Learning ReLU Networks on Linearly Separable Data: Algorithm, Optimality, and Generalization

Introduction

Deep neutral networks have recently boosted the notion of “deep learning from data,” with field-changing performance improvements reported in numerous machine learning and artificial intelligence tasks . Despite their widespread use as well as numerous recent contributions, our understanding of how and why neural networks (NNs) achieve this success remains limited. While their expressivity (expressive power) has been well argued , the research focus has shifted toward addressing the computational challenges of training such models and understanding their generalization behavior.

From the vantage point of optimization, training deep NNs requires dealing with extremely high-dimensional and non-convex problems, which are NP-hard in the worst case. It has been shown that even training a two-layer NN of three nodes is NP-complete , and the loss function associated with even a single neuron exhibits exponentially many local minima . It is therefore not clear whether and how we can provably yet efficiently train a NN to global optimality.

Nevertheless, as often evidenced by empirical tests, these NN architectures can be ‘successfully’ trained by means of simple local search heuristics, such as ‘plain-vanilla’ (stochastic) (S) gradient descent (GD) on real or randomly generated data. Considering the over-parameterized setting in particular, where the NNs have far more parameters than training samples, SGD can often successfully train these networks while exhibiting favorable generalization performance without overfitting . As an example, the celebrated VGG19 net with 20 million parameters trained on the CIFAR-10 dataset of 50 thousand data samples achieves state-of-the-art classification accuracy, and also generalizes well to other datasets . In addition, training NNs by e.g., adding noise to the training samples , or to the (stochastic) gradients during back-propagation , has well-documented merits in training with enhancing generalization performance, as well as in avoiding bad local minima . In this contribution, we take a further step toward understanding the analytical performance of NNs, by providing fundamental insights into the optimization landscape and generalization capability of NNs trained by means of SGD with properly injected noise.

For concreteness, we address these challenges in a binary classification setting, where the goal is to train a two-layer ReLU network on linearly separable data. Although a nonlinear NN is clearly not necessary for classifying linearly separable data, as a linear classifier such as the Perceptron, would do , the fundamental question we target here is whether and how one can efficiently train a ReLU network to global optimality, despite the presence of infinitely many local minima, maxima, and saddle points . Separable data have also been used in recent works . The motivation behind employing separable data is twofold. They can afford a zero training loss, and distinguish whether a NN is successfully trained or not (as most loss functions for training NNs are non-convex, it is in general difficult to check its global optimum). In addition, separable data enable improvement of the plain-vanilla SGD by leveraging the power of random noise in a principled manner, so that the modified SGD algorithm can provably escape local minima and saddle points efficiently, and converge to a global minimum in a finite number of non-zero updates. We further investigate the generalization capability of successfully trained ReLU networks leveraging compression bounds . Thus, the binary classification setting offers a favorable testbed for studying the effect of training noise on avoiding overfitting when learning ReLU networks. Although the focus of this paper is on two-layer networks, our novel algorithm and theoretical results can shed light on developing reliable training algorithms for as, well as on, understanding generalization of deep networks.

In a nutshell, the main contributions of the present work are:

A simple SGD algorithm that can provably escape local minima and saddle points to efficiently train any two-layer ReLU network to attain global optimality;

Theoretical and empirical evidence supporting the injection of noise during training NNs to escape bad local minima and saddle points; and

Tight generalization error bounds and guarantees for (possibly over-parameterized) ReLU networks optimally trained with the novel SGD algorithm.

The remainder of this paper is structured as follows. Section 2 reviews related contributions. Section 3 introduces the binary classification setting, and the problem formulation. Section 4 presents the novel SGD algorithm, and establishes its theoretical performance. Section 5 deals with the generalization behavior of ReLU networks trained with the novel SGD algorithm. Numerical tests on synthetic data and real images are provided in Section 6. The present paper is concluded with research outlook in Section 7, while technical proofs of the main results are delegated to the Appendix.

Notation: Lower- (upper-)case boldface letters denote vectors (matrices), e.g., $\bm{a}$ ($\bm{A}$). Calligraphic letters are reserved for sets, e.g. $\mathcal{S}$, with the exception of $\mathcal{D}$ representing some probability distribution. The operation $\lfloor c\rfloor$ returns the largest integer no greater than the given number $c>0$, the cardinality $|\mathcal{S}|$ counts the number of elements in set $\mathcal{S}$, and $\|\bm{x}\|_{2}$ denotes the Euclidean norm of $\bm{x}$.

Related Work

As mentioned earlier, NN models have lately enjoyed great empirical success in numerous domains . Many contributions have been devoted to explaining such a success; see e.g., . Recent research efforts have focused on the expressive ability of deep NNs , and on the computational tractability of training such models . In fact, training NNs is NP-hard in general, even for small and shallow networks . Under various assumptions (e.g., Gaussian data, and a sufficiently large number of hidden units) as well as different models however, it has been shown that local search heuristics such as (S)GD can efficiently learn two-layer NNs with quadratic or ReLU activations .

Recent efforts have also been centered on understanding generalization behavior of deep NNs by introducing and/or studying different complexity measures. These include Rademacher complexity, uniform stability, and spectral complexity; see for a recent survey. However, the obtained generalization bounds do not account for the underlying training schemes, namely optimization methods. As such, they do not provide tight guarantees for generalization performance of (over-parameterized) networks trained with iterative algorithms . Even though recent work suggested an improved generalization bound by optimizing the PAC-Bayes bound of an over-parameterized network in a binary classification setting , this result is meaningful only when the optimization succeeds. Leveraging standard compression bounds, generalization guarantees have been derived for two-layer leaky ReLU networks trained with plain-vanilla SGD . But this bound does not generalize to ReLU networks, due to the challenge and impossibility of using plain-vanilla SGD to train ReLU networks to global optimum.

Problem Formulation

We deal with single-hidden-layer NNs having $d$ scalar inputs, $k>0$ hidden neurons, and a single output (for binary classification). The overall input-output relationship of such a two-layer NN is

where the ReLU activation $\sigma(\bm{z})$ should be understood entry-wise when applied to a vector $\bm{z}$.

where $\mathds{1}_{\{\cdot\}}$ denotes the indicator function taking value $1$ if the argument is true, and otherwise.

1+1’ (colored red) and one of which belong to class ‘$-1$’ (colored black). Data points with a non-zero classification error (hence non-zero loss) must lie in the negative cone of all hyperplanes. In this paper, we fix the second layer of network $f(\bm{x};\mathcal{W})$ to be some constant vector $\bm{v}$ given a priori, with at least one positive and at least one negative entry. Therefore, training the ReLU network $f(\bm{x};\mathcal{W})$ boils down to learning the weight matrix $\bm{W}$ only. As such, the network is henceforth denoted by $f(\bm{x};\bm{W}):=f(\bm{x};\mathcal{W})$, and the goal is to solve the following optimization problem

Consider minimizing $L_{\mathcal{S}}(\bm{W})$ by means of plain-vanilla SGD with constant learning rate $\eta>0$, as

with the (sub-)gradient of the hinge loss at a randomly sampled datum $(\bm{x}_{i_{t}},y_{i_{t}})\in\mathcal{S}$ given by

where $\text{diag}(\bm{z})$ is a diagonal matrix holding entries of vector $\bm{z}$ on its diagonal, and the indicator function $\mathds{1}_{\{\bm{z}\geq\bm{0}\}}$ applied to $\bm{z}$ is understood entry-wise. For any sub-optimal critical point $\bm{W}^{\dagger}$ incurring a nonzero loss for some $(\bm{x}_{i_{t}},y_{i_{t}})$, it can be readily deduced that $\text{diag}(\mathds{1}_{\{\bm{W}^{\dagger}\bm{x}_{i_{t}}\geq\bm{0}\}})=\bm{0}$ .

Following the convention , we say that a ReLU is active if its output is non-zero, and inactive otherwise. Furthermore, we denote the state of per $j$-th ReLU by its activity indicator function $\mathds{1}_{\{\bm{w}_{j}^{\top}\bm{x}_{i_{t}}\geq 0\}}$. In words, there exists always some data sample(s) for which all hidden neurons become inactive at a sub-optimal critical point. This is corroborated by the fact that under some conditions, plain-vanilla SGD converges to a sub-optimal local minimum with high probability . It will also be verified by our numerical tests in Section 6, that SGD can indeed get stuck in sub-optimal local minima when training ReLU networks.

Main Results

In this section, we present our main results that include a modified SGD algorithm and theory for efficiently training single-hidden-layer ReLU networks to global optimality. As in the convergence analysis of the Perceptron algorithm (see e.g., , [41, Chapter 9]), we define an update at iteration $t$ as non-zero or effective if the corresponding (modified) stochastic gradient is non-zero, or equivalently, whenever one has $\bm{W}^{t+1}\neq\bm{W}^{t}$.

As explained in Section 3, plain-vanilla SGD iterations for minimizing $L_{\mathcal{S}}(\bm{W})$ can get stuck in sub-optimal critical points. Recall from (9) that whenever this happens, it must hold that $\text{diag}(\mathds{1}_{\{\bm{W}\bm{x}_{i_{t}}\geq\bm{0}\}})=\bm{0}$ for some data sample $(\bm{x}_{i_{t}},y_{i_{t}})\in\mathcal{S}$, or equivalently $\bm{w}_{j}^{\top}\bm{x}_{i_{t}}<0$ for all $j=1,\,2\,\ldots,\,k$. To avoid being trapped in these points, we will endow the algorithm with a non-zero ‘(sub-)gradient’ even at a sub-optimal critical point, so that the algorithm will be able to continue updating, and will have a chance to escape from sub-optimal critical points. If successful, then when the algorithm converges, it must hold that $\mathds{1}_{\{1-y_{i}\bm{v}^{\top}\sigma(\bm{W}^{\dagger}\bm{x}_{i})>0\}}=0$ for all data samples $(\bm{x}_{i},y_{i})\in\mathcal{S}$ (cf. (9)), or $1-y_{i}\bm{v}^{\top}\sigma(\bm{W}^{\dagger}\bm{x}_{i})\leq 0$ for all $i=1,\,2,\,\ldots,\,n$, thanks to linear separability of the data. This in agreement with the definition of the hinge loss function satisfies that $L_{\mathcal{S}}(\bm{W}^{\dagger})=0$ in (4), which guarantees that the algorithm converges to a global optimum. Two critical questions arise at this point: Q1) How can we endow a non-zero ‘(sub-)gradient’ based search direction even at a sub-optimal critical point, while having the global minima as limiting points of the algorithm? and Q2) How is it possible to guarantee convergence?

Albeit empirically effective in training ReLU networks, SGD with such architecture-agnostic injected noise into all ReLU activity indicator functions cannot guarantee convergence in general, or convergence is difficult or even impossible to establish. We shall take a different route to bypass this hurdle here, which will lead to a simple algorithm provably convergent to a wanted global optimum in a finite number of non-zero updates. This result holds regardless of the data distribution, initialization, network size, or the number of hidden neurons. Toward, to ensure convergence of our modified SGD algorithm, we carefully design the noise injection process by maintaining at least one non-zero ReLU activity indicator variable at every non-optimal critical point.

For the picked data sample $(\bm{x}_{i_{t}},y_{i_{t}})\in\mathcal{S}$ per iteration $t\geq 0$, we inject Gaussian noise $\epsilon_{j}^{t}\sim\mathcal{N}(0,\,\gamma^{2})$ into the $j$-th ReLU activity indicator function $\mathds{1}_{\{\bm{w}_{j}^{\top}\bm{x}_{i_{t}}\geq 0\}}$ in the SGD update of (9), if and only if the corresponding quantity $y_{i_{t}}v_{j}\geq 0$ holds, and we repeat this for all neurons $j=1,\,2,\,\ldots,\,k$.

Interestingly, the noise variance $\gamma^{2}$, admits simple choices, so long as it is selected sufficiently large matching the size of the corresponding summands $\{|{\bm{w}_{j}^{t}}^{\top}\bm{x}|^{2}\}_{j,t}$. We will build up more intuition and highlight the basic principle behind such a noise injection design shortly in Section 4.2, along with our formal convergence analysis. For implementation purposes, we summarize the novel SGD algorithm with randomly perturbed ReLU activity indicator functions in Algorithm 1. As far as stopping criterion is concerned, it is safe to conclude that the algorithm has converged, if there has been no non-zero update for a succession of say, $np$ iterations, where $p>0$ is some fixed large enough integer. This holds with high probability, which depends on $p$, and $|\mathcal{N}_{v}^{+}|$ ($|\mathcal{N}_{v}^{-}|$), where the latter denotes the number of neurons with $v_{j}>0$ ($v_{j}<0$). We have the following result, whose proof is provided in Appendix Appendix D.4.

Let $\|\bm{w}_{j}^{t}\|_{2}\leq w_{\max}$ for all neurons $j=1,\,2,\,\ldots,\,k$, and all iterations $t\geq 0$, and consider $i_{t}$ cycling deterministically through $\{1,2,\ldots,n\}$. If there is no non-zero update after a succession of $np$ iterations, then Algorithm 1 converges to a global optimum of $L_{\mathcal{S}}(\bm{W})$ with probability at least $1-\left[\Phi\!\left(w_{\max}/\gamma\right)\right]^{p\min\{|\mathcal{N}_{v}^{+}|,\,|\mathcal{N}_{v}^{-}|\}}$, where $\Phi(z):=\left(1/\sqrt{2\pi}\right)\int_{-\infty}^{z}e^{-s^{2}}ds$ is the cumulative density function of the standardized Gaussian distribution $\mathcal{N}(0,1)$.

Observe that the probability in Proposition 1 can be made arbitrarily close to $1$ by taking sufficiently large $p$ and/or $\gamma$. Regarding our proposed approach in Algorithm 1, three remarks are worth making.

With the carefully designed noise injection rule, our algorithm constitutes a non-trivial generalization of the Perceptron or plain-vanilla SGD algorithms to learn ReLU networks. Implementing Algorithm 1 is as easy as plain-vanilla SGD, requiring almost negligible extra computation overhead. Both numerically and analytically, we will demonstrate the power of our principled noise injection into partial ReLU activity indicator functions, as well as establish the optimality, efficiency, and generalization performance of Algorithm 1 in learning two-layer (over-parameterized) ReLU networks on linearly separable data.

It is worth remaking that the random (Gaussian) noise in our proposal is solely added to the ReLU activity indicator functions, rather than to any of the hidden neurons. This is evident from the first indicator function $\mathds{1}_{\{1-y_{i_{t}}\bm{v}^{\top}\sigma(\bm{W}^{t}\bm{x}_{i_{t}})>0\}}$ being the (sub)derivative of a hinge loss, in Step 5 of Algorithm 1, which is kept as it is in the plain-vanilla SGD, namely it is not affected by the noise. Moreover, our use of random noise in this way distinguishes itself from those in the vast literature for evading saddle points (see e.g., , , , ), which simply add noise to either the iterates or to the (stochastic) (sub)gradients. This distinction endows our approach with the unique capability of also escaping local minima (in addition to saddle points). To the best of our knowledge, our approach is the first of its kind in provably yet efficiently escaping local minima under suitable conditions.

Compared with previous efforts in learning ReLU networks (e.g., , , , , , ), our proposed Algorithm 1 provably converges to a global optimum in a finite number of non-zero updates, without any assumptions on the data distribution, training/network size, or initialization. This holds even in the presence of exponentially many local minima and saddle points. To the best of our knowledge, Algorithm 1 provides the first solution to efficiently train such a single-hidden-layer ReLU network to global optimality with a hinge loss, so long as the training samples are linearly separable. Generalizations to other objective functions based on e.g., the $\tau$-hinge loss and the smoothed hinge loss (a.k.a. polynomial hinge loss) , as well as to multilayer ReLU networks are possible, and they are left for future research.

2 Convergence analysis

Before presenting our main convergence results for Algorithm 1, we introduce some notation. To start, let $\mathcal{N}_{y}^{+}\subseteq\{1,2,\ldots,n\}$ ($\mathcal{N}_{y}^{-}$) be the index set of data samples $\{(\bm{x}_{i},y_{i})\}_{1\leq i\leq n}$ belonging to the ‘positive’ (‘negative’) class, namely whose $y_{i}=+1$ ($y_{i}=-1$). It is thus self-evident that $\mathcal{N}_{y}^{+}\cup\mathcal{N}_{y}^{-}=\{1,2,\ldots,n\}$ and $\mathcal{N}_{v}^{+}\cup\mathcal{N}_{v}^{-}=\{1,2,\ldots,k\}$ hold under our assumptions. Putting our work in context, it is useful to first formally summarize the landscape properties of the objective function $L_{\mathcal{S}}(\bm{W})=(1/n)\sum_{i=1}^{n}\max\{0,\,1-y_{i}f(\bm{x}_{i};\bm{W})\}$, which can help identify the challenges in learning ReLU networks.

Function $L_{\mathcal{S}}(\bm{W})$ has the following properties: i) it is non-convex, and ii) for each sub-optimal local minimum (that incurs a non-zero loss), there exists (at least) a datum $(\bm{x}_{i},y_{i})\in\mathcal{S}$ for which all ReLUs become inactive.

The proof of Property i) in Proposition 2 can be easily adapted from that of [5, Proposition 5.1], while Property ii) is just a special case of [23, Thm. 5] for a fixed $\bm{v}$; hence they are both omitted in this paper.

We will provide an upper bound on the number of non-zero updates that Algorithm 1 performs until no non-zero update occurs after within a succession of say, e.g. $np$ iterations (cf. (10)), where $p$ is a large enough integer. This, together with the fact that all sub-optimal critical points of $L_{\mathcal{S}}(\bm{W})$ are not limiting points of Algorithm 1 due to the Gaussian noise injection with a large enough variance $\gamma^{2}>0$ at every iteration, will guarantee convergence of Algorithm 1 to a global optimum of $L_{\mathcal{S}}(\bm{W})$. Specifically, the main result is summarized in the following theorem.

In particular, if $\bm{W}^{0}=\bm{0}$, then Algorithm 1 converges to a global optimum after at most T_{k}^{0}:=\frac{k}{\eta v_{\min}^{2}}\big{(}\eta\|\bm{v}\|_{2}^{2}+2\big{)}\|\bm{\omega}^{\ast}\|_{2}^{2} non-zero updates.

Regarding Theorem 1, a couple of observations are of interest. The developed Algorithm 1 converges to a globally optimal solution of the non-convex optimization (4) within a finite number of non-zero updates, which implicitly corroborates the ability of Algorithm 1 to escape sub-optimal local minima, as well as saddle points. This holds regardless of the underlying data distribution $\mathcal{D}$, the number $n$ of training samples, the number $k$ of hidden neurons, or even the initialization $\bm{W}^{0}$. It is also worth highlighting that the number $T_{k}$ of non-zero updates does not depend on the dimension $d$ of input vectors, but it scales with $k$ (in the worst case), and it is inversely proportional to the step size $\eta>0$. Recall that the worst-case bound for SGD learning of leaky-ReLU networks with initialization $\bm{W}^{0}=\bm{0}$ is [5, Thm. 2]

We briefly present the main ideas behind the proof of Theorem 1 next, but delegate the technical details to Appendix Appendix A.1. Our proof mainly builds upon the convergence proof of the classical Perceptron algorithm (see e.g., [41, Thm. 9.1]), and it is also inspired by that of [5, Thm. 1]. Nonetheless, the novel approach of performing SGD with principled noise injection into the ReLU activity indicator functions distinguishes itself from previous efforts. Since we are mainly interested in the (maximum) number of non-zero updates to be performed until convergence, we will assume for notational convenience that all iterations $t\geq 0$ in (10) of Algorithm 1 perform a non-zero update. This assumption is made without loss of generality. To see this, since after the algorithm converges, one can always re-count the number of effective iterations that correspond to a non-zero update and re-number them by $t=0,1,\ldots.$

It will become clear in the proof that injecting random noise into just a subset of (rather than all) ReLU activity indicator functions enables us to leverage two key inequalities, namely, $\sum_{j=1}^{k}y_{i_{t}}v_{j}\sigma({\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}})<1$ and $y_{i}{\bm{\omega}^{\ast}}^{\top}\bm{x}_{i}\geq 1$ for all data samples $(\bm{x}_{i},y_{i})\in\mathcal{S}$. These inequalities uniquely correspond to whether an update is non-zero or not. In turn, this characterization is indeed the key to establishing the desired lower and upper bounds for the two quantities on the two sides of the Cauchy-Schwartz inequality, a critical ingredient of our convergence analysis.

3 Lower bound

Besides the worst-case upper bound given in Theorem 1, we also provide a lower bound on the number of non-zero updates required by Algorithm 1 for convergence, which is summarized in the following theorem. The proof is provided in Appendix Appendix C.3.

Under the conditions of Theorem 1, consider Algorithm 1 with initialization $\bm{W}^{0}=\bm{0}$. Then for any $d>0$, there exists a set of linearly separable data samples on which Algorithm 1 performs at least {\|\bm{\omega}^{\ast}\|_{2}^{2}}\big{/}\!\left({\eta\|\bm{v}\|_{2}^{2}}\right) non-zero updates to optimally train a single-hidden-layer ReLU network.

The lower bound on the number of non-zero updates to be performed in Theorem 2 matches that for learning single-hidden-layer leaky-ReLU networks initialized from zero [5, Thm. 4]. On the other hand, it is also clear that the worst-case bound established in Theorem 1 is (significantly) loose than the lower bound here. The gap between the two bounds (in learning ReLU versus leaky ReLU networks) is indeed the price we pay for escaping bad local minima and saddle points through our noise-injected SGD approach.

Generalization

In this section, we investigate the generalization performance of training (possibly over-parameterized) ReLU networks using Algorithm 1 with randomly perturbed ReLU activity indicator functions. Toward this objective, we will rely on compression generalization bounds, specifically for the $0/1$ classification error as in (3) .

Suppose now that Algorithm 1 has converged after $\tau_{k}\leq T_{k}$ non-zero updates, as per Theorem 1. And let $(\bm{x}_{i_{1}},\,\bm{x}_{i_{2}},\,\ldots,\,\bm{x}_{i_{\tau_{k}}})$ be the $\tau_{k}$-tuple of training data from $\mathcal{S}$ randomly picked by SGD iterations of Algorithm 1. To exemplify the $\tau_{k}$-tuple used per realization of Algorithm 1, we write $G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0})=\Gamma_{\bm{W}^{0}}(\bm{x}_{i_{1}},\,\bm{x}_{i_{2}},\,\ldots,\,\bm{x}_{i_{\tau_{k}}})$. Since $\tau_{k}$ can be smaller than $n$, function $\Gamma_{\bm{W}^{0}}$ and thus $G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0})$ rely on compressed (down to size $\tau_{k}$) versions of the $n$-tuples comprising the set $\mathcal{H}_{k}$ [41, Definition 30.4]. Let $\mathcal{S}_{\tau_{k}}^{c}:=\{i|i\in\{1,\,2,\,\ldots,\,n\}\backslash\{i_{1},\,i_{2},\,\ldots,\,i_{\tau_{k}}\}\}$ be the subset of training data not picked by SGD to yield $G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0})$; and correspondingly, let $R_{\mathcal{D}}(G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0}))$ denote the ensemble risk associated with $G_{k}^{\rm Alg1}$, and $\hat{R}_{\mathcal{S}_{\tau_{k}}^{c}}(G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0}))$ the empirical risk associated with the complement training set, namely $\mathcal{S}_{\tau_{k}}^{c}$. With these notational conventions, our next result follows from [41, Thm. 30.2].

If $n\geq 2\tau_{k}$, then the following inequality holds with probability of at least $1-\delta$ over the choice of $\mathcal{S}$ and $\bm{W}^{0}$

Regarding Theorem 3, two observations are in order. The bound in (13) is non-asymptotic but as $n\to\infty$, the last two terms on the right-hand-side vanish, implying that the ensemble risk $R_{\mathcal{D}}(G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0}))$ is upper bounded by the empirical risk $\hat{R}_{\mathcal{S}_{\tau_{k}}^{c}}(G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0}))$. Moreover, once the SGD iterations in Algorithm 1 converge, we can find the complement training set $\mathcal{S}_{\tau_{k}}^{c}$, and thus $\hat{R}_{\mathcal{S}_{\tau_{k}}^{c}}(G_{k}(\mathcal{S};\bm{W}^{0}))$ can be determined. After recalling that $\hat{R}_{\mathcal{S}_{\tau_{k}}^{c}}(G_{k}^{\rm Alg1}(\mathcal{S};\bm{W}^{0}))=0$ holds at a global optimum of $L_{\mathcal{S}}$ by Theorem 1, we obtain from Theorems 1 and 3 the following corollary.

If $n\geq 2\tau_{k}$, and all rows of the initialization satisfy $\{\|\bm{w}_{j}^{0}\|_{2}\leq\rho\}_{j=1}^{k}$, then the following holds with probability at least $1-\delta$ over the choice of $\mathcal{S}$

Expressed differently, the bound in (14) suggests that in order to guarantee a low generalization error, one requires in the worst case about $n=\mathcal{O}(k^{2}\|\bm{\omega}^{\ast}\|_{2}^{2})$ training data to reliably learn a two-layer ReLU network of $k$ hidden neurons. This holds true despite the fact that Algorithm 1 can achieve a zero training loss regardless of the training size $n$. One implication of Corollary 1 is a fundamental difference in the sample complexity for generalization between training a ReLU network (at least in the worst case), versus training a $\alpha$-leaky ReLU network ($0<\alpha<1$), which at most needs $n=\mathcal{O}(\|\bm{\omega}^{\ast}\|_{2}^{2}/\alpha^{2})$ data to be trained via SGD-type algorithms.

Numerical Tests

To validate our theoretical results, this section evaluates the empirical performance of Algorithm 1 using both synthetic data and real data. To benchmark Algorithm 1, we also simulated the plain-vanilla SGD. To compare between the two algorithms as fair as possible, the same initialization $\bm{W}^{0}$, constant step size $\eta>0$, and data random sampling scheme were employed. For reproducibility, the Matlab code of Algorithm 1 is publicly available at https://gangwg.github.io/RELUS/.

Figure 2 depicts our results, where we display success rates of the plain-vanilla SGD (top panel) and our noise-injected SGD in Algorithm 1 (bottom panel); each plot presents results obtained from the $100$ experiments. Within each plot, a white square signifies that $100\%$ of the trials were successful, meaning that the learned ReLU network yields a training loss $L_{\{(\bm{x}_{i},y_{i})\}_{i=1}^{n}}(\bm{W}^{T})\leq 10^{-10}$, while black squares indicate $0\%$ success rates. It is evident that the developed Algorithm 1 trained all considered ReLU networks to global optimality, while plain-vanilla SGD can get stuck with bad local minima, for small $k$ in particular. The bottom panel confirms that Algorithm 1 achieves optimal learning of single-hidden-layer ReLU networks on separable data, regardless of the network size, the number of training samples, and the initialization. The top panel however, suggests that learning ReLU networks becomes easier with plain-vanilla SGD as $k$ grows larger, namely as the network becomes ‘more over-parameterized.’

2 Real data

Performance of Algorithm 1 for training (over-)parameterized ReLU networks is further corroborated using two real datasets: iris in UCI’s machine learning repository , and MNIST images Downloaded from http://yann.lecun.com/exdb/mnist/. . The iris dataset contains $150$ four-dimensional feature vectors belonging to three classes. To obtain a two-class linearly separable dataset, the first-class data vectors were relabeled $+1$, while the remaining were relabeled $-1$. We performed $100$ independent experiments over a varying set of $n\in\{30,\,60,\,90,\,120,\,150\}$ training samples using ReLU networks with $k\in\{2,\,4,\,6,\,8,\,10\}$ hidden neurons. Gaussian initialization from $\mathcal{N}(\bm{0},\bm{I})$, step size $\eta=0.1$, noise variance $\gamma=10$, and a maximum of $100$ effective data passes were simulated. Success rates of plain-vanilla SGD are given in Fig. 3 (right). Again, Algorithm 1 achieves a $100\%$ success rate in all simulated settings.

The linearly separable MNIST dataset collects $2,000$ images of digits $3$ (labeled $+1$) and $5$ (labeled $-1$), each having dimension $784$. We performed $100$ independent experiments over a varying set of $n\in\{200,\,400,\,\ldots,\,2,000\}$ training samples using ReLU networks with $k\in\{2,\,4,\,\ldots,\,40\}$ hidden neurons. The constant step size of both plain-vanilla SGD and Algorithm 1 was set to $\eta=0.001$ ($\eta=0.01$) when the ReLU networks have $k\leq 4$ ($k>4$) hidden units, while the noise variance in Algorithm 1 was set to $\gamma=10$. Similar to the first experiment on randomly generated data, we plot success rates of the plain-vanilla SGD (top panel) and our noise-injected SGD (bottom panel) algorithms over training sets of MNIST images in Figure 4. It is self-evident that Algorithm 1 achieved a $100\%$ success rate under all testing conditions, which confirms our theoretical results in Theorem 1, and it markedly improves upon its plain-vanilla SGD alternative.

Conclusions

This paper approached the task of training ReLU networks from a non-convex optimization point of view. Focusing on the task of binary classification with a hinge loss criterion, this contribution put forth the first algorithm that can provably yet efficiently train any single-hidden-layer ReLU network to global optimality, provided that the data are linearly separable. The algorithm is as simple as plain-vanilla SGD, but it is able to exploit the power of random additive noise to break ‘optimality’ of the SGD learning process at any sub-optimal critical point. We established an upper and a lower bound on the number of non-zero updates that the novel algorithm requires for convergence to a global optimum. Our result holds regardless of the underlying data distribution, network/training size, or initialization. We further developed generalization error bounds for two-layer NN classifiers with ReLU activations, which provide the first theoretical guarantee for the generalization behavior of ReLU networks trained with SGD. A comparison of such bounds with those of a leaky ReLU network reveals a key difference between optimally learning a ReLU network versus that of a leaky ReLU network in the sample complexity required for generalization.

Since analysis, comparisons, and corroborating tests focus on single-hidden-layer networks with a hinge loss criterion here, our future work will naturally aim at generalizing the novel noise-injection design to SGD for multilayer ReLU networks, and considering alternative loss functions, and generalizations to (multi-)kernel based approaches.

References

Appendix A.1 Proof of Theorem 1

which is constructed from the optimum $\bm{\omega}^{\ast}$ of the linear classifier, where $v_{\min}:=\min_{1\leq j\leq k}|v_{j}|>0$. Using the definition of $\bm{\Omega}^{\ast}$ and (4), it holds that

For $i\in\mathcal{N}_{y}^{+}$, we clearly have $y=1$, and $\bm{\omega}^{\ast}\bm{x}_{i}>0$. In addition, for $j\in\mathcal{N}_{v}^{+}$, we find $v_{j}>0$ and hence ${\rm sgn}(v_{j})\,{\bm{\omega}^{\ast}}^{\top}\bm{x}_{i}>0$, which implies $\sigma({\rm sgn}(v_{j})\,{\bm{\omega}^{\ast}}^{\top}\bm{x}_{i})={\bm{\omega}^{\ast}}^{\top}\bm{x}_{i}$; similarly, for $j\in\mathcal{N}_{v}^{-}$, we find $v_{j}<0$, and thus ${\rm sgn}(v_{j})\,{\bm{\omega}^{\ast}}^{\top}\bm{x}_{i}<0$, which yields $\sigma({\rm sgn}(v_{j})\,{\bm{\omega}^{\ast}}^{\top}\bm{x}_{i})=0$. These considerations show that for $i\in\mathcal{N}_{y}^{+}$ in (16), only summands with $j\in\mathcal{N}_{v}^{+}$ survive; and arguing along the same lines, we deduce that for $i\in\mathcal{N}_{y}^{-}$, only summands with $j\in\mathcal{N}_{v}^{-}$ should be present. All in all, (16) reduces to

But since $y_{i}\,{\bm{\omega}^{\ast}}^{\top}\bm{x}_{i}\geq 1$, and $\sum_{v_{j}>0}v_{j}/v_{\min}\geq 1$ as well as $\sum_{v_{j}<0}v_{j}/v_{\min}\leq-1$, we infer that $L_{\mathcal{S}}(\bm{\Omega}^{\ast})=0$, and hence $\bm{\Omega}^{\ast}$ is indeed a global minimum of $L_{\mathcal{S}}(\bm{W})$.

The subsequent analysis builds critically on the following two functions (cf. (15))

Using the Cauchy-Schwartz inequality, we can write

We will next derive a lower and an upper bound for the numerator and denominator of (19). Consider an iteration $t$ for which Algorithm 1 admits a non-zero update, meaning that $\mathds{1}_{\{1-y_{i_{t}}\bm{v}^{\top}\sigma(\bm{W}^{t}\bm{x}_{i_{t}})>0\}}=1$, or equivalently, $y_{i_{t}}\bm{v}^{\top}\sigma(\bm{W}^{t}\bm{x}_{i_{t}})=\sum_{j=1}^{k}y_{i_{t}}v_{j}\sigma({\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}})<1$. It will also come handy to rewrite (10) row-wise as

Combining (20) with (18b), we can upper bound $\psi^{2}(\bm{W}^{t})$ in the denominator of (19) as

where $(a)$ follows directly from (20) after expanding the squares; $(b)$ uses the working condition $\|\bm{x}_{i_{t}}\|_{2}\leq 1$ adopted without loss of generality, as well as the fact that $\mathds{1}_{\{\cdot\}}\leq 1$ holds true for any event, and also the inequality $\sum_{j=1}^{k}y_{i_{t}}v_{j}{\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}}\mathds{1}_{\{{\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}}+\epsilon_{j}^{t}\geq 0\}})\leq\sum_{j=1}^{k}y_{i_{t}}v_{j}\sigma({\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}})$ established in Lemma 1 below, whose proof is postponed to Appendix Appendix B.2 for readability. Finally, $(c)$ is due to the non-zero update at iteration $t$, which implies that $\sum_{j=1}^{k}y_{i_{t}}v_{j}\sigma({\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}})<1$.

Writing down (21) for the already executed $t$ non-zero updates and by means of telescoping, we obtain

We now turn to deriving a lower bound for $\phi(\bm{W}^{t})$ in (19), starting with (cf. (18a) and (18b))

where $(a)$ is derived by plugging in (20); $(b)$ uses $v_{j}>0$ ($<0$) if $j\in\mathcal{N}_{v}^{+}$ ($\in\mathcal{N}_{v}^{-}$); and $(c)$ follows from the two critical inequalities: i) $y_{i}\bm{x}_{i}^{\top}\bm{\omega}^{\ast}\geq 1$ for all $(\bm{x}_{i},y_{i})\in\mathcal{S}$, and ii) $(|v_{j}|/v_{\min})\mathds{1}_{\{{\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}}+\epsilon_{j}^{t}\geq 0\}}\geq 1$, because a non-zero update at iteration $t$ asserts that at least one out of the $k$ ReLU activity indicator functions $\{\mathds{1}_{\{{\bm{w}_{j}^{t}}^{\top}\bm{x}_{i_{t}}+\epsilon_{j}^{t}\geq 0\}}\}_{j=1}^{k}$ equals one.

Again, telescoping the $t$ recursions (24) for the non-zero updates to $(t-1)$, yields

Substituting the bounds in (23) and (25) into (19), we have that

Using further that $\sqrt{p^{2}+q^{2}}\leq|p|+|q|$, we arrive at

Using that $[{\rm sgn}(v_{j})]^{2}=1$, it is easy to verify that

Under our assumption that all rows of $\bm{W}^{0}$ satisfy $\|\bm{w}_{j}^{0}\|_{2}\leq\rho$, we have for $\psi(\bm{W}^{0}):=\|\bm{W}^{0}\|_{F}$ that

Using (18a) along with (28) and (29), we find

Substituting the bounds in (28), (29), and (30) into (27) and re-arranging terms, we further arrive at

which upon letting $z:=\sqrt{t}\geq 0$, boils down to the quadratic inequality

where the coefficients are given by $a=\eta v_{\min}>0$, $b=-\|\bm{\omega}^{\ast}\|_{2}\sqrt{k(\eta^{2}\|\bm{v}\|_{2}^{2}+2\eta)}$, and $c=-2k\rho\|\bm{\omega}^{\ast}\|_{2}<0$. Because $c<0$ and $b^{2}-4ac>0$, we have real roots of opposite sign, which implies that (32) is satisfied for

Plugging in those coefficients and appealing again to the inequality $\sqrt{p^{2}+q^{2}}\leq|p|+|q|$, we deduce that

By taking $\rho=0$ in (34), one finally confirms that the maximum number of non-zero updates for Algorithm 1 initialized with $\bm{W}^{0}=\bm{0}$ until convergence, is

Appendix B.2 Proof of Lemma 1

We first prove that the following inequality holds per hidden neuron $j=1,\,2,\,\ldots,\,k$

Depending on whether the $j$-th ReLU is active or not ($\bm{w}_{j}^{\top}\bm{x}\gtreqless 0$) and (Gaussian) noise is injected or not ($v_{j}y\gtreqless 0$), we consider separately the following four cases:

$\bm{w}_{j}^{\top}\bm{x}\geq 0$ and $v_{j}y\geq 0$ (ReLU active and noise injected);

$\bm{w}_{j}^{\top}\bm{x}\geq 0$ and $v_{j}y<0$ (ReLU active and no noise);

$\bm{w}_{j}^{\top}\bm{x}<0$ and $v_{j}y\geq 0$ (ReLU inactive and noise injected); and,

$\bm{w}_{j}^{\top}\bm{x}<0$ and $v_{j}y<0$ (ReLU inactive and no noise).

For c1), the right-hand-side of (36) satisfies

Regarding the left-hand-side, it takes values depending on whether $\epsilon_{j}$ changes the state of the ReLU activity indicator function, it leads to a two-branch inequality

Combining (37) with (40), we deduce that $yv_{j}\bm{w}_{j}^{\top}\bm{x}\mathds{1}_{\{\bm{w}_{j}^{\top}\bm{x}+\epsilon_{j}\geq 0\}}\leq yv_{j}\sigma(\bm{w}_{j}^{\top}\bm{x})$ holds under c1).

For c2), the $j$-th ReLU is active too, but there is no noise injection, namely $\epsilon_{j}=0$. The right-hand-side of (37) still holds however. It is also not difficult to check the left-hand-side term $yv_{j}\bm{w}_{j}^{\top}\bm{x}\mathds{1}_{\{\bm{w}_{j}^{\top}\bm{x}+\epsilon_{j}\geq 0\}}=yv_{j}\bm{w}_{j}^{\top}\bm{x}$. Evidently, the desired inequality holds with equality in this case.

For c3), we have $\bm{w}_{j}^{\top}\bm{x}<0$, meaning that the $j$-th ReLU is inactive, and therefore, the right-hand-side of (36) becomes $yv_{j}\sigma(\bm{w}_{j}^{\top}\bm{x})=0$. However, given $v_{j}y\geq 0$, there is a noise injection. Hence, the left-hand-side can be similarly treated as in c2), to infer that (40) remains valid. Recalling again that $\bm{w}_{j}^{\top}\bm{x}<0$ and $yv_{j}\geq 0$, one deduces that $yv_{j}\bm{w}_{j}^{\top}\bm{x}\mathds{1}_{\{\bm{w}_{j}^{\top}\bm{x}+\epsilon_{j}\geq 0\}}\leq 0$, regardless of $\epsilon_{j}$. Thus, the inequality under consideration is also true under c2).

Finally, for c4), the $j$-th ReLU is inactive, and there is no noise injection. It is straightforward to verify that both the left-hand-side and right-hand-side equal zero, and (36) holds with equality as well.

Putting together c1)-c4), we have established that $yv_{j}\bm{w}_{j}^{\top}\bm{x}\mathds{1}_{\{\bm{w}_{j}^{\top}\bm{x}+\epsilon_{j}\geq 0\}}\leq yv_{j}\sigma(\bm{w}_{j}^{\top}\bm{x})$ for $j=1,\,2,\,\ldots,\,k$. Summing up such inequalities for all $k$ hidden neurons completes the proof.

Appendix C.3 Proof of Theorem 2

Consider the update in (10) initialized with $\bm{W}^{0}=\bm{0}$, and telescope each row to obtain

where $i_{t}$ deterministically cycles through $\{1,\,2,\,\ldots,\,d\}$.

At the global optimum, say $\bm{W}^{\tau}$ for some iteration number $\tau$, it holds for $(\bm{e}_{s},1)\in\mathcal{S}_{1}$ that

Since $|v_{j}|\sigma({\bm{w}_{j}^{\tau}}^{\top}\bm{e})\geq 0$ in (43), a necessary condition for $\bm{W}^{\tau}$ to be a global minimum is (cf. (42))

Assume for simplicity that $\tau-1$ is a multiple of $d$, namely $\tau-1=dp$ for some integer $p\geq 1$. On the other hand, we have from (Appendix C.3) that

where we have used the following inequalities: i) $\mathds{1}_{\left\{{\bm{w}_{j}^{t}}^{\top}\bm{e}_{i_{t}}+\epsilon_{j}^{t}\geq 0\right\}}\leq 1$, ii) $\sum_{i=1}^{\tau-1}\bm{e}_{i_{t}}=\sum_{i=1}^{p}\bm{1}$ with $\bm{1}$ being an all-one vector of suitable dimension that is clear from the context, and iii) $\sum_{j\in\mathcal{N}_{v}^{+}}v_{j}^{2}\leq\|\bm{v}\|_{2}^{2}$.

Combing the bounds in (Appendix C.3) and (45), we obtain that $p\eta\|\bm{v}\|_{2}^{2}\geq 1$, or equivalently $p\geq 1/(\eta\|\bm{v}\|_{2}^{2})$. Hence, to find a global optimum, Algorithm 1 initialized from $\bm{W}^{0}=\bm{0}$ makes at least

Appendix D.4 Proof of Proposition 1

We have also proved in Theorem 1 that Algorithm 1 performs at most $T_{k}$ non-zero updates regardless of $\gamma^{2}$ (cf. (34)). Hence, as long as the initialization $\bm{W}^{0}$ is bounded, all iterates $\bm{W}^{t}$ will be bounded; that is, there exists some constant $w_{\max}>0$ such that $\|\bm{w}_{j}^{t}\|_{2}\leq w_{\max}$ holds for all $j=1,\,2,\,\ldots,\,k$ and $t=0,\,1,\,\ldots,\,T_{k}$.

For notational brevity, we drop the iteration index $t$, and let the current iterate be denoted by $\bm{W}$. If there is no non-zero update with the current sampled data point $(\bm{x}_{i},y_{i})\in\mathcal{S}$, then one of the following two cases must be true: c1) $\mathds{1}_{\{1-y_{i}\bm{v}^{\top}\sigma(\bm{W}\bm{x}_{i})>0\}}=0$, or equivalently, $\max(0,1-y_{i}\bm{v}^{\top}\sigma(\bm{W}\bm{x}_{i}))=0$ implying that $(\bm{x}_{i},y_{i})$ is correctly classified; and, c2) $\mathds{1}_{\{\bm{W}\bm{x}_{i}+\bm{\epsilon}\geq\bm{0}\}}=\bm{0}$, or equivalently, $\epsilon_{j}<-\bm{w}_{j}^{\top}\bm{x}_{i}$ for all neurons $j=1,\,2,\,\ldots,\,k$.

When $i$ cycles through $\{1,\,2,\,\ldots,\,n\}$ in a deterministic manner (with each integer drawn exactly once every $n$ iterations), then within every succession of $np$ iterations, the noise-injected stochastic gradient term $\mathds{1}_{\{1-y_{i}\bm{v}^{\top}\sigma(\bm{W}\bm{x}_{i})>0\}}\cdot y_{i}\bm{v}\,\text{diag}(\mathds{1}_{\{\bm{W}\bm{x}_{i}+\bm{\epsilon}\geq\bm{0}\}})\,\bm{x}_{i}^{\top}$ in (10) will be evaluated exactly $p$ times at every datum $(\bm{x}_{i},y_{i})\in\mathcal{S}$, but with $p$ different random noise realizations. Hence, if there is no non-zero update within a succession of $np$ iterations, the probability of event c2) occurring $np$ times is at most

when there is only one out of the $n$ data points that is left incorrectly classified. Here, by a slight abuse of notation, we use $j\in\min\{\mathcal{N}_{v}^{+},\,\mathcal{N}_{v}^{-}\}$ to mean $j\in\mathcal{N}_{v}^{+}$ if $|\mathcal{N}_{v}^{+}|\leq|\mathcal{N}_{v}^{-}|$, and $j\in\mathcal{N}_{v}^{-}$ otherwise. Furthermore, to obtain the inequality in (48), we have used $\max_{1\leq i\leq n}(-\bm{w}_{j}^{\top}\bm{x}_{i})\leq\|\bm{w}_{j}\|_{2}\|\bm{x}_{i}\|_{2}\leq w_{\max}$ under our assumptions that $\|\bm{w}_{j}\|_{2}\leq w_{\max}$ and $\|\bm{x}_{i}\|_{2}\leq 1$ for all $j=1,\,2,\,\ldots,\,k$ and $i=1,\,2,\,\ldots,\,n$. Therefore, by the total probability theorem, the probability of having c1) hold for all data points is at least

which can be made arbitrarily close to $1$ by taking either a large enough $p$ and/or $\gamma>0$. The case of picking $i_{t}$ uniformly at random from $\{1,\,2,\,\ldots,\,n\}$ can be discussed in a similar fashion, but it is omitted here. This completes the proof.