Kernel and Rich Regimes in Overparametrized Models
 

A string of recent papers study neural networks trained with gradient descent in the “kernel regime.” They observe that, in a certain regime, networks trained with gradient descent behave as kernel methods (Jacot et al., 2018; Daniely et al., 2016; Daniely, 2017). This allows one to prove convergence to zero error solutions in overparametrized settings (Du et al., 2018, 2019; Allen-Zhu et al., 2018), and also implies gradient descent will converge to the minimum norm solution (in the corresponding RKHS) (Chizat and Bach, 2018; Arora et al., 2019b; Mei et al., 2019) and more generally that models inherit the inductive bias and generalization behavior of the RKHS. This suggests that, in a certain regime, deep models can be equivalently replaced by kernel methods with the “right” kernel, and deep learning boils down to a kernel method with a fixed kernel determined by the architecture and initialization, and thus it can only learn problems learnable by some kernel.

This contrasts with other recent results that show how in deep models, including infinitely overparametrized networks, training with gradient descent induces an inductive bias that cannot be represented as an RKHS norm. For example, analytic and/or empirical results suggest that gradient descent on deep linear convolutional networks implicitly biases toward minimizing the $L_p$ bridge penalty, for $p=2/\text{depth}\le 1$, in the frequency domain (Gunasekar et al., 2018b); weight decay on an infinite width single input ReLU implicitly biases towards minimizing the second order total variations $\int \left|f^{\prime \prime }\right(x\left)\right|dx$ of the learned function (Savarese et al., 2019); and gradient descent on a overparametrized matrix factorization, which can be thought of as a two layer linear network, induces nuclear norm minimization of the learned matrix (Gunasekar et al., 2017) and can ensure low rank matrix recovery (Li et al., 2018). All these natural inductive biases ($L_p$ bridge penalty for $p<1$, total variation norm, nuclear norm) are not Hilbert norms, and therefore cannot be captured by a kernel. This suggests that training deep models with gradient descent can behave very differently from kernel methods, and have much richer inductive biases.

One might then ask whether the kernel approximation indeed captures the behavior of deep learning in a relevant and interesting regime, or does the success of deep learning come when learning escapes this regime? In order to understand this, we must first carefully understand when each of these regimes hold, and how the transition between the “kernel” regime and the “rich” regime happens.

Some investigations of the kernel regime emphasized the number of parameters (“width”) going to infinity as leading to this regime. However Chizat and Bach (2018) identified the scale of the model as a quantity controlling entry into the kernel regime. Their results suggest that for any number of parameters (any width), a model can be approximated by a kernel when its scale at initialization goes to infinity (see details in Section 3). Considering models with increasing (or infinite) width, the relevant regime (kernel or rich) is determined by how the scaling at initialization behaves as the width goes to infinity. In this paper we elaborate and expand of this view, carefully studying how the scale of initialization effects the model behaviour for $D$-homogeneous models.

In Section 4 we provide a complete and detailed study for a simple 2-homogeneous model that can be viewed as linear regression with squared parametrization, or as a “diagonal” linear neural network. For this model we can exactly characterize the implicit bias of training with gradient descent, as a function of the scale $\alpha$ of initialization, and see how this implicit bias becomes the ${\ell }_2$ norm in the $\alpha \to \infty$ kernel regime, but the ${\ell }_1$ norm in the $\alpha \to 0$ rich regime. We can therefore understand how, e.g. for a high dimensional sparse regression problem, where it is necessary to discover the relevant features, we can get good generalization when the initialization scale $\alpha$ is small, but not when $\alpha$ is large. Deeper networks corresponds to higher orders of homogeneity, and so in Section 5 we extend our study to a $D$-homogeneous model, studying the effects of $D$. In Sections 6 and 7, we demonstrate similar transitions experimentally in matrix factorization and non-linear networks.

We consider models $f:R^p\times X\to R$ which map parameters $w\in R^p$ and examples $x\in X$ to predictions $f(w,x)\in R$. We denote the predictor implemented by the parameters $w$ as $h_w=F\left(w\right)$ such that $h_w\left(x\right)=f(w,x)$. Much of our focus will on models, such a linear networks, which are linear in $x$ (but not on the parameters $w$!), in which case $F\left(w\right)\in X^{\ast }$ is a linear predictor and can be represented as a vector ${\beta }_w$ with $f(w,x)=⟨{\beta }_w,x⟩$. Such models are essentially alternate parametrizations of linear models, but as we shall see that change of parametrization is crucial.

In this paper, we consider models that are $D$-positive homogeneous in the parameters $w$, for some integer $D\ge 1$, meaning that for any $c\in R_{+}$, $F(c\cdot w)=c^{D}F\left(w\right)$ and $f(c\cdot w,x)=c^{D}f(w,x)$. We refer to such models simply as $D$-homogeneous. Many interesting model classes have this property, including multi-layer ReLU networks with fully connected and convolutional layers, layered linear neural networks, and matrix factorization where $D$ corresponds to the depth of the network.

Consider a training set $\left\{\right(x^{\left(n\right)},y^{\left(n\right)}){\}}_{n=1}^N$ consisting of $N$ examples of input label pairs. For a given loss function $\ell :R\times R\to R$, the loss of the model parametrized by $w$ is $L\left(w\right)=L\left(F\right(w\left)\right)={\sum }_{n=1}^N\ell \left(f\right(w,x^{\left(n\right)}),y^{\left(n\right)})$. We will focus mostly on the squared loss ${\ell }_{\text{sq}}(\hat{y},y)=(\hat{y}-y{)}^2$. We slightly abuse notation and use $f(w,X)\in R^N$ to denote the vector of predictions $\left[f\right(w,x^{\left(1\right)}),\dots ,f(w,x^{\left(N\right)}\left)\right]$ and so for the squared loss we can write $L\left(w\right)={\parallel f(w,X)-y\parallel }_2^2$, where $y\in R^N$ is the vector of target labels.

Minimizing the loss $L\left(w\right)$ using gradient descent amounts to iteratively updating the parameters

We consider gradient descent with infinitesimally small stepsize $\eta$, i.e. gradient flow dynamics

We are particularly interested in the scale of initialization and we capture it through a scalar parameter $\alpha \in R_{+}$. For scale $\alpha$, we will denote by $w_{\alpha }\left(t\right)$ the gradient flow path (2) with the initial condition $w_{\alpha }\left(0\right)=\alpha w_0$ for some fixed $w_0$. We use $h_{\alpha }\left(t\right)=F\left(w_{\alpha }\right(t\left)\right)$, and for linear predictors ${\beta }_{\alpha }\left(t\right)={\beta }_{w_{\alpha }\left(t\right)}$, to denote the dynamics on the predictor $F\left(w\right)$ induced by the gradient flow on $w$.

In many cases, we expect the dynamics to converge to a minimizer of $L\left(w\right)$, though proving this happens will not be our main focus. Rather, we are interested in the underdetermined case, $N\ll p$, where there are generally many minimizers of $L\left(w\right)$, all with $f(w,X)=y$ and $L\left(w\right)=0$. Our main focus is which of the many minimizers does gradient flow converge to. That is, we want to characterize $w_{\alpha }\left(\infty \right)={lim}_{t\to \infty }{w}_{\alpha }\left(t\right)$ or, more importantly, the predictor $h_{\alpha }\left(\infty \right)=F\left(w_{\alpha }\right(\infty \left)\right)$ or ${\beta }_{\alpha }\left(\infty \right)={\beta }_{w_{\alpha }\left(\infty \right)}$ we converge to, and how these depend on the scale $\alpha$. In underdetermined problems, where there are many zero error solutions, simply fitting the data using the model does not provide enough inductive bias to ensure generalization. But in many cases, the specific solution reached by gradient flow (or some other optimization procedure) has special structure, or minimizes some implicit regularizer, and this structure or regularizer provides the needed inductive bias (Gunasekar et al., 2018b, a; Soudry et al., 2018; Ji and Telgarsky, 2018).

Gradient descent/flow considers only the first-order approximation of the model w.r.t. $w$:

That is, locally around any $w\left(t\right)$, gradient flow operates on the model as if it were an affine model $f(w,x)\approx f_0\left(x\right)+⟨w,{\varphi }_{w\left(t\right)}\left(x\right)⟩$ with feature map ${\varphi }_{w\left(t\right)}\left(x\right)={\nabla }_{w}f\left(w\right(t),x)$, corresponding to the tangent kernel $K_{w\left(t\right)}(x,x^{\prime })=⟨{\nabla }_{w}f\left(w\right(t),x),{\nabla }_{w}f\left(w\right(t),x)⟩$ (Jacot et al., 2018; Zou et al., 2018; Yang, 2019; Lee et al., 2019). Of particular interest is the tangent kernel at initialization, $K_{w_{\alpha }\left(0\right)}={\alpha }^{2(D-1)}{K}_0$ where we denote $K_0=K_{w_0}$.

The “kernel regime” refers to a situation in which the tangent kernel $K_{w\left(t\right)}$ does not change over the course of optimization, and less formally to the regime where it does not change significantly, i.e. where ${\forall }_t{K}_{w\left(t\right)}\approx K_{w\left(0\right)}$. In this regime, training the model is exactly equivalent to training an affine model with kernelized gradient descent/flow with the kernel ${\alpha }^{2(D-1)}{K}_0$ and a “bias term” of ${\alpha }^{D}f(w_0,x)$. To avoid handling this bias term, and in particular its scaling, Chizat and Bach (2018) suggest using “unbiased” initializations such that $F\left(w_0\right)=0$, so that the bias term vanishes. This can often be achieved by replicating units or components with opposite signs at initialization, which is the approach we use here (see Sections 4–6 for examples and details).

For underdetermined problem with multiple solutions $f(w,X)=y$, unbiased kernel gradient flow (or gradient descent) converges to the minimum norm solution ${\hat{h}}_K={argmin}_{h\left(X\right)=y}{\parallel h\parallel }_K$, where ${\parallel h\parallel }_K$ is the RKHS norm corresponding to the kernel. And so, in the kernel regime, we will have that $h\left(\infty \right)={\hat{h}}_{K_0}$, and the implicit bias of training is precisely given by the kernel.

When does the “kernel regime” happen? Chizat and Bach (2018) showed that for any homogeneous¹¹1Chizat and Bach did not consider only homogenous models, and instead of studying the scale of initialization they studied scaling the output of the model. For homogeneous models, the dynamics obtained by scaling the initialization are equivalent to those obtained by scaling the output, and so here we focus on homogenous models and on scaling the initialization. model satisfying some technical conditions, the kernel regime is reached as $\alpha \to \infty$. That is, as we increase the scale of initialization, the dynamics converge to the kernel gradient flow dynamics with the kernel $K_0$, and we have ${lim}_{\alpha \to \infty }{h}_{\alpha }\left(\infty \right)={\hat{h}}_K$. In Sections 4 and 5 we prove this limit directly for our specific models, and we also demonstrate it empirically for matrix factorization and deep networks in Sections 6 and 7.

In contrast, and as we shall see in later sections, the $\alpha \to 0$ small initialization limit often leads to a very different and rich inductive bias, e.g. inducing sparsity or low-rank structure (Gunasekar et al., 2017; Li et al., 2018; Gunasekar et al., 2018b), that allows for generalization in many settings where kernel methods would not. We refer to this limit reached as $\alpha \to 0$ as the “rich regime.” This regime is also referred to as the “active” or “adaptive” regime (Chizat and Bach, 2018) since the tangent kernel $K_{w\left(t\right)}$ changes over the course of training, in a sense adapting to the data. We argue that this regime is the one that truly allows us to exploit the power of depth, and thus is the more relevant regime for understanding the success of deep learning.

We study in detail a simple $2$-homogeneous model. Consider the class of linear functions over $X=R^d$, with squared parameterization as follows:

where we use the notation $z^2$ for $z\in R^d$ to denote elementwise squaring. We consider initializing all weights equally with $w_0=1$.

This is nothing but a linear regression model, except with an unconventional parametrization. The model can also be thought of as a “diagonal” linear neural network (i.e. where the weight matrices have diagonal structure) with $2d$ units. A standard diagonal linear network would have $d$ units, with each unit connected to just a single input unit with weights $u_i$ and the output with weight $v_i$, thus implementing the model $f\left(\right(u,v),x)={\sum }_i{u}_i{v}_i{x}_i$. But if at initialization $\left|u_i\right|=\left|v_i\right|$, their magnitude will remain equal and their signs will not flip throughout training, and so we can equivalently replace both with a single weight $w_i$, yielding the model $f(w,x)=⟨w^2,x⟩$.

The reason for using both $w_{+}$ and $w_{-}$ (or $2d$ units) is two-fold. First, it ensures that the image of $F\left(w\right)$ is all (signed) linear functions, and thus the model is truly equivalent to standard linear regression. Second, it allows for initialization at $F\left(\alpha w_0\right)=0$ without this being a saddle point from which gradient flow will never escape.²²2Our results can be generalized to non-uniform initialization, “biased initiliation” (i.e. where $w_{-}\ne w_{+}$ at initialization), or the asymmetric parameterization $f\left(\right(u,v),x)={\sum }_i{u}_i{v}_i{x}_i$, however this complicates the presentation without adding much insight.

The model (4) is perhaps the simplest non-trivial $D$-homogeneous model for $D>1$, and we chose it for this reason, as it already exhibits distinct and interesting kernel and rich regimes. Furthermore, we can completely understand both the implicit regularization driving this model and the transition between the regimes analytically.

Consider the behavior of the limit of gradient flow (2) as a function of the initialization, in the underdetermined $N\ll d$ case where there are many possible solutions $X\beta =y$. The tangent kernel at initialization is $K_0(x,x^{\prime })=8{\alpha }^2⟨x,x^{\prime }⟩$, i.e. a scaling of the standard inner product kernel, so ${\parallel \beta \parallel }_{K_0}\propto {\parallel \beta \parallel }_2$. Thus, in the kernel regime, gradient flow leads to the minimum ${\ell }_2$ norm solution, ${\beta }_{L2}^{\ast }\doteq {argmin}_{X\beta =y}{\parallel \beta \parallel }_2$. Following Chizat and Bach (2018) and the discussion in Section 3, we thus expect that ${lim}_{\alpha \to \infty }{\beta }_{\alpha }\left(\infty \right)={\beta }_{L2}^{\ast }$, and we also show this below.

In contrast, Gunasekar et al. (2017) shows that as $\alpha \to 0$, gradient flow leads instead to the minimum ${\ell }_1$ norm solution ${lim}_{\alpha \to 0}{\beta }_{\alpha }\left(\infty \right)={\beta }_{L1}^{\ast }\doteq {argmin}_{X\beta =y}{\parallel \beta \parallel }_1$. This is the “rich regime.” Comparing this with the kernel regime, we already see two very distinct behaviors and, in high dimensions, two very different inductive biases. In particular, the rich regime’s bias is not an RKHS norm for any choice of kernel. Can we charactarize and understand the transition between the two regimes as $\alpha$ transitions from very small to very large? The following theorem does just that.

In the previous Section, we considered a 2-homogeneous model, corresponding to a simple depth-2 “diagonal” network. Deeper models correspond to higher order homogeneity (a depth-$D$ ReLU or linear network is $D$-homogeneous), motivating us to understand the effect of the order of homogeneity on the transition between the regimes. We therefore generalize our model and consider:

We again consider initializing all weights equally so $w_0=1$. As before, this is just a linear regression model with an unconventional parametrization. It is equivalent to a depth-$D$ matrix factorization model with commutative measurement matrices, as studied by Arora et al. (2019a), and can be thought of as a depth-$D$ diagonal linear network.

We can again study the effect of the scale of the initialization $\alpha$ on the implicit bias. Let ${\beta }_{\alpha ,D}\left(\infty \right)$ denote the limit of gradient flow on $w$ when initialized at $\alpha 1$ for the $D$-homogeneous model. Using the same approach as in Section 4, in Appendix C we show:

In the two extremes we see that we again get the minimum ${\ell }_2$ solution in the kernel regime, and more interestingly, for any depth $D\ge 2$, we get the same minimum ${\ell }_1$ norm solution in the rich regime, as has also been observed by Arora et al. (2019a). The fact that the rich regime solution does not change with depth is perhaps surprising, and does not agree from what is obtained with explicit regularization (regularizing ${\parallel w\parallel }_2$ is equivalent to ${\parallel \beta \parallel }_{2/D}$ regularization), nor with implicit regularization on logistic-type loss Gunasekar et al. (2017).

Although the two extremes do not change as we go beyond $D=2$, what does change is the intermediate regime (the regularizer $Q_{\alpha }^D$ is unique and cannot be obtained with any other order $D^{\prime }\ne D$), as well as the sharpness of the transition into the extreme regimes, as illustrated in Figures 2(a)-2(c). The most striking difference is that for order $D>2$ the scale of $\alpha$ needed to approximate the ${\ell }_1$ is polynomial rather then exponential, yielding a much quicker transition to the “rich regime”, and allowing near-optimal sparse regression with reasonable initialization scales. Increasing $D$ further hastens the transition. This might also help explain some of the empirical observations about the benefit of depth in deep matrix factorization Arora et al. (2019a).

We now turn to a more complex depth two model, namely a matrix factorization model, and demonstrate similar transitions empirically. Specifically, we consider the model over matrix inputs $X\in R^{d\times d}$ defined by $f\left(\right(U,V),X)=⟨U{V}^{\top },X⟩$, where $U,V\in R^{d\times k}$. This corresponds to linear predictors over matrix arguments specified by $F(U,V)=U{V}^{\top }$. In overparameterized regime with $k\ge d$, the parameterization itself does not introduce any explicit rank constraints. We consider here a random low rank matrix completion problem where $X_n=e_{i_n}{e}_{j_n}^{\top }$ represents an uniform random observation of entry $(i_n,j_n)$ of a planted low rank matrix $M^{\ast }$ of rank $r^{\ast }\ll d$: $y_n=⟨M^{\ast },X_n⟩=M_{ij}^{\ast }$. For underdetermined problems where $N\ll d^2$, there are many trivial global minimizers, most of which are not low rank and hence will not guarantee recovery. As was demonstrated empirically by Gunasekar et al. (2017) and also proven rigorously for Gaussian measurements by Li et al. (2018), as $\alpha \to 0$ gradient flow implicitly regularizes the nuclear norm, which for random measurements leads to recovery of ground truth (Candès and Recht, 2009; Recht et al., 2010): these are very different and rich implicit biases that are not RKHS norms.

Crucially, the reconstruction results in Gunasekar et al. (2017) and Li et al. (2018) are dependent on initialization with scale $\alpha \to 0$. Here we further explore the role of initialization. Similar to Section 4, in order to get unbiased $0$ initialization, we consider $k\ge 2d$ and initialization of the form $U\left(0\right)=\alpha [U_0,-U_0]$ and $V\left(0\right)=\alpha [V_0,V_0]$, where $U_0,V_0\in R^{d\times k/2}$. We will study implicit bias of gradient flow over the factorized parameterization with above initialization.

For matrix completion problems with $X=e_{i_X}{e}_{j_X}^{\top }$, the tangent kernel at initialization is given by $K_0(X,X^{\prime })\propto ⟨U_0[i_X,:],U_0[i_{X^{\prime }},:]⟩1(j_X=j_{X^{\prime }})+⟨V_0[j_X,:],V_0[j_{X^{\prime }},:]⟩1(i_X=i_{X^{\prime }})$. This defaults to the trivial delta kernel $K(X,X^{\prime })=1(i_X=i_{X^{\prime }})\cdot 1(j_X=j_{X^{\prime }})$ for the two special cases (a) $U_0,V_0$ have orthogonal columns (e.g. $U_0=V_0=I$), or (b) $U_0,V_0$ have independent Gaussian entries and $k\to \infty$. In these cases, minimizing the RKHS norm of the tangent kernel corresponds to returning a zero imputed matrix (minimum Frobenius norm solution). Figure 4 demonstrates the behaviour of gradient flow updates in the “rich" regime (where for $\alpha \to 0$ recovers the ground truth) and in the “kernel" regime (where for large $\alpha$, there are no updates to the unobserved entries).

In the preceding sections, we intentionally focused on the simplest possible models in which a kernel-to-rich transition can be observed, in order to isolate this phenomena and understand it in detail. In those simple models, we were able to obtain a complete analytic description of the transition. Obtaining such a precise description in more complex models is somewhat optimistic at this point, as we do not yet have a satisfying description of even just the rich regime. Instead, we now provide empirical evidence suggesting that also for non-linear and realistic networks, the scale of initialization induces a transition into and out of a “kernel” regime, and that to reach good generalization we must operate outside of the “kernel” regime. To track whether we are in the “kernel” regime, we track how much the gradient ${\nabla }_{w}f\left(w\right(t),x)$ changes throughout training. In particular, we define the gradient distance to be the cosine distance between the initial tangent kernel feature map ${\nabla }_{w}f\left(w\right(0),x)$ and the final tangent kernel feature map ${\nabla }_{w}f\left(w\right(T),x)$.

In Figures 4(a) and 4(b), we see that also for a non-linear ReLU network, we remain in the kernel regime when the initialization is large, and that exiting from the kernel regime is necessary in order to achieve small test error on the synthetic data. Interestingly, when ${\alpha }^D\approx 1$, the models achieve good test error but have smaller gradient distance which, not coincidentally, corresponds to using the out-of-the-box Uniform He initialization. This lies on the boundary between the rich and kernel regimes, which is desirable due to the learning vs. optimization tradeoffs discussed in Section 4. On MNIST data, Figure 4(e) shows that previously published successes with training overly wide depth-2 ReLU networks without explicit regularization (e.g. Neyshabur et al., 2014) relies on the initialization being small, i.e. being outside of the “kernel regime”. In fact, the 2.4% test error reached for large initialization is no better than what can be achieved with a linear model over a random feature map. Turning to a more realistic network, 4(f) shows similar behavior when training a VGG11-like network on CIFAR10.

So far, we attempted to use the best fixed stepsize for each initialization (i.e. achieving the best test error). But as demonstrated in Figures 4(c) and 4(d), the stepsize choice can also have a significant effect, with larger stepsizes allowing one to exit the kernel regime even at an initialization scale where a smaller stepsize would remain trapped in the kernel regime. Further analytic and empirical studies are necessary in order to understand the joint behavior of the stepsize and initialization scale.

The main point of this paper is to emphasize the distinction between the “kernel” regime in training overparametrized multi-layered networks, and the “rich” (active, adaptive) regime, show how the scaling of the initialization can transition between them, and understand this transition in detail. We argue that rich inductive bias that enables generalization may arise in the rich regime, but that focusing on the kernel regime restricts us to only what can be done with an RKHS. By studying the transition we also see a tension between generalization and optimization, which suggests we would tend to operate just on the edge of the rich regime, and so understanding this transition, rather then just the extremes, is important. Furthermore, we see that at the edge of the rich regime, the implicit bias of gradient descent differs substantively from that of explicit regularization. Although in our theoretical study we focused on a simple model so that we can carry out a complete and exact analysis analytically, our experiments show that this is representative of the behaviour also in other homogeneous models, and serve as a basis of a more general understanding.