A V ARIATIONAL I NEQUALITY P ERSPECTIVE ON G ENERATIVE A DVERSARIAL N ETWORKS
Anonymous authors
Paper under double-blind review
A BSTRACT
Generative adversarial networks (GANs) form a generative modeling approach known for producing appealing samples, but they are notably difﬁcult to train.
One common way to tackle this issue has been to propose new formulations of the GAN objective.
Yet, surprisingly few studies have looked at optimization methods designed for this adversarial training.
In this work, we cast GAN optimization problems in the general variational inequality framework.
Tapping into the mathematical programming literature, we counter some common misconceptions about the difﬁculties of saddle point optimization and propose to extend techniques designed for variational inequalities to the training of GANs.
We apply averaging , extrapolation and a novel computationally cheaper variant that we call extrapolation from the past to the stochastic gradient method (SGD) and Adam.
I NTRODUCTION
Generative adversarial networks (GANs) (Goodfellow et al., 2014) form a generative modeling approach known for producing realistic natural images (Karras et al., 2018) as well as high quality super-resolution (Ledig et al., 2017) and style transfer (Zhu et al., 2017).
Nevertheless, GANs are also known to be difﬁcult to train, often displaying an unstable behavior (Goodfellow, 2016).
Much recent work has tried to tackle these training difﬁculties, usually by proposing new formulations of the GAN objective (Nowozin et al., 2016; Arjovsky et al., 2017).
Each of these formulations can be understood as a two-player game, in the sense of game theory (Von Neumann and Morgenstern, 1944), and can be addressed as a variational inequality problem (VIP) (Harker and Pang, 1990), a framework that encompasses traditional saddle point optimization algorithms (Korpelevich, 1976).
Solving such GAN games is traditionally approached by running variants of stochastic gradient descent (SGD) initially developed for optimizing supervised neural network objectives.
Yet it is known that for some games (Goodfellow, 2016, §8.2) SGD exhibits oscillatory behavior and fails to converge.
This oscillatory behavior, which does not arise from stochasticity, highlights a fundamental problem: while a direct application of basic gradient descent is an appropriate method for regular minimization problems, it is not a sound optimization algorithm for the kind of two-player games of GANs.
This constitutes a fundamental issue for GAN training, and calls for the use of more principled methods with more reassuring convergence guarantees.
Contributions.
We point out that multi-player games can be cast as variational inequality problems and consequently the same applies to any GAN formulation posed as a minimax or non-zero-sum game.
We present two techniques from this literature, namely averaging and extrapolation , widely used to solve variational inequality problems (VIP) but which have not been explored in the context of GANs before.
We extend standard GAN training methods such as SGD or Adam into variants that incorporate these techniques (Alg. 3, 4 are new).
We also explain that the oscillations of basic SGD for GAN training previously noticed (Goodfellow, 2016) can be explained by standard variational inequality optimization results and we illustrate how averaging and extrapolation can ﬁx this issue.
We introduce a new technique, called extrapolation from the past , that only requires one gradient computation per iteration compared to extrapolation which requires to compute the gradient twice.
We prove its convergence in the stochastic variational inequality setting, i.e. when applied to SGD.
Finally, we test these techniques in the context of standard GAN training.
We observe a 4%-6% improvement on the inception score (Salimans et al., 2016) of WGAN (Arjovsky et al., 2017) and
WGAN-GP (Gulrajani et al., 2017) on the CIFAR-10 dataset.
Outline.
§2 presents the background on GAN and optimization, and shows how to cast this optimization as a VIP.
§3 presents standard techniques to optimize variational inequalities in a batch setting as well as our new one, extrapolation from the past .
§4 considers these methods in the stochastic setting, yielding three corresponding variants of SGD, and provides their respective convergence rates.
§5 develops how to combine these techniques with already existing algorithms.
§6 discusses the related work and §7 presents experimental results.
GAN OPTIMIZATION AS A VARIATIONAL INEQUALITY PROBLEM 2. GAN FORMULATIONS
The purpose of generative modeling is to generate samples from a distribution q θ that matches best the true distribution p of the data.
The generative adversarial network training strategy can be understood as a game between two players called generator and discriminator .
The former produces a sample that the latter has to classify between real or fake data.
The ﬁnal goal is to build a generator able to produce sufﬁciently realistic samples to fool the discriminator.
In the original GAN paper (Goodfellow et al., 2014), the GAN objective is formulated as a zero-sum game where the cost function of the discriminator D ϕ is given by the negative log-likelihood of the binary classiﬁcation task between real or fake data generated from q θ by the generator, min θ max ϕ L ( θ , ϕ ) where L ( θ , ϕ ) def = − E x ∼ p
[log D ϕ ( x )] − E x (cid:48) ∼ q θ [log(1 − D ϕ ( x (cid:48)
[Equation]
However Goodfellow et al.
(2014) recommends to use in practice a second formulation, called non-saturating GAN .
This formulation is a non-zero-sum game where the aim is to jointly minimize:
L ( θ ) [Equation] ϕ ) def [Equation]
∼ q θ
[Equation]
ϕ [Equation] ϕ
[Equation]
The dynamics of this formulation have the same stationary points as the zero-sum one (1) but are claimed to provide “much stronger gradients early in learning”
(Goodfellow et al., 2014) .
2. E QUILIBRIUM
The minimax formulation (1) is theoretically convenient because a large literature on games studies this problem and provides guarantees on the existence of equilibria.
Nevertheless, practical considerations lead the GAN literature to consider a different objective for each player as formulated in (2).
In that case, the two-player game problem (Von Neumann and Morgenstern, 1944) consists in ﬁnding the following Nash equilibrium :
[Equation]
Only when L ( θ ) =
−L ( ϕ ) is the game called a zero-sum game and (3) can be formulated as a minimax problem.
One important point to notice is that the two optimization problems in (3) are coupled and have to be considered jointly from an optimization point of view.
Standard GAN objectives are non-convex (i.e. each cost function is non-convex), and thus such (pure) equilibria may not exist.
As far as we know, not much is known about the existence of these equilibria for non-convex losses (see Heusel et al.
and references therein for some results).
In our theoretical analysis in §4, our assumptions (monotonicity (24) of the operator and convexity of the constraints set) imply the existence of an equilibrium.
In this paper, we focus on ways to optimize these games, assuming that an equilibrium exists.
As is often standard in non-convex optimization, we also focus on ﬁnding points satisfying the necessary stationary conditions .
As we mentioned previously, one difﬁculty that emerges in the optimization of such games is that the two different cost functions of (3) have to be minimized jointly in θ and ϕ .
Fortunately, the optimization literature has for a long time studied so-called variational inequality problems , which generalize the stationary conditions for two-player game problems.
2. V ARIATIONAL INEQUALITY PROBLEM FORMULATION
We ﬁrst consider the local necessary conditions that characterize the solution of the smooth two-player game (3), deﬁning stationary points , which will motivate the deﬁnition of a variational inequality.
In the unconstrained setting, a stationary point is a couple ( θ ∗ , ϕ ∗ ) with zero gradient:
[Equation]
When constraints are present, 2 a stationary point ( θ ∗ , ϕ ∗ ) is such that the directional derivative of each cost function is non-negative in any feasible direction (i.e. there is no feasible descent direction): ∇ θ L ( θ ) ( θ ∗ , ϕ ∗ ) (cid:62) ( θ − θ ∗ ) ≥ 0 and ∇ ϕ L ( ϕ ) ( θ ∗ , ϕ ∗ ) (cid:62) ( ϕ − ϕ ∗ ) ≥ 0 , ∀ ( θ , ϕ
) ∈
Θ × Φ .
Deﬁning ω
def =
( θ , ϕ ) , ω ∗ def = ( θ ∗ , ϕ ∗ ) , Ω def =
Θ × Φ , Eq.
(5) can be compactly formulated as:
F ( ω ∗ ) (cid:62) ( ω − ω ∗ ) ≥ 0 , ∀ ω ∈ Ω where F ( ω ) def = (cid:104) ∇ θ L ( θ ) ( θ , ϕ ) ∇ ϕ L ( ϕ ) ( θ , ϕ ) (cid:105) (cid:62) .
These stationary conditions can be generalized to any continuous vector ﬁeld: let Ω ⊂ R d and
F : Ω → R d be a continuous mapping.
The variational inequality problem (Harker and Pang, 1990) (depending on F and Ω ) is:
ﬁnd ω ∗ ∈
Ω such that F ( ω ∗ ) (cid:62) ( ω − ω ∗ ) ≥ 0 , ∀ ω ∈ Ω .
(VIP)
We call optimal set the set Ω ∗ of ω ∈ Ω verifying (VIP).
The intuition behind it is that any ω ∗ ∈
Ω ∗ is a ﬁxed point of the constrained dynamic of F (constrained to Ω ).
We have thus showed that both saddle point optimization and non-zero sum game optimization, which encompasses the large majority of GAN variants proposed in the literature, can be cast as Variational
Inequality Problems.
In the following section, we turn to suitable optimization techniques for such problems.
O PTIMIZATION OF V ARIATIONAL I NEQUALITIES ( BATCH SETTING )
Let us begin by looking at techniques that were developed in the optimization literature to solve (VIP).
We present the intuitions behind them as well as their performance on a simple bilinear problem (see
Fig. 1).
Our goal here is to provide mathematical insights into the techniques of averaging (§3.1) and extrapolation (§3.2), to inspire their application to extending other optimization algorithm.
We then propose a novel variant of the extrapolation technique in §3.3 extrapolation from the past .
We here treat the batch setting, i.e. considering that the operator F ( ω ) as deﬁned in Eq.
6 yields an exact full gradient.
We will present extensions of these techniques to the stochastic setting later in §4.
The two standard methods studied in the VIP literature are the gradient method (Bruck, 1977) and the extragradient method (Korpelevich, 1976).
The iterates of the basic gradient method are given by ω t
+1 = P Ω
[ ω t − ηF ( ω t )] where P Ω [ · ] is the projection onto the constraints set (if constraints are present) associated to (VIP).
These iterates are known to converge linearly under an additional assumption on the operator 3 (Chen and Rockafellar, 1997), but oscillate for a bilinear operator as shown in Fig.
On the other hand, the uniform average of these iterates converge for any bounded monotone operator with a O (1 / √ t ) rate (Nedi´c and Ozdaglar, 2009), motivating the presentation of averaging in §3.1.
By contrast, the extragradient method (extrapolated gradient) does not require any averaging to converge for monotone operators (in the batch setting), and can even converge at the faster O (1 /t ) rate (Nesterov, 2007).
The idea of this method is to compute a lookahead step (see intuition on extrapolation in §3.2) in order to compute a more stable direction to follow.
3. A VERAGING
More generally, we consider a weighted averaging scheme with weights ρ t ≥ 0 .
This weighted averaging scheme have been proposed for the ﬁrst time for (batch) VIP by Bruck (1977),
[Equation]
Averaging schemes can be efﬁciently implemented in an online fashion noticing that,
[Equation]
For instance, setting ˜ ρ t = 1 t provides uniform averaging ( ρ t = 1 ) and ˜ ρ t = 1 − β < 1 provides geometric averaging also known as exponential moving averaging ( ρ t = β t ).
Averaging is experimentally compared with the other techniques presented in this section in Fig.
In order to illustrate how averaging tackles the oscillatory behavior in game optimization, we consider a toy example where the discriminator and the generator are linear: D ϕ ( x ) =
ϕ
T x and G θ ( z ) =
θ z (implicitly deﬁning q θ ).
By replacing these expressions in the WGAN objective, 4 we get the following bilinear objective:
[Equation]
A similar task was presented by Nagarajan and Kolter (2017) where they consider a quadratic discriminator instead of a linear one, and show that gradient descent is not necessarily asymptotically stable.
The bilinear objective has been extensively used (Goodfellow, 2016; Mescheder et al., 2018;
Yadav et al., 2018) to highlight the difﬁculties of gradient descent for saddle point optimization.
Yet, ways to cope with this issue have been proposed decades ago in the context of mathematical programming.
Simplifying further by setting the dimension to 1 and centering the equilibrium to the origin, Eq. 9 becomes:
[Equation]
max
[Equation]
and
[Equation]
The operator associated with this minimax game is F ( θ, φ ) =
( φ, − θ ) .
There are several ways to compute the discrete updates of this dynamics.
The two most common ones are the simultaneous and the alternating gradient update rules,
[Equation]
Interestingly, these two choices give rise to have a completely different behavior.
The norm of the simultaneous updates diverges geometrically whereas the alternating iterates are bounded but do not converge to the equilibrium.
As a consequence, their respective uniform average have a different behavior, as highlighted in the following proposition (more details and proof in §B.1):
Proposition 1.
The simultaneous iterates diverge geometrically and the alternating iterates deﬁned in (11) are bounded but do not converge to 0 as
Simultaneous:
[Equation]
where u t =
Θ( v t ) ⇔ ∃ α, β > 0 : αv t ≤ u t ≤ βv t .
The uniform average (¯ θ t , ¯ φ t )
def = 1 t (cid:80) t − 1 s =
( θ s , φ s ) of the simultaneous updates (resp.
the alternating updates ) diverges (resp.
converges to 0) as,
Simultaneous:
[Equation]
This sublinear convergence result, proved in §B, underlines the beneﬁts of averaging when the sequence of iterates is bounded (i e. for alternating update rule).
When the sequence of iterates is not bounded (i e. for simultaneous updates) averaging fails to ensure convergence. This theorem also shows how alternating updates may have better convergence properties than simultaneous updates
3. E XTRAPOLATION
Another technique used in the variational inequality literature to prevent oscillations is extrapolation .
This concept is anterior to the extragradient method since Korpelevich (1976) mentions that the idea of extrapolated “prices” to give “stability” had been already formulated by Polyak (1963, Chap. II).
The idea behind this technique is to compute the gradient at an (extrapolated) point different from the current point from which the update is performed, stabilizing the dynamics:
[Equation]
Note that, even in the unconstrained case , this method is intrinsically different from Nesterov’s momentum 5 (Nesterov, 1983, Eq. 2.2.9) because of this lookahead step for the gradient computation:
Nesterov’s method:
[Equation]
Nesterov’s method does not converge when trying to optimize (10).
One intuition explaining why extrapolation provides better convergence properties than the standard gradient method comes from Euler’s method framework (see for instance (Atkinson, 2003) for more details on that topic).
Actually, if we consider a ﬁrst order approximation of ω t +1 / 2 , we have ω t +1 / 2 ≈ ω t
+
+ o ( η ) and consequently, the update step (15) is close to an implicit method step:
[Equation]
[Equation]
In the literature on Euler’s method, implicit methods are known to be more stable and to beneﬁt from better convergence properties (Atkinson, 2003) than explicit methods .
They are not often used in practice though since they require to solve a potentially non-linear system at each iteration.
Taking back the simpliﬁed WGAN toy example (10) from §3.1 we get the following update rules,
Implicit:
[Equation]
In the following proposition, we will see that the respective convergence rates of the implicit method and extrapolation are highly similar.
Keeping in mind that the latter has the major advantage of being more practical, this proposition clearly underlines the beneﬁts of extrapolation (more details and proof in §B.2),
Proposition 2.
The squared norm of the iterates N t def = θ 2 t + φ 2 t , where the update rule of θ t and φ t are deﬁned in (18) , decreases geometrically for any η < 1 as,
[Equation]
3. E XTRAPOLATION FROM THE PAST
One issue with extrapolation is that the algorithm “wastes” a gradient (14).
Indeed we need to compute the gradient at two different positions for every single update of the parameters.
We thus propose a new technique that we call extrapolation from the past which only requires to compute one gradient for every update.
The idea of this technique is to store and re-use the previous extrapolated gradient to compute the new extrapolation point:
[Equation]
This update scheme can be related to (Chiang et al., 2012, Alg. 1) in the context of online learning and to the optimistic mirror descent (Rakhlin and Sridharan, 2013; Daskalakis et al., 2018). In the unconstrained case, (20) and (21) reduce to:
[Equation]
However our technique comes from a different perspective, it was motivated by VIP and inspired from the extragradient method.
Furthermore our technique extends to constrained optimization as shown in (20) and (21).
It is not clear whether or not a single projection added to (22) provides a provably converging algorithm.
Using the VIP point of view we are able to prove a linear convergence rate for a projected version of the extrapolation from the past (see details and proof of Theorem 1 in §B.3).
We also extend these results to the stochastic operator setting in §4.
−  2 θ − 1 .− 0 .0 .0 .1 .φ
Start
Adam with γ = 0 .
Gradient method γ = 0 .
Averaging γ = 2 .
Extrapolation from the past γ = 0 .
Extrapolation γ = 0 .
Algorithm 1 AvgSGD
Let ω 0 ∈ Ω for t = 0 . . .
T − 1 do ξ t ∼ P (minibatch) d t ← F ( ω t , ξ t ) ω
t +1 ← P Ω [ ω t − η t d t ] end for
Return ¯ ω T ← (cid:80) T − 1 t =
η t ω t (cid:80) T −t =
η t
Algorithm t = 0 . . .
T − 1 do ξ t , ξ (cid:48) t ∼ P (minibatches) d t ← F ( ω t , ξ t ) ω (cid:48)
t ← P Ω [ ω t − η t d t ] d (cid:48)
t ← F ( ω (cid:48) t , ξ (cid:48) t ) ω
t +1 ← P Ω [ ω t − η t d (cid:48) t ] end for
Return ¯ ω T ← (cid:80) T − 1 t =
η t ω (cid:48) t (cid:80) T −t =
η t
Let ω 0 ∈ Ω for t = 0 . . .
T − 1 do ξ t ∼ P (minibatch) ω (cid:48) t ← P Ω [ ω t − η t d t − 1 ] d t ← F ( ω (cid:48) t , ξ t )
ω t +1 ← P Ω [ ω t − η t d t ] end for
Return ¯ ω T ← (cid:80) T − {+1 t =
+}
{+η t ω (cid:48) t (cid:80) T −t =
 η t+}
Theorem 1 (Linear convergence of extrapolation from the past ) .
If F is µ -strongly monotone (see §A for the deﬁnition of strong monotonicity) and L -Lipschitz, then the updates (20) and (21) with η = 14 L provide linearly converging iterates,
[Equation]
In comparison to the results from (Daskalakis et al., 2018) that hold only for a bilinear objective, we provide a faster convergence rate (linear vs sublinear) on the last iterate for a general (strongly monotone) operator F and any projection on a convex Ω .
One thing to notice is that the operator of a bilinear objective is not strongly monotone, but in that case one can use the standard extrapolation method (14) which converges linearly for a (constrained or not) bilinear game (Tseng, 1995, Cor. 3.3).
O PTIMIZATION OF VIP WITH STOCHASTIC GRADIENTS
In this section, we consider extensions of the techniques presented in section §3 for optimizing (VIP), to the context of a stochastic operator.
In this case, at each time step we no longer have access to the exact gradient F ( ω ) but to an unbiased stochastic estimate of it F ( ω , ξ ) where ξ ∼ P and
F ( ω ) :
=
E ξ ∼ P [ F ( ω , ξ )] .
This is motivated from the GAN formulation where we only have access to a ﬁnite sample estimate of the expected gradient, computed on a mini-batch.
For GANs, ξ is thus a mini-batch of points coming from the true data distribution p and the generator distribution q θ .
For our analysis, we require at least one of the two following assumptions on the stochastic operator:
Assumption 1.
Bounded variance by σ 2 : E ξ [ (cid:107) F ( ω ) − F ( ω , ξ ) (cid:107) 2 ] ≤ σ 2 , ∀ ω ∈ Ω .
Assumption 2.
Bounded expected squared norm by M 2 : E ξ [ (cid:107) F ( ω , ξ ) (cid:107) 2 ] ≤ M 2 , ∀ ω ∈ Ω .
Assump. 1 is standard in stochastic variational analysis, while Assump. 2 is a stronger assumption sometimes made in stochastic convex optimization.
To illustrate how strong Assump.
2 is, note that it does not hold for an unconstrained bilinear objective like in our example 10 in §3.
It is thus mainly reasonable for bounded constraint sets.
Note that in practice we have σ (cid:28) M .
We now present and analyse three algorithms, variants of SGD that are appropriate to solve (VIP).
The ﬁrst one Alg. 1 (AvgSGD) is the stochastic extension of the gradient method for solving (VIP);
Alg. 2 (AvgExtraSGD) uses extrapolation and Alg. 3 (AvgPastExtraSGD) uses extrapolation from the past .
A fourth variant (Alg.5) is proposed in §D. These three algorithms return an average of the iterates.
The proofs of the theorems presented in this section are in §F.
To handle constraints such as parameter clipping (Arjovsky et al., 2017), we present a projected version of theses algorithms, where P Ω [ ω (cid:48) ] denotes the projection of ω (cid:48) onto Ω (see §A).
Note that when Ω = R d , the projection is the identity mapping (unconstrained setting).
In order to prove the convergence of these three algorithms we will assume that F is monotone:
[Equation]
If F can be written as (6), it implies that the cost functions are convex 6 .
Note however that in general
GANs parametrized with neural networks lead to non-monotone VIPs.
Assumption 3.
F is monotone and Ω is a compact convex set, such that
[Equation]
In that setting the quantity g ( ω ) := max ω ∈ Ω F ( ω ∗ ) (cid:62) (¯ ω ∗ − ω ) is well deﬁned and is equal to
if and only if ω ∗ is a solution of (VIP).
Moreover, if we are optimizing a zero-sum game , we have ω
= ( θ , ϕ ) , Ω =
Θ × Φ and F ( θ , ϕ ) =
[ ∇ θ L ( θ , ϕ ) −∇ ϕ L ( θ , ϕ )] (cid:62) .
Hence, the quantity h ( θ , ϕ ) :
= max θ ∈ Θ L ( θ , ϕ ∗ ) − min
ϕ ∈
Φ L ( θ ∗ , ϕ ) is well deﬁned and equal to
if and only if ( θ ∗ , ϕ ∗ ) is a Nash equilibrium of the game.
The two functions g and h are called merit functions (more details on the concept of merit functions in §C).
In the following, we call,
[Equation]
Averaging.
Alg.
1 (AvgSGD) presents the stochastic gradient method with averaging , which reduces to the standard (simultaneous) SGD updates for the two-player games used in the GAN literature, but returning an average of the iterates.
Theorem 2.
Under Assump. 1, 2 and 3, SGD with averaging (Alg. 1) with a constant step-size gives,
[Equation]
where
[Equation]
Thm. 2 uses a similar proof as (Nemirovski et al., 2009).
The constant term η ( M 2 + σ 2 ) / 2 in (26) is called the variance term .
This type of bound is standard in stochastic optimization.
We also provide in §F a similar rate with an extra log factor when η t = η √ t .
We show that this variance term is smaller than the one of SGD with prediction method (Yadav et al., 2018) in §E.
Extrapolations.
Alg.
2 (AvgExtraSGD) adds an extrapolation step compared to Alg.
1 in order to reduce the oscillations due to the game between the two players.
A theoretical consequence is that it has a smaller variance term than (26).
As discussed previously, Assump. 2 made in Thm.for the convergence of Alg. 1 is very strong in the unbounded setting.
One advantage of SGD with extrapolation is that Thm.
does not require this assumption.
Theorem 3.
(Juditsky et al., 2011, Thm. 1) Under Assump.
1 and 3, if E ξ [ F ] is L -Lipschitz, then
SGD with extrapolation and averaging (Alg. 2) using a constant step-size η ≤ 1 √ 3 L gives,
[Equation]
where
[Equation]
Since in practice σ (cid:28) M , the variance term in (27) is signiﬁcantly smaller than the one in (26).
To summarize, SGD with extrapolation provides better convergence guarantees but requires two gradient computations and samples per iteration.
This motivates our new method, Alg.
3 (AvgPastExtraSGD) which uses extrapolation from the past and achieves the best of both worlds .
Theorem 4.
Under Assump. 1 and 3, if E ξ [ F ] is L -Lipschitz then SGD with extrapolation from the past using a constant step-size η ≤ 1 2 √ 3 L , gives that the averaged iterates converge as,
[Equation]
where
[Equation]
The bounds is similar to the one provided in Thm.
3 but each iteration of Alg.
3 is computationally half the cost of an iteration of Alg. 2.
C OMBINING THE TECHNIQUES WITH ESTABLISHED ALGORITHMS
In the previous sections we presented several techniques that converge on a simple bilinear example.
These techniques can be combined in practice with existing algorithms.
We propose to combine them to two standard algorithms used for training deep neural networks: the Adam optimizer (Kingma and
Ba, 2015) and the SGD optimizer (Robbins and Monro, 1951).
Note that in the case of a two-player game (3), the previous results can be generalized to gradient updates with a different step-size for each player by simply rescaling the objectives L ( θ ) and L ( ϕ ) by a different scaling factor.
A detailed pseudo-code for Adam with extrapolation step (Extra-Adam) is given in Algorithm 4.
[Table]
R ELATED W ORK
The extragradient method is the standard algorithm to optimize variational inequalities.
This algorithm has been originally introduced by Korpelevich (1976) and extended by Nesterov (2007) and Nemirovski (2004).
Stochastic versions of the extragradient have been recently analyzed (Juditsky et al., 2011;
Youseﬁan et al., 2014; Iusem et al., 2017) for stochastic variational inequalities with bounded constraints .
A linearly convergent variance reduced version of the stochastic gradient method has been proposed by Palaniappan and Bach (2016) for strongly monotone variational inequalities.
Several methods to stabilize GANs consist in transforming a zero-sum formulation into a more general game that can no longer be cast as a saddle point problem.
This is the case of the non-saturating formulation of GANs (Goodfellow et al., 2014; Fedus et al., 2018), the DCGANs (Radford et al., 2016), the gradient penalty 7 for WGANs (Gulrajani et al., 2017).
Yadav et al.
(2018) propose an optimization method for GANs based on AltSGD using a momentum based step on the generator.
Daskalakis et al.
(2018) proposed a method inspired from game theory.
Li et al.
(2017) suggest to dualize the GAN objective to reformulate it as a maximization problem and Mescheder et al.
(2017) propose to add the norm of the gradient in the objective and provide an interesting perspective on
GANs, interpreting the training as the search of a two-player game equilibrium.
A study of the continuous version of two player games has been conducted by Ratliff et al.
Interesting non-convex results were proved, for a new notion of regret minimization, by Hazan et al.
(2017) and in the context of GANs by Grnarova et al.
The technique of unrolling steps proposed by Metz et al.
(2017) can be confused with extrapolation but is actually fundamentally different: the perspective is try to construct the “true generator objective function" unrolling for K steps the updates of the generator and then update the discriminator.
Nevertheless the fact that this “true generator function" may not be found with a satisfying accuracy may lead to a different behavior than the one expected.
Regarding the averaging technique, some recent work appear to have already successfully used geometric averaging (7) for GANs in practice, but only brieﬂy mention it (Karras et al., 2018;
Mescheder et al., 2018).
By contrast the present work formally motivates and justiﬁes the use of averaging for GANs by relating them to the VIP perspective, and sheds light on its underlying intuitions in §3.1.
Another independent work (Yazıcı et al., 2018) made a similar attempt but in the context of regret minimization in games.
Mertikopoulos et al.
(2018) also independently explored extrapolation providing asymptotic convergence results (i.e. without any rate of convergence) in the context of coherent saddle point .
The coherence assumption is slightly weaker than monotonicity.
E XPERIMENTS
Our goal in this experimental section is not to provide new state-of-the art results with architectural improvements or a new GAN formulation but to show that using the techniques (with theoretical guarantees in the monotone case) that we introduced earlier allow us to optimize standard GANs in a better way.
These techniques, which are orthogonal to the design of new formulations of GAN optimization objectives, and to architectural choices, can potentially be used for the training of any type of GAN.
We will compare the following optimization algorithms: baselines are SGD and Adam using either simultaneous updates on the generator and on the discriminator (denoted SimAdam and SimSGD ) or k updates on the discriminator alternating with 1 update on the generator (denoted
AltSGD { k } and AltAdam { k } ) 8 .
Variants that use extrapolation are denoted ExtraSGD (Alg. 2) and ExtraAdam (Alg. 4).
Variants using extrapolation from the past are PastExtraSGD (Alg. 3) and PastExtraAdam (Alg. 4).
We also present results using as output the averaged iterates, adding
Avg as a preﬁx of the algorithm name when we use (uniform) averaging .
7. B ILINEAR SADDLE POINT ( STOCHASTIC )
[Figure]
We evaluate the performance of the various stochastic algorithms ﬁrst on a simple ( n = 10 3 , d = 10 3 ) ﬁnite sum bilinear objective (a monotone operator) constrained to [ − 1 , 1] d :
[Equation]
(29) solved by ( θ
[Equation]
, ϕ
[Equation]
) s.t.
[Equation]
where ¯ a def = 1 n (cid:80) ni
=1 a ( i ) , ¯ b def = 1 n (cid:80) ni
=1 b ( i ) and ¯ M def = 1 n (cid:80) ni
=1 M ( i ) .
Marices and vectors M ( i ) kj , a ( i ) k , b ( i ) k ; 1 ≤ i ≤ n , 1 ≤ j, k ≤ d were randomly generated, but ensuring that ( θ ∗ , ϕ ∗ ) would belong to [ − 1 , 1] d .
Results are shown in Fig.
We can see that AvgSGD and AvgPastExtraSGD perform the best on this task.
7. WGAN AND WGAN-GP ON CIFAR
We now evaluate the proposed techniques in the context of GAN training, which is a challenging stochastic optimization problem where the objectives of both players are non-convex.
We propose to evaluate the Adam variants of the different optimization algorithms (see Alg. 4 for Adam with extrapolation ) by training two different architectures on the CIFAR10 dataset (Krizhevsky and Hinton, 2009). First we consider a constrained zero-sum game by training the DCGAN architecture (Radford et al., 2016) with the WGAN objective and weight clipping as proposed in Arjovsky et al.
[Table]
[Figure]
[Figure]
n c o r e
[Figure]
Then we compared the different methods on a state of the art architecture, by training a ResNet with the WGAN-GP objective similar to Gulrajani et al.
Models are evaluated using the inception score (Salimans et al., 2016) computed on 10,000 samples.
For each algorithm we did an extensive search over the hyperparameters of Adam, we ﬁxed β 1 = 0 .and
β 2 = 0 . 9 for all methods as they seemed to perform well. We want to emphasize that as proposed by Heusel et al. (2017) it is quite important to set different learning rates for the generator and the discriminator.
Experiments were run with 5 random seeds for 500,000 updates of the generator.
Table 1 reports the best inception score achieved on this problem by each considered method.
We see that the techniques of extrapolation and averaging consistently enable improvements over the baselines (see §G.4 for more experiments on averaging ).
Fig.
4 shows training curves for each method (for their best performing learning rate), as well as samples from a ResNet generator trained with ExtraAdam on a WGAN-GP objective.
For both tasks, using an extrapolation step and averaging with Adam (ExtraAdam) outperformed all other methods.
ExtraAdam with averaging is able to reach state of the art results on CIFAR10 similar to Miyato et al.
We also observed that methods based on extrapolation are less sensitive to the choice of learning rate and can be used with higher learning rates with less degradation; see App.
§G.3 for more details.
C ONCLUSION
We newly addressed GAN objectives in the framework of variational inequality.
We tapped into the optimization literature to provide more principled sound techniques to optimize such games.
We leveraged these techniques to develop practical optimization algorithms suitable for a wide range of GAN training objectives (including non-zero sum games and projections onto constraints).
We experimentally veriﬁed that this could yield better trained models, achieving to our knowledge the best inception score when optimizing a WGAN objective on the reference unmodiﬁed DCGAN architecture (Radford et al., 2016).
The presented techniques address a fundamental problem in
GAN training in a principled way, and are orthogonal to the design of new GAN architectures and objectives.
They are thus likely to be widely applicable, and beneﬁt future development of GANs.
A D EFINITIONS
In this section we recall usual deﬁnitions and lemmas from convex analysis.
We start with the deﬁnitions and lemmas regarding the projection mapping.
A. P ROJECTION MAPPING
Deﬁnition 1.
The projection P Ω onto Ω is deﬁned as,
[Equation]
When Ω is a convex set this projection is unique.
This is a consequence of the following lemma that we will use in the following sections: the non-expansiveness of the projection onto a convex set.
Lemma 1.
Let Ω a convex set, the projection mapping P Ω : R d → Ω is nonexpansive, i.e.,
[Equation]
This is standard convex analysis result which can be found for instance in (Boyd and Vandenberghe, 2004).
The following Lemma is also standard in convex analysis and its proof uses similar arguments as the proof of Lemma 1.
Lemma 2.
Let ω ∈ Ω and ω
+ def = P Ω ( ω + u ) then for all ω (cid:48) ∈ Ω we have,
[Equation]
Proof of Lemma 2 .
We start by simply developing,
[Equation]
Then since ω + is the projection onto the convex set Ω of ω + u we have that ( ω + − ( ω + u )) (cid:62) ( ω + − ω (cid:48) ) ≤ 0 , ∀ ω (cid:48)
∈ Ω , leading to the result of the Lemma.
A. S MOOTHNESS AND M ONOTONICITY OF THE OPERATOR
Another important property used is the Lipschitzness of an operator.
Deﬁnition 2.
A mapping F : R p
→ R d is said to be L -Lipschitz if,
[Equation]
Deﬁnition 3.
A differentiable function f : Ω → R is said to be µ -strongly convex if
[Equation]
Deﬁnition 4.
A function ( θ , ϕ ) (cid:55)→ L ( θ , ϕ ) is said convex-concave if L ( · , ϕ ) is convex for all ϕ ∈ Φ and L ( θ , · ) is concave for all θ ∈ Θ .
An L is said to be µ -strongly convex concave if ( θ , ϕ )
(cid:55)→ L ( θ , ϕ ) − µ 2 (cid:107) θ
(cid:107) 22 + µ 2 (cid:107) ϕ (cid:107)
is convex concave
Deﬁnition 5.
For µ > 0 , an operator F : Ω → R d is said to be µ -strongly monotone if
[Equation]
B G RADIENT METHODS ON UNCONSTRAINED BILINEAR GAMES
In this section we will prove the results provided in §3, namely Proposition 1, Proposition 2 and
Theorem 1.
For Proposition 1 and 2 let us recall the context.
We wanted to derive properties of some gradient methods on the following simple illustrative example
[Equation]
B. P ROOF OF P ROPOSITION
Let us ﬁrst recall the proposition:
Proposition’ 1.
The simultaneous iterates diverge geometrically and the alternating iterates deﬁned in (11) are bounded but do not converge to 0 as
[Equation]
where u t =
Θ( v t ) ⇔ ∃ α, β > 0 : αv
[Equation]
≤ u t ≤ βv t .
The uniform average (¯ θ t , ¯ φ t )
def = 1 t (cid:80) t − 1 s =
( θ s , φ s ) of the simultaneous updates (resp.
the alternating updates ) diverges (resp.
converges to 0) as,
[Equation]
Proof.
Let us start with the simultaneous update rule:
[Equation]
Then we have,
[Equation]
The update rule (39) also gives us,
[Equation]
Summing these equation for 0 ≤ t ≤ T − 1 we get,
[Equation]
Let continue start with the alternating update rule
[Equation]
Then we have
[Equation]
by simple linear algebra
[Equation]
for η < [Equation] the matrix M
[Equation]
has two complex conjugate eigenvalues which are
[Equation]
and their squared magnitude is equal to det( M ) = 1 − η 2 + η 2 = 1 .
We can diagonalize M meaning that there exist P an invertible matrix
such that M = P − 1 diag( λ + , λ − ) P .
Then, we have
[Equation]
49) and consequently,
[Equation]
where (cid:107) · (cid:107) C is the norm in C 2 and (cid:107) P (cid:107) :
= max u
[Equation]
[Equation]
C is the induced matrix norm.
The same way we have
[Equation]
Hence, if θ 20 + φ 20 > 0 the sequence ( θ t , φ t ) is bounded but do not converge to 0.
Moreover the update rule gives us,
[Equation]
[Equation] Consequently, since θ 2 t + φ 2 t = O ( θ 20 + φ 20 ) ,
[Equation]
B. I MPLICIT AND EXTRAPOLATION METHOD
In this section we will prove a slightly more precise proposition than Proposition 2,
Proposition’ 2.
The squared norm of the iterates N t def = θ 2 t + φ 2 t , where the update rule of θ t and φ t deﬁned in (18) , decrease geometrically for any 0 < η < 1 as,
[Equation]
Proof.
Let us recall the update rule for the implicit method
[Equation]
Then,
[Equation] [Equation]
Implying that
[Equation]
For the extrapolation method we have the update rule
[Equation]
Implying that,
[Equation] [Equation]
[Equation]
B. E XTRAPOLATION FROM THE PAST
Let us recall what we call projected extrapolation form the past , where we used the notation ω (cid:48)
t =
ω t +1 / 2 for compactness,
Extrapolation from the
[Equation]
where P Ω [ · ] is the projection onto the constraint set Ω .
An operator F : Ω → R d is said to be µ -strongly monotone if
[Equation]
If F is strongly monotone we can prove the following theorem:
Theorem’ 1.
If F is µ -strongly monotone (see §A for the deﬁnition of strong monotonicity) and
L -Lipschitz, then the updates (20) and (21) with η = 14 L provide linearly converging iterates,
[Equation]
Proof.
In order to prove tis theorem we will prove a slightly more general result,
[Equation]
implying that
[Equation]
with the convention that ω (cid:48)
[Equation]
ω
[Equation]
Let us ﬁrst proof three technical lemmas,
Lemma 3.
If F is µ -strongly monotone we have,
[Equation]
Proof.
By strong monotonicity and optimality of ω ∗ ,
[Equation]
and then we use the inequality 2 (cid:107) ω (cid:48)
t − ω ∗ (cid:107) 22 ≥
[Equation]
ω t − ω ∗
[Equation]
22 − 2 (cid:107) ω (cid:48)
t − ω t (cid:107) 22 to get the result claimed.
Lemma 4.
If F is L -Lipschitz, we have for any ω ∈ Ω ,
[Equation]
Proof.
Applying
Lemma for ( ω , u , ω
[Equation]
, ω
[Equation]
) and
[Equation]
, we get
[Equation] [Equation]
72) and
[Equation] [Equation]
Summing (72) and (73) we get,
[Equation]
[Equation]
Then, we can use the Young’s inequality 2 a
[Equation]
22 to get,
[Equation]
Lemma 5.
For all t ≥ 0 , if we set ω (cid:48)− 2 = ω
(cid:48)− 1 = ω (cid:48)
we have
[Equation]
ω (cid:48)
[Equation]
1 − ω (cid:48)
[Equation]
Proof.
We start with (cid:107) a + b (cid:107) 22 ≤ 2 (cid:107) a (cid:107) 2 + 2 (cid:107) b (cid:107) 2 .
[Equation]
Moreover
[Equation]
since the projection is contractive we have that
[Equation]
Combining (80) and (83) we get,
[Equation]
Proof of Theorem 1.
Let ω ∗ ∈
Ω ∗ be an optimal point of (VIP).
Combining Lemma 3 and Lemmawe get,
[Equation]
leading to,
[Equation]
Now using Lemma 5 we get,
[Equation]
Now with η t = 14 L ≤ 14 µ we get,
[Equation]
Hence, using the fact that µ 4 L ≤ 14 we get,
[Equation]
C M ORE ON MERIT FUNCTIONS
In this section, we will present how to handle unbounded constaints set Ω with a more reﬁned merit function than (25) used in the main paper.
Let F the continuous operator and Ω the constraints set associated with the VIP
[Equation]
When the operator F is monotone, we have that F ( ω ∗ ) (cid:62) ( ω − ω ∗ ) ≤ F ( ω ) (cid:62) ( ω − ω ∗ ) , ∀ ω , ω ∗ .
Hence, in this case (VIP) implies a stronger formulation sometimes called Minty variational inequality (Crespi et al., 2005):
[Equation]
that F
[Equation]
ω
[Equation]
This formulation is stronger in the sense that if (MVI) holds for some ω ∗ ∈ Ω , then (VIP) holds too.
A merit function useful for our analysis can be derived from this formulation.
Roughly, a merit function is a convergence measure.
More formally, a function g : Ω → R is called a merit function if g is non-negative such that g ( ω ) = 0 ⇔ ω ∈ Ω ∗ (Larsson and Patriksson, 1994).
A way to derive a merit function from (MVI) would be to use g ( ω ∗ )
= sup ω ∈ Ω F ( ω ) (cid:62) ( ω ∗ − ω ) which is zero if and only if (MVI) holds for ω ∗ .
To deal with unbounded constraint sets (leading to a potentially inﬁnite valued function outside of the optimal set), we use the restricted merit function (Nesterov, 2007):
[Equation]
This function acts as merit function for (VIP) on the interior of the open ball of radius R around ω 0 , as shown in Lemma 1 of Nesterov (2007).
That is, let Ω R def = Ω ∩ { ω : (cid:107) ω − ω
(cid:107) < R } .
Then for any point ˆ ω ∈ Ω R , we have:
[Equation]
The reference point ω 0 is arbitrary, but in practice it is usually the initialization point of the algorithm.
R has to be big enough to ensure that Ω R contains a solution.
Err R measures how much (MVI) is violated on the restriction Ω R .
Such merit function is standard in the variational inequality literature.
A similar one is used in (Nemirovski, 2004; Juditsky et al., 2011).
When F is derived from the gradients (5) of a zero-sum game, we can deﬁne a more interpretable merit function.
One has to be careful though when extending properties from the minimization setting to the saddle point setting (e.g. the merit function used by Yadav et al.
(2018) is vacuous for a bilinear game as explained in
App C.2).
In the appendix we adopt a set of assumptions a little more general than the one in the main paper
Assumption 4.
• F is monotone and Ω is convex and closed.
• R is set big enough such that R > (cid:107) ω 0 − ω ∗ (cid:107) and F is a monotone operator.
Contrary to Assumption 3, in Assumption 4 the constraints set in no longer assumed to be bounded.
Assumption 4 is more general than Assumption 3 since that if Ω is bounded then with R set to the diameter of Ω , Assumption 4 is true.
C. M ORE GENERAL MERIT FUNCTIONS
In the appendix we will note Err ( VI ) R the restricted merit function deﬁned in (90).
Let us recall its deﬁnition,
[Equation]
When the objective is a saddle point problem i.e.
[Equation]
and L is convex-concave (see Deﬁnition 4 in §A), we can use another merit function than (92) on Ω R that is more interpretable and more directly related to the cost function of the minimax formulation:
[Equation]
In particular, if the equilibrium ( θ ∗ , ϕ ∗ ) ∈
Ω ∗ ∩ Ω R and we have that L ( · , ϕ ∗ ) and −L ( θ ∗ , · ) are µ -strongly convex (see §A), then the merit function for saddle points upper bounds the distance for ( θ , ϕ ) ∈
Ω R to the equilibrium as:
[Equation]
In the appendix, we provide our convergence results with the merit functions (92) and (94), depending on the setup:
[Equation]
C. O N THE IMPORTANCE OF THE MERIT FUNCTION
In this section, we illustrate the fact that one has to be careful when extending results and properties from the minimization setting to the minimax setting (and consequently to the variational inequality setting).
Another candidate as a merit function for saddle point optimization would be to naturally extend the suboptimality f ( ω ) − f ( ω ∗ ) used in standard minimization (i.e. ﬁnd ω ∗ the minimizer of f ) to the gap P ( θ , ϕ ) =
L ( θ , ϕ ∗ ) − L ( θ ∗ , ϕ ) .
In a previous analysis of a modiﬁcation of the stochastic gradient descent (SGD) method for GANs, Yadav et al.
(2018) gave their convergence rate on P that they called the “primal-dual“ gap.
Unfortunately, if we do not assume that the function L is strongly convex-concave (a stronger assumption deﬁned in §A and which fails for bilinear objective e.g.), P may not be a merit function .
It can be 0 for a non optimal point, see for instance the discussion on the differences between (94) and P in (Gidel et al., 2017, Section 3).
In particular, for the simple 2D bilinear example L ( θ , ϕ )
= θ · ϕ , we have that θ ∗ = ϕ ∗ = 0 and thus P ( θ , ϕ ) =
0 ∀ θ , ϕ .
C. V ARIATIONAL INEQUALITIES FOR NON - CONVEX COST FUNCTIONS
When the cost functions deﬁned in (3) are non-convex, the operator F is no longer monotone.
Nevertheless, (VIP) and (MVI) can still be deﬁned, though a solution to (MVI) is less likely to exist.
We note that (VIP) is a local condition for F (as only evaluating F at the points ω ∗ ).
On the other hand, an appealing property of (MVI) is that it is a global condition.
In the context of minimization of a function f for example (where F = ∇ f ), if ω ∗ solves (MVI) then ω ∗ is a global minimum of f (and not just a stationary point for the solution of (MVI); see Proposition 2.2 from Crespi et al.
A less restrictive way to consider variational inequalities in the non-monotone setting is to use a local version of (MVI).
If the cost functions are locally convex around the optimal couple ( θ ∗ , ϕ ∗ ) and if our iterates eventually fall and stay into that neighborhood, then we can consider our restricted merit function Err R ( · ) with a well suited constant R and apply our convergence results for monotone operators.
D A NOTHER WAY OF IMPLEMENTING EXTRAPOLATION TO SGD
We now introduce another way to combine extrapolation and SGD.
This extension is very similar to Alg.
5, the only difference is that it re-uses the minibatch sample of the extrapolation step for the update of the current point.
The intuition is that it correlates the estimator of the gradient of the extrapolation step and the one of the update step leading to a better correction of the oscillations which are also due to the stochasticity.
One emerging issue (for the analysis) of this method is that since ω (cid:48) t depend on ξ t , the quantity F ( ω (cid:48) t , ξ t ) is a biased estimator of F ( ω (cid:48)
Algorithm 5 Re-used minibatches for stochastic extrapolation (ReExtraSGD) 1: Let ω 0 ∈ Ω 2: for t = 0 . . .
T − 1 do 3:
Sample ξ t ∼ P 4: ω (cid:48) t def = P Ω [ ω t −
[Equation]
F (
[Equation]
t )]
(cid:46) Extrapolation step 5: ω t +1 def = P Ω [ ω t − η
[Equation]
F (cid:0) ω (cid:48) t , ξ t (cid:1) ]
(cid:46) Update step wit the same sample 6: end for
: Return ¯ ω T = (cid:80) T − 1 t =
η t ω (cid:48)
t / (cid:80) T − 1 t =
η t
Theorem 5.
Assume that (cid:107) ω (cid:48)
t − ω 0 (cid:107) ≤ R, ∀ t ≥ 0 where ( ω (cid:48) t ) t ≥ 0 are the iterates of Alg. 5.
Under
Assumption 1 and 4, for any T ≥ 1 , Alg. 5 with constant step-size η ≤ 1 √ 2 L has the following convergence properties:
[Equation]
where ¯
[Equation]
def
[Equation]
Particularly, η t = η √ T gives E [Err R (¯ ω T )] ≤ O (1) √ T .
The assumption that the sequence of the iterates provided by the algorithm is bounded is strong, but has been also done for instance in (Yadav et al., 2018).
The proof of this result is provided in §F.
E V ARIANCE COMPARISON BETWEEN A VG SGD AND SGD WITH PREDICTION METHOD
To compare the variance term of AvgSGD in (26) with the one of the SGD with prediction method (Yadav et al., 2018), we need to have the same convergence certiﬁcate.
Fortunately, their proof can be adapted to our convergence criterion (using Lemma 6 in §F), revealing an extra σ 2 / 2 in the variance term from their paper.
The resulting variance can be summarized with our notation as ( M 2 (1 + L ) + σ 2 ) / 2 where the L is the Lipschitz constant of the operator F .
Since M (cid:29) σ , their variance term is then 1 + L time larger than the one provided by the AvgSGD method.
F P ROOF OF T HEOREMS
This section is dedicated on the proof of the theorems provided in this paper in a slightly more general form working with the merit function deﬁned in (96).
First we prove an additional lemma necessary to the proof of our theorems.
Lemma 6.
Let F be a monotone operator and let ( ω t ) , ( ω (cid:48) t ) , ( z t ) , (∆ t ) , ( ξ t ) and ( ζ t ) be six random sequences such that, for all t ≥
[Equation]
ω ) , where N t = N ( ω t , ω (cid:48)
t − 1 , ω (cid:48)
t − 2 ) ≥
and we extend ( ω (cid:48) t ) with ω (cid:48)− 2 = ω (cid:48)− 1 = ω (cid:48) 0 .
Let also assume that with N 0 ≤ R , E
[ (cid:107) ∆ t (cid:107) 22 ] ≤ σ 2 , E [∆ t | z t , ∆ 0 , . . .
, ∆ t − 1 ] = 0 , E [ M 1 ( ω t , ξ t )] ≤ M 1 and E [ M 2 ( ω (cid:48) t , ζ t )] ≤ M 2 , then,
[Equation]
(97) where ¯ ω T def = (cid:80) T − 1 t =
η t ω (cid:48)
t /S T and S T def =
(cid:80) T − 1 t =0 η t .
Proof of Lemma 6 .
We sum (6) for 0 ≤ t ≤ T − 1 to get,
[Equation]
We will then upper bound each sum in the right-hand side,
[Equation]
where u t +1 def = P Ω ( u t − η t ∆ t ) and u 0 def = ω 0 .
Then,
[Equation]
leading to
[Equation]
Then noticing that z 0 def = ω 0 , back to (98) we get a telescoping sum,
[Equation]
If F is the operator of a convex-concave saddle point (5), we get, with ω (cid:48)
t
= ( θ t , ϕ t )
[Equation]
( by convexity and concavity
[Equation]
then by convexity of L ( · , ϕ ) and concavity of L ( θ , · ) , we have that,
[Equation]
Otherwise if the operator F is just monotone since F ( ω (cid:48) t ) (cid:62) ( ω (cid:48)
t − ω ) ≥ F ( ω (cid:48) ) (cid:62) ( ω (cid:48)
t − ω ) we have that
[Equation]
In both cases, we can now maximize the left hand side respect to ω (since the RHS does not depend on ω ) to get,
[Equation]
Then taking the expectation, since E [∆ t | z t , u t ]
[Equation]
E ζ t [
[Equation]
∆
[Equation]
(cid:107) 22 ] ≤ σ 2 , E ξ t [ M 1 ( ω t , ξ t )] ≤ M 1 and E ζ t
[ M 2 ( ω (cid:48) t , ζ t )] ≤ M 2 , we get that,
[Equation]
F. P ROOF OF T HM .
First let us state Theorem 2 in its general form,
Theorem’ 2.
Under Assumption 1, 2 and 4, Alg. 1 with constant step-size η has the following convergence rate for all T ≥ 1 ,
[Equation]
where
[Equation]
Particularly, η =
R √ T ( M 2 + σ 2 ) gives E
[Err R (¯ ω T )] ≤ R
[Equation]
2 +
[Equation]
√
[Equation]
Proof of Theorem 2 .
Let any ω ∈ Ω such that (cid:107) ω 0 − ω (cid:107) 2 ≤ R ,
[Equation]
( projections are non-contractive, Lemma 1 )
[Equation]
Then we can make appear the quantity F ( ω t ) (cid:62) ( ω t − ω ) on the left-hand side,
[Equation]
[Equation]
we can sum (106) for 0 ≤ t ≤ T − 1 to get,
[Equation]
where we noted ∆ t def = F ( ω t ) − F ( ω t , ξ t ) .
By monotonicity, F ( ω t ) (cid:62) ( ω t − ω ) ≥ F ( ω ) (cid:62) ( ω t − ω ) we get,
[Equation]
[Equation]
where S T def = (cid:80) T − 1 t =
η t and ¯ ω T def = 1 S T (cid:80) T − 1 t =
η t ω t .
We will then upper bound each sum in the right hand side,
[Equation]
) where u t +1 def = P Ω ( u t − η t ∆ t ) and u 0 = ω 0 .
Then,
[Equation]
leading to
[Equation]
Then noticing that u 0 def = ω 0 , back to (108) we get a telescoping sum,
[Equation]
Then the right hand side does not depends on ω , we can maximize over ω to get,
[Equation]
Noticing that E [∆ t | ω t , u t ] =
(the estimates of F are unbiased), by Assumption 2 E
[( (cid:107) F ( ω t , ξ t ) (cid:107) 22 ] ≤ M 2 and by Assumption 1 E [ (cid:107) ∆ t (cid:107) 22 ] ≤ σ 2 we get,
[Equation]
111) particularly for η t = η and η t
=
η √ t +1 we respectively get,
[Equation]
112) and
[Equation]
F. P ROOF OF T HM .
Theorem’ 3.
Under Assumption 1 and 4, if E ξ [ F ] is L -Lipschitz, then Alg. 2 with a constant step-size η ≤ 1 √ 3 L has the following convergence rate for any T ≥ 1 ,
[Equation]
where
[Equation] [Equation]
Particularly, η = √ 2 R σ √ 7 T gives E [Err R (¯ ω T )] ≤ √ 14 Rσ √ T .
Proof of Thm.3 .
Let any ω ∈ Ω such that (cid:107) ω 0 − ω (cid:107) 2 ≤ R .
Then, the update rules become ω t +1 = P Ω ( ω t − η t F ( ω (cid:48) t , ζ t )) and ω (cid:48) t
= P Ω ( ω t − ηF ( ω t , ξ t )) .
We start by applying Lemma 2 for ( ω , u , ω (cid:48) , ω + ) =
( ω t , − ηF ( ω (cid:48) t , ζ t ) , ω , ω t +1 ) and ( ω , u , ω (cid:48) , ω + ) =
( ω t , − η t F ( ω t , ξ t ) , ω t +1 , ω (cid:48) t ) ,
[Equation]
Then, summing them we get
[Equation]
22 − (cid:107) ω t +1 − ω (cid:48) t (cid:107)(115) leading to (cid:107) ω t +1 − ω
(cid:107) 22 ≤ (cid:107) ω t − ω
(cid:107) 22 − 2 η t F ( ω (cid:48) t , ζ t ) (cid:62) ( ω (cid:48)
t − ω ) + 2 η t ( F ( ω (cid:48) t , ζ t ) − F ( ω t , ξ t ))
(cid:62) ( ω (cid:48)
t − ω t +1 ) − (cid:107) ω
t − ω (cid:48) t (cid:107) 22 − (cid:107) ω
t +1 − ω (cid:48) t (cid:107)
Then with 2 a (cid:62) b ≤ (cid:107) a (cid:107) 22 + (cid:107) b (cid:107) 22 we get (cid:107) ω t +1 − ω (cid:107) 22 ≤ (cid:107) ω t − ω
(cid:107) 22 − 2 η t F ( ω (cid:48) t , ζ t ) (cid:62) ( ω (cid:48)
t − ω ) − (cid:107) ω
t − ω (cid:48) t (cid:107)
+ η 2 t (cid:107) F ( ω (cid:48) t , ζ t ) − F ( ω t , ξ t ) (cid:107)
Using the inequality (cid:107) a + b + c (cid:107) 22 ≤ 3( (cid:107) a (cid:107)
+ (cid:107) b (cid:107)
+ (cid:107) c (cid:107) 22 ) we get, (cid:107) ω
t +1 − ω
(cid:107) 22 ≤ (cid:107) ω t − ω
(cid:107) 22 − 2 η t F ( ω (cid:48) t , ζ t ) (cid:62) ( ω (cid:48)
t − ω ) − (cid:107) ω
t − ω (cid:48)
[Equation] [Equation]
Then we can use the L -Lipschitzness of F to get,
[Equation]
As we restricted the step-size to η t ≤ 1 √ 3 L we get,
[Equation]
We get a particular case of (6) so we can use Lemma 6 where N t
[Equation]
M
[Equation]
) =
[Equation] [Equation]
[Equation]
and z t
[Equation]
By Assumption 1, M
[Equation]
3 σ 2 and by the fact that E
[Equation]
the hypothesis of Lemma 6 hold and we get,
[Equation]
F. P ROOF OF T HM .
Theorem’ 4.
Under Assumption
[Equation]
if E ξ [ F ] is L -Lipschitz
[Equation]
then AvgPastExtraSGD (Alg. 3) with a constant step-size η ≤ 1 2 √ 3 L has the following convergence rate for any T ≥ 1 ,
[Equation]
where
[Equation]
Particularly, η = √ 2 R σ √ 7 T gives E [Err R (¯ ω T )] ≤ √ 14 Rσ √ T .
First let us recall the update rule
[Equation]
Lemma 7.
We have for any ω ∈ Ω ,
[Equation]
Proof.
Applying Lemma 2 for ( ω , u , ω + , ω (cid:48) )
= ( ω t , − η t F ( ω (cid:48) t , ξ t ) , ω t +1 , ω ) and ( ω , u , ω + , ω (cid:48) )
= ( ω t , − η t F ( ω (cid:48)
t − 1 , ξ t − 1 ) , ω (cid:48) t , ω t +1 ) , we get, (cid:107) ω
t +1 − ω
(cid:107) 22 ≤ (cid:107) ω t − ω
(cid:107) 22 − 2 η t F ( ω (cid:48) t , ξ t ) (cid:62) ( ω t +1 − ω ) − (cid:107) ω
t +1 − ω t (cid:107)(120) and (cid:107)
[Equation]
Summing (120) and (121) we get,
[Equation]
Then, we can use the inequality of arithmetic and geometric means 2 a (cid:62)
[Equation]
≤
[Equation]
a
[Equation]
22 +
[Equation]
b
[Equation]
22 to get,
[Equation] [Equation]
[Equation] [Equation]
Using the inequality (cid:107) a + b + c (cid:107) 22 ≤ 3( (cid:107) a
[Equation]
22 +
[Equation]
22 + (cid:107) c
[Equation]
22 ) we get,
[Equation] [Equation]
[Equation]
where we used the L -Lipschitzness of F for the last inequality.
Combining (128) with (130) we get,
[Equation]
Lemma 8.
For all t ≥ 0 , if we set ω (cid:48)− 2 = ω
(cid:48)− 1 = ω (cid:48)
we have (cid:107)
[Equation]
Proof.
We start with (cid:107) a + b (cid:107) 22 ≤ 2 (cid:107) a (cid:107) 2 + 2 (cid:107) b (cid:107) 2 .
(cid:107) ω (cid:48)
t
[Equation]
1 − ω (cid:48) t (cid:107) 22 ≤ 2 (cid:107)
[Equation]
Moreover
[Equation]
since the projection is contractive we have that
[Equation] [Equation]
where in the last line we used the same inequality as in (130)
[Equation]
Combining (132) and (136) we get,
[Equation]
Proof of Theorem 4 .
Combining Lemma 8 and Lemma 7 we get,
[Equation]
[Equation]
Then for η t ≤
[Equation]
we have 36 η 2 t
[Equation]
We can then use Lemma 6 where
[Equation]
By
Assumption
[Equation]
and by the fact that
[Equation]
the hypothesis of Lemma 6 hold and we get,
[Equation]
F. P ROOF OF T HEOREM
Theorem 5 has been introduced in §D. This theorem is about Algorithm 5 which consists in another way to implement extrapolation to SGD.
Let us ﬁrst restate this theorem,
Theorem’ 5.
Assume that (cid:107) ω (cid:48)
t − ω 0 (cid:107) ≤ R, ∀ t ≥ 0 where ( ω (cid:48) t ) t ≥ 0 are the iterates of Alg. 5.
Under
Assumption 1 and 4, for any T ≥ 1 , Alg. 5 with constant step-size η ≤ 1 √ 2 L has the following convergence properties:
[Equation]
where
[Equation]
Particularly, η t = η √ T gives E [Err R (¯ ω T )] ≤ O (1) √ T .
Proof of Thm.5 .
Let any ω ∈ Ω such that (cid:107) ω 0 − ω (cid:107) 2 ≤ R .
Then, the update rules become ω t +1 = P Ω ( ω t − η t F ( ω (cid:48) t , ξ t )) and ω (cid:48) t
= P Ω ( ω t − ηF ( ω t , ξ t )) .
We start the same way as the proof of Thm. 3 by applying Lemma 2 for ( ω , u , ω (cid:48) , ω ∗ ) =
( ω t , − ηF ( ω (cid:48) t , ξ t ) , ω , ω t +1 ) and ( ω , u , ω (cid:48) , ω + ) =
[Equation]
ω t , −
[Equation]
t F
[Equation]
ω t ,
[Equation]
t ) , ω t +1 , ω
[Equation]
t ) ,
[Equation]
Then, summing them we get (cid:107) ω t
[Equation]
1 − ω
[Equation]
22 ≤
[Equation]
ω t − ω
[Equation]
22 −
[Equation]
η t F
[Equation]
t
[Equation] [Equation]
leading to
[Equation]
Then with 2 a (cid:62) b ≤ (cid:107) a (cid:107) 22 + (cid:107) b (cid:107) 22 we get
[Equation]
Using the Lipschitzness assumption we get
[Equation]
Then we add 2 η t F ( ω (cid:48) t ) (cid:62) ( ω (cid:48)
t − ω ) in both sides to get,
[Equation]
η t F ( ω (cid:48) t ) (cid:62) ( ω (cid:48)
[Equation] [Equation]
Here, unfortunately we cannot use Lemma 6 because F ( ω (cid:48) t , ξ t ) is biased.
We will then deal with the quantity
A =
( F ( ω (cid:48) t , ξ t )
[Equation]
F ( ω
[Equation] [Equation]
( ω (cid:48)
t − ω ) .
We have that,
[Equation]
( Using Cauchy-Schwarz and the L -Lip of F )
Then using 2 (cid:107) a (cid:107)(cid:107) b (cid:107) ≤ δ (cid:107) a (cid:107)
+ 1 δ (cid:107) b (cid:107) 22 , for δ = 4 , − 2 η t
[Equation] [Equation]
[Equation] [Equation]
leading to,
[Equation]
If one assumes ﬁnally that (cid:107) ω (cid:48) t − ω 0 (cid:107) 2 ≤ R (assumption of the theorem) and that η t ≤ 12 L we get,
[Equation]
where we used that (cid:107) ω (cid:48)
t − ω
[Equation] [Equation]
Once again this equation is a particular case of Lemma 6 where N t = (cid:107)
[Equation]
ω (cid:107) 22 ,
M
[Equation]
(cid:107)
[Equation]
M
[Equation] [Equation]
(cid:48)
[Equation]
z t = ω t and ∆ t = F ( ω t , ξ t ) − F ( ω t ) .
By Assumption 1 E [ M 1 ( ω t , ξ t )] ≤ 16 L 2 R 2 + 4 σ 2 and E [∆ t | ω t , ∆ 0 , . . . , ∆ t − 1 ] =
E [ E [∆ t | ω t ] | ∆ 0 , . .
. , ∆ t − 1 ] =
so we can use Lemma 6 and get,
[Equation]
G A DDITIONAL EXPERIMENTAL RESULTS G. T OY NON - CONVEX GAN (2D AND DETERMINISTIC )
We now consider a task similar to (Mescheder et al., 2018) where the discriminator is linear
D ϕ ( ω ) =
ϕ T ω , the generator is a Dirac distribution at θ , q θ = δ θ and the distribution we try to match is also a Dirac at ω ∗ , p = δ ω ∗ .
The minimax formulation from Goodfellow et al.
(2014) gives:
[Equation]
Note that as observed by Nagarajan and Kolter (2017) [Figure] this objective is concave-concave, making it hard to optimize.
We compare the methods on this objective where we take ω ∗ = − 2 , thus the position of the equilibrium is shifted towards the position ( θ , ϕ )
= ( − 2 , 0) .
The convergence and the gradient vector ﬁeld are shown in Figure 5.
We observe that depending on the initialization some methods can fail to converge but extrapolation (18) seem to perform better than the rest of the methods.
[Figure]
G. DCGAN WITH WGAN-GP OBJECTIVE
We also trained the DCGAN architecture of the WGAN experiments but this time with the WGAN-GP objective.
The results are shown in Table 2.
The best results are achieved with uniform averaging of AltAdam5, However its iterations require to update the discriminator 5 times for every generator update.
With a small drop in best ﬁnal score, ExtraAdam can train WGAN-GP signiﬁcantly faster (see Fig.
6 right) as the discriminator and generator are updated only twice.
[Table]
[Figure]
G. C OMPARISON OF THE METHODS WITH THE SAME LEARNING RATE
In this section we compare how the methods presented in §7 perform with the same step-size.
We follow the same protocol as in the experimental section §7.
In Figure 7 we plot the inception score provided by each training method as a function of the number of generator updates.
Note that these plots advantage AltAdam5 a bit because each iteration of this algorithm is a bit more costly (since it perform 5 discriminator updates for each generator update).
Nevertheless, the goal of this experiment is not to show that AltAdam5 is faster but to show that that ExtraAdam is less sensitive to the choice of learning rate and can be used with higher learning rates with less degradation.
The same way in Figure 8 we compare the sample quality on the WGAN-GP experiment (DCGAN architecture) of AltAdam5 and AvgExtraAdam for different step-sizes.
We notice that for AvgExtraAdam the sample quality do not signiﬁcantly change whereas the sample quality of AltAdam seems to be really sensitive to step-size tunning.
We think that robustness to step-size tuning is a key property for an optimization algorithm in order to save as much time as possible to tune other hyperparameters of the learning procedure such as regularization.
[Figure]
[Figure]
G. C OMPARISON OF THE METHODS WITH AND WITHOUT AVERAGING
In this section we compare how uniform averaging improve the performance of the methods presented in §7.
We follow the same protocol as in the experimental section §7.
In Figure 9 and 10, we plot the inception score provided by each training method as a function of the number of generator updates with and without averaging
[Figure] We notice that uniform averaging seems to improve the inception score, nevertheless it looks like the sample are a bit more blurry (see Figure 11) this is conﬁrmed by our result (Figure 12) on the Fréchet
Inception Distance (FID) which is more sensitive to blurriness.
[Figure] [Figure]
[Figure]
[Figure]