Gradient descent GAN optimization is locally stable
Vaishnavh Nagarajan, J. Zico Kolter
Introduction
Since their introduction a few years ago, Generative Adversarial Networks (GANs) (Goodfellow et al., 2014) have gained prominence as one of the most widely used methods for training deep generative models. GANs have been successfully deployed for tasks such as photo super-resolution, object generation, video prediction, language modeling, vocal synthesis, and semi-supervised learning, amongst many others (Ledig et al., 2017; Wu et al., 2016; Mathieu et al., 2016; Nguyen et al., 2017; Denton et al., 2015; Im et al., 2016).
At the core of the GAN methodology is the idea of jointly training two networks: a generator network, meant to produce samples from some distribution (that ideally will mimic examples from the data distribution), and a discriminator network, which attempts to differentiate between samples from the data distribution and the ones produced by the generator. This problem is typically written as a min-max optimization problem of the following form:
For the purposes of this paper, we will shortly consider a more general form of the optimization problem, which also includes the recent Wasserstein GAN (WGAN) (Arjovsky et al., 2017) formulation.
Despite their prominence, the actual task of optimizing GANs remains a challenging problem, both from a theoretical and a practical standpoint. Although the original GAN paper included some analysis on the convergence properties of the approach (Goodfellow et al., 2014), it assumed that updates occurred in pure function space, allowed arbitrarily powerful generator and discriminator networks, and modeled the resulting optimization objective as a convex-concave game, therefore yielding well-defined global convergence properties. Furthermore, this analysis assumed that the discriminator network is fully optimized between generator updates, an assumption that does not mirror the practice of GAN optimization. Indeed, in practice, there exist a number of well-documented failure modes for GANs such as mode collapse or vanishing gradient problems.
In this paper, we consider the “gradient descent” formulation of GAN optimization, the setting where both the generator and the discriminator are updated simultaneously via simple (stochastic) gradient updates; that is, there are no inner and outer optimization loops, and neither the generator nor the discriminator are assumed to be optimized to convergence. Despite the fact that, as we show, this does not correspond to a convex-concave optimization problem (even for simple linear generator and discriminator representations), we show that:
Under suitable conditions on the representational powers of the discriminator and the generator, the resulting GAN dynamical system is locally exponentially stable.
That is, for some region around an equilibrium point of the updates, the gradient updates will converge to this equilibrium point at an exponential rate. Interestingly, our conditions can be satisfied by the traditional GAN but not by the WGAN, and we indeed show that WGANs can have non-convergent limit cycles in the gradient descent case.
Our theoretical analysis also suggests a natural method for regularizing GAN updates by adding an additional regularization term on the norm of the discriminator gradient. We show that the addition of this term leads to locally exponentially stable equilibria for all classes of GANs, including WGANs. The additional penalty is highly related to (but also notably different from) recent proposals for practical GAN optimization, such as the unrolled GAN (Metz et al., 2017) and the improved Wasserstein GAN training (Gulrajani et al., 2017). In practice, the approach is simple to implement, and preliminary experiments show that it helps avert mode collapse and leads to faster convergence.
Background and related work
Although the theoretical analysis of GANs has been far outpaced by their practical application, there have been some notable results in recent years, in addition to the aforementioned work in the original GAN paper. For the most part, this work is entirely complementary to our own, and studies a very different set of questions. Arjovsky and Bottou (2017) provide important insights into instability that arises when the supports of the generated distribution and the true distribution are disjoint. In contrast, in this paper we delve into an equally important question of whether the updates are stable even when the generator is in fact very close to the true distribution (and we answer in the affirmative). Arora et al. (2017), on the other hand, explore questions relating to the sample complexity and expressivity of the GAN architecture and their relation to the existence of an equilibrium point. However, it is still unknown as to whether, given that an equilibrium exists, the GAN update procedure will converge locally.
From a more practical standpoint, there have been a number of papers that address the topic of optimization in GANs. Several methods have been proposed that introduce new objectives or architectures for improving the (practical and theoretical) stability of GAN optimization (Arjovsky et al., 2017; Poole et al., 2016). A wide variety of optimization heuristics and architectures have also been proposed to address challenges such as mode collapse (Salimans et al., 2016; Metz et al., 2017; Che et al., 2017; Radford et al., 2016). Our own proposed regularization term falls under this same category, and hopefully provides some context for understanding some of these methods. Specifically, our regularization term (motivated by stability analysis) captures a degree of “foresight” of the generator in the optimization procedure, similar to the unrolled GANs procedure (Metz et al., 2017). Indeed, we show that our gradient penalty is closely related to -unrolled GANs, but also provides more flexibility in leveraging this foresight. Finally, gradient-based regularization has been explored for GANs, with one of the most recent works being that of Gulrajani et al. (2017), though their penalty is on the discriminator rather than the generator as in our case.
Finally, there are several works that have simultaneously addressed similar issues as this paper. Of particular similarity to the methodology we propose here are the works by Roth et al. (2017) and Mescheder et al. (2017). The first of these two present a stabilizing regularizer that is based on a gradient norm, where the gradient is calculated with respect to the datapoints. Our regularizer on the other hand is based on the norm of a gradient calculated with respect to the parameters. Our approach has some strong similarities with that of the second work noted above; however, the authors there do not establish or disprove stability, and instead note the presence of zero eigenvalues (which we will treat in some depth) as a motivation for their alternative optimization method. Thus, we feel the works as a whole are quite complementary, and signify the growing interest in GAN optimization issues.
Thus, to understand stability of the stochastic approximation algorithm, it suffices to understand the stability and convergence of the deterministic differential equation. Though such analysis is typically used to show global asymptotic convergence of the stochastic approximation algorithm to an equilibrium point (assuming the related ODE also is globally asymptotically stable), it can also be used to analyze the local asymptotic stability properties of the stochastic approximation algorithm around equilibrium points.Note that the local analysis does not show that the stochastic approximation algorithm will necessarily converge to an equilibrium point, but still provides a valuable characterization of how the algorithm will behave around these points. This is the technique we follow throughout this entire work, though for brevity we will focus entirely on the analysis of the continuous time ordinary differential equation, and appeal to these standard results to imply similar properties regarding the discrete updates.
Given the above consideration, our focus will be on proving stability of the dynamical system around equilbrium points, i.e. points for which .Note that this is a slightly different usage of the term equilibrium as typically used in the GAN literature, where it refers to a Nash equilibrium of the min max optimization problem. These two definitions (assuming we mean just a local Nash equilibrium) are equivalent for the ODE corresponding to the min-max game, but we use the dynamical systems meaning throughout this paper, that is, any point where the gradient update is zero. Specifically, we appeal to the well known linearization theorem (Khalil, 1996, Sec 4.3), which states that if the Jacobian of the dynamical system evaluated at an equilibrium point is Hurwitz (has all strictly negative eigenvalues, ), then the ODE will converge to for some non-empty region around , at an exponential rate. This means that the system is locally asymptotically stable, or more precisely, locally exponentially stable (see Definition A.1 in Appendix A).
Thus, an important contribution of this paper is a proof of this seemingly simple fact: under some conditions, the Jacobian of the dynamical system given by the GAN update is a Hurwitz matrix at an equilibrium (or, if there are zero-eigenvalues, if they correspond to a subspace of equilibria, the system is still asymptotically stable). While this is a trivial property to show for convex-concave games, the fact that the GAN is not convex-concave leads to a substantially more challenging analysis.
In addition to this, we provide an analysis that is based on Lyapunov’s stability theorem (described in Appendix A). The crux of the idea is that to prove convergence it is sufficient to identify a non-negative “energy” function for the linearized system which always decreases with time (specifically, the energy function will be a distance from the equilibrium, or from the subspace of equilibria). Most importantly, this analysis provides insights into the dynamics that lead to GAN convergence.
GAN optimization dynamics
This section comprises the main results of this paper, showing that under proper conditions the gradient descent updates for GANs (that is, updating both the generator and discriminator locally and simultaneously), is locally exponentially stable around “good” equilibrium points (where “good” will be defined shortly). This requires that the GAN loss be strictly concave, which is not the case for WGANs, and we indeed show that the updates for WGANs can cycle indefinitely. This leads us to propose a simple regularization term that is able to guarantee exponential stability for any concave GAN loss, including the WGAN, rather than requiring strict concavity.
For the remainder of the paper, we consider a slightly more general formulation of the GAN optimization problem than the one presented earlier, given by the following min/max problem:
Assuming the generator and discriminator networks to be parameterized by some set of parameters, and respectively, we analyze the simple stochastic gradient descent approach to solving this optimization problem. That is, we take simultaneous gradient steps in both and , which in our “ODE method” analysis leads to the following differential equation:
2 Why is proving stability hard for GANs?
Before presenting our main results, we first highlight why understanding the local stability of GANs is non-trivial, even when the generator and discriminator have simple forms. As stated above, GAN optimization consists of a min-max game, and gradient descent algorithms will converge if the game is convex-concave – the objective must be convex in the term being minimized and concave in the term being maximized. Indeed, this was a crucial assumption in the convergence proof in the original GAN paper. However, for virtually any parameterization of the real GAN generator and discriminator, even if both representations are linear, the GAN objective will not be a convex-concave game:
The GAN objective in Equation 2 can be a concave-concave objective i.e., concave with respect to both the discriminator and generator parameters, for a large part of the discriminator space, including regions arbitrarily close to the equilibrium.
To see why, consider a simple GAN over 1 dimensional data and latent space with linear generator and discriminator, i.e. and . Then the GAN objective is:
Because is concave, by inspection we can see that is concave in and ; but it is also concave (not convex) in and , for the same reason. Thus, the optimization involves concave minimization, which in general is a difficult problem. To prove that this is not a peculiarity of the above linear discriminator system, in Appendix B, we show similar observations for a more general parametrization, and also for the case where (which happens in the case of WGANs).
However, as we will show, gradient descent GAN optimization is locally asymptotically stable, even for natural parameterizations of generator-discriminator pairs (which still make up concave-concave optimization problems). Furthermore, at equilibrium, although the zero-discriminator property means that the generator is not stable “independently”, the joint dynamical system of generator and discriminator is locally asymptotically stable around certain equilibrium points.
3 Local stability of general GAN systems
This section contains our first technical result, establishing that GANs are locally stable under proper local conditions. Although the proofs are deferred to the appendix, the elements that we do emphasize here are the conditions that we identified for local stability to hold. Indeed, because the proof rests on these conditions (some of which are fairly strong), we want to highlight them as much as possible, as they themselves also convey valuable intuition as to what is required for GAN convergence.
To formalize our conditions, we denote the support of a distribution with probability density function (p.d.f) by and the p.d.f of the generator by . Let denote the Euclidean -ball of radius of . Let and denote the largest and the smallest non-zero eigenvalues of a non-zero positive semidefinite matrix. Let and denote the column space and null space of a matrix respectively. Finally, we define two key matrices that will be integral to our analyses:
Here, the matrices are evaluated at an equilibrium point which we will characterize shortly. The significance of these terms is that, as we will see, is proportional to the Hessian of the GAN objective with respect to the discriminator parameters at equilibrium, and is proportional to the off-diagonal term in this Hessian, corresponding to the discriminator and generator parameters. These matrices also occur in similar positions in the Jacobian of the system at equilibrium.
We now discuss conditions under which we can guarantee exponential stability. All our conditions are imposed on both and all equilibria in a small neighborhood around it, though we do not state this explicitly in every assumption. First, we define the “good” equilibria we care about as those that correspond to a generator which matches the true distribution and a discriminator that is identically zero on the support of this distribution. As described next, implicitly, this also assumes that the discriminator and generator representations are powerful enough to guarantee that there are no “bad” equilibria in a local neighborhood of this equilibrium.
and , .
The assumption that the generator matches the true distribution is a rather strong assumption, as it limits us to the “realizable” case, where the generator is capable of creating the underlying data distribution. Furthermore, this means the discriminator is (locally) powerful enough that for any other generator distribution it is not at equilibrium (i.e., discriminator updates are non-zero). Since we do not typically expect this to be the case, we also provide an alternative non-realizable assumption below that is also sufficient for our results i.e., the system is still stable. In both the realizable and non-realizable cases the requirement of an all-zero discriminator remains. This implicitly requires even the generator representation be (locally) rich enough so that when the discriminator is not identically zero, the generator is not at equilibrium (i.e., generator updates are non-zero). Finally, note that these conditions do not disallow bad equilibria outside of this neighborhood, which may potentially even be unstable.
Assumption I. (Non-realizable) The discriminator is linear in its parameters and furthermore, for any equilibrium point , , .
Our goal next is to identify strong curvature conditions that can be imposed on the objective (or a function related to the objective), though only locally at equilibrium. First, we will require that the objective is strongly concave in the discriminator parameter space at equilibrium (note that it is concave by default). However, on the other hand, we cannot ask the objective to be strongly convex in the generator parameter space as we saw that the objective is not convex-concave even in the nicest scenario, even arbitrarily close to equilbrium. Instead, we identify another convex function, namely the magnitude of the update on the equilibrium discriminator i.e., , and require that to be strongly convex in the generator space at equilibrium. Since these strong curvature assumptions will allow only systems with a locally unique equilibrium, we will state them in a relaxed form that accommodates a local subspace of equilibria. Furthermore, we will state these assumptions in two parts, first as a condition on , second as a condition on the parameter space.
First, the condition on is straightforward, making it necessary that the loss be concave at ; as we will show, when this condition is not met, there need not be local asymptotic convergence.
The function satisfies , and
Next, to state conditions on the parameter space while also allowing systems with multiple equilibria locally, we first define the following property for a function, say , at a specific point in its domain: along any direction either the second derivative of must be non-zero or all derivatives must be zero. For example, at the origin, is flat along , and along any other direction at an angle with the axis, the second derivative is . For the GAN system, we will require this property, formalized in Property I, for two convex functions whose Hessians are proportional to and . We provide more intuition for these functions below.
Here is an intuitive explanation of what these two non-negative functions represent and how they relate to the objective. The first function is a function of which measures how far is from an all-zero state, and the second is a function of which measures how far is from the true distribution; at equilibrium these functions are zero. We will see later that given , the curvature of the first function at is representative of the curvature of in the discriminator space; similarly, given the curvature of the second function at is representative of the curvature of the magnitude of the discriminator update on in the generator space. The intuition behind why this particular relation holds is that, when moves away from the true distribution, while the second function in Assumption III increases, also becomes more suboptimal for that generator; as a result, the magnitude of update on increases too. Note that we show in Lemma C.2, that the Hessian of the two functions in Assumption III in the discriminator and the generator space respectively, are proportional to and .
The above relations involving the two functions and the GAN objective, together with Assumption III, basically allow us to consider systems with reasonable strong curvature properties, while also allowing many equilibria in a local neighborhood in a specific sense. In particular, if the curvature of the first function is flat along a direction (which also means that ) we can perturb slightly along and still have an ‘equilibrium discriminator’ as defined in Assumption I i.e., , . Similarly, for any direction along which the curvature of the second function is flat (i.e., ), we can perturb slightly along that direction such that remains an ‘equilibrium generator’ as defined in Assumption I i.e., . We prove this formally in Lemma C.2. Perturbations along any other directions do not yield equilibria because then, either is no longer in an all-zero state or does not match the true distribution. Thus, we consider a setup where the rank deficiencies of , if any, correspond to equivalent equilibria (which typically exist for neural networks, though in practice they may not correspond to ‘linear’ perturbations as modeled here).
Our final assumption is on the supports of the true and generated distributions: we require that all the generators in a sufficiently small neighborhood of the equilibrium have distributions with the same support as the true distribution. Following this, we briefly discuss a relaxation of this assumption.
such that , .
This may typically hold if the support covers the whole space ; but when the true distribution has support in some smaller disjoint parts of the space , nearby generators may correspond to slightly displaced versions of this distribution with a different support. For the latter scenario, we show in Appendix C.1 that local exponential stability holds under a certain smoothness condition on the discriminator. Specifically, we require that be zero not only on the support of but also on the support of small perturbations of as otherwise the generator will not be at equilibrium. (Additionally, we also require this property from the discriminators that lie within a small perturbation of in the null space of so that they correspond to equilibrium discriminators.) We note that while this relaxed assumption accounts for a larger class of examples, it is still strong in that it also restricts us from certain simple systems. Due to space constraints, we state and discuss the implications of this assumption in greater detail in Appendix C.1.
The dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 3 is locally exponentially stable with respect to an equilibrium point when the Assumptions I, II, III, IV hold for and other equilibria in a small neighborhood around it. Furthermore, the rate of convergence is governed only by the eigenvalues of the Jacobian of the system at equilibrium with a strict negative real part upper bounded as:
If , then
If , then
The vast majority of our proofs are deferred to the appendix, but we briefly describe the intuition here. It is straightforward to show that the Jacobian of the system at equilibrium can be written as:
Recall that we wish to show this is Hurwitz. First note that (the Hessian of the objective with respect to the discriminator) is negative semi-definite if and only if . Next, a crucial observation is that i.e, the Hessian term w.r.t. the generator vanishes because for the all-zero discriminator, all generators result in the same objective value. Fortunately, this means at equilibrium we do not have non-convexity in precluding local stability. Then, we make use of the crucial Lemma G.2 we prove in the appendix, showing that any matrix of the form is Hurwitz provided that is strictly negative definite and has full column rank.
However, this property holds only when is positive definite and is full column rank. Now, if or do not have this property, recall that the rank deficiency is due to a subspace of equilibria around . Consequently, we can analyze the stability of the system projected to an subspace orthogonal to these equilibria (Theorem A.4). Additionally, we also prove stability using Lyapunov’s stability (Theorem A.1) by showing that the squared distance to the subspace of equilibria always either decreases or only instantaneously remains constant.
In order to illustrate our assumptions in Theorem 3.1, in Appendix D we consider a simple GAN that learns a multi-dimensional Gaussian using a quadratic discriminator and a linear generator. In a similar set up, in Appendix E, we consider the case where i.e., the Wasserstein GAN and so , and we show that the system can perennially cycle around an equilibrium point without converging. A simple two-dimensional example is visualized in Section 4. Thus, gradient descent WGAN optimization is not necessarily asymptotically stable.
4 Stabilizing optimization via gradient-based regularization
Motivated by the considerations above, in this section we propose a regularization penalty for the generator update, which uses a term based upon the gradient of the discriminator. Crucially, the regularization term does not change the parameter values at the equilibrium point, and at the same time enhances the local stability of the optimization procedure, both in theory and practice. Although these update equations do require that we differentiate with respect to a function of another gradient term, such “double backprop” terms (see e.g., Drucker and Le Cun (1992)) are easily computed by modern automatic differentiation tools. Specifically, we propose the regularized update
The intuition of this regularizer is perhaps most easily understood by considering how it changes the Jacobian at equilibrium (though there are other means of motivating the update as well, discussed further in Appendix F.2). In the Jacobian of the new update, although there are now non-antisymmetric diagonal blocks, the block diagonal terms are now negative definite:
As we show below in Theorem 3.2 (proved in Appendix F), as long as we choose small enough so that , this guarantees the updates are locally asymptotically stable for any concave . In addition to stability properties, this regularization term also addresses a well known failure state in GANs called mode collapse, by lending more “foresight” to the generator. The way our updates provide this foresight is very similar to the unrolled updates proposed in Metz et al. (2017), although, our regularization is much simpler and provides more flexibility to leverage the foresight. In practice, we see that our method can be as powerful as the more complex and slower 10-unrolled GANs. We discuss this and other intuitive ways of motivating our regularizer in Appendix F.
The dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 4, is locally exponentially stable at the equilibrium, under the same conditions as in Theorem 3.1, if . Further, under appropriate conditions similar to these, the WGAN system is locally exponentially stable at the equilibrium for any . The rate of convergence for the WGAN is governed only by the eigenvalues of the Jacobian at equilibrium with a strict negative real part upper bounded as:
If , then
If , then
Experimental results
We very briefly present experimental results that demonstrate that our regularization term also has substantial practical promise.We provide an implementation of this technique at https://github.com/locuslab/gradient_regularized_gan In Figure 1, we compare our gradient regularization to -unrolled GANs on the same architecture and dataset (a mixture of eight gaussians) as in Metz et al. (2017). Our system quickly spreads out all the points instead of first exploring only a few modes and then redistributing its mass over all the modes gradually. Note that the conventional GAN updates are known to enter mode collapse for this setup. We see similar results (see Figure 2 here, and Figure 4 in the Appendix for a more detailed figure) in the case of a stacked MNIST dataset using a DCGAN (Radford et al., 2016) i.e., three random digits from MNIST are stacked together so as to create a distribution over 1000 modes. Finally, Figure 3, presents streamline plots for a 2D system where both the true and the latent distribution is uniform over $D(x)=w_{2}x^{2}G(z)=az$. Observe that while the WGAN system goes in orbits as expected, the original GAN system converges. With our updates, both these systems converge quickly to the true equilibrium.
Conclusion
In this paper, we presented a theoretical analysis of the local asymptotic stability of GAN optimization under proper conditions. We further showed that the recently proposed WGAN is not asymptotically stable under the same conditions, but we introduced a gradient-based regularizer which stabilizes both traditional GANs and the WGANs, and can improve convergence speed in practice.
The results here provide substantial insight into the nature of GAN optimization, perhaps even offering some clues as to why these methods have worked so well despite not being convex-concave. However, we also emphasize that there are substantial limitations to the analysis, and directions for future work. Perhaps most notably, the analysis here only provides an understanding of what happens locally, close to an equilibrium point. For non-convex architectures this may be all that is possible, but it seems plausible that much stronger global convergence results could hold for simple settings like the linear quadratic GAN (indeed, as the streamline plots show, we observe this in practice for simple domains). Second, the analysis here does not show the equilibrium points necessarily exist, but only illustrates convergence if there do exist points that satisfy certain criteria: the existence question has been addressed by previous work (Arora et al., 2017), but much more analysis remains to be done here. GANs are rapidly becoming a cornerstone of deep learning methods, and the theoretical and practical understanding of these methods will prove crucial in moving the field forward.
We thank Lars Mescheder for pointing out a missing condition in the relaxed version of Assumption IV (see Appendix C.1) in earlier versions of this manuscript.
References
Appendix
Appendix A Preliminaries
In this section, we present preliminaries from non-linear systems theory [Khalil, 1996]. In particular, we formally define local stability of dynamic systems, and then present an important theorem that helps us study stability of non-linear systems. Finally, we present a modification of this result that will be crucial in proving stability of GANs under our assumptions.
Without loss of generality let the origin be an equilibrium point of this sytem. That is, . Let denote the state of the system at some time . Then, we have the following definition of local stability:
(Definition 4.1 from Khalil ) The origin of the system in Equation 5 is
stable if for each , there is such that
asymptotically stable if it is stable and can be chosen such that
exponentially stable if it is asymptotically stable and can be chosen such that
The system is stable if for any chosen ball around the equilibrium (of radius ), one can initialize the system anywhere within a sufficiently small ball around the equilibrium (of radius ) such that the system always stays within the ball. Note that such a system may either converge to equilibrium or orbit around equilibrium perennially within the ball. In contrast, a system is unstable if there are initializations that are arbitrarily close to the equilibrium which can escape the -ball. Finally, asymptotic stability is a stronger notion of stability, which implies that there is a region around the equilibrium such that any initialization within that region will converge to the equilibrium (in the limit ). For example, as we saw, GANs are always stable; however, WGANs are stable but not asymptotically stable.
Note that since a GAN system might have multiple arbitrarily close equilibria, or a subspace of equilibria, we will define asymptotic stability to imply convergence to any of the equilibria in the neighborhood of a considered equilibrium. That is, where is either the considered equilibrium point at the origin or any other equilibrium point that is within some small neighborhood around origin.
We now present Lyapunov’s stability theorem which is used to prove locally asymptotic stability of a given system. The basic idea is that a system is asymptotically stable if we can find a scalar “energy” function (also called a Lyapunov function) that i) is positive definite which means, positive everywhere and zero at the equilibrium ii) its time derivative is strictly negative around the equilibrium.
it is positive definite i.e., and for
for
then the origin is asymptotically stable.
We next present an important tool that simplifies the study of stability of non-linear systems. The result is that one can “linearize” any non-linear system near an equilibrium and analyze the stability of the linearized system to comment on the local stability of the original system.
(Theorem 4.5 from Khalil ) Let be the Jacobian of the system in Equation 5 at its origin i.e.,
The origin is locally exponentially stable if is Hurwitz i.e., for all eigenvalues of .
The origin is unstable if for all eigenvalues of .
The key idea in the proof for this result is that the system can be written as , where , the remainder of the linear approximation is bounded as sufficiently close to equilibrium. Now, it turns out that when is Hurwitz, one can find a quadratic Lyapunov function for the original system whose rate of decrease is also quadratic in . Since, is only a quadratic remainder term, one can show that the remainder term only adds a cubic term to the change in the Lyapunov function. This is however smaller than a quadratic change near the equilibrium, and therefore the quadratic Lyapunov function for the linearized system works as a Lyapunov function for the original system too.
In all our analyses, we will linearize our system and show that the Jacobian is Hurwitz. However, it is often useful to identify the quadratic Lyapunov function for the (linearized) system. Unfortunately, for some of the Jacobians we will encounter, it is hard to come up with a quadratic Lyapunov function that always strictly decreases. Instead, we will identify a function that either strictly decreases or sometimes remains constant but only instantenously. While Lyapunov’s stability theorem does not help us conclude anything about stability for this case, the following corollary of LaSalle’s theorem (we do not state the theorem here) is sufficient to prove asymptotic stability in this case.
if and only if and for such that .
for
Let . There is no trajectory that identically stays in except for the trajectories at equilibrium points.
then the system is locally asymptotically stable with respect to and other equilibria in its neighborhood.
Finally, we prove an extension of the linearization theorem that helps us deal with analyzing the stability of a special kind of non-linear systems, specifically those with multiple equilibria in a local neighborhood of a considered equilibrium. The theorem, though inuitively follows from the original linearization theorem itself, is not a standard theorem in non-linear systems, to the best of our knowledge.
Formally, we consider a case where the system consists of two sets of parameters and such that from the equilibrium, any small perturbation along preserves the equilibrium. We show that it is enough to show that the Jacobian with respect to is Hurwitz to prove stability.
Consider a non-linear system of parameters ,
with an equilibrium point at the origin. Let there exist such that for any , is an equilibrium point. Then, if
is a Hurwitz matrix, the non-linear system in Equation 6 is exponentially stable.
The proof for this statement is quite similar to the proof of the original theorem for linearization. The high level idea is that if is exponentially stable, then there exists a quadratic Lyapunov function that is always decreasing for the system . Then, we show that the same quadratic function works for the original non-linear system too in a small neighborhood around equilibrium for which the non-linear remainder terms are sufficiently small. In particular, we show that converges to zero, and converges to a value less than .
A subtle point however, is that this quadratic function would decrease only when it is within a particular neighborhood of around origin, and also a particular neighborhood of around origin. However, within this neighborhood, say , we can only guarantee that exponentially approaches the origin; might move away from the -neighborhood around origin, and if it does, the system may exit and the system may not even converge! We carefully overcome this, by first identifying , and then identifying a smaller space within where does not vary too much over the course of convergence, so that the system stays within forever – until convergence.
The first crucial step is to show that for any constant , for a sufficiently small neighborhood around the equilibrium, we will have . To show this, consider the Taylor series expansion for the remainder around equilibrium. Clearly, the expansion would not have a constant term because . It would not have a linear term in because that is accounted for already. Finally, it will not have any term that is purely a function of , because in a small neighborhood around equilibrium (since are all equilibria). Therefore, we can write:
where only consists of linear or higher degree terms in and consists only of terms that are quadratic or higher degree terms in (and any arbitrary degree of ). Therefore, we have that:
Then, for an arbitrarily chosen small constant , for a sufficiently close neighborhood around the equilibrium, we can say that and . Thus,
We will use this property soon for a cleverly chosen value of . Now, by Theorem 4.6 in Khalil , we have that for any positive definite symmetric matrix , there exists a positive definite matrix such that . Then, if we choose as the quadratic Lyapunov function for the linearized system , the rate of its decrease is given by which is negative at all points except at .
Now, if we use the same Lyapunov function for the whole system as , the rate of its decrease near the origin would be . If we choose a sufficiently small neighborhood such that for , , then we have that,
Now, as long as we ensure that the trajectory of the system remains in the neighborhood around origin for which and , this system would then exponentially converge to one of the equilibria near origin. Let us call this neighborhood i.e., within this neighborhood of and , the Lyapunov function strictly decreases for the non-linear system.
This brings us to the second crucial part of this proof, which is to ensure that we always stay in . Let contain a ball of radius . We will show that for sufficiently close initializations which are within a ball of radius , the displacement of is at most . Since only approaches origin, this means that the system never exited .
To bound how much changes with time, let us consider the Taylor series expansion of . First of all, there is no constant term. Next, there is no term that is purely a function of because . Then, we can say that:
Since is finite, in a small neighborhood around equilibrium, there exists a fixed constant such that . Then, .
Now, if the trajectory indeed always remained in , we know that for some constant . Assume we initialize within a radius of . The rate at which changes at any point is,
Then, the maximum displacement in can be,
Thus, the trajectory always lies in , which implies exponential convergence along to a point where and . Thus the system exponentially converges to an equilibrium.
Appendix B GANs are not concave-convex near equilibrium
In this section, we consider a more general system than the one considered in the main paper to demonstrate that GANs are not concave-convex near equilibrium. In particular, consider the following discriminator and generator pair learning a distribution in 1-D:
where and . Let the distribution to be learned be arbitrary. Let the latent distribution be the standard normal. Then, the gradient of the objective with respect to the generator parameters is:
Now, consider the case where . For points in the discriminator parameter space where but for all , the term above simplifies to the following when :
which is clearly negative i.e., the objective is concave in most of the generator parameters, and this holds for parameters arbitrarily close to the all-zero discriminator parameter (as ).
If for all (which is true in the case of WGANs), then in the region the above term is negative i.e., the GAN objective is concave in terms of the generator parameters.
Appendix C Local exponential stability of GANs
In this section, we provide the full proof for our result about the local stability of GANs through the following lemmas. First, we derive the Jacobian at equilibrium.
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 3, the Jacobian at an equilibrium point , under the Assumptions I and IV is:
Throughout this paper we will use the notation to denote the row vector corresponding to the gradient that is being computed. Now, let be the number of discriminator parameters and the number of generator parameters. Then the first block in , which we will denote by is:
The subsequent matrix, which we will denote by is:
It is easy to see that the lower matrix is :
Furthermore, the lower matrix turns out to be zero. Here, we will use an implication of Assumption IV. More specifically, generators that are within a sufficiently small radius around the equilibrium have the same support and therefore i) for in this support. Furthermore for all generators within a radius , any perturbation of the generator is not going to change the support, and therefore ii) for that is not in this support. We can consider only perturbations and not perturbations because for that is away from , perturbing it a little further might potentially change its support as a result of which may not necessarily be zero for all
Now, to show that is zero, we take any vector that is a perturbation in the generator space and show that . Here, we will use the limit definition of the derivative along a particular direction .
To prove that the system is stable we will need to show that this matrix is Hurwitz. We show later in Lemma G.2 that when i) and furthermore ii) is full column rank, then is indeed Hurwitz. However from , we only have that . For these two conditions to be met, we will need and to be full rank, which you may recall from our discussion in the main paper below Assumption III, is met only when there is a unique equilibrim locally.
Now, we show why this is the case – by establishing a relation between the matrices and and the curvature of functions in Assumption III – and further show how the null spaces of these matrices correspond to a subspace of equilibria. Then, we show in Lemma C.3, how to consider a rotation of the system and project to a space that is orthogonal to this subspace of equilibria. Then from the Theorem A.4 that we have proved in Appendix A, it is sufficient to show that the Jacobian of the projected system is Hurwitz.
In the following discussion, we will use the term “equilibrium discriminator” to denote a discriminator that is identically zero on the support and “equilibrium generator” to denote a generator that matches the true distribution, as defined in Assumption I. Note that for an equilibrium discriminator, the generator updates are zero and vice versa for an equilibrium generator.
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 3, under Assumptions I and III, there exists such that for all and , and for any unit vectors , is an equilibrium point as defined in Assumption I.
In other words, is an equilibrium discriminator which when paired with any generator results in zero updates on the generator.
In summary, for all slight perturbations along we have established that the discriminator and generator individually satisfy the requirements of an equilibrium discriminator and generator pair, and therefore the system is itself in equilibrium for these perturbations. ∎
Now, we show how to rotate and project the system to get a Hurwitz Jacobian matrix.
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 3, consider the eigenvalue decompositions and . Let and such that and . Consider the projections, and . Then, the block in the Jacobian at equilibrium that corresponds to the projected system has the form:
Under Assumption II, we have that and is full column rank.
Note that the columns of and correspond to eigenvectors, and furthermore, the rows of and are the eigenvectors that correspond to zero eigenvalues. These eigenvectors correspond to a local subspace of equilibria and the above lemma considers a projection of the system to a space orthogonal to this subspace.
We first address a corner case where either or (the eigenvectors with non-zero eigenvalues) is empty. In the case that is empty, it means that all discriminators in a neighborhood of the considered equilibrium are identically zero on the support of the true distribution (as proved in Lemma C.2). Then, for any generator, the discriminator update would be zero (because moving the discriminator in any direction locally does not result in a change in the objective). At the same time, the generator update would be zero too because these are all equilibrium discriminators. This means that the considered point is surrounded by a neighborhood of equilibria. Then, the system is trivially exponentially stable since any sufficiently close initialization is already at equilibrium.
Similarly when is empty it means that all generators in a small neighborhood have the same distribution, namely the true underlying distribution (as proved in Lemma C.2). Then, the generator update for any discriminator would be zero (changing the generator slightly in any direction does not change the generated distribution, and hence the objective). Furthermore, since these are equilibrium generators, the discriminator updates would be zero too, for any discriminator. Thus, again we are situated in a neighborhood of equilibria and the system is trivially exponentially stable.
Now we handle the general case. First note that, the Jacobian block of the projected variables must be
where is the Jacobian of the original system which we derived in Lemma C.1. Now note that, which is a diagonal matrix with only the positive eigenvalues. Therefore, since , .
Next, in a similar manner we can show that , which is a diagonal matrix with only positive eigenvalues. Thus, is full column rank. The non-trivial step here is to show that the matrix which has fewer rows is full column rank too. This will follow if we showed that for any such that , too. That is, the left null space of is a subset of the left null space of and therefore projecting to the row span of does not hurt the row rank of .
To see why this is true, observe that from Lemma C.2 for any small perturbation along such a , since we are always at an equilibrium discriminator i.e., for in the true support, it must be that . Furthermore, recall from our derivation of the Jacobian that for outside of this support. Then,
Therefore, since , this means is full column rank.
The main theorem then follows from the above lemmas.
We have from Lemma C.2 that the considered equilibrium point lies in a subspace of equilibria in a small neighborhood. Then, we have from Lemma C.3 that the Jacobian block corresponding to the subspace orthogonal to this, satsifies properties from Lemma G.2 which make it Hurwitz. We can then conclude exponential stability of the system from Theorem A.4. The eigenvalue bounds presented in the theorem follow from Lemma G.2. ∎
Finally, we show that we can indeed find a Lyapunov function that satisfies LaSalle’s principle for the projected linearized system.
For the linearized projected system with the Jacobian , we have that is a Lyapunov function such that for all non-equilbrium points, it either always decreases or only instantaneously remains constant.
Note that the Lyapunov function is zero only at the equilibrium of the projected system. Furthermore, it is straightforward to verify that the rate at which this changes is given by . Observe that the generator terms have canceled out. Clearly this is zero only when because is positive definite; otherwise it is strictly negative. Now, when this rate is indeed zero, we have that because the other term in the update which is proportional to is zero. Now, again, this term is zero only when because is full column rank. Thus, when we are not at equilibrium which means , the update on the discriminator parameters is nonzero i.e., . In other words, it does not identically stay in the manifold on which the energy does not decrease. ∎
In this section, we will relax Assumption IV and prove stability under certain conditions. Specifically, recall that originally we required the equilibrium generator to share the same support with any perturbation of the generator. Now, we will allow the generator to have different supports when perturbed, and instead impose conditions on the discriminator.
Our first condition is that the equilibrium discriminator must be zero not only on the support of but also on the supports of small perturbations of . If this were not true, may not be at equilibrium as the slope of the discriminator function may be non-zero at the boundaries of in , thus potentially encouraging the generator to push data points away from the true support.
To motivate our second condition, recall from Assumption III, we have that there could be directions along which we can perturb , while ensuring that the discriminator still outputs zero on . The intention behind allowing this was that these directions could allow other equivalent equilibrium discriminators in the neighborhood of . However, under the relaxation of Assumption IV that we are now aiming for, these perturbations will correspond to equilibrium discriminators only if they satsify the above condition i.e., that they are zero on the support of perturbations of too. We need to explicitly assume that this holds as we describe below. Thanks to Lars Mescheder for identifying that such a condition was missing in earlier versions of this paper.
Formally, we can state these assumptions as follows:
Assumption IV (Relaxed) such that for all :
for all , .
We now show that if these conditions hold, local exponentially stability holds too.
Most of the original proof holds as it is because all we needed was that the equilibrium discriminator be identically zero on the true support. We will prove only parts of the proof that required more than just this.
In this case, we only have a weaker guarantee that for a generator within a perturbation of from , the support is contained in the combined support . But then, i) for all in the combined support we have that and ii) for all not in the combined support and for any generator , . Then, the generator updates are:
The second part of Lemma C.2 holds similarly.
We need to make a similar argument to prove that the generator’s Hessian at equilibrium.
A similar modification of the proof can be done for Lemma C.3 where we show that has the same column rank as . The rest of the proof follows as it did. ∎
C.2 The non-realizable case
In this section, we extend our results about local stability of GANs to the case in which the true distribution can not be represented by any generator in the generator space. While this is a hard problem in general, we consider a specific case in which the discriminator is linear in its parameters and show that the system is locally stable at any equilibrium and its surrounding equilibria (none of which may correspond to the true distribution). More formally, consider a discriminator of the form:
where is any feature mapping. For example, could be a polynomial basis or the representation learned by a neural network (which we assume is not trained during the updates near equilibrium). Thus, the objective in this case is:
We consider a generator space that does not necessarily contain the true distribution, but however contains a generator that is an equilibrium point when paired with a discriminator that is zero on the support of the true data and the generated data. It must be noted that is not necessarily the only equilibrium discriminator. Especially, if lies in a lower dimensional manifold, there could be a subspace of all-zero discriminators. Now, for such a generator to exist, we need:
In other words, we want the means of the generated distribution and the true distribution in the representation to be identical. For a given generator space, this essentially is a restriction on the representation that has been learned/chosen for the discriminator. If was a richer representation that computes many higher order moments of the data, we may never find an equilibrium generator.
We now prove Theorem 3.1 for the non-realizable case. Our main idea is identical to that of the proof in the realizable case. However, we need to be careful in a number of steps. We first prove a result similar to Lemma C.1 that derives the Jacobian of the system at equilibrium.
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 3, the Jacobian at an equilibrium point , under the Assumptions I (for the non-realizable case) and IV is:
First we show that has a similar form which is still negative semi-definite when :
The most crucial step here is that we were able to ignore the terms corresponding to because the discriminator is linear in its parameters i.e., and thus the Hessian is zero.
All other terms in the Jacobian are identical to the realizable case because we assume that at equilibrium the discriminator must be identically zero.
Now, we again show that the equilibrium point in consideration lies in a subspace of equilibria.
Under Assumptions I (Non-realizable), III, and IV there exists such that for all and , and for any unit vectors , is an equilibrium point.
is independent of the discriminator variables (Here, we have used the fact that the discriminator is linear in its parameters.) . This means that for these generators along , the discriminator update must be zero. In other words, these generators are equilibrium generators in the non-realizable sense, that their representation matches with the true distribution.
In summary, for all slight perturbations along we have established that the discriminator and generator individually satisfy the requirements of an equilibrium discriminator and generator pair, and therefore the system is itself is in equilibrium for these perturbations. ∎
It turns out that given these two lemmas, Lemma C.3 follows as it did earlier, and therefore the main theorem follows too.
Appendix D Linear Quadratic GAN – Gaussian example
In order to illustrate our assumptions in Theorem 3.1, consider a simple GAN that learns an -dimensional Gaussian distribution , where . Let the latent variable be drawn from the standard normal, . Consider a quadratic discriminator , and a linear generator . We call the resulting system LQ (linear-quadratic). Let be the unique real positive definite matrix such that . Then we have the following:
In LQ, and corresponds to an equilibrium that is locally exponentially stable provided and .
Since the system consists of parameters arranged in the form of matrices, we will need vectorization calculus [Magnus et al., 1995] to arrange these parameters as a vector and differentiate them/with respect to them.
To verify that the given point is indeed an equilibrium, let us look at the GAN objective:
The updates in Equation 3 for LQ can be written as :
Clearly, when , we have that , therefore at and , all the above updates become zero, implying that it is an equilibrium for which the generator has converged to the true distribution. To prove that it is locally stable, we need to examine the Jacobian at that point. Note that since the Jacobian is a matrix with one cell for each pair of discriminator-generator parameters, we need to calculate second-order derivatives after vectorizing the parameter matrices and .
We first calculate the derivative of the discriminator updates with respect to the discriminator itself.
Recall that the Jacobian can then be written as:
We can show that is negative definite because it is a moment matrix with a negative multiplicative factor. This is proved in Theorem D.2. Recall that as long as , and is full column rank (in this case full rank because is a square matrix), the matrix has eigenvalues whose real components are strictly negative.
To show that is full column rank, first observe that the last few columns corresponding to are linearly independent because, if belongs to its null space, then
which implies that .
To verify whether the first few columns corresponding to are linearly independent or not, consider any . Then, we want to verify whether the following term is always non-zero or not:
which is equivalent to testing whether is non-zero.
Now, we will show that if , then . Recall that . Then,
Observe that the left hand side is positive semi-definite while the right hand side is negative semi-definite. Therefore these terms must be equal to zero, which would then imply that i.e., . Thus the Jacobian is indeed Hurwitz.
We now prove that is negative definite.
Let be any arbitrary matrix and be an arbitrary vector. Then,
Now, forms a quadric -dimensional hypersurface in dimensions, and therefore is of measure zero. For all other points, and therefore the above expectation is strictly positive.
Appendix E WGANs are not necessarily asymptotically stable
We consider a specific case of the LQ WGAN that learns a zero mean gaussian distribution, and show that there exists points near certain equilibria such that if the system is initialized to that point, it will periodically come back to that initial point rather than converge to the equilibrium.
The LQ WGAN system for learning a zero mean Gaussian distribution () is not asymptotically stable at the equilibrium corresponding to and .
In order to show that the system is not asymptotically stable, we show that there are initializations of the system that are arbitrarily close to the equilibrium such that the system goes orbits around the equilibrium forever. For simplicity, we first prove this for the one-dimensional gaussian and later extend it to the multi-dimensional case. Let the quadratic discriminator be and the linear generator be . Then the WGAN objective in Equation 2 for the LQ system is:
The updates in Equation 3 for LQ simplify as follows:
The system has two equilibria, . We will assume that the system is initialized with , which means that the system will forever have because the respective updates are zero too. Hence, we only need to focus on the variables and .
Now, it can be shown that if is initialized to , never becomes negative (and similarly for ). Therefore, we will focus on the equilibrium where , and assuming examine how the distance from the equilibrium changes with time. The rate of change of this quantity is given by . Observe that when , this term is non-negative i.e., the system never gets closer to the equilibrium. Thus, when the system is in the “bad” half-space , the only hope for it to converge is to exit this half-space so that becomes negative. However, we show that there exists initializations that are close to the equilibrium such that even if it does exit the bad half-space it eventually re-enters it, going in a perpetual loop.
More specifically, let denote the system at time . Let the initialization satisfy and . We will now analyze the trajectory of this system. First note that , which means the system enters the bad half-space after immediately . Thus, if the system had to converge to the considered equilibrium, it would have to reach again at some time . First observe that at this time because we need at this time. (In fact we can say that because we know that the radius never decreased until time .) Now, we claim that the system simply retraces back its path along and reaches at time . More clearly, we claim that the system at time can be described in terms of what it was at time as .
To prove this observe that this statement is true for because . Then we only need to show that at any , if , then and . This is indeed true because and . Therefore, applying , we get i.e., the system has looped back to its original state by following its old path mirrored across the line . Since this holds for initializations that are arbitrarily close to the equilibrium (i.e., can be arbitrarily close to ), the system is not asymptotically stable.
We extend this argument to the higher dimensional case as follows. Again, we initialize the system so that and , then we can only focus on the updates on and :
As before, we initialize . We will also consider a more sophisticated initialization compared to . Since is positive definite, let . We initialize such that has at least one diagonal element that is positive but strictly less than the corresponding diagonal element in (where ).
Now, we first establish that all the updates and the variables in the system remain in the eigenspace defined by . That is, at any point in time , the variables can be expressed as and for some real diagonal matrices and . Clearly, this is true for time . Assuming this is true for arbitrary time , observe that the updates are
Thus this is true for any time . Therefore, we can analyze the system in terms of and the constant as though there are independent 1-dimensional Gaussian systems. Then, the orbiting systems from the 1-dimensional updates must manifest here too. More specifically, these cycles would correspond to the diagonal in which was initialized to be less than . ∎
Appendix F Gradient-based regularization
In Section F.1, we prove how our gradient-based regularizer stabilizes the both the GAN and the WGAN system. Besides this property, in Section F.2 we provide an alternative mathematical intuition that is based on arg-max differentiation, to motivate our regularization term. Finally, in Section F.3, we discuss how our regularizer addresses mode collapse and 1-unrolled GAN updates.
We first restate our main result below. See 3.2
To prove this result, we first present the Jacobian of the system at equilibrium in the presence of the gradient penalty. Recall that the penalty basically adds an extra to the generator’s update.
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 4, the Jacobian at an equilibrium point , under the Assumptions I and IV is:
where and are terms in the Jacobian corresponding to the original updates, as described in Theorem 3.1.
Note that the only change to the Jacobian would be in the rows corresponding to the generator parameters. Therefore, we will focus only on the additional terms in these rows.
The additional term added to is:
Now, the additional term added to is:
Now, we will prove stability of the regularized system for conventional GANs. Observe that Lemmas C.2 regarding the subspace of equilibria holds in this case too. Again, we can project the system as follows:
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 4, consider the eigenvalue decompositions and . Let and such that and . Consider the projections, and . Then, the block in the Jacobian at equilibrium that corresponds to the projected system has the form:
Under Assumption II, we have that and is full column rank and .
It is straightforward to extend the proof of Lemma C.3 to prove this lemma. Now, recall from Theorem A.4 that if we show is Hurwitz the original system is exponentially stable. In the non-regularized system, we showed this by making use of the structure of the matrix. For this system, we will design a quadratic Lyapunov function that strictly decreases at non-equilibria points.
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 4, if the linearization of the system projected to a subspace orthogonal to the subspace of equilibria is exponentially stable with the Lyapunov function where,
and is . The function strictly decreases with time except at the equilibrium .
Note that when , therefore the Lyapunov function is indeed positive definite. Furthermore, note that the rate of decrease is given by where . To show that this is strictly decreasing, we only need to show that . First of all, note that
Here, the off-diagonal terms are and its negative. This can be equated to zero because,
Then, the above matrix is equal to the diagonal matrix:
Note that by our choice of , . Therefore, and share the same set of eigenvectors. Thus, the null space of and the term are the same, specifically orthogonal to . In other words, the top-left block above is a diagonal matrix with strictly negative eigenvalues. Similarly, we also know that is a diagonal matrix with negative values. Hence, the above matrix is negative definite. ∎
We now proceed to the Wasserstain GAN scenario. First we lay down equivalent assumptions for the WGAN under which we can guarantee exponential stability in the regularized case. Note that even under these conditions, the unregularized update does not ensure asymptotic stability.
First, we note that due to the linearity of the loss function, it is not necessary that the discriminator be only identically zero on the support for the system to be at equilibrium — it could also be constant on the support. Thus, we relax Assumption I for this case to accommodate this.
Note that in effect, we get rid of the assumption on the other function and introduce a different function here; in either case, the original system is not asymptotically stable due to zero diagonal blocks in its Jacobian. Next, we retain Assumption IV as it is. These are the only three assumptions we will need.
We will now begin with a lemma similar to Lemma C.2
For the dynamical system defined by the WGAN objective in Equation 2 and the updates in Equation 4, under Assumptions I and III under the WGAN case, there exists such that for all and , and for any unit vectors , is an equilibrium point.
Note that is the Hessian of the function at equilibrium, namely the magnitude of the generator update. Then, by Assumption III, this function is locally constant along any unit vector . That is, for sufficiently small , if , the function value is equal to the value at equilibrium which is zero, because by definition at equilibrium the generator update is zero. Now at , the discriminator update is zero too since the generator matches the true distribution. Then by Assumption I, it means that is identical over the true support. Then, it is an equilibrium discriminator such that the update for any generator would be zero.
In summary, for all slight perturbations along we have established that the discriminator and generator individually satisfy the requirements of an equilibrium discriminator and generator pair, and therefore the system is itself is in equilibrium for these perturbations. ∎
Now, we show that this system can again be projected to a subspace orthogonal the equilibrium subspace such that the resulting Jacobian of the reduced system is Hurwitz. While earlier we chose based on the matrix now we will choose it based on .
For the dynamical system defined by the GAN objective in Equation 2 and the updates in Equation 4, consider the eigenvalue decompositions and . Let and such that and . Consider the projections, and . Then, the block in the Jacobian at equilibrium that corresponds to the projected system has the form:
Furthermore and is full column rank.
Observe that the form of follows from Lemma F.1 by substituting . Furthermore, like we have seen before, observe that is a diagonal matrix with positive eigenvalues and therefore . Similarly, is a full column rank matrix because we have projected it to the subspace orthogonal to its null space. However, we need to show that which may have fewer rows, did not reduce in its rank. This is indeed true, since this is effectively a projection onto the subspace orthogonal to its left null space. ∎
We now compile the above lemmas to prove our main result in Theorem 3.2.
The first part of the theorem statement for the conventional GAN follows from Lemma F.1, F.2, F.3.
To prove the second part it is sufficient to show that the projected Jacobian of the linearized system in Lemma F.5 is Hurwitz, from which exponential stability of the original system follows from Theorem A.4. The fact that this is Hurwitz follows as usual from Lemma G.2 after we flip the discriminator and generator variables:
The Jacobian is thus Hurwitz because is negative definite and is full column rank. Now, for the eigenvalue bounds we have from Lemma G.2 that:
If , then
If , then
However, this can be further simplified to arrive at the given bound by noting that all the non-zero eigenvalues of any matrix is also equal to the non-zero eigenvalues of the matrix . Therefore, we can replace every occurrence of with in the above inequality.
Additionally, we show that we can find a Lyapunov function that satisfies LaSalle’s principle for the projected linearized system.
For the linearized projected system with the Jacobian , we have that is a Lyapunov function such that for all non-equilbrium points, it either always decreases or only instantaneously remains constant.
Note that the Lyapunov function is zero only at the equilibrium of the projected system. Furthermore, it is straightforward to verify that the rate at which this changes is given by which is non-positive. Clearly this is zero only when because is positive definite. When this rate is indeed zero, we have that for the linearized system, because the other term becomes zero. For the system to identically stay on the manifold we need , which happens only when because is full column rank. When that is the case, we are at equilibrium.
F.2 Intuition based on arg-max differentiation
Observe that the last term is zero because, for the optimal discriminator . However, in practice, we would not be at the optimal discriminator and therefore this term may be non-zero. Our hypothesis is that, instead of ignoring this term like it is done for the conventional updates, retaining this term may prove to be useful. To do so, we simply plug in the current discriminator for while computing this term, and furthermore, estimate the value of using the following equation:
In the second step above, we apply the chain rule. Rearranging, we get:
Since we hope the objective to be concave in the discriminator parameters, we can approximate the Hessian as . Plugging this into the update equation of and also replacing the optimal discriminator with the current discriminator, we get the following update rule which is equivalent to the original one presented in Equation 4:
F.3 Mode Collapse and Relation to 1-unrolled updates
Our regularization term also has natural and intuitive connections to an important issue that arises in GAN optimization called mode collapse. Mode collapse is a situation where a GAN may enter an irrecoverable failure state where the generator incorrectly assigns all its probability mass to a small region in space. This arises because a globally optimal strategy for the generator is to push all its mass towards the single point that the discriminator is the most confident about being a real data point. To overcome this the generator needs more “foresight” – it must know that when it collapses all the mass, the discriminator will subsequently label the collapsed point as fake data. Our penalty indeed encodes this foresight, because the discriminator’s ability to outdo the generator is quantified by the magnitude of the discriminator’s gradient. More clearly, our generator seeks a state where it can spread data out enough, to make sure the discriminator has no obvious countermeasure (i.e., no big gradients).
In fact, we can show how our penalty term and 1-unrolled GANs have very similar structure because intuitively both provide a one-step lookahead to the generator. More precisely, we can arrive at 1-unrolled updates if we simplify our updates further and replace by an “unrolled” .
We begin by simplifying the 1-unrolled updates. The key idea of a 1-unrolled update is to allow the generator to explicitly foresee how the discriminator would react to its update, and optimize accordingly:
In the first step, we compute gradient with respect to as the sum of the gradients with respect to the two instances of that occur in , the first one that occurs as the second argument to , and the second one that occurs in the unrolled update of the first argument. In the second step, we apply the chain rule on the second gradient.
We can compare our updates in Equation 10 with the above to show how our updates are more flexible in terms of using the lookahead. While both have two similar terms, a crucial difference is that in the latter, every occurrence of the discriminator parameters (except one) has an additional unrolled update, namely . Clearly, this should provide more power to the latter; however in practice, we observe that our technique can be more powerful than -unrolled or even -unrolled updates (which are in fact much slower to run). The reason is that the unrolled updates constrain to be small, typically of the order which is the step size. It would not be possible to increase to greater magnitudes as it would be equivalent to a coarse step size in the unrolling. Our method on the other hand, allows for larger because the discriminator is retained as it is; in some sense, our penalty provides a way of extracting and leveraging the unrolled update more flexibly.
Appendix G Eigenvalue bounds
In this section, we prove one of the most useful lemmas that we used in our proofs, that matrices of the form are Hurwitz when and is full column rank. We also prove eigenvalue bounds for such a matrix. To do so, we begin with a simple fact:
For be a real symmetric matrix. If , then .
Let be the eigenvalue decomposition of . Let . Then, or in other words, . Similarly, which differs from by a multiplicative factor within . ∎
where is a symmetric real positive definite matrix and is a full column rank matrix. Then, for every eigenvalue of . In fact,
We consider a generic eigenvector equation and equate the real and complex parts together so as to arrive at our bounds. Consider the following eigenvector equation:
where are all real-valued. We assume that the vector is normalized i.e., . So, in case , we assume that . We want to show that . Let us first rewrite the above equation as follows:
We can then equate the real and imaginary parts.
We now multiply the above equations by respectively and add them:
As a result, only square terms and terms remain:
Now observe that because . If , it would immediately imply that .
However this may not be true, and that would happen only when and because . We will show that this case would not occur. First of all, this would force to ensure the above equality. By applying the Equations 12 and 14, we can conclude that and . Since one of this implies that too. Now, by applying Equation 11 and 13, we have that and . Since one of (if they were both zero, our eigenvector would itself be zero), this implies that is not a full column rank matrix, which is a contradiction of our assumption. Therefore, it cannot be the case that both and .
Now, we prove our bounds on . (Note that an easy lower bound follows as but we are interested in an upper bound). In order to prove the upper bound, we multiply Equations 11 and Equations 13 by and respectively and sum them up, and Equations 12 and Equations 14 by and respectively and sum them up.
From the above we have that either or . Now, if , since , we immediately get a bound .
In the former case, since the imaginary part of the eigenvalue is zero i.e., , the imaginary part of the eigenvector must be zero too i.e., . Then, we have the equations:
Rearranging and squaring the first equation we get:
In the first step, we make use of the fact that . In the second step, we use . In the third step, we use Lemma G.1 i.e., . We also use the fact that since , .
How do we upper bound using this inequality?
Let us examine the quadratic in in the above expression. Since the discriminant of this quadratic is , the quadratic always takes the same sign, specifically positive. Next, note that the quadratic reaches its minimum at .
Now, can either satisfy or . Since the former is already an upper bound, we will derive an upper bound in the latter case. Now, in the interval , the quadratic in increases, and therefore for this interval, the above inequality can be rewritten by plugging in inside the quadratic. On plugging it, we will get:
Observe that the term on the right here lies in . This is because the fraction that is besides in this term lies in . Thus, we will use this term as our bound on . ∎
Now, we provide a similar upper bound result, though only partially, for eigenvalues of matrices that have the same structural properties as the Jacobian of our regularized system. Note that we have upper bounds only for eigenvalues that are complex (we have not used them anywhere in the main paper though).
where is a real symmetric positive definite matrix and is a full column rank matrix. Let .
Then, if for any eigenvalue of ,
Consider the following eigenvector equation:
where are all real-valued. We want to show that . Let us first rewrite the above equation as follows:
We can then equate the real and imaginary parts.
We now multiply the above equations by respectively and add them:
Above, we can substitute for and in using the previous equations.
We could do the above only because and therefore the denominator .
In order to prove our upper bound, we first note the following inequality:
We now multiply the first and third equations by and respectively and sum them up, and second and fourth by and respectively and sum them up. Then, we get:
Then, either or when , we have . This translates to the inequality:
By adding everywhere and using the fact that , the above inequality becomes:
The above inequalities yield an immediate lower bound on the magnitude in the latter case by plugging them in Equation 20:
Since, we know is negative, this implies an upper bound on .