of the notation for the sigmoid loss,

We follow the same proof technique of [FCB22], whereby in order to control the ratio of the sigmoid losses we show instead that the ratio of the exponential losses is small and that this suffices for showing the sigmoid losses is small. As we mention above, we generalize their analysis to emphasize that near-orthogonality of gradients and gradient persistence suffice for showing the loss ratio does not grow significantly.

Denote $R:=R_{\text{max}}/R_{\text{min}}$ where $R_{\text{max}}=\max_{i}\|x_{i}\|$ and $R_{\text{min}}=\min_{i}\|x_{i}\|$, and let $\phi$ be an arbitrary 1-Lipschitz and $H$-smooth activation. Suppose that near-orthogonality of gradients (Condition E.5) holds for some $C^{\prime}>1$ and gradient persistence (Condition E.6) hold at time $t$ for some $c>0$. Provided $\alpha\leq[5HR_{\text{max}}^{2}n(10R^{2}/c+10)]^{-1}$ and $C^{\prime}\geq 25R^{2}/c+25$, then for any $i,j\in[n]$ we have,

It suffices to consider $i=1$ and $j=2$. For notational simplicity denote

We now calculate the exponential loss ratio between two samples at time $t+1$ in terms of the exponential loss ratio at time $t$.

Inequality $(i)$ uses Lemma E.2 while $(ii)$ uses the definition of $A_{t}$. We now proceed in a manner similar to [FCB22] to bound each of the three terms in the product separately. For the first term, since gradient persistence (Condition E.6) holds at time $t$, we have for any $i\in[n]$,

On the other hand, since $\phi$ is 1-Lipschitz we also have

Putting the preceding two displays together, we get

Inequality $(i)$ uses \tagform@18, and the equality uses the definition $R=R_{\text{max}}/R_{\text{min}}$. This bounds the first term in \tagform@17.

For the second term, we use the fact that the gradients are nearly orthogonal at time $t$ (Condition E.5) and the lemma’s assumption on $C^{\prime}$ to get for any $i\neq k$,

Inequality $(i)$ uses the triangle inequality. Inequality $(ii)$ uses \tagform@20. The inequality $(iii)$ uses \tagform@18.

Finally, for the third term of \tagform@17, we have

Inequality $(i)$ uses Lemma E.3, while $(ii)$ uses the lemma’s assumption that $\alpha$ is smaller than $[5HR_{\text{max}}^{2}n(10R^{2}/c+10)]^{-1}$. Putting \tagform@19, \tagform@21 and \tagform@22 into \tagform@17, we get

Lemma E.8 shows that if the sigmoid loss ratio $g_{i}^{(t)}/g_{j}^{(t)}$ is large, then for a small-enough step-size, the exponential loss ratio will contract at the following interation. This motivates understanding how the exponential loss ratios relate to the sigmoid loss ratios. We recall the following fact, shown in [FCB22, Fact A.2 ].

and if $z_{1},z_{2}>0$, then we also have

This fact demonstrates that if we can ensure that the inputs to the losses is positive, then we can essentially treat the sigmoid and exponential losses interchangeably. Thus, if the network is able to interpolate the training data at a given time $t$, we can swap the sigmoid loss ratio appearing in Lemma E.8 with the exponential loss, and argue that if the exponential loss is too large at a given iteration, it will contract the following one. This allows for the exponential losses to be bounded throughout gradient descent. We formalize this in the following lemma.

Denote $R:=R_{\text{max}}/R_{\text{min}}$ where $R_{\text{max}}=\max_{i}\|x_{i}\|$ and $R_{\text{min}}=\min_{i}\|x_{i}\|$. Let $\phi$ be an arbitrary 1-Lipschitz and $H$-smooth activation. Suppose that,

Gradient persistence (Condition E.6) holds at time $t$ for some $c>0$, and

Near-orthogonality of gradients (Condition E.5) holds at time $t$ for some $C^{\prime}>25R^{2}/c+25$,

For some $\rho\geq 5R^{2}/c+5$, an exponential loss ratio bound holds at time $t$ with,

The network interpolates the training data at time $t$: $y_{i}f(x_{i};W^{(t)})>0$ for all $i$.

Then, provided the learning rate satisfies $\alpha\leq[5HR_{\text{max}}^{2}n(10R^{2}/c+10)]^{-1}$, we have an exponential loss ratio bound at time $t+1$ as well,

As in the proof of Lemma E.8, it suffices to prove that the ratio of the exponential loss for the first sample to the exponential loss for the second sample is bounded by $\rho$. Let us again denote

Above, inequality $(i)$ follows since $g_{1}^{(t)}/g_{2}^{(t)}>0$. The equality uses that $R=R_{\text{max}}/R_{\text{min}}$. Inequality $(ii)$ uses that $g_{i}^{(t)}<1$, the lemma’s assumption on the step-size, $\alpha\leq[5HR_{\text{max}}^{2}n(10R^{2}/c+10)]^{-1}$, and that $\exp(0.1)\leq 1.2$. The inequality $(iii)$ uses the proposition’s assumption that the network interpolates the training data at time $t$, so that the ratio of exponential losses is at most twice the ratio of the sigmoid losses by Fact E.9. The final inequality $(iv)$ follows by the case assumption that $g_{1}^{(t)}/g_{2}^{(t)}\leq\frac{2}{5}\rho$.

Inequality $(i)$ uses the Case 2 assumption that $g_{1}^{(t)}/g_{2}^{(t)}>\frac{2}{5}\rho$. Inequality $(ii)$ uses the proposition’s assumption that the exponential loss ratio at time $t$ is at most $\rho$, so that the sigmoid loss ratio is at most $2\rho$ by Fact E.9 (note that the sigmoid loss ratio is at least $2\rho/5>2$ by the case assumption and as $\rho>5$). The equality uses that $R=R_{\text{max}}/R_{\text{min}}$. The final inequality $(iii)$ follows as we can write

The first inequality above uses that $|x/(1+x)|\leq 1$ for $x>0$, and the final inequality follows by the assumption that $\rho\geq 5R^{2}/c+5>5R^{2}/c$. This proves $(iii)$ above, so that in Case 2, the exponential loss ratio decreases at the following iteration.

In summary, the preceding proposition demonstrates that a loss ratio bound can hold for general Lipschitz and smooth activations provided the following four conditions hold for some time $t_{0}$:

an exponential loss ratio bound holds at time $t_{0}$;

near-orthogonality of the gradients holds for all times $t\geq t_{0}$;

gradient persistence holds at all times $t\geq t_{0}$; and

the network interpolates the training data for all times $t\geq t_{0}$.

This is because the proposition guarantees that once you interpolate the training data, if the gradients are nearly-orthogonal and gradient persistence holds, the maximum ratio of the exponential losses does not become any larger than the maximum ratio at time $t_{0}$. Note that the above proof outline does not rely upon the training data being nearly orthogonal, nor that the activations are ‘leaky’, and thus may be applicable to more general settings than the ones we consider in this work.

On the other hand, when the training data is nearly-orthogonal and the activations are $\gamma$-leaky and $H$-smooth activations, Fact E.7 shows that (2) and (3) above hold for all times $t\geq 0$. Thus, to show a loss ratio bound in this setting, the main task is to show items (1) and (4) above. Towards this end, we present the final auxiliary lemma that will be used in the proof of Lemma E.4. A similar lemma appeared in [FCB22, Lemma A.3 ], and our proof is only a small modification of their proof. For completeness, we provide its proof in detail here.

Let $\phi$ be a $\gamma$-leaky, $H$-smooth activation. Then the following hold with probability at least $1-\delta$ over the random initialization.

An exponential loss ratio bound holds at initialization:

If there is an absolute constant $C_{R}^{\prime}>1$ such that at time $t$ we have $\max_{i,j}\{g_{i}^{(t)}/g_{j}^{(t)}\}\leq C_{R}^{\prime}$, and if for all $k\in[n]$ we have

then for $\alpha\leq\gamma^{2}/(2HC_{R}^{\prime}R^{2}R_{\text{max}}^{2}n)$, we have

If for all $k\in[n]$ we have $\|x_{k}\|^{2}\geq 8\gamma^{-2}n\max_{i\neq k}|\langle x_{i},x_{k}\rangle|$, then under Assumptions (A1) and (A2), at time $t=1$ and for all samples $k\in[n]$, we have $y_{k}f(x_{k};W^{(t)})>0$.

We shall prove each part of the lemma in sequence.

Part (a).

Since $\phi$ is 1-Lipschitz and $\phi(0)=0$, Cauchy–Schwarz implies

Applying this bound to the network output for each sample at initialization, we get

Inequality $(i)$ uses Lemma E.1, while inequality $(ii)$ and $(iii)$ follow by Assumptions (A2) and (A1), respectively. We therefore have,

Part (b).

Let $k\in[n]$. Let us re-introduce the notation $\nabla f_{i}^{(t)}:=\nabla f(x_{i};W^{(t)})$. By Lemma E.2, we know

where Inequality $(i)$ uses Lemma E.3. Continuing we get that

Inequality $(i)$ uses the lemma’s assumption that $\max_{i,j}\{g_{i}^{(t)}/g_{j}^{(t)}\}\leq C_{R}^{\prime}$. Inequality $(ii)$ uses that $\phi$ is $\gamma$-leaky and 1-Lipschitz (see eq. \tagform@27). Inequality $(iii)$ uses that the assumption that the samples are nearly-orthogonal,

Inequality $(iv)$ uses the definition $R_{\text{min}}=\min_{i}\|x_{i}\|$. Inequality $(v)$ again uses the lemma’s assumption of a sigmoid loss ratio bound, so that

The final inequality $(vi)$ follows since the step-size $\alpha\leq\gamma^{2}/(2HC_{R}^{\prime}R^{2}R_{\text{max}}^{2}n)$ is small enough. This completes part (b) of this lemma.

Part (c).

Note that by \tagform@25, $|f(x_{k};W^{(0)})|\leq 1/50$. Since $g$ is monotone this implies the sigmoid losses at initialization satisfy $g_{i}^{(0)}\in[(1+\exp(0.02))^{-1},(1+\exp(-0.02))^{-1}]\subset[0.49,0.51]$ and so

Thus, the assumption that $\|x_{k}\|^{2}\geq 8\gamma^{-2}n\max_{i\neq k}|\langle x_{i},x_{k}\rangle|$ and Assumption (A1) allow for us to apply part (b) of this lemma as follows,

We now have all of the pieces necessary to prove Lemma E.4.

In order to show that the ratio of the $g(\cdot)$ losses is bounded, it suffices to show that the ratio of exponential losses $\exp(-(\cdot))$ is bounded, since by Fact E.9,

We will prove the lemma by first showing an exponential loss ratio holds at time $t=0$ and $t=1$, and then use an inductive argument based on Proposition E.10 with $\rho=5R^{2}/\gamma^{5}+5=\frac{1}{2}C_{R}$.

By part (a) of Lemma E.11, the exponential loss ratio at time $t=0$ is at most $\exp(2)$. To see the loss ratio holds at time $t=1$, first note that by assumption, we have that the samples satisfy,

Because $\phi$ is a $\gamma$-leaky, $H$-smooth activation, by Fact E.7 this implies that gradient persistence (Condition E.6) holds with $c=\gamma^{2}$ and near-orthogonality of gradients (Condition E.5) holds for all times $t\geq 0$ with $C^{\prime}>2(25R^{2}\gamma^{-2}+25)$. By Assumption (A1), we can therefore apply Lemma E.8 at time $t=0$, so that we have for any $i,j$,

Inequality $(i)$ uses that $g_{i}^{(t)}<1$, while inequality $(ii)$ uses that the step-size is sufficiently small $\alpha\leq 1/20R_{\text{max}}^{2}$ by Assumption (A1). Therefore, the exponential loss ratio at times $t=0$ and $t=1$ is at most $9\leq 5R^{2}/\gamma^{2}+5$.

Now suppose by induction that at times $\tau=1,\dots,t$, the exponential loss ratio is at most $5R^{2}/\gamma^{2}+5$, and consider $t+1$. (The cases $t=0$ and $t=1$ were just proved above.) By the induction hypothesis and \tagform@29, the sigmoid loss ratio from times $0,\dots,t$ is at most $10R^{2}/\gamma^{2}+10$. By Assumption (A1), the step-size satisfies

Further, the samples satisfy \tagform@30, so that

Thus all parts of Lemma E.11 hold with $C_{R}^{\prime}=C_{R}=10R^{2}\gamma^{-2}+10$. By part (b) of that lemma, the unnormalized margin for each sample increased for every time $\tau=0,\dots,t$: