E scaping limit cycles : G lobal convergence for constrained nonconvex - nonconcave minimax problems
A bstract
This paper introduces a new extragradient-type algorithm for a class of nonconvex-nonconcave minimax problems.
It is well-known that ﬁnding a local solution for general minimax problems is computationally intractable.
This observation has recently motivated the study of structures su ﬃ cient for convergence of ﬁrst order methods in the more general setting of variational inequalities when the so-called weak Minty variational inequality (MVI) holds.
This problem class captures non-trivial structures as we demonstrate with examples, for which a large family of existing algorithms provably converge to limit cycles.
Our results require a less restrictive parameter range in the weak MVI compared to what is previously known, thus extending the applicability of our scheme.
The proposed algorithm is applicable to constrained and regularized problems, and involves an adaptive stepsize allowing for potentially larger stepsizes.
Our scheme also converges globally even in settings where the underlying operator exhibits limit cycles.
Moreover, a variant with stochastic oracles is proposed—making it directly relevant for training of generative adversarial networks.
For the stochastic algorithm only one of the stepsizes is required to be diminishing while the other may remain constant, making it interesting even in the monotone setting.
I ntroduction
Many machine learning applications, from generative adversarial networks (GANs) to robust reinforcement learning, result in nonconvex-nonconcave constrained minimax problems, which pose notorious di ﬃ culties to the scalable (stochastic)
ﬁrst order methods.
Indeed, there is no shortage of results illustrating divergent or cycling behavior when going beyond minimization problems (Benaım & Hirsch, 1999; Hommes & Ochea, 2012; Mertikopoulos et al., 2018b; Hsieh et al., 2021).
Traditionally, minimax problems have been studied for more than half a century under the umbrella of the variational inequalities (VIs).
The extragradient-type algorithms from the VI literature was recently brought to the awareness of the machine learning community (Mertikopoulos et al., 2018a;
Gidel et al., 2018; Böhm et al., 2020), and have provided a principled way of stabilizing training and avoiding Poincaré recursions.
However, these results mostly concern the convex-concave setting.
In nonconvex-nonconcave minimax problems, or more generally nonmonotone variational inequalities (VIs), even ﬁnding a local solution is in general intractable.
This has been made precise through exponential lower bound of the classical optimization type (Hirsch & Vavasis, 1987) and computational complexity results (Papadimitriou, 1994; Daskalakis et al., 2021b).
This is in sharp contrast to minimization problems, where only ﬁnding a global solution is intractable.
The recent result of (Hsieh et al., 2021) provides some intuition behind this di ﬀ erence by showing that the asymptotic limits of most schemes, including extragradient, can converge to attracting limit cycles.
To make progress in lieu of these negative results, Diakonikolas et al.
(2021) proposes a simple generalization of extragradient, called (EG + ), that can converge to a stationary point even for a class of nonmonotone problems provided that the weak Minty variational inequality (MVI) holds.
This problem class is parametrized by a constant ρ , which controls the degree of nonconvexity.
However, given the range of ρ in Diakonikolas et al.
(2021), the new class is still too small to include even the simplest counterexample of Hsieh et al.
(2021) for the general Robbins-Monro schemes.
Contributions Building on the analysis in Diakonikolas et al.
(2021), we propose a new adaptive scheme, called (CurvatureEG + ), that converges even in the di ﬃ cult counter example of Hsieh et al.
(2021) as illustrated in Fig.
Our main contributions are summarized below.
We propose an adaptive extragradient-type algorithm that converges for a larger range of ρ , the parameter in the weak MVI assumption (cf.
Assumption I (iii) ) than previously known.
More importantly, we show that convergence is ensured if 2 ρ ` γ k ą 0, where γ k is the extrapolation stepsize.
This is crucial since by selecting γ k through a backtracking procedure larger stepsizes are allowed, which in turn implies convergence for more negative values of ρ , thus capturing a larger class of problems.
In addition, we show that the linesearch eventually passes without triggering any backtrack if initialized based on the Jacobian of F (cf.
Section 4).
We present a non-adaptive variant of our algorithm (CEG + ), and show that for particular parameter choices (EG + ) of Diakonikolas et al.
(2021), and when ρ “ 0 the
celebrated forwardbackward-forward (FBF) algorithm of Tseng (2000) are recovered, thus unifying and generalizing both methods.
We improve upon Diakonikolas et al.
(2021) by not only relaxing the problem class but also the stepsize range.
We show that our results are tight by providing a matching lower bound, thus providing a complete picture of (EG + ) under weak MVI.
In the stochastic setting, similarly to Hsieh et al.
(2020), we consider two separate stepsizes.
Whereas they require both to be diminishing we show that the extrapolation step can in fact be picked constant—a modiﬁcation which is critical for convergence when only the weak MVI holds.
Related work The community has resorted to various approaches to make progress for nonconvex-nonconcave minimax problems.
One line of work focuses on deriving local convergence results (Mazumdar et al., 2019; Fiez & Ratli ﬀ , 2020; Heusel et al., 2017).
For global results, the two primary approaches have been to either assume a global oracle for the inner problem (Jin et al., 2019; Davis & Drusvyatskiy, 2018) or assume particular problem structure such as the PolyakŁojasiewicz condition (Nouiehed et al., 2019; Yang et al., 2020) or concavity for the inner problem (Raﬁque et al., 2019).
We follow the same tradition of assuming structure, but from the general perspective of operator theory.
The idea of studying minimax and related problems through the lens of variational inequality has a long history (Minty, 1962; Rockafellar, 1976; Polyak, 1987; Bertsekas, 1997), with recent renewed interest due to its relevance for minimax formulations (Mertikopoulos et al., 2018a;
Gidel et al., 2018; Azizian et al., 2020).
One relaxation of the monotone case for which we have positive results is that of Minty variational inequalities (MVI) (Mertikopoulos et al., 2018a; Song et al., 2021;
Zhou et al., 2017), which includes all quasiconvex-concave and starconvex-concave problems.
Diakonikolas et al.
(2021) introduced the relaxed condition of weak MVI.
In the unconstrained setting they showed non-asymptotic convergence results under a restricted problem constant ρ .
Similarly to us, Lee & Kim (2021a) extends the regime but under the stronger condition of cohypomonotonicity.
They do so by studying a more evolved variant of extragradient building on anchoring techniques.
We instead directly improve upon (EG + ) and generalize it to new settings.
In the stochastic setting, usually the stepsize for the extrapolation step is diminishing.
This is the case in Böhm et al.
(2020) where they consider a forward-backward-forward type scheme.
However, they remain in the monotone setting, where the limit cycles are non-attracting, as exempliﬁed by a bilinear game.
Hsieh et al.
(2021) recently showed that a large family of algorithms, which includes the extragradient method with diminishing stepsize, can converge to attracting limit cycles.
Going beyond this restriction, Hsieh et al.
(2020) interestingly considers two separate stepsizes similar to us.
However, only the more restrictive setting where MVI is satisﬁed is considered, and both of the stepsizes are required to be diminishing.
P roblem formulation and preliminaries
In this paper we are interested in ﬁnding zeros of an operator (or
set-valued mapping) T : (cid:146) n Ñ (cid:146) n that is written as the sum of a Lipschitz continuous (but possibly nonmonotone) operator F and a maximally monotone operator A .
That is, we wish to ﬁnd z P (cid:146) n such that the general inclusion
[Equation]
holds.
The set of all such points is denoted by zer T (cid:66) t z P R n |
P Tz u .
Throughout the paper problem (2.1) is studied under the following assumptions (deﬁnitions can be found in Appendix A).
Assumption I.
In problem (2.1) ,
(i) Operator A : (cid:146) n Ñ (cid:146) n is a maximally monotone operator.
(ii) Operator F : (cid:146) n Ñ (cid:146) n is L-Lipschitz continuous.
(iii) Weak Minty inequality (MVI) holds, i e., there exists a nonempty set S ‹ Ď zer T such that for all z ‹ P S ‹ and some ρ P p´
[Equation]
L , 8q
[Equation]
Generally, we do not require the weak Minty assumption to hold at every z ‹ P zer T .
In fact, as shown in Theorem 3.1 nonemptiness of S ‹ is su ﬃ cient for ensuring that the limit points belong to zer T .
Interestingly, despite nonmonotonicity of F , global (as opposed to subsequential) convergence can be established when S ‹ “ zer T , an assumption that is still weaker than cohypomonotonicity.
VIs provide a convenient abstraction for a range of problems.
We mention some central examples below but otherwise defer to the overview in Facchinei & Pang (2007).
Subsequently, we provide examples where the weak MVI holds.
Example 1 : (minimax optimization).
A comprehensive way to capture a wide range of applications in machine learning is to consider structured minimax problems of the form minimize
[Equation]
where ϕ is not necessarily convex in x or concave in y .
Functions g and h are proper extended real-valued lower semicontinuous and convex, with easy to compute proximal maps.
Common examples for g and h involve regularizers such as (cid:96) 1 , (cid:96) 2 norms, or indicator functions of sets allowing us to capture constrained minimax problems.
The ﬁrst order optimality condition associated with this problem may be written in the form of the structured inclusion (2.1) by letting
Fz “ p ∇ x ϕ p x , y q , ´ ∇ y ϕ p x , y qq , Az “ pB g p x q , B h p y qq .
As it will become clear in the next section (cf.
Algorithm 1), the main computations involved in the proposed scheme are evaluations of F and resolvent J A “ p id ` A q ´ 1 .
Recall that the resolvent of a maximally monotone operator is ﬁrmly nonexpansive with full domain (cf.
(Bauschke & Combettes, 2017, Sect. 23)).
If A “ B f is the subdi ﬀ erential operator of a convex function f , then its resolvent is the proximal mapping.
For instance when A is as in Example 1, then its resolvent is given by
J A p x , y q “ p prox g p x q , prox h p y qq .
Example 2 : ( N -player games).
More generally, we can consider a continuous game of N players in normal form.
Denote the decision variables z : “ p z i ;
z ´ i q : “ p z 1 , ..., z N q and let the loss incurred by the i th player be L i p
z i ;
z ´ i q “ ϕ i p
z q ` g i p
z i q where ϕ
i is the payo ﬀ function and g i typically enforce constraints on z i .
Then we seek a Nash equilibrium, which is any decision which is unilaterally stable, i.e.,
L i p z ‹ i ; z ‹´ i q ď L i p
z i ; z ‹´ i q @
z i and i P r N s (cid:66) t 1 , . . .
, N u .
The corresponding ﬁrst order optimality conditions may be written as Az “ pB g 1 p z 1 q , . . .
, B g N p z N qq
and Fz “ p ∇ z 1 ϕ 1 p z q , . .
. , ∇ z N ϕ N p z qq .
A solution to (2.1) thus returns a candidate for which the ﬁrst order condition of the above problems is satisﬁed.
In the monotone case these two solution concepts coincide, while in the more general case of weak MVI, we provide examples where this still holds.
In particular, we introduce in Section 5 a nonconvex-nonconcave minimax game which additionally exhibits limit cycles for Fz .
As a consequence most schemes including gradient descent ascent, extragradient and optimistic gradient descent ascent do not converge to a stationary point globally (Hsieh et al., 2021).
However, the global Nash equilibrium satisﬁes Assumption I (iii) with ρ ą ´ 1 { 2 L , which we show is su ﬃ cient for global convergence of (CEG + ).
The weak MVI condition is satisﬁed in certain reinforcement learning settings.
Speciﬁcally, Diakonikolas et al.
; Daskalakis et al.
(2021a) considers a two-player zero-sum game where the weak MVI holds, while neither MVI nor cohypomonotonicity holds.
Interestingly, the formulation requires constraint—a condition they do not handle.
We thus provide the ﬁrst provable algorithm for this setting.
Weak MVI also contains all quasiconvex-concave and starconvex-concave problems.
For further examples, the literature on cohypomonotonicity (Bauschke et al., 2020) is relevant since it implies weak MVI, see for instance Lee & Kim (2021b, Example 1).
G eneralizing E xtragradient+
Our starting point is the Extragradient + (EG + ) algorithm of Diakonikolas et al.
(2021) which is identical to extragradient (Korpelevich, 1976) except for the second stepsize being smaller.
They only treat the inclusion (2.1) when A ” 0, and in our notation require ρ P p´ 1 { 8 L , 0 s .
Speciﬁcally,
[Equation]
where they choose γ k “ 1 { L and ¯ α k “ 1 { 2 (Diakonikolas et al., 2021, Thm. 3.2).
We generalize (EG + ) in Algorithm 1 to take the operator A into account—consequently we capture constraint and regularized problems as well.
In addition, the scheme is adaptive in ¯ α k .
We will show that the weaker requirement of ρ P p´ 1 { 2 L , 8q su ﬃ ces even for the more general inclusion (2.1).
The main convergence results of Algorithm 1 are established in the next theorem.
The proof is largely inspired by recent developments in operator splitting techniques in the framework of monotone inclusions (Latafat & Patrinos, 2017; Giselsson, 2021).
The key idea lies in interpreting each iteration of the algorithm as a projection onto a certain hyperplane, an interpretation that dates back to Solodov & Tseng (1996);
Solodov & Svaiter (1999).
Theorem 3.1.
Suppose that Assumption I holds, and let λ k P p 0 , 2 q , γ k P ` t ´ 2 ρ u ` , 1 { L ‰ where t x u ` (cid:66) max t 0 , x u , δ k P p [Equation]
γ k { 2 , [Equation]
s , lim
[Equation]
k
[Equation]
0 , and lim inf k Ñ8 p δ k ` γ k { 2 q ą 0 . Consider the sequences p z k q k P (cid:142) , p ¯ z k q k P (cid:142) generated by Algorithm 1. Then for all z ‹ P S ‹ ,
[Equation]
where κ “ lim inf k Ñ8 λ k p 2 ´ λ k
qp δ k ` γ k { 2 q 2 .
Moreover, the following holds
(i) p ¯ z k q k P (cid:142) is
bounded
and
its limit points belong to zer T
; (ii) if in addition lim sup k Ñ8 γ k ă 1 { L and S ‹ “ zer T, then p z k q k P (cid:142) , p ¯ z k q
k P
(cid:142) both converge to some z P zer T.
Algorithm
I nitialize z 0 “ z init P (cid:146) n , λ k P p 0 , 2 q , γ k P ` t ´ 2 ρ u ` , 1 { L ‰ , δ k P p ´ γ k { 2 , ρ s ,
R epeat for k “ 0 , 1 , . . .
until convergence 1 . 1 : Let ¯ z k “ ` id ` γ k A ˘ ´ 1 ` z k
[Equation]
˘ 1 . 2 :
Compute stepsize
[Equation]
, where H “ id ´ γ k F .
1 . 3 : Update the vector z k ` 1 “ z k ` λ k α k p
H ¯ z k ´
Hz k q
R eturn z k `
Note that whenever lim sup k Ñ8 γ k ă 1 L , Lemma A.3 (ii) may be used to derive a similar inequality in terms of } ¯ z k ´ z k } by lower bounding } H ¯ z k
´
Hz k } in (3.1).
We also remark that tighter rates may be obtained in the regime ρ ě 0, however, this will not be pursued in this work.
3.1 N on - adaptive stepsize variant
Although we do not incur additional costs for evaluating the adaptive stepsize α k in step 1. 2 , it proves instructive to present a variant with constant stepsize.
As a result we compare the range of our stepsizes against Diakonikolas et al.
(2021) showing an improvement by a factor of 3 { 2 .
Moreover, in the monotone case ( ρ “ 0), with a certain choice of stepsizes the algorithm reduces to the celebrated forward-backward-forward
(FBF) algorithm of Tseng (2000).
We remark that the relation of FBF to projection-type algorithms was noted in Tseng (2000), (Giselsson, 2021, Sect.
To this end, in this subsection consider the following non-adaptive variant of Algorithm 1 that generalizes (EG + ).
Letting ¯ α k P p 0 , 1 ` 2 δ k { γ k q :
[Equation]
The convergence of this algorithm is an immediate byproduct of Theorem 3.1.
To see this, note that the z k ` 1 update in step 1.
3 may be written as z k ` 1 “ z k ` 2 η k α k p
H ¯ z k ´
Hz k q , for η k P p 0 , 1 q .
Therefore, convergence is still ensured for any ¯ α k
ă 2 α k as the di ﬀ erence may be absorbed by the relaxation parameter η k .
Note that by 1 { 2 -cocoercivity of H (cf.
Lemma A.3 (i) )
[Equation]
establishing the validity of the prescribed stepsize range.
The convergence of the non-adaptive variant is summarized in the next corollary that for simplicity is stated with constant parameters (dropping subscripts k ).
Corollary 3.2 (Constant stepsize) .
Suppose that Assumption I holds, and let γ P ` t ´ 2 ρ u ` , 1 { L ‰ , δ P p ´ γ { 2 , ρ s , and ¯ α P p 0 , 1 ` 2 δ { γ q .
Consider the sequences p
z k q k P (cid:142) , p ¯ z k q k P (cid:142) generated according to the update rule (CEG + ) .
Then,
[Equation]
where κ “ ¯ α p 1 ` 2 δγ ´ ¯ α q .
Moreover, the claims of Theorems 3.1(i) and 3.1(ii) hold true.
The setting of Diakonikolas et al.
(2021) in (EG + ) involves the stepsizes γ k “ 1 { L , α k “ 1 { 2 .
Note that when restricting to A ” 0, the iterates (CEG + ) simplify to this form owing to the fact that
H ¯ z k ´
Hz k “ ¯ z k ´ γ F ¯ z k
´
Hz k “ ´ γ F ¯ z k .
In comparison, in our setting if δ “ ρ “ ´ 1 { 8 L (the smallest ρ permitted in Diakonikolas et al. (2021)) is selected, then based on our analysis in Corollary 3.2 we may select γ k “ 1 { L , and ¯ α k P p 0 , 3 { 4 q , thus the upper bound for the second stepsize is 3 { 2 times that of Diakonikolas et al.
Remark 3.3 (relation to FBF) .
In Corollary 3.2 the range of stepsizes γ , ¯ α may alternatively be set as γ P
[Equation]
, ¯ α P p 0 , 1 ` 2 δ { γ s .
This is due to the fact that if γ ă 1 { L (strictly), then H is strictly 1 { 2 -cocoercive.
Therefore, in (3.2), 1 ` 2 δγ ă 2 α k holds, and thus the stepsize ¯ α “ 1 ` 2 δγ is permitted.
Although this may appear to be of little practical signiﬁcance, by setting γ P p 0 , 1 { L q , δ “ ρ “ 0, and ¯ α “ 1 in (CEG + ), we obtain z k ` 1 “ ¯ z k ` γ Fz k ´ γ F ¯ z k , which is the forward-backward-forward (FBF) algorithm of Tseng (2000), (Bauschke & Combettes, 2017, Thm. 26.17)).
3.2 L ower bounds
We show that the result in Corollary 3.2 is tight by providing a matching lower bound when A ” 0.
We do so by ﬁxing ¯ α k and showing a stepsize dependent lower bound.
In particular, note that if ¯ α k “ 1 { 2 as in Diakonikolas et al.
(2021, Thm. 3.2), then Theorem 3.4 implies a lower bound of ρ ą ´ 1 { 4 L for the (EG + ) scheme.
The lower bound is contextualized in Fig.
2 by relating it to our convergence results and existing results in the literature.
Theorem 3.4.
Consider a sequence p z k q k P N generated according to (EG + ) ﬁxing γ k “ γ “ 1 { L and ¯ α k “ ¯ α P p 0 , 1 q .
Let ´ ρ L ě 1 ´ ¯ α 2 .
Then, there exists an F : (cid:146) n Ñ (cid:146) n , n ą 1 , satisfying Assumption I(ii) and Assumption I(iii) for which the sequence will not converge.
3.3 S tochastic setting
In this section we assume that one has access only to independent samples of F that will be denoted by ˆ F p¨ , ξ q depending on some random variable ξ whose distribution is revealed online by observations of i.i.d.
copies of ξ .
We make the following standard assumptions.
Assumption II.
For all z P (cid:146) n
[Equation]
We proceed by presenting Algorithm 2 which is the stochastic variant of (CEG + ).
Interestingly, our analysis does not impose further assumptions on γ k (may be selected constant), while it is only α k that must satisfy the classical conditions ř 8 k “ 1 α k “ 8 , and ř 8 k “ 1 α 2 k ă 8 to guarantee convergence.
Theorem 3.5.
Suppose that Assumptions I and II hold, γ k P p t ´
ρ u ` , 1 { L s , and let α k P p 0 , 12 ` ργ k q .
The sequence p z k q k P (cid:142) generated by Algorithm 2 satisﬁes
[Equation]
where η k “ 2 α k ` p 12 ` ργ k q ´ α k ˘ and H “ id ´ γ k F.
Set constant stepsize γ k “ γ P p t ´
ρ u ` , 1 { L s and α k “ β k p 12 ` ργ q for some p β k q k P (cid:142) with β k P p 0 , 1 q .
Then the following holds
[Equation]
where k ‹ is chosen from t 0 , 1 , . .
, m u according to probability P r k ‹ “ k s “
[Equation]
β k .
When γ ă 1 L , Lemma A.3 (ii) may be used to derive a similar inequality in terms of E
[Equation]
Algorithm 2 (SEG + ) Stochastic algorithm for problem (2.1)
R equire
[Equation]
R epeat for k “ 0 , 1 , . . .
until convergence
[Table]
: Sample ξ k „ P and let F k “ ˆ F p z k , ξ k q
[Table]
[Equation]
p id
[Table]
[Equation]
[Table]
and let
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
[Table]
[Equation]
Next, we derive complexity bounds based on Theorem 3.5.
Set β k “ ξ ?
m ` 1 for some constant ξ .
Assume that m is large enough such that β k ă 1.
By minimizing the left-hand-side in (3.5) we obtain ξ “ 1 2 ? 2 σγ p 12 ` ργ q ´ 1 } z 0 ´ z ‹ } .
Therefore, the algorithm will reach E
[Equation]
(cid:15) accuracy for some k P t 0 , 1 , . . .
, m u after at most m “ r 8 (cid:15) 4 σ 2 γ 2 p 12 ` ργ q 2 } z 0 ´ z ‹ } 2 s iterations.
We remark that stochastic variant of (EG + ) was also studied in (Diakonikolas et al., 2021, Thm.
4.4(i)) under a more restricted range of parameters ρ P p 1 8 ?
2 L , 0 s and γ “ 1 2 ? 2 L .
A daptively taking larger stepsizes using local curvature
As made apparent in the analysis in Section 3 (cf.
Appendix B.1) the bound on the smallest weak
MVI constant ρ in Assumption I (iii) may be replaced with the requirement that ρ ą ´ γ k { 2 for all k P (cid:142) .
Therefore, larger stepsizes γ k would guarantee global convergence for an even larger class of problems.
Since a global Lipschitz constant is inherently pessimistic the natural question then becomes how to locally choose a maximal stepsize without diverging.
The proposed scheme involves a backtracking linesearch that uses the local curvature for its initial guess.
The reason being that this will immediately pass, close enough to the solution z ‹ , by argument of continuity.
More precisely, we will set the initial guess to something slightly smaller than }
JF p z k q} ´ 1 , where JF p z q denotes the Jacobian of F at z and } ¨ } is the spectral norm.
Note that, despite the use of second order information, the scheme remains e ﬃ cient since }
JF p z q} only requires one eigenvalue computation performed through Jacobian-vector product (Pearlmutter, 1994).
Given an initial point z 0 “ z init and ν P p 0 , 1 q , the ﬁnal scheme which we denote (CurvatureEG + ) proceeds for k “ 0 , 1 , . . .
as follows:
Obtain γ k and ¯ z k according to Algorithm 3 with γ init “ ν } JF p z k q} ´2 .
Compute z k ` 1 according to steps 1. 2 and 1. 3 of Algorithm
[Equation]
CurvatureEG + )
The above intuitive reasoning is made precise in the next lemma where it is shown that backtracking linesearch will terminate in ﬁnite time and that γ init will be immediately accepted asymptotically.
Lemma 4.1 (Lipschitz constant backtracking) .
Suppose that F : (cid:146) n Ñ (cid:146) n is a L-Lipschitz continuous operator.
Consider the linesearch procedure in Algorithm 3.
Then,
(i)
The linesearch terminates in ﬁnite time with γ ě min t γ init , ντ { L u ; (ii) Suppose that p z k q k P (cid:142) converges to z ‹ P zer
T.
If F is continuously di ﬀ erentiable, and γ init P p 0 , ν
} JF p z k q} ´ 1 q with ν P p 0 , 1 q , then eventually the backtrack will never be invoked ( γ init would be accepted).
Algorithm
I nitialize z k P (cid:146) n , τ P p 0 , 1 q , ν P p 0 , 1 q 3 . 1 :
Set initial guess γ “ γ init , and let G γ p z k q (cid:66) ` id ` γ A ˘ ´ 1 ` z k ´ γ Fz k ˘ while γ } F p G γ p
z k qq ´
Fz k } ą ν } G γ p
z k q ´ z k } do γ Ð τγ
R eturn γ k “ γ and ¯ z k “ G γ p z k q
The convergence results for (CurvatureEG + ) are deduced based of the above lemma and Theorem 3.1 and are provided in Corollary B.1 in Appendix B.2.
We illustrate the behavior of (CurvatureEG + ) in Fig.
1 and in Section 6.
C onstructing toy examples
When Assumption I (iii) holds for negative ρ , limit cycles of the underlying operator Fz can emerge.
We illustrate this with simple polynomial examples for which all the properties of interest can be computed in closed form.
Deﬁnition 1 (PolarGame) .
A PolarGame denotes a two-player game whose associated operator F has limit cycles at } z } 2 “ c i for all i P r k s where c i ‰ 0.
This turns out to be particularly easy to construct in polar coordinates as the name suggests (see
Appendix C.1).
Apart from introducing arbitrary number of limit cycles it also gives us control over ρ .
This is illustrated in the following instantiations capturing three important cases.
Example 3 : (PolarGame).
Consider Fz “ p ψ p x , y q ´ y , ψ p y , x q ` x q where } z } 8 ď 11 { 10 and ψ p x , y q “ 116 ax p´ 1 ` x 2 ` y 2 qp´ 9 ` 16 x 2 ` 16 y 2 q .
We have the following three cases:
[Equation]
then
[Equation] [Equation]
[Equation]
iii
[Equation]
13 then
[Equation]
P
[Equation]
18 L ,
[Equation]
110 L q where L denotes the Lipschitz constant of F restricted to the constraint set.
For all cases F exhibits limit cycles at } z } “ 1 and } z } “ 3 { 4 .
Proof is deferred to Appendix C.2.
Example 4 : (minimax).
In the particular case of constrained minimax problem we introduce the following polynomial game:
[Equation]
where ψ p z q “ 2 z 6 21 ´ z 4 3 ` z 2 3 .
We provide proof of the following properties in Appendix C.3:
(i)
There exists a repellant limit cycle and an attracting limit cycle of F .
(ii) z ‹ “ p 0 , 0 q is a global Nash equilibrium for which Assumption I (iii) holds inside the constraint with ρ ą ´ 1 { 2 L , where L denotes the Lipschitz constant of F restricted to the constraint set.
E xperiments
The algorithms considered in the experiments include the adaptive Algorithm 1, (CurvatureEG + ), and constant stepsize methods that can be seen as instances of (CEG + ) for various choices of γ k and ¯ α k .
When γ k “ 1 { L and ¯ α k “ 1 we recover a constrained variant of extragradient, which we denote CEG.
When ¯ α k “ 1 { 2 we denote the scheme CEG + , which is the direct generalization to the constraint setting of the (EG + ) scheme studied in Diakonikolas et al.
(2021, Thm. 3.2).
Note that this choice of ¯ α k restricts the problem class for which we otherwise can have guaranteed convergence according to Corollary 3.2.
When ¯ α k is chosen adaptively according to Algorithm 1 we refer to it as
AdaptiveEG + .
Finally, when γ k is additionally chosen adaptively we use the name (CurvatureEG + ).
In the stochastic setting, we consider Algorithm 2.
When γ k “ 1 { k and α k “ 1, e ﬀ ectively both stepsizes diminish, and we recover a constrained variant of the popular stochastic extragradient scheme (see e.g. Hsieh et al.
(2021, Algorithm 3)), which we refer to as SEG.
When γ k “ 1 { L as suggested by Theorem 3.5, and only α k is decreasing, we refer to it as SEG + .
We test the algorithms on the constructed examples and conﬁrm their convergence guarantees.
Speciﬁcally, we apply the algorithms to the minimax problem in Example 4, the PolarGames in
Example 3, and a worst case construction, Example 5, from the proof of the lower bound (cf.
Appendix B.3).
For Example 5 we choose the problem parameters such that ρ “ ´ 1 { 3 L according to (B.15), and additionally add an (cid:96) 8 -ball constraint to keep the iterates bounded.
To simulate the stochastic setting we add Gaussian noise to calls of F .
Results for the deterministic setting and stochastic setting can be found in Fig.
3 and Fig.
4 respectively.
[Figure] [Figure]
C onclusion
This paper introduced an EG-type algorithm for a class of nonconvex-nonconcave minimax problems that satisfy the weak Minty variational inequality (MVI).
The range of parameter in the weak MVI was extended compared to EG + of Diakonikolas et al.
and tightness of our results were demonstrated through construction of a counter example.
In addition, EG + (Diakonikolas et al., 2021), as well as the forward-backward-forward algorithm (Tseng, 2000) were all shown to be special cases of our scheme.
Furthermore, (CurvatureEG + ) was proposed that performs a backtracking linesearch on the extrapolation stepsize γ k allowing for larger stepsizes and relaxes the condition ρ ą 12 L to ρ ą ´ γ k { 2 which is often a much weaker condition.
More importantly, it is shown that asymptotically the linesearch always passes with γ k “ ν } JF p z k q} ´ 1 for any ν P p 0 , 1 q , thus ratifying the name (CurvatureEG + ).
In the stochastic setting, unlike what is common in the literature, it was shown that it is only the second stepsize that must be diminishing.
Future direction include exploring applications of the proposed algorithm in particular in the setting of GANs.
It is also interesting to develope a variance reduced variant of the algorithm for ﬁnite sum minimax problems.
A cknowledgments and disclosure of funding
This project has received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research and innovation programme (grant agreement n° 725594 - time-data).
This work was supported by the Swiss National Science Foundation (SNSF) under grant number 200021_205011.
The work of the second and third author was supported by the Research Foundation Flanders (FWO) postdoctoral grant 12Y7622N and research projects G081222N, G0A0920N,
G086518N and G086318N; Research Council KU Leuven C1 project
No. C14 / 18 / 068; Fonds de la Recherche Scientiﬁque – FNRS and the Fonds Wetenschappelijk Onderzoek – Vlaanderen under EOS project no 30468160 (SeLMA); European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 953348.
The work of Olivier
Fercoq was supported by the Agence National de la Recherche grant ANR-20-CE40-0027, Optimal
Primal-Dual Algorithms (APDO).
A P reliminary definitions
Notationally we will use t x u ` (cid:66) max t 0 , x u throughout.
We additionally recall some standard deﬁnitions and results and refer to Bauschke & Combettes (2017); Rockafellar (1970)) for further details.
An operator or set-valued mapping A : (cid:146) n Ñ (cid:146)
d maps each point x P (cid:146) n to a subset Ax of (cid:146) d .
We will use the notation A p x q and Ax interchangably.
We denote the domain of A by
[Equation]
its graph by
[Equation]
[Equation]
, and the set of its zeros by zer A (cid:66) t x P (cid:146) n | 0 P Ax u .
The inverse of A is deﬁned through its graph: gph A ´ 1 (cid:66) tp y , x q | p x , y q P gph A u .
The resolvent of A is deﬁned by J A (cid:66) p i
d ` A q ´ 1 , where id denotes the identity operator.
Deﬁnition A.1 ((co)monotonicity Bauschke et al.
An Operator A : (cid:146) n Ñ (cid:146) n is said to be ρ -monotone for some ρ P (cid:146) , if for all p x , y q , p x 1 , y 1 q P gph A
[Equation]
2 ď x
[Equation]
´ x 1 , y ´ y 1 y , and it is said to be ρ -comonotone if for all p x , y q , p x 1 , y 1 q P gph A ρ } y ´ y 1 } 2 ď x x ´ x 1 , y ´ y 1 y .
The operator A is said to be maximally (co)monotone if its graph is not strictly contained in the graph of another (co)monotone operator.
We say that A is monotone if it is 0-monotone.
When ρ ă 0, ρ -comonotonicity is also referred to as | ρ | -cohypomonotonicity.
Deﬁnition A.2 (Lipschitz continuity and cocoercivity) .
Let D Ď (cid:146) n be a nonempty subset of (cid:146) n .
A single-valued operator A : D Ñ (cid:146) n is said to be L-Lipschitz continuous if for any x , x 1 P D } Ax ´ Ax 1 } ď L } x ´ x 1 } , and β -cocoercive if
[Equation]
Moreover, A is said to be nonexpansive if it is 1 -Lipschitz continuous, and ﬁrmly nonexpansive if it is 1 -cocoercive.
The resolvent operator J A is ﬁrmly nonexpansive (with dom J A “ (cid:146) n ) if and only if A is (maximally) monotone.
The following lemma plays an important role in our convergence analysis.
Lemma A.3.
Let A : (cid:146) n Ñ (cid:146) n denote a single valued operator.
Then,
(i)
A is 1 -Lipschitz if and only if T “ id ´ A is 1 { 2 -cocoercive.
(ii)
If A is L-Lipschitz, then T “ id ´ η A, η P p 0 , 1 { L q , is p 1 ´ η
L q -monotone , and in particular
} Tu ´ Tv } ě p 1 ´ η L q} u ´ v } for all u , v P (cid:146) n .
Proof.
The ﬁrst claim follows directly from (Bauschke & Combettes, 2017, Prop.4.11).
That T is strongly monotone is a consequence of the Cauchy Schwarz inequality and Lipschitz continuity of
A :
[Equation]
In turn, the last claim follows
B P roofs and further results B.1 P roofs of S ection
Proof of Theorem 3.1 .
Let H “ id ´ γ k F .
By Step 1. 1 Hz k P
¯
z
k
` γ k A ¯ z k .
Therefore,
[Equation]
p
[Equation]
In what follows we will show that Algorithm 1 is equivalent to taking a forward-backward step followed by a correction step.
Consider the updates
[Equation]
where D
[Equation]
Note that
[Equation]
where in the inequality Lemma A.3 (i) was used.
Hence, by (B.3) the stepsize α k is positive and bounded away from zero.
Moreover, if z k P D k , then from (B.3) we may conclude that } H ¯ z k ´ Hz k } ď 0 which implies that the generated sequence remains constant and ¯ z k P zer T (cf.
(B.1)).
The projection onto D k for any v R D k is given by
[Equation]
Moreover, (B.1) together with Assumption I (iii) at ¯ z k yields
[Equation]
thus ensuring z ‹ P S ‹ Ď D k .
The projection onto D k is then given by Π D k p z k q “ z k ` α k p
H ¯ z k ´
Hz k q , where α k is as in step 1. 2 .
Finally, since the projection Π D k is ﬁrmly nonexpansive, it follows from (Bauschke & Combettes, 2017, Cor. 4.41) that the mapping p 1 ´ λ k q id ` λ k Π D k is λ k { 2 -averaged.
Consequently, we may conclude that p z k q k P (cid:142) is Fejér monotone relative to S ‹ (Bauschke & Combettes, 2017, Prop. 4.35(iii)).
That is for all z ‹ P S ‹
[Equation]
where ε k (cid:66) λ
k p 2 ´
λ k qp γ k 2 ` δ k q 2 .
The convergence rate in (3.1) is obtained by telescoping (B.5).
Since lim inf k Ñ8 ε k ą 0, p 1 γ 2 k } H ¯ z k ´ Hz k } 2 q k P (cid:142) converges to zero.
Moreover, p} z k ´ z ‹ } 2 q k P (cid:142) converges and the sequence p
z k q k P (cid:142) is bounded.
Since γ k is bounded, and F and the resolvents p id ` γ k A q ´ 1 are Lipschitz continuous (cf.
(Bauschke & Combettes, 2017, Cor. 23.9)), so is their composition.
Hence, p ¯ z k q k P (cid:142) is also bounded.
Let p ¯ z k q k P K be a subsequence converging to some ¯ z P (cid:146) n .
Combined with the fact that p 1 γ 2 k } H ¯ z k ´ Hz k } 2 q k P (cid:142) converges to zero, we may conclude from (B.1) along with (Bauschke & Combettes, 2017, Prop. 20.38) and Lipschitz continuity of F that ¯ z P zer T .
Finally, if in addition γ “ lim sup k Ñ8 γ k ă 1 { L , then p 1 ´ γ L q} ¯ z k ´ z k } ď } H ¯ z k ´ Hz k } (invoke Lemma A.3 (ii) ).
Therefore, p} ¯ z k ´ z k }q k P (cid:142) converges to zero, which in turn implies that a subsequence p z k q k P K 1 converges to a point
z 1 i ﬀ so does the subsequence p ¯ z k q k P K 1 .
Hence, p z k q k P K also converges to ¯ z P zer T .
Consequently, if Assumption I (iii) holds at all of the zeros of T , i.e., if S ‹ “ zer T , then the second claim follows by invoking (Bauschke & Combettes, 2017, Thm. 5.5).
Proof of Corollary 3.2 (Constant stepsize).
The proof of convergence was already given prior to the statement of the corollary.
It remains to derive (3.3).
By Assumption I (iii) and owing to 1 { 2cocoercivity of H (cf.
Lemma A.3 (i) )
[Equation]
B [Equation]
Therefore, provided that ¯ α ą 0 we have
[Equation]
Telescoping the above inequality yields the claimed inequality
[Equation]
Proof of Theorem 3.5 .
Let w k
“ ˆ F p z k , ξ k q ´ F p z
[Equation]
q , ¯ w
[Equation]
“ ˆ F p ¯ z k , ¯ ξ k q ´ F p ¯ z k q denote the additive noise associated with the stochastic oracle access at z k and ¯ z k , respectively.
Then, letting ˆ H p z k q “
H p z k q ´ γ k w k , ˆ H p ¯ z k q “ H p ¯ z k q ´ γ k ¯ w k , it holds that
[Equation]
where the Young inequality was used in both inequalities.
Using the law of total expectation, for z ‹ P zer T
[Equation]
where F k and ¯ F k represent the information available at z k and ¯ z k updates, repectively.
It follows from the above two inequalities that
[Equation]
B.8)
[Equation]
(B.7), Assumption II (ii
[Equation]
ď
[Equation]
as claimed.
Next, with constant stepsize γ and p α k q k P (cid:142) as prescribed in the lemma, η k “ 2 β k p 1 ´ β k q ` 12 ` ρ γ ˘ 2 ě 2 β k
` 12 ` ρ γ ˘ 2 .
Using this and summing (3.4) over k we obtain
[Equation]
Dividing both sides by 2 p 12 ` ρ γ q 2 ř mk “ 0 β k yields
[Equation]
establishing the claimed inequality.
B.2 C onvergence results and proofs of S ection
The convergence results for (CurvatureEG + ) are provided in the next corollary where ρ in Assumption I (iii) is allowed to take potentially larger values provided that ρ ą ´ γ k { 2 .
Note that owing to the lower bound on γ k (cf.
Lemma 4.1 (i) ), the weak MVI assumption in the corollary is always satisﬁed if ρ P p´ ντ { 2 L , 8q , however, in practice γ k may take larger values.
Corollary B.1.
Suppose that Assumptions I(i) and I(ii) hold, and consider the sequences p
z k q k P (cid:142) , p ¯ z k q k P (cid:142) generated by (CurvatureEG + ) .
Suppose that Assumption I(iii) holds for some ρ P (cid:146) satisfying γ k ` 2 ρ ą 0 , and let δ k P p ´ γ k { 2 , ρ s , λ k P p 0 , 2 q , lim inf k Ñ8 λ k p 2 ´ λ k q ą 0 , and lim inf k Ñ8 p
δ k ` γ k { 2 q ą 0 .
Then,
(i)
The sequence p} ¯ z k ´ z k } 2 q k P (cid:142) vanishes; (ii) p ¯ z k q k P (cid:142) , p z k q k P (cid:142) are bounded, and have the same limit points belonging to zer T; (iii) if in addition S ‹ “ zer T, then p z k q k P (cid:142) , p ¯ z k q k P (cid:142) both converge to some z ‹ P zer T.
Moreover, if z k , ¯ z k Ñ z ‹ P zer T (as is the case in B.1(iii)), and F is continuously di ﬀ erentiable, then eventually the backtrack will never be invoked.
Proof.
Observe that in the proof of Theorem 3.1 1-Lipschitz continuity of γ k F is only used at the generated points ¯ z k and z k (see (B.3)), and is thus ensured by the linesearch Algorithm 3.
Therefore, it is easy to see that α k is positive and bounded away from zero provided that ρ ą ´ γ k { 2 , see (B.3).
Moreover, since γ k } F ¯ z k ´ Fz k } ď ν } ¯ z k ´ z k } , arguing as in Lemma A.3 (ii) we obtain } H ¯ z k ´ Hz k } ě p 1 ´ ν q} ¯ z k ´ z k } .
Hence, it follows from (B.5)
that
[Equation]
By telescoping the inequality and noting that γ k is bounded, we obtain ř k P (cid:142) }
¯ z k ´ z k } 2 ă
[Equation]
implying B.1 (i) .
Noting this and arguing as in the last part of the proof of Theorem 3.1 establishes
B.1 (ii) , B.1 (iii) .
The last claim is the direct consequence of Lemma 4.1 (ii) .
Proof of Lemma 4.1 (Lipschitz constant backtracking).
4.1 (i) :
Since F is L -Lipschitz continuous the linesearch would terminate in ﬁnite steps.
Either γ init satisﬁes the condition, or else the backtrack procedure is invoked, which in turn implies the previous candidate γ { τ should have violated the condition leading the the claimed lower bound.
4.1 (ii) :
Since the resolvent p id ` γ A q ´ 1 and F are Lipschitz continous, so is their composition.
Hence,
G γ p
z k q Ñ G γ p
z ‹ q .
Furthermore, by deﬁnition z ‹ ´ γ Fz ‹ P G γ p z ‹ q ` γ A p G γ p
z ‹ qq .
Consequently, using monotonicity of A at G γ p
z ‹ q and z ‹ , and that ´ Fz ‹ P Az ‹ yields
ď x z ‹ ´ γ Fz ‹ ´ G γ p
z ‹ q `
Fz ‹ , G γ p z ‹ q ´ z ‹ y “ ´ } z ‹ ´ G γ p
z ‹ q} 2 .
Thus G γ p z ‹ q “ z ‹ .
Using the fact that both p G γ p z k qq k P (cid:142) and p z k q k P (cid:142) converges to z ‹ P zer T :
[Equation]
where (Rockafellar & Wets, 2009, Thm. 9.7) was used.
The claim follows from continuity of JF and the fact that p z k q k P (cid:142) converges to z ‹ .
B.3 P roofs of S ection 3.
To prove the lower bound we introduce the following unconstrained bilinear minimax problem with an unstable critical point.
Example 5 : Consider the following minimax problem:
[Equation]
where b ă 0 and a ą 0.
Proof of Theorem 3.4 .
The associated operator of Example 5 can easily be computed,
[Equation]
where z “ p x , y q .
In this particular case, both L and ρ turn out to be constants.
By simple calculation we have,
[Equation]
where } ¨ } is the spectral norm.
Since the norm of the Jacobian is constant it equates the global
Lipschitz constant, L “ } JF p z q} .
By linearity of F , one step of (EG + ) is conveniently also a linear operator.
Speciﬁcally, z k ` 1 “ Tz k with T : “ ¨
[Equation]
We know that a linear dynamical system is globally asymptotically stable if and only if the spectral radius of the linear mapping is strictly less than 1.
Let λ 1 , λ 2 be the eigenvalues of T .
Then the spectral radius is the largest absolute value of the eigenvalues.
For T this becomes,
[Equation]
So we can ask what c in ρ “ ´ cL needs to be for the sequence p z k q k P N to converge.
Solving for c in this equality with max i | λ i | ă 1, we obtain,
[Equation] [Equation]
provided that we pick
[Equation]
Equation (B.15) provides a speciﬁcation for Example 5. As long as (B.14) is satisﬁed, (EG + ) is guaranteed to converge for γ k “ 1 { L .
On the other hand, since (B.12) is a linear system, we simultaneously learn that picking c any larger would imply non-convergence through max i | λ i | ě 1 (given z 0 ‰ 0).
We can trivially embed problem (B.12) into a higher dimension to generalize the result.
Noting that c “ ´ ρ L completes the proof.
We provide Mathematica code to verify each step of the above proof.
C T oy examples
In the following appendix, L denotes the Lipschitz constant of F restricted to the constraint set and ρ is the parameter of the weak MVI (Assumption I (iii) ) when restricted to the constraint set.
This restriction of the deﬁnitions is warranted, since z k remains within the constraint set in all simulations, while ¯ z k is guaranteed to stay within by deﬁnition of Step 1. 1 in Algorithm 1 (and likewise for all other considered method treating problem (2.1)).
All computer-assisted calculations can be found in the supplementary code.
C.1 C onstructing a P olar G ame (D efinition 1)
Recall Deﬁnition 1 which considers a vectorﬁeld F : R n Ñ R n with limit cycles at r P t c 1 , ..., c k u where c
i ‰ 0 for all i P r k s .
Such a vectorﬁeld can be constructed for n “ 2 by departing from the following dynamics in polar coordinates,
[Equation]
with a , b ‰ 0.
Transforming this dynamics into cartesian coordinates yields the desired vectorﬁeld,
F , while subsequently integrating with respect to x and y yields the two potentials associated with the two players.
Note that the roots t´ c
i u ki “ 1 for the polynomial deﬁning 9 r are not strictly necessary for showing existence of limit cycles, but leads to a simpler form for Fz .
We illustrate the construction in Fig.
Proposition 1.
Let Fz “ p 9 x , 9 y q be the evolution in cartesian coordinates of the associated vectorﬁeld in polar coordinates deﬁned by (C.1).
Then the only stationary point of F is at the origin p 0 , 0 q
and there exists a limit cycle at r “ c i for all i P r k s .
Proof.
Let r “ a x 2 ` y 2 .
It is easy to see from (C.1) that the only stationary point is at r “ 0.
By construction, 9 r is a polynomial with roots
c i for all i P r k s , so any trajectory starting on the circle deﬁned by r “ c i remains in that set.
However, 9 θ is strictly nonzero.
As a consequence Fz is nonzero, so r “ c i must deﬁne a limit cycle, which proofs the claim.
[Figure] [Figure]
C.2 P roof for properties of E xample
The operator F : R 2 Ñ R 2 deﬁned in Example 3 is obtained by constructing the associated dynamics in polar coordinates,
[Equation]
This can easily be veriﬁed by a change of variables.
From Proposition 1 it then follows, that there must exist a limit cycle at } z } “ 1 and } z } “ 3 { 4 .
To verify the conditions on ρ we compute the closed form solution to ρ and L in Mathematica:
(i) For a “ 1 we have ρ “ ´ 501761050977 and L “
[Figure]
(ii) For a “ 3 { 4 we have ρ “ ´ 60211216798825 and L “
[Figure]
(iii)
For a “ 1 { 3 we have ρ “ ´ 1505289439585 and L “
[Table]
It can easily be veriﬁed that the stated conditions for ρ in Example 3 are met for the values above.
This completes the proof.
We provide Mathematica code verifying the construction of F and the closed form solutions to L and ρ .
[Figure]
C.3 P roof for properties of E xample
Under the deﬁnitions of ρ and L in Appendix C, we claim that the origin p 0 , 0 q in (GlobalForsaken) is a global Nash equilibrium and satisﬁes Assumption I (iii) with ρ ą ´ 1 { 2 L .
To verify that p 0 , 0 q is indeed a global Nash equilibrium we need to check that the solution cannot be unilaterally improved.
In other words, the solution should coincide with p x ‹ , y ‹ q where
[Equation]
We can easily verify this with Minimize in Mathematica, since the functions are polynomial for which a closed form solutions to the global optimization problem will be returned.
To ﬁnd ρ for z ‹ “ p 0 ,
q we solve the global minimization problem, minimize
[Equation]
for which a closed form solution can be found with Mathematica, which when numerically evaluated is approximately ´ 0 .
We need to compute L to ensure ρ ą ´ 1 { 2 L .
In our case of convex constraints, C , we have that
L “ sup z P C } JF p z q} where } ¨ } denotes the spectral norm (Rockafellar & Wets, 2009, Thm. 9. and 9.7).
Under our constraint } z } 8 ď 4 { 3 , this can similarly be computed in closed form, yielding
L “
[Table]
12 p
[Table]
q{
[Table]
So ´ 12 L « ´ 0 .
165432 which satisfy the condition ρ ą ´ 12 L .
This completes the proof.
Proposition 2.
Let F be the associated operator of φ in (GlobalForsaken) deﬁned as Fz “ p ∇ x φ p x , y q , ´ ∇ y φ p x , y qq .
Deﬁne the radius as r “ } z } .
Then, Fz has a stable critical point at the origin p 0 , 0 q , at least one attracting limit cycle in the region deﬁned by a
[Equation]
2 ă r ă 2 and at least one repellant limit cycle within r ď
[Equation]
Proof.
We follow a similar argument as in Hsieh et al.
(2021, D.2).
We can compute the associated operator F ,
[Equation]
With a change of variables into polar coordinates p r , θ q we get that r “
[Equation]
as
[Equation]
When r “ a 3 { 2 this reduces to
[Equation]
and we observe that 9 r ą 0 for any θ .
Likewise for r “ 2, we have that 9 r “ ´ 421 p 22 cos p 4 θ q` 25 q which implies 9 r ă 0.
Since there is no stationary point in the region S “ !
p r , θ q : a
[Equation] [Equation]
) it then follows from the Poincaré-Bendixson theorem (Teschl, 2012, Thm. 7.16) that there must exist at least one attracting limit cycle in S .
Further, it is easy to see that p 0 ,
q is a critical point and that it is stable by inspection of the Jacobian JF p z q .
Since S is trapping, it follows from Poincaré–Hopf index theorem, that there must exist a repellant limit cycles in the region deﬁned by r ă a 3 { 2 .
This completes the proof.
C.4 P roof of properties for (H sieh et al ., 2021, E xample 5.2)
This section considers (Hsieh et al., 2021, Example 5.2) on the constraint domain D “ t z P (cid:146) n | } z } 8 ď 3 { 2 u .
We show that the unique critical point z ‹ does not satisﬁes the weak MVI for ρ ą ´ 1 { 2 L even when restricted to the constraint set z P D .
We restate the example with the additional constraint for convenience.
Example 6 : (Hsieh et al., 2021, Example 5.2) minimize | x |ď 3 {maximize | y |ď 3 {φ p x , y q : “ x p y ´ 0 .
45 q ` ψ p x q ´ ψ p y q , (Forsaken) where ψ p z q “ 14 z 2 ´ 12 z 4 ` 16 z 6 .
By using Mathematica, we can obtain a closed form solution of the Lipschitz constant L of F restricted to the constraint set, which we ﬁnd to be L “b ` 1089 ?
801761 ` 993841 ˘ .
Mathematica can solve approximately for the critical point, yielding z ‹ “ p 0 . 0780267 , 0 .
411934 q .
To ﬁnd ρ we want to globally minimize ρ p z q : “ x Fz , z ´ z ‹ y } Fz }for z P D .
Mathematica ﬁnds the candidate z 1 “ p´ 1 . 01236 , ´ 0 .
104749 q for which ρ p z 1 q “ ´ 0 .
So ρ must be at least this small, i.e. ρ ă ´ 0 .
Since ´ 1 { 2 L « ´ 0 . 04, this implies that ρ ă ´ 1 { 2 L .
See Forsaken.nb for
Mathematica-assisted computations.
This rules out convergence guarantees for both (CEG + ) and AdaptiveEG + (Algorithm 1), which is supported by the simulation in Figure 8.
However, as observed, (CurvatureEG + ) converges in the simulations.