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We investigate empirically the effectiveness of PNM and show that it can indeed deal well with typical GAN problems such as mode collapse and forgetting, especially in distributions with asymmetric structure to their modes. We show that the found solutions are much less susceptible to being exploited by an adversary, a... |
about the conditions at a Nash equilibrium. Background We defer a more detailed treatment of related work on GANs and recent game-theoretic approaches until Section Here, we introduce some basic game-theoretic notation. |
Definition 1 (‘game’) two-player strategic game , which we will simply call ‘game’, is a tuple [MATH] where [MATH] is the set of players |
[MATH] is the set of pure strategies (actions) for player [MATH] and [MATH] is [MATH] payoff function defined on the set of pure strategy profiles |
[MATH] When the action sets are finite, the game is finite We also write [MATH] and [MATH] for the strategy of agent [MATH] and its opponent respectively. |
A fundamental concept is the Nash equilibrium (NE), which is a strategy profile such that no player can unilaterally deviate and improve his payoff. |
Definition 2 (Pure Nash equilibrium) A pure strategy profile [MATH] is an NE if and only if [MATH] for all players [MATH] and [MATH] |
A finite game may not possess a pure NE. A mixed strategy [MATH] of player [MATH] is a probability distribution over [MATH] ’s pure strategies |
[MATH] . The set of such probability distributions is denoted by [MATH] . The payoff of a player under a profile of mixed strategies |
[MATH] is defined as the expectation: [MATH] Then an NE in mixed strategies is defined as follows. Definition 3 (Mixed Nash equilibrium) |
[MATH] is an NE if and only if [MATH] for all players [MATH] and potential unilateral deviations [MATH] Every finite game has at least one NE in mixed strategies (Nash, 1950 In this paper we deal with two-player zero-sum games, where |
[MATH] for all [MATH] The equilibria of zero-sum games, also called saddle points have several important properties, as stated in the Minmax theorem. |
Theorem 1 von Neumann ( 1928 In a finite zero-sum game, [MATH] where [MATH] is the value of the game. All equilibria have payoff [MATH] |
and equilibrium strategies are interchangeable: if [MATH] and [MATH] are equilibria, then so are [MATH] and [MATH] (Osborne and Rubinstein, 1994 This means that in zero-sum games we do not need to worry about equilibrium selection: any equilibrium strategy for a player is guaranteed to achieve the value of the game. Mo... |
We also employ the standard, additive notion of approximate equilibrium: Definition 4 A pair of (possibly pure) strategies [MATH] is an [MATH] -NE if |
[MATH] In other words, no player can gain more than [MATH] by deviating. In the literature, GANs have not typically been considered as finite games. The natural interpretation of the standard setup of GANs is of an infinite game where payoffs are defined over all possible weight parameters for the respective neural net... |
In finite games every LNE is an NE, as, whenever there is a global deviation (i.e., a better response), one can always deviate locally in the space of mixed strategies towards a pure best response (by playing that better response with [MATH] higher probability). |
GANGs In order to capitalize on the insight that we can escape local equilibria by switching to mixed strategy space for a finite game, we formalize adversarial networks in a (finite) games setting, that we call (finite) Generative Adversarial Network Games (GANGs) |
We start with making explicit how general (infinite) GANs correspond to strategic games, via the GANG framework: Definition 5 (GANG) |
GANG is a tuple [MATH] with [MATH] is the distribution over (‘true’ or ‘real’) data points [MATH] [MATH] is a neural network class parametrized by a parameter vector [MATH] |
and [MATH] outputs, such that [MATH] denotes the (‘fake’ or ‘generated’) output of [MATH] on a random vector [MATH] drawn from some distribution [MATH] |
[MATH] is a neural network class parametrized by a parameter vector [MATH] and a single output, such that the output [MATH] indicates the ‘realness’ of [MATH] according to [MATH] |
[MATH] is a measuring function (Arora et al., 2017 —e.g., [MATH] for GANs, the identity mapping for WGANs—used to specify game payoffs, explained next. |
A GANG induces a zero-sum game in an intuitive way: Definition 6 The induced zero-sum strategic-form game of a GANG is [MATH] with: |
[MATH] the set of strategies [MATH] [MATH] the set of strategies [MATH] [MATH] I.e., the score of [MATH] is the expected ‘measured realness’ of the real data minus that of the fake data; |
[MATH] As such, when using [MATH] , GANGs employ a payoff function for [MATH] that use Goodfellow et al. ( 2014 ’s trick to enforce strong gradients early in the training process, but it applies this transformation to [MATH] too, in order to retain the zero-sum property. The correctness of these transformations are pro... |
In practice, GANs are represented using floating point numbers, of which, for a given setup, there is only a finite (albeit large) number. In GANGs, we formalize this: |
Definition 7 Any GANG where [MATH] are finite classes—i.e., classes of networks constructed from a finite set of node types (e.g., {Sigmoid, ReLu, Linear})—and with architectures of bounded size, is called a finite network class GANG. |
A finite network class GANG in which the sets [MATH] are finite too, it is called a finite GANG. From now on, we will focus on finite GANGs. We emphasize this finiteness, because this is exactly what enables us to obtain the desirable properties mentioned in Section existence of (one or infinitely many) mixed NEs with ... |
Parallel Nash Memory (PNM) (Oliehoek et al., 2006 , which can be seen as a generalization (to non-exact best responses) of the double oracle method |
(McMahan et al., 2003 Resource-Bounded GANGs While finite GANGs do not suffer from LNEs, solving them is non-trivial: even though we know that there always is a direction (i.e., a best response) that we can play more frequently to move towards an NE, computing that direction itself is a computationally intractable task... |
Resource-Bounded Best-Responses (RBBR). A Nash equilibrium is defined by the absence of better responses for any of the players. As such, best response computation is a critical tool to verify whether a strategy profile is an NE, and is also a common subroutine in algorithms that compute an NE. |
However, since computing an [MATH] -)best response will generally be intractable for GANGs, we examine the type of solutions that we actually can expect to compute with bounded computational power by the notion of |
resource-bounded best response and show how it naturally leads to the resource-bounded NE (RB-NE) solution concept. We say that [MATH] is the subset of strategies of player [MATH] , that [MATH] can compute as a best response, given its bounded computational resources. This computable set is an abstract formulation that... |
Definition 8 A strategy [MATH] of player [MATH] is a resource-bounded best-response (RBBR) against a (possibly mixed) strategy [MATH] , if [MATH] |
That is, [MATH] only needs to be amongst the best strategies that player [MATH] can compute in response to [MATH] . We denote the set of such RBBRs to [MATH] by |
[MATH] Definition 9 resource-bounded best-response function [MATH] is a function that maps from the set of possible strategies of player [MATH] to an RBBR for [MATH] , s.t. [MATH] |
Using RBBRs, we define an intuitive specialization of NE: Definition 10 [MATH] is a resource-bounded NE (RB-NE) iff [MATH] That is, an RB-NE can be thought of as follows: we present [MATH] to each player [MATH] and it gets the chance to switch to another strategy, for which it can apply its bounded resources (i.e., use... |
Clearly, an RB-NE can be linked to the familiar notion of [MATH] -NE by making assumptions on the power of the best response computation. |
Theorem 2 If both players are powerful enough to compute [MATH] -best responses, then an RB-NE is an [MATH] -NE. Proof. Starting from the RB-NE [MATH] , assume an arbitrary [MATH] By definition of RB-NE [MATH] .∎ |
Non-deterministic Best Responses. The above definitions assumed deterministic RBBR functions [MATH] . However, in many cases the RBBR function can be non-deterministic (e.g., due to random restarts), which means that the sets [MATH] are non-deterministic. This is not a fundamental problem, however, and the same approac... |
In particular, now let [MATH] be a non-deterministic function, and define [MATH] as the range of this function. That is, we define [MATH] as that set of strategies that our non-deterministic RBBR function might deliver. Given this modification the definition of the RB-NE remains unchanged: a strategy profile [MATH] is ... |
non-deterministic RB-NE if each player [MATH] uses all its computational resources by calling [MATH] once, and no player finds a better strategy to switch to. |
Solving GANGs Treating GANGs as finite games in mixed strategies permits building on existing tools and algorithms for these classes of games (Fudenberg and Levine, 1998 ; Rakhlin and Sridharan, 2013a ; Foster et al., 2016 In this section, we describe how to use the Parallel Nash Memory (Oliehoek et al., 2006 which exp... |
converges to such an equilibrium. In the following, we give a concise description of a slightly simplified form of Parallel Nash Memory (PNM) and how we apply it to GANGs. For ease of explanation, we focus on the setting with deterministic best responses. |
1: [MATH] 2: [MATH] [MATH] set initial mixtures 3: while True do 4: [MATH] [MATH] get new bounded best resp. 5: [MATH] 6: // Expected payoffs of these ‘tests’ against mixture: |
7: [MATH] 8: if [MATH] then 9: break 10: end if 11: [MATH] 12: [MATH] 13: end while 14: return [MATH] [MATH] found an BR-NE Algorithm 1 |
Parallel Nash Memory for GANGs with deterministic RBBRs The algorithm is shown in Algorithm Intuitively, PNM incrementally grows a strategic game [MATH] , over a number of iterations, using the AugmentGame function. It uses SolveGame to compute (via linear programming, see, e.g., Shoham and Leyton-Brown ( 2008 a mixed ... |
of this smaller game at the end of each iteration. In order to generate new candidate strategies to include in [MATH] , at the beginning of each iteration the algorithm uses a ‘search’ heuristic to deliver new promising strategies. In our GANG setting, we use the resource-bounded best-response (RBBR) functions of the p... |
In order to augment the game, PNM evaluates (by simulation) each newly found strategy for each player against all of the existing strategies of the other player, thus constructing a new row and column for the maintained payoff matrix. |
In order to implement the best response functions, any existing neural network architectures can be used. However, we need to compute RBBRs against mixtures of networks of the other player. For [MATH] this is trivial: we can simply generate a batch of fake data from the mixture [MATH] Implementing an RBBR for [MATH] ag... |
[MATH] Intuitively, one can think of a combined network consisting of the [MATH] network with its outputs connected to every [MATH] (see Figure ). |
The predictions [MATH] of these components [MATH] are combined in a single linear output node [MATH] This allows us to evaluate and backpropagate through the entire network. A practical implementation that avoids memory concerns instead loops through each component [MATH] and does the evaluation of the weighted predict... |
[MATH] and subsequent backpropagation per component. Intuitively, it is clear that PNM converges to an RB-NE, which we now prove formally. |
Theorem 3 If PNM terminates, it has found an RB-NE. Proof. We show that [MATH] implies we have an RB-NE: [EQUATION] Note that, per Def. |
[MATH] [MATH] [MATH] for all computable [MATH] (and similar for [MATH] ). Therefore, the only way that [MATH] [MATH] [MATH] could fail to hold, is if [MATH] would include some strategies that are not computable (not in [MATH] ) that provide higher payoff. However, as the support of [MATH] is composed of strategies comp... |
[MATH] [MATH] [MATH] and similarly [MATH] [MATH] [MATH] Together with ( ) this directly implies [MATH] and [MATH] , indicating we found an RB-NE. |
Corollary 1 Algorithm terminates and monotonically converges to an equilibrium. Proof. This follows directly from the fact that there are only finitely many RBBRs and the fact that we never forget RBBRs that we computed before, thus the proof for PNM (Oliehoek et al., 2006 extends to Algorithm |
The PNM algorithm for GANGs is parameter free, but we mention two adaptations that are helpful. Interleaved training of best responses. |
In order to speed up convergence of PNM, it is possible to train best responses of [MATH] and [MATH] in parallel, giving [MATH] access to the intermediate results of |
[MATH] . This reduces the number of needed PNM iterations, but does not seem to affect the quality of the found solutions, as demonstrated in Appendix The results shown in the main paper do not employ this trick. |
Regularization of classifier best responses. In initial experiments, we found that the best-responses by [MATH] tended to overfit—see the supplement for an extensive analysis. In order to regularize best responses for [MATH] , in each iteration, in the experiments below we sample additional data points [MATH] uniformly... |
Experiments Here we report on experiments that aim to test if searching in mixed strategies with PNM-GANG can help in reducing problems with training GANs, and if the found solutions (near-RB-NEs) provide better generative models and are potentially closer to true Nash equilibria than those found by GANs (near-LNEs). S... |
Experimental setup. We compare our PNM approach (‘PNM-GANG’) to a vanilla GAN implementation and state-of-the-art MGAN (Hoang et al., 2018 . Detailed training and architecture settings are summarized in the appendix. |
The mixture components comprise grids and annuli with equal-variance components, as well as non-symmetric cases with randomly located modes and with a random covariance matrix for each mode. For each domain we create test cases with 9 and 16 components. In our plots, black points are real data, green points are generat... |
Found solutions. The results produced by regular GANs and PNM-GANGs are shown in Figure and clearly convey three main points: 1. |
The PNM-GANG mixed classifier has a much flatter surface than the classifier found by the GAN. Around the true data, the GANG classifier outputs around 0.5 indicating indifference, which is in line with the theoretical predictions about the equilibrium |
(Goodfellow et al., 2014 2. We see that this flatter surface is not coming at the cost of inaccurate samples. In contrast: nearly all samples shown are hitting one of the modes and thus the PNM-GANG solutions are highly accurate much more so than the GANs’ solutions. |
3. Finally, the PNM-GANGs, unlike GANs, do not suffer from mode omission ; they leave out no modes. We also note that PNM-GANG typically achieved these results with fewer total parameters than the regular GAN, e.g., 1463 vs. 7653 for the random 9 task in Figure |
This shows that, qualitatively, the use of multiple generators seems to lead to good results. This is corroborated by results we obtained using MGANs (illustrations and complete description can be found in Appendix C.2 ), which only failed to represent one mode in the ‘random’ task. |
Impact of generator learning rate. The above results show that PNM-GANG can accurately cover multiple modes, however, not all modes are fully covered. As also pointed out by Arjovsky et al. ( 2017 , the best response of [MATH] |
against [MATH] is a single point with the highest ‘realness’, and therefore the WGAN they introduced uses fewer iterations for [MATH] than for [MATH] . Inspired by this, we investigate if we can reduce the mode collapse by reducing the learning rate of [MATH] . The results in Figure show that more area of the modes are... |
Exploitability of solutions. Finally, to complement the above qualitative analysis, we also provide a quantitative analysis of the solutions found by GANs, MGANs and PNM-GANGs. We investigate to what extent they are exploitable by newly introduced adversaries with some fixed computational power (as modeled by the compl... |
In particular, for a given solution [MATH] we use the following measure of exploitability: [EQUATION] where ‘RBmax’ denotes an approximate maximization performed by an adversary of some fixed complexity. |
That is, the ‘RBmax’ functions are analogous to the [MATH] functions employed in PNM, but the computational resources of ‘RBmax’ could be different from those used for the [MATH] Intuitively, it gives a higher score if [MATH] is easier to exploit. However, it is not a true measure of distance to an equilibrium: it can ... |
Our exploitability is closely related to the use of GAN training metrics (Im et al., 2018 , but additionally includes the exploitability of the classifier. This is important: when only testing the exploitability of the generator, this does give a way to compare generators, but it does not give a way to assess how far f... |
We perform two experiments on the 9 random modes task; results for other tasks are in Appendix C.2 First, we investigate the exploitability of solutions delivered by GANs, MGANs and GANGs of different complexities (in terms of total number of parameters used). For this, we compute ‘attacks’ (approximate best responses)... |
These results are shown in Figure (left and middle). The left plot shows the exploitability of PNM-GANG after different numbers of iterations, as well as the number of parameters used in the solutions found in those iterations (a sum over all the networks in the support of the mixture). Error bars indicate standard dev... |
The middle plot shows those same results together with exploitability results we obtained for GANs and MGANs of different complexities (all were trained to convergence). Note that here the x-axis shows the complexity in terms of total parameters. The figure shows an approximately monotonic decrease in exploitability fo... |
Secondly, we investigate what happens for the converged GAN / PNM-GANG solution of Figure , which have comparable complexities, when attacked with varying complexity attackers. We also employ an MGAN which has a significantly larger number of parameters (see Appendix C.2 ). These results are shown in Figure (right). Cl... |
Overall, these results demonstrate that GANGs can provide more robust solutions than GANs/MGANs with the same number of parameters, suggesting that they are closer to a Nash equilibrium and provide better generative models. |
Related work Progress in zero-sum games. Bosanský et al. ( 2014 devise a double-oracle algorithm for computing exact equilibria in extensive-form games with imperfect information. Their algorithm uses best-response oracles ; PNM does so too, though in this paper using resource-bounded rather than exact best responses. |
Lanctot et al. ( 2017 generalize to non-exact sub-game routines. Inspired by GANs, Hazan et al. ( 2017 deal with general zero-sum settings with non-convex loss functions. They introduce a weakening of local equilibria known as smoothed local equilibria and provide algorithms with guarantees on the smoothed local regret... |
GANs. The literature on GANs has been growing at an incredible rate, and a full overview of all the related works such as those by |
Arjovsky et al. ( 2017 ); Arjovsky and Bottou ( 2017 ); Huszár ( 2015 ); Nowozin et al. ( 2016 ); Dai et al. ( 2017 ); Zhao et al. ( 2017 ); Arora and Zhang ( 2017 ); Salimans et al. ( 2016 ); Gulrajani et al. ( 2017 ); Radford et al. ( 2015 |
is beyond the scop of this paper. Instead we refer to Unterthiner et al. ( 2018 for a reasonable comprehensive recent overview. Unterthiner et al. ( 2018 introduce Coulomb GANs and show convergence for them but only under the strong assumption that the “generator samples can move freely” (which is not the case when tra... |
Karras et al. ( 2018 , like PNM, incrementally grow the complexity of the maintained models, but, unlike PNM, do not maintain a mixture. Many of the techniques designed to improve GAN training, e.g., |
(Salimans et al., 2016 ; Gulrajani et al., 2017 ; Sønderby et al., 2017 ; Creswell et al., 2018 concern modifications to the update for one or both players. Such techniques can directly be used in GANGs by adapting the best response computation. |
Explicit representations of mixtures of strategies. Recently, more researchers have investigated the idea of (more or less) explicitly representing a set or mixture of strategies for the players. For instance, Jiwoong Im et al. ( 2016 retains sets of networks that are trained by randomly pairing up with a network for t... |
MAD-GAN (Ghosh et al., 2017 uses [MATH] generators, but one discriminator. MGAN (Hoang et al., 2018 proposes mixtures of [MATH] generators, a classifier and a discriminator with weight sharing; and presents a theoretical analysis similar to |
Goodfellow et al. ( 2014 assuming infinite capacity densities. Unlike PNM, none of these approaches have convergence guarantees. |
Generally, explicit mixtures can bring advantages in two ways: (1) Representation : intuitively, a mixture of [MATH] neural networks could better represent a complex distribution than a single neural network of the same size, and would be roughly on par with a single network that is [MATH] times as big. Arora et al. ( ... |
(2) Training : Arora et al. use an architecture that is tailored to representing a mixture of components and train a single such network. We, in contrast, explicitly represent the mixture; given the observation that good solutions will take the form of a mixture. This is a form of domain knowledge that facilitates lear... |
A closely related paper is the work by Grnarova et al. ( 2017 , which also builds upon game-theoretic tools to give certain convergence guarantees. The main differences are as follows: |
1. We provide a more general form of convergence (to an RB-NE) that is applicable to all architectures, that only depends on the power to compute best responses, and show that PNM-GANG converges in this sense. We also show that if agents can compute an [MATH] -best response, then the procedure converges to an [MATH] -N... |
2. Grnarova et al. ( 2017 show that for a quite specific GAN architecture their first algorithm converges to an [MATH] -NE. On the one hand, this result is an instantiation of our more general theory: they assume they can compute exact (for [MATH] ) and |
[MATH] -approximate (for [MATH] ) best responses; for such powerful players our Theorem provides that guarantee. On the other hand, their formulation works without discretizing the spaces of strategies. |
3. The practical implementation of their algorithm does not provide guarantees. Finally, Ge et al. ( 2018 propose a method similar to ours that uses fictitious play |
(Brown, 1951 ; Fudenberg and Levine, 1998 rather than PNM. Fictitious play does not explicitly model mixed strategies for the agents, but interprets the opponent’s historical behavior as such a mixed strategy. The average strategy played by the ‘Fictitious GAN’ approach converges to a Nash equilibrium |
assuming that “the discriminator and the generator are updated according to the best-response strategy at each iteration” , which follow from the result by |
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