Paper ID: | 4162 |
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Title: | Competitive Gradient Descent |

This paper deals with the computation of Nash Equilibria in competitive two-player games where the $x$ player is minimizing a function $f(x,y)$ and the $y$ player is minimizing a function $g(x,y)$. Such problems arise in a wide variety of domains, notably in training GANs, and there has been much recent interest in developing algorithms for solving such problems. Gradient Descent Ascent (GDA) is a natural candidate algorithm for finding Nash Equilibrium, but it will provably oscillate or diverge even in simple settings. As such, many recent works have modified GDA or proposed different algorithms or schemes to guarantee convergence. This paper proposes a new algorithm called Competitive Gradient Descent (CGD), which updates each player's iterates by adding the Nash Equilibrium of a regularized bilinear approximation of the game at the current iterates. CGD requires Hessian-vector products, making it a second-order algorithm. The authors prove that their algorithm is locally stable around critical points for zero-sum games. They discuss how their algorithm relates to other recent algorithms for finding Nash Equilibria, and they also provide a number of experimental comparisons. Unlike other algorithms, CGD does not need to reduce its step-size parameter $\eta$ when the ``interaction'' between the players is large, as signified by the magnitude of $D^2_{xy} f$ being large. On the positive side, this paper proposes a novel algorithm that seems reasonably well-motivated. Moreover, the paper gives a nice interpretation of how this algorithm compares to other existing algorithms. Finally, the experimental results are reasonably extensive and seem to indicate that CGD may have good empirical performance. On the negative side, the theoretical results are fairly weak, since they just prove stability near critical points. The authors emphasize how CGD improves when $D^2_{xy}$ is large, whereas other algorithms have poor performance for fixed step-size as $D^2_{xy}$ grows. However, this comparison isn't exactly fair since CGD uses second-order information to explicitly regularize step-sizes by the off-diagonal term, whereas OGDA or extragradient use only first-order information. Moreover, CGD still requires that the step-size is bounded by one over the max diagonal entry of the Hessian, so at least from the theory, CGD will still require step-size tuning. Finally, while it is true that Hessian-vector products are theoretically as fast as gradient calls for neural networks and explicitly defined functions, it is misleading to simply say ``Hessian vector products can be computed at minimal overhead over the cost of computing gradients'' because this is not true in general, for instance when one only has oracle access to the function and gradient. Overall, this paper is an accept. The algorithm is novel and well-motivated, and the authors do a good job of comparing it to existing results. While the theoretical results are minimal, the experimental results seem well-done and fairly thorough. The paper is clear. --- After Rebuttal --- I have read the other reviews and the rebuttal. I had no major concerns for the paper, so my rating is unchanged.

The writing of this paper is quite good, and the game-theoretic perspective is very much welcome. Theorem 2.2 is the main theoretical contribution of this work. The convergence analysis is nice, but it is not very surprising, nor is the proof technique itself very novel. The experiments are fair, but they are not the strength of this work. I have the following major concerns. 1. If the authors are willing to use a matrix inverse each iteration, why not consider the full 2nd order Taylor expansion and perform a (damped) Newton method? What benefit is there to only consider the quadratic interaction terms (bilinear terms) and not the individual quadratic terms? 2. Glaringly absent from this paper is the discussion of whether computing the matrix inverse (via iterative methods) is computationally worth it compared to existing first-order methods that do not require a matrix inverse. The computational experiments only demonstrate stability and rate of convergence as a function of the iteration count. For the x-axis, the wall-clock time should have been used instead of the iteration count. I imagine the method does not scale very well as the dimension of the variable gets large. Overall, I believe the theoretical contribution is nice but not very strong. As the experiments are minimal and do not address the concerns about the computational cost, they fail to provide a proof-of-concept of the practical value of the proposed method. Minor issues. line 181, the inline equation seems wrong. line 235 " are not break down" line 250, the idea of computing Hessian vector products with automatic differentiation is not an original idea of this work. Balduzzi et al, or whoever thought of it first should be credited. line 280 "an reasonable" -> "a reasonable" For GAN and covariance matrix estimation experiments, you should state in the text or within the figure captions indicating which figure corresponds to which experiment. Supplement. The theorem numberings are off by one. -------------------------------------------------------------------------------------------- Post rebuttal comments: The authors’ response has resolved my concerns regarding practical computational costs. I now see that the experiments do provide a proof of concept for the practical effectiveness of the proposed method. I still think my #1 concern of “why not Newton?” is still a significant weakness of this work, and I do not expect further discussion without any analysis or experiments to resolve this issue. Nevertheless, I think the presented results are interesting enough to be published.

A new algorithm, competitive gradient descent (CGD), is proposed to solve for the Nash equilibrium of two-player zero-sum games. It is derived from the closed form solution of a local bilinear approximation to the game at each learning step. Given assumptions such as bounds on the player Hessians, exponential convergence is proven. CGD is compared to several other algorithms in the literature both empirically and based on mathematical form. A link to levels of intentionality is also made using the Taylor series expansion of the inverse. Finally, experiments fitting a GAN to a Gaussian mixture (bi-modal) distribution show CGD converges faster than other methods. I believe the perspective of solving for the Nash equilibrium of a bilinear approximation is novel, however, an argument for the specific bilinear approximation proposed is lacking. Moreover, if the bilinear approximation were to retain the diagonal Hessian terms, e.g., 0.5*x^T (D^2_xx) x, the derived update becomes the standard regularized Newton method (based on a quick calculation): new preconditioning matrix from line 147 = [[I+eta*Dxx f, eta*Dxy f],[eta*Dyx g, I+eta*Dyy g]]^{-1} = (I + eta*Jacobian)^{-1}. This is regularized in the sense that identity better conditions the matrix for inversion. Therefore, the update presented on line 147 is nearly Newton's method applied to fixed point iteration, i.e., Delta = -J^{-1}F where J is the Jacobian. Theorem 2.2 requires eta*||D^2_xx f|| <= 1 -- does this condition arise because the diagonal terms in line 147 are replacing eta*(D^2_xx f) from Newton's method with identity? A discussion around these similarities would be helpful. It is known that Newton's method is attracted to critical points generally, not just minima (see "Identifying and attacking the saddle point problem in high-dimensional non-convex optimization"). By replacing the diagonal terms in line 147 with identity, CGD can avoid converging to some unstable fixed points -- does this prevent all failure modes? What are some of CGD's failure modes if any? Doesn't the requirement in Theorem 2.2 (eta*||D^2_xx f|| <= 1) suggest that a large enough step size, eta, may prevent convergence? The work is generally high quality and clear, and many of the perspectives are original. I think the community would benefit from some of the ideas in this paper. The paper would benefit from a discussion of the weaknesses of CGD. Main concerns: 1) CGD is very similar to Newton's method, but there is no rationale mentioned for dropping the diagonal terms of Jacobian. 2) No argument is presented for the specific bilinear form proposed as the fundamental local approximation to the game. 3) Complexity of inverting a matrix is dealt with using conjugate gradient but comparisons are done using "simple" GANs. Would this method scale well to higher dimensions? Do you believe there is a threshold at which the matrix inversion method becomes intractable? Have you tried CGD on more standard GAN tasks, e.g., MNIST, Celeb-A, CIFAR10? Do you expect problems at that scale? Also, SGA [Balduzzi et al, 2018] was independently derived in "Global Convergence to the Equilibrium of GANs using Variational Inequalities" [Gemp et al, 2018]. This work provides a comparison similar to Figure 1 in this paper. It should probably be cited along with Balduzzi '18 and Letcher '19.