Convergences for Minimax Optimization Problems over Infinite-Dimensional Spaces Towards Stability in Adversarial Training
Authors: Takashi Furuya
Satoshi Okuda
Kazuma Suetake
Yoshihide Sawada
Venue: Accepted by TMLR ()

Abstract: Training neural networks that require adversarial optimization, such as generative adversarial networks (GANs) and unsupervised domain adaptations (UDAs), suffers from instability. This instability problem comes from the difficulty of the minimax optimization, and there have been various  approaches in GANs and UDAs to overcome this problem. In this study, we tackle this problem theoretically through a functional analysis. Specifically, we show the convergence property of the minimax problem by the gradient descent over the infinite-dimensional spaces of continuous functions and probability measures under certain conditions.
Using this setting, we can discuss GANs and UDAs comprehensively, which have been studied independently.
In addition, we show that the conditions necessary for the convergence property are interpreted as stabilization techniques of adversarial training such as the spectral normalization and the gradient penalty.

Published in Transactions on Machine Learning Research (06/2024)
Convergences for Minimax Optimization Problems over Infinite-Dimensionaa Spaces Towards Stability in Adversarial Training
Takashi Furuya" Shimane University
aakashi.furuycli @gmail com
Satoshi Okuda"
vatoshi..okadda@aisii.coo
Tokyo Research Center, Aisin Corporation
Kazuma Suetake
AISIN SOFTWARE, Japan
Yoshihide Sawada
beshihide.suwaddee Doissi.cco.pp
Tokyo Research Center, Aisin Corporation
These two authors contributed equally this work
Reviewed on OpenReview:
Abstract
Training neural networks that require adversarial optimization, such generative adversarial networks (GANs) and unsupervised domain adaptations (UDAs). suffers from instability. This instability problem comes from the difficulty of the minimax optimization, and there have been various approaches in GANs and UDAs lo overcome this problem. In this study, we tackle this problem theoretically through a functional analysis. Specifically, we show the convergence property of the minimax problem by the gradient descent over the infinitediimensionall spaces of continuous functions and probability measures under certain conditions. Using this setting, can discuss GANs and UDAs comprehensively, which have been studied independently. In addition, we show that the conditions necessary for the convergence property are interpreted as stabilization techniques of adversarial training such as the spectral normalization and the gradient penalty.
Keywords: Minimax, Non-convex Optimization, Convergence Analysis, Adversarial Training, Functional Analysis.
1 Introduction
With the increased computational resources and available data, neural networks (NNs) trained by adversarial training have emerged prominently in various fields. An example is the application of generative adversarial networks (GANs) in generative tasks. GANs train the generator to capture the data distribution in an adversarial manner against the discriminator, which distinguishes between data generated by the generator and the dataset (Goodfellow et al.) 2014). Another example is the utilization of adversarial training in unsupervised domain adaptations (UDAs) as generalization techniques. UDAs transfer knowledge from source domains to the target domain by extracting domain-invariant features against. the domain critic that distinguish between data from source and target domains an adversarial manner (Ganin and Lempitsky 2015 Tzeng et al. 2017). Despite the effectiveness of GANs and UDAs, both pose challenges as nonconvex- nonconcave minimax problems, leading to inherent instability (Salimans al. 2016). This instability, though insufficiently explored ihelletically, complicates the widespread deployment of these models and hinders their

---

Published in Transactions on Machine Learning Research (06/2024)
practical application. To address and pave the way for more robust applications, we analyze the instability problem from a functional analysis perspective.
As instability is related to the convergence properties of the gradient descent algorithm (Chu et al. 2020 we aim to clarify the convergence conditions for adversarial optimization problems. To facilitate the derivation of these conditions from the functional analysis perspective, we begin by considering the ideal setting. In our study, the ideal setting is derived from the dual formula of the minimization of a functional over probability distributions, leading to the minémaz problem over infinite-dimensiinaal spaces of continuous functions prove the
probability measures. By exploring this minimax problem over infiiiteddimesionna. spaces, convergence to minimax solution for convex-concave setting (Section 5.1 and stationary point for a nonconvex-concave setting (Section 5.2 under appropriate assumptions.
Throughout the convergence analyses, we maintain the assumption that the discrepancy measure, appearing in both GANs and UDAs, strongly convex and L-smooth for the convergence. Achieving strong convexity involves confining the discriminator to a suitable subset within Lipschitz continuous function spaces. This concept aligns with the spectral normalization (Miyato et all 2018). To ensure L-smoothness, we utilize the inf-convolution with a regularizer, such the squared maximum mean discrepancy (MMD) in the reproducing kernel Hilbert spaces (RKHS) with the Gaussian kernel. This process corresponds to the gradient penalty [Gulrajani et al., 2017). Therefore, we can theoretically interpret widely-used stabilization techniques in adversarial training as the desired condition for achieving convergence properties.
Contributions
(A) We show the convergence to the minimax solution for convex-concave setting and the stationary point for a nonconvex-concave setting over infinite-dimensional spaces of continuous functions or probability measures. This analysis is motivated by adversarial training in the scheme of the gradient.
descent (Section 5
(B) We verify the fulfillment of sufficient conditions for the convergence properties in certain GANs and UDAs settings (Section G providing a theoretical interpretation of existing techniques such as the spectral normalization and gradient penalty.
2 Related Work
GAN training often exhibits an unstable trajectory, resulting in poor solutions (Goodfellow et al. 2014 Melz et al. 2016). To address this instability, various stabilization techniques have been proposed, including the Wasserstein GAN (Arjovsky et al., 2017), gradient penalty (Gulrajani et al. 2017), and spectral normalization (Miyato et al. 2018). The effectiveness of these techniques in stabilizing GAN training has been theoretically demonstrated (Chu et al. 2020). This theoretical result implies that the instability of GANs is due to adversarial training. Thus, UDAs with adversarial training are expected to encounter similar instability during training. Although methods to introduce various distances into UDAs to stabilize them have been
investigated (Shen et al. 2018 Acuna et al.) 2021 Wu et al. 2022 Chen and Marchand 2023). theoretical analysis encompassing CDAs and GANs with respect to instability has not yet. been proposed. Notably, Chu et al. (220)) provides theoretical insight into GANs, interpreting stabilization techniques as conditions from the perspective of the minimization problem. On the other hand, our work provides similar theoretical insight from the viewpoint of the minimaz problem rather than minimization, as the actual setting in GANs and UDAs employs the convex duality of the objective function to be minimized (Ganin and Lempitsky 2015 Shen al.) 2018 Acuna et al. 2021). Firstly, we provide insights similar to Chu et al. (2020), which interpret stability techniques often used in GANs, such spectral normalization and gradient penalty, via the sufficient conditions for minimax convergence, even though Chu et al. (2020) discussed the minimization problem. Secondly, we provide insight for UDAs, emphasizing that joint convexity is an important factor for stabilization, which could be achieved by strongly convex regularization for the predictor and generator. This is a new insight for UDAs, which Chu et al. (2020) never included, as they only considered minimization of divergence, not including source risk. It should be noted that while the existence of equilibrium in the GAN and UDA minimax problems has been discussed (Arora et al. 2017 Acuna et al.) 2022). we discuss the convergence of appropriate algorithms to the equilibrium point.
2

---

Published in Transactions on Machine Learning Research (06/2024)
Numerous references delve into the minimax optimization problem over finite-dimensional spaces, often
treated as specific cases of Hilbert spaces. For instance, Cherukuri et al. (2017) Mokhtari et al. [202 Du
and Hu (2019) explore the convex-concave setting, while Huang et al. [0021] ; Thekumparampil et al. 2019
Lin et al. (2020a) focus on the nonconvex-concave setting. Although the minimax problem over Hilbert
spaces has received extensive altention, with works such as Bauschke el al. (2017 : Bot et al. (2022) Bol
ct al. 2023), the exploration of the minimax problem over spaces of probability measures or continuous
functions as opposed to Hilbert spaces remains relatively limited. On the other hand, our work delves into the
minimax optimization problem for infinite-dimensione spaces of probability measures contimuous functions Nedić and Ozdaglar (2009); Lin et al. (2020a). Moreover, we generalize the analysis from minimax to miniminimax (minimizing with respect to the predictor and generator and maximizing with respect to the classifier).
by generalizing results of
3 Preliminary
This section describes the mathematical tools required in this paper.
Let No be the set of natural numbers including zero, X C Rd be compact set, and R [- +oo} be the extended real number. We denote by M(X), M (X), and P(X) the set of all finite signed measures X, the set of all non-negative finite measure X, and the set of Borel probability measures X, respectively. Let C(X) be the set of all continuous functions X + R. As shown in Aliprantis and Border (2006) Section 5.14), C(X)) is a dual pair equipped with the bilinear functional
(n, 1p) := =f pdd,, 1 EE M M(X), P PEC(X,
and the topological dual of M(X) with respect to weak topology is C(X) (Aliprantis and Border 2006 Theorem 5.93). In the context of machine learning, we restrict M(X) to P(X). As X is a compact subset of RA, P(X) is a compact subset in M(X) (see e.g., Aliprantis and Border (2006 Theorem 15.11)).
Let M|x)) and Il-leex) be norms induced by inner products M(X) and C(X), respectively. Then, we
first define the dual norms, convex conjugation, and strong convexity as follows:
Dual Norms We denote dual norms Il-l and I1-llxx)) of Il-llmxx) and |Hllxx by, respectively,
|| sup {// pdul |lll S 1, a E M(x) &EC(X),
(1)
| sup { pdp
I|ellcxx $1, p e (xx)] M(X).
(2)
Convex Conjugation The convex conjugates F* and G* of each functionals F C(X) + R and G M(X)= + R are defined by, respectively,
/ (()), MEM(X),
(*(n) = sup pEC(X)
(3)
/ |pdd G(H), & PEC(X).
(x))) sup HEM(X)
(4)
Strong Convexity Tet Se C C(X) and SM C M(X). We say that F: C(X) + Te and G (MM(X) - R2
are ß-strongly convex (B > 0) with respect to [(-l(((x) and [I-llwax over Se and SM, respectively, holds that for any a E 10
+ a) F(ar + (1 - a)p) s (1- - a(1 - a)
|b - pllcx)) v. E Sc.
(5)
2
a(1 ())
Glau+ + (1- aG(p) (1-a)G(()
|| M| n, E SM.
(6)
2
3

---

Published in Transactions on Machine Learning Research (06/2024)
Next, review Gâteaux differentials, Bregman divergences, and L-smoothness in order.
The Gâteaux differential is a generalization of the concept of directional derivative in finite- dimte-nlieeiional differential calculus. Let F C(X) + R and G M(X) + R. then Gâteaux differentials are defined as follows.
Definition 3.1. We define Gâteaur differentials dF :((X) + and dG : :M(X) - R of the functionals F and G at up E C() and μ. E M(X) in the direction X E (() and x E M(X) by, respectively,
F(g + eλ) - F({)
dFş(A) = lim c++0
G(u + ex) G(u)
dG,(x)=
lim +0+
We note that F and G are proper convex functionals, then for yp E C(X) and μ E M there exist Gâteaux differentials d. C(X) + and dG,, MMX) + R, respectively (Aliprantis and Border 2006 Lemma. 7.14).
Then, review the Bregman divergences. The Bregman divergences functions measure between two points defined in terms of convex functions.
spaces of measures and continuous
Definition 3.2. Let F C(X) + R and G MMX) R be proper, lower semd-continnuuss and conver functionals. Then, F-Bregman divergence Dp .M(X) M(X) + R+ and G-Bregman divergence DG C(X) x C(X) + R4 are defined by, respeclively,
Dr(vlu)= = F(v) F()) dFF 11), H. M(X),
Do(b)p) = G(b) G(p) dG 1),, p. C(X).
Finally, we review the L-smoothness. The L-smoothness over spaces of measures and continuous functions are defined using the Bregman divergence as follows.
Definition 3.3. Let Se c C(X) and SM C (MX) be subsets, and F C(X) - R and G MM(X - R be proper, lower semé-conlinuous, and conver. Then, we say that F and G are T-smoooth (I. > 0) with respect to [l-leexx) and (1-Ilxx(x) over Se and SM if it holds that, respectively,
L 11 - pl(c()): 8, Sc
Dr(()p)
2
L
Da(vlu) < || ---lllxxyy n2 E SM-
2
4 Problem Setting
This section describes the problem setup of GAN and UDA training, building upon the reformulation introduced by Chu et al. (2019) as the foundation for our thcoretical framework.
In their work, Chu et al. (2019) reformulated GAN training as a minimization problem with an objective function Jnh (4) over the set of probability measures, which represents discrepancy measure between generated distribution f and unknown true distribution Po- Morcover, the adversarial loss can be obtained Uhrough the Fenchel-Moreau theorem. Consequently, they showed that various GAN models can be constructed by identifying particular discrepancy measures on an infinite dimensional space, such as the ordinal GAN (Goodfellow et al., 2014), maximal mean discrepancy (MMD) GAN (Li et al. (2015), f-GAN (Nooozin et al. 2016), and Wasserstein GAN (Arjovsky et al. 2017). Building upon this formulation, we extend it. to unsupervised domain adaptation by adversarial training.
The UDA can be regarded as a simultaneous optimization problem for source risk R : C(X) P(X) + R and discrepancy measure Jmn (u) P(X) + R between source distribution μ and fixed target distribution Vo. Then, the optimization problem for the UDA can be expressed
min (x)x)c(())x((X))
R(v.,) + Jo, (14)
(7)
4

---

Published in Transactions on Machine Learning Research (06/2024)
Here, the first variable & in R corresponds to the predictor. A typical example of R is that R(v; μ1) - (x)))
+ W()) where vo is the true predictor, and V C(X) + R and W P(X) R are certain regularization terms. The particular discrepancy measures lead to the well-known models of domain adversarial ncural networks (DANNs) (Ganin and Lempitsky 2015), such as DANNs with its extensions with Wasserstein-1 distance (Shen et al. 2018), J-divergence (Acuna et al. 2021), , and MMD (Wu el al. 2022). As
in the case of GANs (Chu et al. 2019). the Fenchel-Moreau theorem yields the following formulation equal
max R((,, (4) + / sdu J% (#w))C(())x(() PEC(X)
min
(8)
This objective function is convex for and μ4, and concave for Φ, where & corresponds to the domain classifier in the UDA. In Section 5.1 we delve into the convergence of this objective function in the general setting.
By omitting the source risk R, the formulation { reduces to that of GAN
/ pdy J (9).
min max PEP(X) PEC(X)
where io corresponds to the discriminator in the GAN. This allows us to analyze the convergence properties in GANs and UDAs in unified manner. In other words, the findings of GANs, which have been extensively studied for stability, could be used for UDAs. In fact, the assumptions used in this paper are related to the constraints of the GANs (see Section 6
However, the formulation of S which extends the reformulation of Chu et al. 2019 deviates from minimax optimization in actual GANs and UDAs such as Goodfellow al. (2014); Ganin and Lempitsky (2015), as it does not directly optimize the distribution μ.. To get more practical situations, we consider the source distribution μ as pushforward measure fıEo of fixed probability measure to E P(Z) by continuous function f e C(Z; X), which corresponds to generator in GANs, feature extractor in UDAs. Then, the problem
g is reformulated as
min min max R(v, (156) pd([{So)= (%, VEC(X) SEC(Z.X) (EC(())))))))
(sp).
(10)
This objective function generally nonconvex for and f. In Section 5.2 we explore the convergence of this objective function in the general setting.
5 Minimax Analysis
Our goal in this section is to prove the convergence of the minimax optimization problem in the scheme of the gradient descent under appropriate assumptions. In Section 5.1 will consider the convex-concave problem over spaces of continuous functions and probability measures, and prove that the sequence obtained by a certain gradient descent. converges to the optimal minimax solution. While, in Section 5.2 we will consider the nonconvex-concave problem over spaces of continuous functions, and show that the sequence obtained by a certain gradient descent converges to stationary point.
Note that the objective functions in Sections 5.1 and
5.2 are general forms of and 10 respectively.
5.1 Convex-concave setting
This section considers the following minimax problem:
min max K(U,, p), (WH))S X PE
(11)
where S C P(X) and S2 C C(X) are compact convex subsets and K. C(X) x M(X) x C(X) - is supposed to be an objective function of GANs or UDAs. The typical example IC is the objective function in that is,
R(b, K(W,p,)) -)(())))-(())))))))))))))))))))) + +/ pdd (p).
5

---

Published in Transactions on Machine Learning Research (06/2024)
We show that the sequence obtained by the gradient descent converges to the optimal solution of appropriate assumplions. To do this, we first consider the joint convexity as follows:
11
under
Assumption 5.1. Assume the following:
(1) ()) is proper, lower semi-conlinuous, and convex over S1 x S' for each up E S2. (ti) XC(,,, ...)) is proper, upper semi-continuous, and concave over S2 for each & E S1 and S is convex-concave problem. Under this assumption, Sion's
This assumption means that the problem 11 minimax theorem (iion 1958 guarantees that
min (V.))SS X PE.2
max K( = max min K(U, p). PES (v.p)
Morcover, there exists at least one minimax solution (i, P.) in our minimax problem 11
K(y sP.) 2 H(,4,) & E Ki P.) & ((V,,, (W, x
(12)
Note that this assumption is in line with practical settings. Indeed, as shown in Section 6.1.1 the source risk R of UDAs can be joint convex by adding both reproducing kernel Hilbert space (RKHS) AAvarez et al., 2012 and maximal mean discrepancy (MMD) (Gretton et al. 2012) constraints.
Next, we put the assumptions related to the Gâteaux differentials. Let ||lloxx), and ||-le(x),2 be norms induced by inner products in C(X), and let |l-lMy( be norm induced by an inner product M(X). Note that both the first variable v and the third variable y in K. are contimuous functions, but the inner product space (C(X), (1)))(()).) for & is different from the inner product space (C(X), (l-)le(x),) for sp Assumption 5.2. We assume as follows:
(i) For each St, μ S' and up S2, there exist the following arguments of the mazimum:
{ vdd K(,, (((.)))))()
Ne argmax vEM(X)
{ odd
Parni.pp = argmax
(EC(X)
{
Av.p. = argmaxx
pdx - K(V,
XEM(X)
(ti) Norupp Puiiee and Agp are bounded with respect to dual norms I1-léc(), II-10xxx) and (1-1
, that is, there exists B > O such that, for (v, 9) E S, x S' x S2,
|l $ B, P li(x) S B, || l2 $ B.
(13)
Here, ((.,H1,))) s K(),,,)) and (().,,)) are convex conjugates of K(+, p), K(b, ' pp), and K(b,,,,)) respectively. The above assumption guarantees that the existence of Gâteaux differentials of K,, and provide the form their Gâteaux differentials as following lemma. The proof is given by the similar arguments in Chu et al. (2019 Theorem 2).
Lemma 5.3. Let Assumption 5.2 hold.
Then, for each &p S1, M S', and v S2, there erist Gâteaur differentials dIC(, u, 1)s, dKC and dK((,, u.)) of K(,H,,)), ((b,,))) and K(V, u. -) at to E St, E S', and E S2, and they are expressed as follows:
dK((,1,))(()) ydNe..ies
dK(b) (0) / odde

---

Published in Transactions on Machine Learning Research (06/2024)
In addition to these assumptions, assume the L-smoothness of K. for each variable to show the convergence to minimax solution.
Assumption 5.4. Let T. > 0. Then, we assume as follows:
(i) For each S' and yp E S2, (() is L-smoolh with respect to ||-le(x),1 over St.
(t) For each v E St and E S2, (((b,,,)))) is L-smooth with respect to II- (lx)) over S'
(18) For each v E St and μ E -K(v,,
-ECS""KK(V,,, (1.)) L-smooth with respect to ||-lle(x),2 over S2.
This assumption also aligns with practical settings, e.g., (iii corresponds to the case where the f-divergence Ali
and Silvey 1966 Csiszár 1967) or integral probability metric (IPM) (Müller 1997) is utilized as a discrepancy measure.
Here, we define the gradient descent for solving minimax optimization problem 11
Definition 5.5. Let the S1, no E so E S2 be initial guesses. (V Mmy (n)))ant S1 8 S2 by
We define the gradient descent
Vnl1 argmin WE81
dlC(, Moi Unlon (vb vnn) +
|1 vollecx1.11
20nn
1
Pnt1 argmin peg'
dKC Pan (10 - -(n) +
|| - M (mll2(x))
20
1
11p - -pxll pPllc(x)11 )
Pat1 argmax PE82
dKCWns Pay )o (1 - -vn)=
2an
where on 0 is the step size of the update rule.
If subsets S,, S", and S2 are subspaces, then update can be expressed a sum of a previous step and a gradient term, form that is commonly encountered in the gradient descent algorithm (see e.g., Chong et al. (2023) ). However, in the general case of subsets S1, S', and S2, the argmin and argmax in Definition 5.5 may not exist. Therefore, in this paper, the following assumption established to ensure the existence of the gradient descent of Definition 5.5)
Assumption 5.6. Assume that there erists sequence {(vn. Pny PA)DE C S1 S' S2 defined in Definition 5.5
Building upon the background established above,
ready to present our main theorem of this section:
5.4 and 5.6 hold, and let on 1/T Let ((ynn Pus (n) IeNoo S1 x S' x S2 be the gradient descent defined in Definition 5.5 Let (1h (Wn, be minimar solution for 11). Then, for any N E N, we have
Theorem 5.7. Let Assumptions 5. 5.2
N-1 Σ On ==
C... 6B
N
K(sb,, S
Σ
(14)
n=0
where
C. = sup || sup
voll(())11 + sup || Hollaxx) west
+ sup se
1| - po poll 2
(15)
SES.
Here, UN, DuN and GN are weighted averages given by
-N antin
N- Onlfi
>N N aniin
EN
ĀN =
GN =
(16)
DN11 an
N an
-N-1 an
Proof- See Appendix A
7
7

---

Published in Transactions on Machine Learning Research (06/2024)
We note that the the constant C. is finite due to the compactness of S', S1, and S2. We observe that the upper bounds 14 with different choices of step sizes on E (0, 1] are as follows:
If the step sizes are constant, denoted by On a, then the right-hand side (RHS) of 14 is expressed as
C,
RHS
14
+ 6B²a.
2a
which does not converges zero as N + oc. Therefore, in this case, weighted averages 16 provides an approximate solution to the minimax problem. The first term converges to zero as N 30 with an order of (1/NN) The second term can be reduced as a - 0, despite the first term diverging. This is a trade-off relationship with respect to the step size a. A similar observation was made in Nedié and Ozdaglar (2009) Proposition 3.1), which studied the minimax problem finite dimensional space using subgradient methods.
If step sizes decay as an = =a/vi v where a is constant, then the right-hand side of 14 is expressed as
C.
6B (1 + log N),
RHS of
2avN
VN
which converges to zero as N + oo with an order of Olog N/vN). Therefore, in this case, weighted averages 116 provide an exact solution to the minimax problem.
5.2 Nonconvex-concave setting
Unlike the previous section, which considered the convex-concave minimax problem expressed section considers the nonconvex-concave minimax problem.
11 this
Let Z C Rgd be a compact set, and let X(Z;X) be the set of all continuous functions X, and let S" c C( and S2 C C(X) be subspaces. Then, consider the following minimax problem:
min min max G(V, f.4),
(17)
#EEL,e FESy #E82.
where 5% S", Sr, C S1, and S2,0 C S2 are convex subsets, and g C(X) x x(Z; + R is supposed to be an objective function of GANs or UDAs. The typical example of G is the objective function in 100 that is,
G(v.f.p R(v, feEo) / pd((fis)) Ji, (1))
The difference with Section 5.1 is that 17 does not assume the convexity for G(-, f. x) and (()), Thus, since there may not exist Nash equilibrium point for problem [17 in general, it is difficult to prove that the sequence obtained by some gradient descent converges to the optimal minimax solution.
We show that the sequence oblained by a certain gradient descent converges to a stalionary point of 17 under appropriate assumptions.
Throughout this section, let (,,)),, (s,),,, and (s-ss, be inner products S1, s", and S2, respectively. We denote || ||s || Ils,, and || llss by norms induced by thier inner products.
First, put the following assumption for S1, S" and S2.
Assumption 5.8. Assume that S1, S",, and S2 are closed subspace with respect to norms || ||s || Ils and || lls, in C(X), C(Z;X), and C(X), respectively.
Under Assumption 5.8 S1, S", and S2 are Hilbert spaces equipped with inner products (s))s, (s)),,, and (s)) respectively. This assumption implies that convex subsets 5% St,e, and S2,e in continuous function spaces (where gradient descent updates are actually performed) must be contained in Hilbert spaces S", Si, and Sz.
Next, put the assumption about the I-strongly concavity.
8

---

Published in Transactions on Machine Learning Research (06/2024)
Assumption 5.9. Let B > O. Assume that for each vb S1,e and Sh g(b, f..) is B-strongly concave with respect to || lls, over S2.c.
Note that this assumption is related to the gradient penalties (Gulrajani et al. 2017), widely-used as the stabilization techniques in adversarial training, as detailed in Scction 6.2.1
Under this assumption, we can define for & E S1,6 and S"
( f) == argmax G(V, f1)),
pESa,
G f) == max 4E822.
= -.
(18)
Hereby, the minimax problem (1 is equivalent to minimization of 188 under Assumption 5.9
Then, we put the following assumption related to the Gâteaux differentials, which is associated with the spectral normalization MMyato et al. 2018) widely-used as stabilization techniques for GANs.
Assumption 5.10. Assume that, for each vp Si,r f E and up E S2.0, there exist Gâteaur differentials f. f, (g(,,,,))) and (v f. of g(. ((b,,,)), and ((D,f.) at E S1,05 f S",, and &p E S2,0, respeclively.
Under Assumptions 5.8 and 5.10 and dg f.)) S2 R are identified with some elements in Hilbert spaces S1, S", and S2, referred as f.(, E Si, Vg(w, ;)) E S", and VG(W, E S2, respectively. Furthermore, by Riesz representation theorem, we have the following:
Gâteaux differentials dgC, f. f) S1 + R, dG(w, sppss S + +,
ag(:f.p) f.9); h889
(g (g(.
(g(w,,))) (Vg(1), ( (1)),
(d)()), 11 ; f. = [ )plss.
f. (v f, fp:
In addition to these assumptions, we assume the L-smoothness of & for each variable v, f, and ip show the convergence to stationary point.
Assumption 5.11. Let L > 0. Then, we assume the following: for v, v1, the E S1, S,J1, E S", and s,, $1,92 E S2,
(4) VG(: vg(:f.4) Ll 4/21ss
(6) 19((-1.9)) FG(-ff2.)ools Lllfi f2lls",
(c): (G(. vG(-ff.91vvls. & Ll #2l1s
(d) (9)6),)) vg(d,, & L|]Л - f211s",
(e) [V(())))))) r) L|l - d2llss
(19)
(f) [Vg(b,,)1) v I Lllş1 42 92
(g): (V VG(0, f. I Lllęi P2llsa.
(h) (V vg(v2, 5 L|| +22lss
(i): (V f1,)p (2 < LIfi fzll
Here, we define the projected gradient descent for solving minimax optimization problem 17

---

Published in Transactions on Machine Learning Research (06/2024)
Definition 5.12. Let Yu E S1,e, fo S. yo E S2,c be initial guesses. Then, we define the projected gradient descenl ((Wn, Jas fnnN C S1 So x S2 by
Pa+1 = argmin #ESI
dd[(, fn: (o vn) + 20 11 1/ 11
Unt = Ps (1n+1))
Ja+1 = argmin JES"
dg
Sn) +
|1 Men li
20xg.
fat = Psy (Frti),
1
Pnt1 = argmax DES2
fwr )) (1p ppn)
111 Pnllss
20ppm
Pn+1 Psz.c (F
where Psr. psy, and Psa.ee projection operators Si,e, Sh and S2,0, and auj > 0, affn > 0, and Oxp, > O are step sizes.
Remark that, using Riesz representation theorem under Assumption 5.8 the above update rule is equivalent to the following:
Vn+l PsL. (Vnn - Own VG(,, Pn.
fnt1 = Pss (fn - ann (
Pn+1 PSs,, (Pn + a ( ) )).)
We also assume small step sizes to show the convergence of the gradient descent algorithm as follows: Assumption 5.13. Assume that there exists Co. C and Co. Cj,7 E (0,1) such that for all n No,
(i) Co < a < min (t:
(1) ++ Lg p.n + (1 + L)(1 + Lz)a + LL2 - 17(1410)07 < C,
(tt)) (1 - 2 a 1)
2L B
1 +
1
(a nn 5
3C%
(10) Oe < Law
L(1 + L)(1 + Lo)arga Lggap.n
22."Caw (1 (ta))))
33(1-3)
(v) Cf 51 Logg
Elllaalasa 2
2120asa LACEF
(1+
2
('11-))
where denote by Lg := L (1 +1)
These assumptions imposes small step sizes agins af.n app,n depending on constants L and B. Roughly speaking, Assumption 5.13 requires that step sizes be chosen small enough depending followings are the details First, need to choose small aip, satisfying (i). However, asp, should not converge to zero as - co. Therefore, its lower bound is set to Co 0. Next, we need to choose small asn. and Of,. satisfying (ii) (v). Note that the smallness of Owi... and a depends on lower bound Co > 0 (see (iii)-(v)) In other words, Oggin and a should be chosen to be smaller than 0tp.n- Similar assumptions are often made in the context of nonconvex-concav minimax problems, as observed works such as Huang et al. (2021): Lin al. 2020a).
and L. The
Building upon the background established above, we ready to present main theorem of this section:
10

---

Published in Transactions on Machine Learning Research (06/2024)
Theorem 5.14. Let Assumptions 5.8
5.9
5.
5.10
5.11
and 5.13
hold. Let {( Sns pn) C S2,e Then, for N E N, we have
be the projected gradient descent defined in Definilion
5.12
N-1
11/
VGv.N
Σ
(20)
8
==
N-1 Σ
-1/2
VG
ff
(21)
1=0
where
C11 s0o 2
1/2
Ö = (G Jo)
inf
f)
(w.f)e x
1 1-7
Here, VG and JGN, weighted averages given by
V. En= av,n fnly
En a VG(ens )
VGIN :=
(22)
NN-1 apn
ΣN-11 afnn
Proof. See Appendix
The idea of the proof is generalize Lin et al. 2020a Theorem 4.4), which studied the convergence of the nonconvex-coneave minimax problem in the finite dimensional setting, to the infinite dimensional function spaces, and to generalize two variables to three variables. Note that if step sizes are chosen as constants
satisfying Assumption 5.13 then the hand sides of 20 and 21
are expressed as
RHS of 20 and
21 O(1/VN)
which converges to zcro N +000 In other words, we have proved that the gradient decent defined by Definition 5.12 converges to stationary point. Finally, we note that the order O(1/VN) agrees with the result obtain by Lin et al. 20200a Theorem 4.4), though we have adopted the infinite dimensional setting.
6 Examples of Relationship Between Objective Functions for GANs and UDAs and Assumptions for Convergence
In this section, confirm that certain objective functions of GANs and UDAs fulfill the conditions for guaranteed convergence described in Section 5 In Section 6.1 we will verify Assumptions 5.1 (i) and 5.4
& (iii) for the problem (8). and in Section 6.2 we will verify Assumptions 5.9 and 5.10 for the problem [10 because we can immediately confirm that the remaining assumptions hold for our objective function.
6.1
Convex-concave setting
In this section, we will verify Assumptions
5.1 (i) and 5.4 (i) and (iii) for the convex-concave setting (
6.1.1
Assumption 5.1 (i) (Joint convexity of (1) + R(v,
Assumption 5.1 requires the joint convexity of source risk R(v', μ) J E(bb, do with respect to (vh u1) e C(X) x P(X) for the existence of minimax solution. Here, μ is marginal distribution of source domain, dh is a predictor to be optimized for task (which may be implemented as neural network), and vh is the true predictor for task. Also, E(p, tho) denotes loss function for task in the source domain. In general, it is obvious that the source risk does not possess the joint convexity. Therefore, we need to introduce some regularization terms to the source risk such as
R(v, p) = / ((b, +
+ W(4).
(23)
11

---

Published in Transactions on Machine Learning Research (06/2024)
where : C(X) - R and W P(X) - R are regularization terms. The next proposition gives the sufficient conditions of the joint convexity for the source risk:
Proposition 6.1. Let Il-1,(x), and 11,le be norms in C(X). Let P.Y > with 7 2 p. Let be a loss function & C(X) x ((X) + I.. Assume that.
(i) db + E(bb, vb) is conver for each Vo E C(X).
(8) vb + l(b, vo) is p-Lipschitz with respect to Il-,, and Il.le(x),2 for any to C(X), that is,
[1((,1, vb) l(v2, Vollo, S slll (21le(x),2) v1 C(X).
(il) V and W are y-strongly conver with respect to Illlc(x),2 and I1-1 respectively.
Then, the source risk R(b, pu) is joint conver with respect to (w. p))
Proof. See Appendix C.1 for the proof.
Assumption (i) and (ii) are the convexity and Lipschitz contimuity for the loss function. For example, the squared error loss satisfies these assumptions. The example for (iii) is that V V= - Il- and W 1-11 where (H,, (1, ) is reproducing kernel Hilbert space (RKHS) with a positive definite kernel K; X x + R. Note that the dual norm I|-,l,,, of the RKHS norm II-1 x,, corresponds to maximal mcan discrepancy (MMD). As both I1-1l,, and II-Il,,, are norms induced by inner products, V and W are 1-strongly convex with respect to I1, and I| respectively.
6.1.2 Assumption 5.4 (i) (Smoothness of + f( ddp)
Assumption 5.4 i) demands that the source risk R(,,, n1) - E(s,, do is T-smooth for sp E C(X). Let consider the following general functional In.p C(X) + R for the later convenience:
Th = (000)=(MM())) MV(x))dp()) & E(()),
where h: IR + R and P(X). Then, the next lemma guarantees I-smoothness of ( Lemma 6.2. Let a, b oo] and h E (a, Then, we denote
Sc,a = (w E((). a as v(x) s x EX
Assume that h (a, + R L-smooth, that is,
Dn(slt) & 2 t/. s,t (a,b),
where (s(s) = h(s) - h(t) - (((t((s t). Then, In, is L-smooth with respect to ||- (xx over Sc.,..
Proof. See Appendix C.2 for the proof.
If loss function f(,, vu) L-smooth (e.g., squared error loss), then by Lemma 6.2 & R(b. p1) E(W ((v. is T-smooth with respect to || (1))))))))
12

---

Published in Transactions on Machine Learning Research (06/2024)
6.1.3 Assumption 5.4 (iii) (Smoothness of yp + Jo (1))
Assumption 5.4 (iii) imposes the L-smoothness condition on the convex conjugate Jrh (pp of the discrepancy measure Joo (1) to ensure convergence for both GAN and UDA. The representative discrepancy measures are f-divergence (Ali and Silvey 1966 Csiszár 1967) and integral probability metric (IPM) (Miller) [197]] which respectively unity different divergences between probability mcasures with various applications such GAN and UDA. The f-divergence includes Kullback-Liebler divergence, Jensen-Shannon divergence, and Pearson x² divergence, while the IPM includes Wasserstein-1 distance, Dudley metric, and maximum mean discrepancy.
In the subsequent, we provide several examples of Jr that satisly the L-smoothness for (A) f-divergence and (B) IPM.
(A) f-divergence Let f: dom, R4 +R R be proper, lower semi-contimuous and convex function. Then, the f-divergence (y(u)) between μ E P(X) and v E P(X) is defined as
S
du de
du
p
Dy(1l))
(24)
+00
otherwise
where μ denotes that a is absolutely continuous with respect to
The f-divergence is joint convex with respect to (H,)) (as the mapping (p.9) + (f(p/4) is joint convex) and non-negative for all and y but not symmetric with respect to and v in general. In our case, we set Jy (1)) = Dy(((p)) with a fixed measure yo which implies a true distribution for GAN and a target distribution for UDA. The convergence theorem demands the L-smoothness for the convex conjugate Jj., (4) of J (p).
The following lemma provide the representation of the convex conjugate J1 (4p of J..vo(u):
Lemma 6.3. Assume that e (dom)), and there exists the inverse (f) -1 of f' Then, the conver conjugate Jim of Js. is given by Jj.no (4) f spdvo for & E Se,f == 1ip C(X) p(z) dom where
(*) == sup sup f(t)}
f s dom
Proof. See Appendix C.3 for the proof.
In the context of fddivergence, it is sufficient to confirm the smoothness of the convex conjugate f* in the sense of the real function. We can take Jensen-Shannon divergence and Pearson x² divergence examples and confirm that Ji. (14 satisfies the T-smoothness.
Example 6.4 (Jensen -hhannon divergence). The
divergence
Lin 1991 is defined as
Dıs(ulv)= - + DKL(ul))
1 DKL(vIp),
(25)
where ( v)/2, and (KI( is the Kullback Leibler divergence between μ and defined by (KL(Hl))= =
dp
log du dv. du du
Here, f(t) is represented as
1+t
fus(t) = - (++ 1) log
+ logt, te (0,
(26)
2
2
The convex conjugate (s)
1 fis(s)= log(1 (1
1 _2 log2, 2,
1
SE (-00, 5 log log2). 2).
(27)
2
The conver conjugate fis is L-smooth over (a, b) with some L 0, and a, E (-00, 1 log Therefore, by using Lemma 6.2 J Jis.. is T.ssmooth with respect to ||
13

---

Published in Transactions on Machine Learning Research (06/2024)
Example 6.5 (Pearson x² divergence). The Pearson x² divergence is defined
Dx-(ul)) = / -1
f(t) is represented
[p(1) (1 ()² LER. R.
The convex conjugate fp(8) is
f5()) 5² + SER.
4
The conver conjugate fp is L-smooth over (a.b) with some > O, and a, b E RR Therefore, by using Lemma 6.2 Jpp0 is L-smooth with respect to 1|22(X
(B) Integral Probability Metric (IPM) Let IF be class of real-valued bounded measurable functions X. The IPM associated with F is defined
{// gdp f gdv
dr(w,v) = sup OEF
(28)
for all pairs of measures (14, v) E P(X) x P(X) such that all functions in F are absolutely μr- and u-integrable. The typical examples are Wasserstein-1 distance for == {g E Lip(X) IIgllpip I 1} where Lip(X) is the class of the real-valued Lipschitz functions on X and I1-Ilaip the Lipschitz norm, and the MMD for F= == (g E H: 1|gl S where (H, Il+1lx) is an RKHS with positive definite kernel K X X
In our obtain the following Lemma.
case, set JIPM, (1) = (p, vo) with fixed measure Py- Then,
Lemma 6.6. Assume that F include the zero function. Then, the convex conjugate JipM of JIPM is given by
JiPM. / pdvo + xxpp F}.
where the indicator function is give by
0
if A is true
x{A}
(29)
oo
if A is false
Proof. See Appendix C.4 for the proof.
From Lemma 6.6 the differential d(JiPMMoo of JiPM. at 4p F is given by
d(JipM..o) = J A xec(X).
which implics that, by Definition 3.2
Dijos (w/() = U.EFF
Therefore, we obtain the following proposition.
Proposition 6.7. For any L > O, and any norn || lle(x) induced by inner products, JipM. L-smooth with respect || (lc(x) over
6.1.4 Examples of KCth u, (p) simultaneously satisfying all assumptions in Section 5.1
In this section, we provide an example of our objective function S that simultancously satisfies all the assumptions (Assumptions and 5.4 for the convex-concave structure and smoothness.
Let tho C(X) be a true predictor, and let vo E P(X) be a true distribution. We consider an objective function 1C, S 5' S2 R defined
! f [ (w- vo) I
1 + 2 [llll
#2 + sdp 1
-olu-/aa k(w))do..
14

---

Published in Transactions on Machine Learning Research (06/2024)
which corresponds to the problem 18 with R(), f 2o) 3 | H 2 I| and J.g is a regularization parameter, and k: (a, b) + R is a convex and
cither IPMs or f-divergences. Here, Co-function with some a, b E R, introduced to encompass more general situations including both IPMs and f-divergences. If the function & takes the form k(s) = 8, then the discrepancy measure Jro corresponds IPMs. If the function & takes the form k(s) = (s), then the discrepancy measure Jnd corresponds to f-divergences. Also, (Ho,ll-llxy I1,)ll,)) is a RKHS with Gaussian kernel (2 y) = (2TO²) -d/2 --=--==== /20" with variance o²,
We choose convex subsets S1, S", and S2 as
= (v (X): 1102 C. for all a E N
S' = ( P(X) m(A) 3 (A(A) for all measurable sets A in X): S2 = FPEF as ((z) s b, x xE
where CJ (X) is the space of 2 functions with compact support X, subset C(X), and Mu E MM(() is non-negative mcasure. Then, Proposition 6.9 is obtained if the following assumption is satisfied.
Assumption 6.8. We assume the following:
1/B
2 40/02 where Co == EjEN, (4o²)3
vo E S1 and vo St.
(a, + R is Lssoohhh in the sense of the real function.
Proposition 6.9. Let Assumption
6.8 hold. Then, the following statements hold:
(1) [Assumption 5.1
(1)] (1) Kr (1,,,, (p) is conver.
(2) [Assumption 5.1
(ii)] p (1b, vp) is concave.
(8) [Assumption 5.4
(1) + K2 (v), pr p) is 1-smooth with respect to
?
1
11 + 11-11 Hovz
(4) [Assumption 3.4
1/2 Il (ii) p KT((b, p4, (p) is 1-smooth with respect to (3) (iii) xp + (1h (1,), is L4-smooth with respect to || -llx(()))))
(5) [Assumption 5.4
Proof. See Appendix C.5
Note that we immediately confirm that the remaining assumptions hold for objective function. That. is, Theorem 5.7 is satisfied by the setting of this scction.
6.2 Nonconvex-concave setting
In this section, will verify Assumptions 5.9 and 5.10 for the nonconvex-concave setting 10
15

---

Published in Transactions on Machine Learning Research (06/2024)
6.2.1 Assumption 5.9 (Strong convexity of (p H J (4))
Assumption 5.9 requires the strong concavity for the problem 10 with respect to y. This requirement is
equivalent to the strong convexity of discrepancy measure Jun We realize this by adding some regularization to the discrepancy measure.
Assume that the discrepancy measure Jno proper, lower semi-continuous, and convex. We define inf- convolution Jro I(j)
Jup @ I(m)= inf JE)(E) + I(u - ))
SEM(X)
where I (M + E is regularizer function, which is proper, lower semi-continoous. and convex. Then, the following Lemma holds.
Lemma 6.10. Let 3 > and || (l(x)) be a norm induced by an inner product in M(X). Then, if I: M(X) R (1/8)-smooth with respect to || [lM(x)), then (Jvo I) C(X) - is B-strongly conver with respect to || (li)(x)-
Proof. See Appendix D.I for the proof.
Thanks to this Lemma, we can attain the strong convexity by introducing the regularizer, such as the squared MMD in the RKHS with the Gaussian kernel, which corresponds to the gradient penalty (Gulrajani al., 2017
6.2.2 Assumption 5.10 (Gâteaux differentiability with respect to f)
Here, we consider the case when the source risk has the form R(v,p) = E(v, do where the loss function {(b,, vo) is convex with respect to v. Then, the minimax problem (10) is translated into
/ ((p o fo o f)dpo + / op fduo - J (1p).
min min max VEC(X) (EC(Z.)) (EC(X)
(30)
It is obvious that the above objective function is Gâtcaux differentiable with respect &i and ip because the above objective function is convex and concave for and y,, respectively. As functions ei and ip are composed with f, some regularity for vo and &o is required to hold Gâtcaux differentiable with respect to
Let us consider the following general functional Jn,e C) + :
In,(())= /no fEC(Z:X),
where h e C(X) and & E P(Z) are fixed. Then, the following lemma guarantees the Gâteaux differentiability of Jn.g-
Lemma 6.11. Assume that h Lip(X), C(Z;X), and μ where m is the Lebesgue measure. Then, In is Gâleaur differentiable al. f. Furthermore, ils Gâleaur differential is given by d d(Jn.e)
d(Jn.es (Vhof)-00 gdg.
Proof. See Appendix D.2 for the proof.
Thanks to this Lemma, we can attain the Lipschitzness interpreted applying the spectral normaliza- tion MMyatto et al. 2018), widely-used stabilization technique for GANs.
16

---

Published in Transactions on Machine Learning Research (06/2024)
6.2.3 Examples of G f. () simultaneously satisfying all assumptions in Section 5.2
this section, we provide an example of our objective function 10 to simultaneously satisfy all the
assumptions (Assumptions 5.9 5.10 and 5.11).
Let th C(X) be true predictor, Po P(X) be true distribution, and B >> 0. We then consider objective function G1 S1 S" x S2 + R defined
1 / / (1o to f) + 1o fdEo M())dm I1pplbu.
9. (().f.))=
which corresponds to the problem 10 with R((, = s( dd,, and discrepancy measure Jvo replaced with inf-convolution Jzh, + (23 (la;a) where Jng either IPMs or f-divergences. Here, (Ho. l1-11x,,) RKHS
with Gaussian kernel Ko (z. y) (2rra))<//e-baa
/20² with variance o². In the same way of Section 6.1.4 introduce (a,b) R. which convex and Cl-function with some a, R, to encompass general situations including both IPMs and f-divergences.
We choose norms || lls. || lls,, and || lls, as
lls, = ||. ((xx)); 11lss == |1 11A((2XX.00) I = || 11
(31)
and subset SI, St.c, S", Sh S2, and S2,6
lls.
S = (v E (X) is and V are Lipschitz continuous]
[v(x)), sup [v)(t)) S SC2 S1, = $ E Si (ipt,,, Ci, sup =ex
xEX
d(s{So)
S" = 1 (c); [1 < oo, 1á50 < m, sup
< 00
dm
d(ftEo)
8" = EEs": sup
$ CCay.
dm
11
S2 = (0 E co (X) Ho nF and Vi are Lipschitz continuous)
S2. = 10 E S2 Lip & Ca. a ((r) < b, E X},
with some constants C1, C2, C3 C4 0, where Lip(v) the Lipschitz constant for function sh, and F is a subset in C(X). Then, Proposition 6.13 is obtained if the following assumption is satisfied.
Assumption 6.12. We assume the following:
vo E S1,c-
The derivalive k' of k is Tippippchila continuous.
m, and supzex dra (x) <0.
Proposition 6.13. Let Assumplion
6. 12
hold. Then, the following slatements hold:
(1) [Assumption 5.9
+
01(1,, f.) is B-strongly concave with respect to || lls..
(2) [Assumption 5.10
f.4) is Gâteaur differentiable for each variable.
(3) [Assumption 5.11
fi(bff.p satisfies the condition
19
Proof. See Appendix D.3 for the proof.
As well as Section 6.1.4 the remaining assumptions hold for our objective function. Therefore, Theorem 1 is satisfied by the setting of this section.
17

---

Published in Transactions on Machine Learning Research (06/2024)
6.3 Interpretations of our analysis
Throughout Sections 6. and 6.2 we have verified that certain objective functions for ideal settings of GANs
and UDAs satisfy the sufficient conditions for the convergences discussed in Section 5 Both objective functions for GANs and UDAs involve the discrepancy measure, and its convex conjugate need to be strongly convex and L-smooth.
An example for achieving strong convexity is through the inf-convolution with a discrepancy measure Jro and regularizer such as the squared MMD -1:2 in the RKHS H. with Gaussian kernel (o(x,y) (2xo")-d/aa-lk--ff having variance o² (Lemma 6.10 The convex conjugate of this inf-convolution can
be expressed as
(Jw, + 11 )' (xp = Jin (p) + 11
Also, the RKHS norm I| in this equation is represented as CChu et al. 2020 Proposition 14))
1 HLe
Σ aa/=k
10 a!
and minimizing this RKHS norm involves constraining the gradient of discriminator to be small. This can be interpreted applying gradient penalties (Gulrajani et al. 2017), common stabilization techniques in adversarial training, to penalize gradients with large norm values. Note that the gradient penalty [Gulrajani et al. 2017) is regularization technique to add the gradient norm Egrr/ - 1|2] to the discriminator's loss function.
On the other hand, when considering the discrepancy measure as IPMs, the convex conjugate of IPMs is given by
Po / padvo + x(p E F},
which L-smoothness for y0 F (Leminna 6.6 The function class F should be the subsel of Lipschitz continuous function spaces Lip(X) due the Gâteaux differentiabiiliyy of objective functions the nonconvex- concave problem (17) (Lemma 6.11). The restriction of J C Lip(X) can be interpreted as applying the spectral normalization (Myyato et al. 2018), widely-used stabilization technique, to enforce the discriminator to be Lipschitz continuous. The spectral normalization (Miyato et al. 2018) is a normalization technique for weights of neural networks so that the Lipschitz norm ||pll of the discriminator is bounded above by 1.
In addition, our sufficient conditions involve the joint convexity of the objective functions for UDAs. An example of achieving joint convexity is adding the strongly convex regularization source risk, that is, considering R
R(v, p)) = ((v). do)du + 1 H2 +
||
+2
2
which is joint convex with respect (1), (Proposition 6.1 Here, (H4, 11-11 is RKHS with a positive definite kernel K; : X X R. The RKHS norm is used as regularization for NNs, such as in Bictti al. (2019).
7 Conclusion and future work
We provided the rigorous framework for the convergence analysis of the minimax problem in the infinite- dimensional spaces of continuous functions and probability measures. We discussed GANs and UDAs comprehensively and interpreted the assumptions for the convergences stabilization techniques.
The following is the list of future work:
It would be interesting to verify whether the H-divergence satisfics the assumption. our current. formulation, we utilized the convex conjugates of the divergence, implying that the classifier yp p::0 X + R belongs to continuous function space C(X). However, the formulation of the H-divergence (e.g.,
18

---

Published in Transactions on Machine Learning Research (06/2024)
Ben- David et al. (2010 2006)) employs the hypothesis spaces, where the corresponding classifier y X + (0,1) is labeling function. Therefore, integrating the H-divergence would need different. formulation.
We employed gradient descent (GD), the simplest algorithm that suffices to achieve our aim. However, Tormulating the stochastic gradient descent (SGD) within the Iramework of our work would be an interesting future direction. Notably, Lān al. (220aa) studied the convergence in fiite-dimennionall minimax problems using both GD and SGD. Our work in Section 5.2 extended the results from GD presented in Lin et al. (2020a) to infinite dimensional spaces. Similarly, could generalize the results for SGD well. In addition, sophisticated minimax algorithms to obtain better convergence rates have been proposed (see, e.g.; Lin et al. (202bb)) It would be also interesting to formulate such algorithus in our framework.
We optimized the distribution μ or the transport map f as variables and fixed vy as the true distribution, which deviates from the actual setting UDAs. In a more realistic scenario for UDAs, μ and Po would be formulated as μ fal and P0 = -r where and g are, for instance, neural networks optimizing their weight parameters, and &. and & are fixed source and target distributions. However, this scenario is highly nonconvex-nonconeent and this setting is more challenging to show convergence.
It would be highly interesting to experimentally confirm how convergence properties in this study relate to the stability of GANs and UDAs. However, due to the fact that the optimization steps outlined in Definitions 5.5 and 1.12 entail proximal steps over infiniteddimensional spaces, practical implementations pose significant. challenges.
Acknowledgments
The research was jointly funded by AISIN and AISIN SOFTWARE. TF was partially supported by JSPS KAKENHI Grant Number JP24K16949.
References
David Acuna, Guojun Zhang, Marc T. Law, and Sanja Fidler. f-domain adversarial learning: Theory and algorithms. Proceedings of the 38th International Conference Machine Leorning, 139:66-75, 18-24 Jul 2021.
David Acuna, Mare T Law, Guojun Zhang, and Sanja Fidler. Domain adversarial training: A game perspective. arXiv preprint arXiv:2202. 05352, 2022.
Syed Mumtaz Ali and Samuel D Silvey. A general class of coefficients of divergence of one distribution from another. Journal of the Royal Stalistical Sociely: Series (Methouologicuu)) 28(1):131-14) 1966.
Charalambos D. Aliprantis and Kim Border. Infinite dimensional analysis. Springer, 2006.
Mauricio A Alvarez, Lorenzo Rosasco, Neil D Lawrence, al. Kernels for vector-valued functions: A review.
Foundations and Trends3 in Machine Learning, 4(3):195 266, 2012.
Martin Arjovsky, Soumith Chintala, and Léon Bottou. Wasserstein generative adversarial networks. In International conference on machine learning, pages 214-223. PMLR, 2017.
Sanjeev Arora, Rong Ge, Yingyu Liang, Tengyu Ma, and Yi Zhang. Generalization and equilibrium in generative adversarial nets (gans). In International conference on machine learning, pages 224 232. PMLR, 2017.
Pierre- Firree-yrrll Fubnn-Frankowski, Anna Korba, and Flavien Léger. Mirror descent with relative smoothness in measure spaces, with application to sinkhorn and em. arXiv preprint or Xrxi:22006.0 08873, 2022.
19

---

Published in Transactions on Machine Learning Research (06/2024)
Heinz H Bauschke, Patrick L Combettes, Heinz H Bauschke, and Patrick L Combettes. Correction to: Conver Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, 2017.
Shai Ben-David, John Blitzer, Koby Crammer, and Fernando Pereira. Analysis of representations for domain adaptation. Advances in neural information processing systems, 19, 2006.
Shai Ben-David, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan. A theory of learning from different domains. Machine learning, 79:151 175, 2010.
Alberto Bietti, Grégoire Mialon, Dexiong Chen, and Julien Mairal. A kernel perspective for regularizing deep neural networks. In International Conference on Machine Learning, pages 664-674. PML.R, 2019.
Radu T Bot, Michael Sedlmayer, and Phan Tu Vuong. relaxed inertial forward-backward-forwaa algorithm for solving monotone inclusions with application to gans, Journal of Machine Learning Research, 24, 2023.
Radu Toan Bot, Ernö Robert Csetnek, and Michael Sedlmayer. An accelerated minimax algorithm for convex-concave saddle point problems with nonsmooth coupling function. Computational Optimization and Applications, pages 1-42, 2022.
Sébastien Bubeck et al. Convex optimization: Algorithms and complexity. Foundations and Trends& in Machine Learning, ((3-4):231-355) 2015.
Gong Chen and Mare Teboulle. Convergence analysis of proximal-like minimization algorithm using bregman functions. SIAM Journal on Oplimizalion, 3(3):538-55) 543, 1993.
Qi Chen and Mario Marchand. Algorithm-depenndett bounds for representation learning of multi-source domain adaptation. In International Conference on Artificial Intelligence and Statistics, pages 10368-10394 PMLR, 2023.
Ashish Cherukuri, Bahman Gharesifard, and Jorge Cortes. Saddle-point dynamics: conditions for asymptotic stability of saddle points. SIAM Journal on Control and Optimization, 55(1):486-511, 2017.
Edwin KP Chong, Wu-Sheng Lu, and Stanislaw H Žak. An Introduction to Optimization: With Applications to Machine Learning. John Wiley & Sons, 2023.
Casey Chu, Jose Blanchet, and Peter Glynn. Probability functional descent: A unifying perspective on gans, variational inference, and reinforcement learning. International Conference on Machine Leurning, pages 1213-1222, 2019.
Casey Chu, Kentaro Minami, and Kenji Fukumizu. Smoothness and stability in gans. arXiv preprint arXiv:2002.04185 185, 2020.
Imre Csiszár. On information-type measure of difference of probability distributions and indirect observations. Studia Sei. Math. Hungar., 2:299-39 318, 1967.
Simon S Du and Wei Hu. Linear convergence of the primal-dual gradient method for convex-concave saddle point problems without strong convexity. In The 22nd International Conference on Artificial Intelligence and Statistics, pages 196-205. PMLR, 2019.
Yaroslav Ganin and Victor Lempitsky. Unsupervised domain adaptation by backpropagation. In International conference on machine learning, pages 1180-1189. PMLR, 2015.
Ian Goodfellow, Jean Pouget-Abadic, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. Advances in Newral Information Processing Systems, 27, 2014.
Arthur Gretton, Karsten M Borgwardt, Malte J Rasch, Bernhard Schölkopf, and Alexander Smola. A kernel two-sample test. The Journal of Machine Learning Research, 13(1):723-777 -73,, 2012.
Ishaan Gulrajani, Faruk Ahmed, Martin Arjovsky, Vincent Dumoulin, and Aaron C Courville. Improved training of wasserstein gans. Advances in neural information processing systems, 30, 2017.
20

---

Published in Transactions on Machine Learning Research (06/2024)
Feihu Huang, Xidong Wu, and Heng Huang. Efficient mirror descent ascent methods for nonsmooth minimax problems. Advances in Newral Information. Processing Systems, 34:10431-10443, 2021.
Yujia Li, Kevin Swersky, and Rich Zemel. Generative moment matching networks. In International conference machine learning, pages 1718-1722 PMLR, 2015.
Jianhua Lin. Divergence measures based on the shannon entropy. TEEE Transactions on Information theory, 37(1):145 1511445-5511111111111 1991.
Tianyi Lin, Chi Jin, and Michael Jordan. On gradient descent ascent for nonconvex-concave minimax problems. In International Conference on Machine Learning, pages 6083 6093. PMLR, 2020a.
Tianyi Lin, Chi Jin, and Michael Jordan. Near-optimal algorithms for minimax optimization. In Conference Learning Theory, pages 2738-2779. PMLR, 2020b.
Luke Metz, Ben Poole, David Pfau, and Jascha arXiv preprinl arxi::1611. 1.02163, 2016.
Unrolled generative adversarial networks.
Paul Milgrom and Ilya Segal. Envelope theorems for arbitrary choice sets. Econometrica, (00)), 2002.
Takern Miyato, Toshiki Kataoka, Masanori Koyama, and Yuichi Yoshida. Spectral normalization for generative adversarial networks. arXiv preprint arXiv:1802.088: 05957, 2018.
Aryan Mokhtari, Asuman Ozdaglar, and Sarath Pattathil. A unified analysis extra-gradient and optimistic gradient methods for saddle point problems: Proximal point approach. In International Conference on Artificial Intelligence and Statistics, pages 1497-1507. PMLR, 2020.
Alfred Müller. Integral probability metrics and their generating classes of functions. Advances in applied probabilily, 29(2):429 443, 1997.
Angelia Nedić and Asuman Ozdaglar. Subgradient methods for saddle-point problems. Journal of optimization theory and applications, 142:205-201 228, 2009.
Yurii Nesterov. Introductory lectures on conver oplimizalion: basic course, volume 87. Springer Science & Business Media, 2003.
Sebastian Nowozin, Botond Cseke, and Ryota Tomioka. f-gan: Training generative neural samplers using variational divergence minimization. Advances in neural information processing systems, 29, 2016.
Tim Salimans, lan Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen. Improved techniques for training gans. Advances in neural information processing systems, 29, 2016.
Jian Shen, Yanru Qu, Weinan Zhang, and Yong Yu. Wasscrstein distance guided representation learning for domain adaptation. In Proceedings of the AAAI Conference Artificial Intellignnce, volume 32, 2018.
Manrice Sion. On general minimax thcorems. Pacífic Journal of mathematics, 8(1):171-176, 1958.
Kiran K Thekumparampil, Prateek Jain, Praneeth Netrapalli, and Sewoong Oh. Efficient algorithms for smooth minimax optimization. Advances in Neural Information Processing Systems, 32, 2019.
Eric Tzeng, Judy Hoffman, Kate Sacnko, and Trevor Darrell. Adversarial discriminative domain adaptation. In Proceedings of the IEEE conference compuler vision and pallern recognition, pages 7167-7176, 2017.
Jun un Wu, Jingrui He, Sheng Wang, Kaiyu Guan, and Elizabeth Ainsworth. Distributtion-informeel neural networks for domain adaptation regression. In Advances in Neural Information Processing Systems, 2022.
21

---

Published in Transactions on Machine Learning Research (06/2024)
Appendix
Notation
Meaning
No
the set of natural numbers including zero extended real number
F
x
compact set in Rd compact set in Rd"
Z
M(X)
the set of all finite signed measures on X
M+(X)
the set of all non-negative finite measure on X
P(X)
the set of Borel probability measures on
C(X)
the set of all continuous functions X +
C(Z:X)
the set of all continuous functions Z X norms induced by inner products in M(X)
HHlex)
norms induced by inner products in C(X)
H
dual norm of (|-l)x())
H)(((
dual norm of [|-lo(x)
dF
Gâteaux differentials of F: C(X) + at y o C()
Dr
Bregman divergence associated with F C(X) R
R
source risk
vo
fixed larget distribution
vo
true predictor
Jun
discrepancy measure convex conjugate of Jvp
J
K
convex-concave objective function in Section 5.1
S'
subset in P(X) (for source distribution μ in Section 5.1
S1
subset in C(X) (for predictor &i in Section
5.1
S2
subset in C(X) (for classifier yo in Section 5.1
step size in Definition 5.5
g
nonconvex-conce objective function in Scction 5.2
S
convex subset C(Z; X) (for generator f in Section 5.2
S1.c
convex subset. C(X) (for predictor 1/ in Section 5.2
S2,
convex subset in C(X) (for classifier y in Section 5.2
afn
step size (for generator f in Section 5.2 in Definition 5.12
agn
step size (for predictor v in Section
5.2 in Definition
5.12
step size (for classifier yp in Section f-divergence
5.2 in Definition 5.12
Dj
F
the class of real-valued bounded measurable functions on integral probability metric associated with F
dy
Lip(X)
the class of the real-valued Lipschitz functions on X reprodueing kernel Hilbert. space
(H, (1-1k)
Table 1: Table of Notations
A Proof of Theorem 5.7
Before the proof of Theorem 5.7 we review the three-point inequality.
Mtree-ppintt inequality The three-point inequality is a key ingredient for the proof of Theorem 5.7 which was first introduced by Chen and Teboulle (1993). We introduce the three-point incquality the space of measures and continuous function spaces without the proof. See Subin-Frankowske al. (2022) for the proof.
22

---

Published in Transactions on Machine Learning Research (06/2024)
Lemma A.1 (Three-point incquality for the space of measures). Let SM C. MM(X), and let G MM()) - R be proper, lower semi-conlinuoue and conver function. Let C.M > O. For given MXX), let.
G(v) + 2am 1|0 A
V == argmin
vESM
where || (lM)x is norm induced by inner products in M(X). Then,
G(V) + 20m
|l mlBxxxi 2 G(V) + 20m |V pll3(x) + 2am || ullu(x) for all E SM-
Lemma A.2 (Three-point inequality for continuous function space). Let Se C C(X), and let F C(X) + R be proper, lower semi-cone continuous, and convex function. Let oc > 0. For a given f E C(X), let
g = argmin
F(g)
- l1g f
gESe
2ac
where ||- (le(x) is norm induced by inner products in C(X). Then,
1 lly g glaox) for all Sc. |1g Slle(x) + 20re
F(g)= + 2ac
|1g fl(x(x) F(g) + 20c
The proof of Theorem 5.7 essentially basod on the L-smoothness of K(b,, μt, (p) for cach variables and the three-point inequality in Lemma A.1 and A.2 associated with the update rules of the gradient descent in Definition 5.5
Proof of Theorem 5.7 First, evaluate the lower bound of K(y Mny Pa+1). The following holds for any $p E Sg:
K(vnn Mns Pn+1)
K(yn Mny Pa) + dKC(nn Pai een (pn+1 - ppn) - fllp++1 9n -nnll
2
1
K(y Mns pn) + dKC Pre )m, (Pn+1 - -Pn)
|| - -
20n
1
2 K(yns Mny Pa) + dKC Mn, ),,, (6p - -pp) - 20
1110 - pntt (2()22) (1/pp - p0lle(1)22
2 K(yns Hny pp)
(1pp 2 101l21xx1. - 11g 1nt1 (((x)22())22)
2an
where the first inequality follows from the L-smoothness of y + K(), and the second inequality follows from 0 an S 1/L, and the last inequality results from the concavity of yp - K(v, p, ()) for each eb and μ. Also, the third incquality follows from the three-point inequality in Lemma A.2 with F(y) = -dK(th Mar )) (yp pn) and ʃ Pn:
dK(Wnn )en (Pat1 - pPn)= - 20n |1 - Pn (
< ddC(in pao )),, (1p -vn)
- (1lv pn 12 - 11pp- 100-+0+1llexppp 112(0))))) E S2.
2an
Furthermore, by using the the concavity of i0 + K(inn Mni (p) for K(ynn Pn41) at the first line, we have
an K( Mnr Pn) a, Mn: :p)
1 (1l &n (2 114e Pn+ 1 (l(xx) + andKK( Uny Mnr )e, (Pa41 pp))
(32)
2
23

---

Published in Transactions on Machine Learning Research (06/2024)
By taking the summation of 32
from nt 0 to N 1, we obtain
N-1
ant( Mxx Pn) +
Σ an n=0
K( - - 2 polléxxx + CX
(33)
N Here, we used En=o On K(kK Mas (p)
mn= On anK((N, (p) by Jensen's inequality where the weighted
sums (#N. PN, PN) are defined by 16 Here, we introduced C defined as
N
C =
and Pas Den (fn+1 Pn) 2 O.
(34)
The non-negativity follows from the update rule defined by Definition 5.5
When xp PN in
33 have
N-1
N-1
anKiWnsM +
Σ On
K(WN,NN. FN) 1l vol( + CR.
(35)
n=
When 90 = pr in 33 we have
Minn +
Σ On
K(vi, 2 5110. 10.11..-olle -Polle(x),2 + CR
(36)
where (o:, is a saddle point defined at 12 satisfying KC(,, Mn, (x, sp.) I K( (N.)
Second, we evaluate the upper bound of K( Pn). The following holds for any (1b, 1) E SM x Se:
K(vnn 1n pPn)
L
S K(NnsMn PP) dIC(1/ss is Pn)n, (pnt1 - (km) + "1llun11 - Mn (2
2
1
< K(Pnn Pn) dIC (KC Pn)un (Mnt1 Mmn +
|| Mn (1ll2(x)
2an
dKC,, Pay Pn)) N. (1/+1 - vn
1 |0a41 en 12(x),1
dKC(, Par Pn) Nm (Vn+1 1b) + 2an
S K(Pni Man (Pn) + dlC(: Mnn Pnnvnnn (10 vn)) ddC(enn ( (H pn)
dK((, Mni v. (Vn+1 - 1n))
(11s - 4n (1bb-anleesx) ()(1(()))1 - 1//b - *)n+2lle(x), + 11 nn mullurxx) - 11 (n+el)((()))
+ 20n
I K(1,, M4 (p)) dl(,, Mn, pn)si. (+hn41 - v/))
+ 2an (11b - Vn 12 - 11 wntt llxx + 1| pn Hallinx)) - 11 (n+1ll((x)))
where the first inequality follows from the L-smoothness of μ + K(v,, p4, ()) for each e and yp, the second inequality follows from 0 an < 1/L, and the last inequality is the result of the joint convexity of (tb. ()) + K(b, p4, up) for any Also, the third inequality results from the three-point inequality in Lemma pa)),, (v (
A.1 with G(v) = dlC(vins
pn) and = Pa,
aK(Nni " pn)p, (Hn+1 - Mn)
|| H Mmlli3cc)
20n
1
s diC(iin ' (pp (u- mm)) +
(1l Hullircx) 1lll-- Mntallm(x)) M
20n
24

---

Published in Transactions on Machine Learning Research (06/2024)
and Lemma A.2 with F(t) dlc('s Mni Pnnvn (vb vn) and f ehn
dKC:: par Paler (1h++1 vnn) +
20n I1 vnn (
s dK((s Mas pa)va 2 (v va)) 2an (10b vllle(x).1 11 Vnttllee)).) / E S1.
Furthermore, by using the the convexity of > K( Pn) for K(Vn; Mnti, Pn) at the first line, we have anK(pnn Mn: Pn) ank((,, n4, (Pn)
$ 2 (116 vn 12),1 - 11 - th + l| uullu(x)) - -Ipp -u+zlllmxx)) aandK(;; Mni v. (1h+1 vhn) and(((nn :Pn) (Mn+1 Mn).
(37)
We perform the summation of (37 over the interval n 1
N
N-1
OnK(Wnn Ma, ppn)
Σ On
(( S 2 (II+ tho v0012(x),1 + 11 -
(10l-ollm(xx)) (ollx(x)) + +CR,
(38)
n=0
where we used DnH (1,, S we introduced ON defined as
(INio on) K(V, n, RN) following from the Jensen's inoquality and
CN
On (dKCs Mn: (n)vn(Un4111 vha) + dK(unss 's Pr)un (Mn+1 Un)) 2 O.
(39)
The non-negativity follows from the update rule defined by Definition 5.5 By substituting &b obN and PN into
38 have
N an Knnns Mre (P++1)
NN-1
Σ an
KCGNNAN,
n=0
(40)
I1 voolexx) + Ilā no (Ollxx)) + C
2
Also, by taking (vh, μ) = (v., in 38 which a saddle point for the minimax solution for minimax problem
K(v,,,)) on (S, x 5') S2 such that (1/ (x)) satisfies the KC Me, @N) I KC(s), Max Ppx,, we have
N Σ
N-1
Kn Pni (n+1))
Σ
K(W,, May spa)
n=0
(41)
1 [
violleexx + llpe - p0ll2()))) + CN.
Third, let combine all the results we have obtained. Summiing up 36 and 40
yields
N Σ
May - C(NN,,AN.
On
カ==
I1 2bollxx2111 11ī no boll + 11s. - - + +C + +ON
Also, summing up
35 and gives
Σ an
PN) - K(rbe;
カ=ル
1 ( (IV-- - voll + l1 - MullMixy + ll ( + + + +GN
$ 2
25

---

Published in Transactions on Machine Learning Research (06/2024)
Therefore, we obtain
N-1 Σ an (2c.+00+
Ki, PPN) Me, pa) s
+08
(42)
where C, a finite constant defined in 15 This means that the value the object function (ON, PN, PN) approximately converges to saddle point under the gradient descent update rule, if the suins C C of the Gâteaux differentials in (34) and (39) are finite.
Finally, we prove that the term Cº + CK defined 34 and 39) are bounded from above by the norms ||, - pm Mux))(()) || Va lloxx), and [| - (n (x)c(())2 under Assumptions 5.2 Actually, each Gâteaux differential is bounded from above follows:
ddC(' Mn, Pnlen (#at1 vna) = /(fttt wn )a)dNac on Minin
(43)
S I1 Vullo(x),1 (n
and
ddC((bas (Pn),, (pn+1 - Mm) = =- Φ Wantios d(ux+1 (ha)
(44)
1
Wnn l'(x) || Mnl)
<B
and
f pn+11 Pn.
dKC(1/ May pm (fnt11 - Pn)) :n)==
(45)
$ 11 - PPlllex),2 (
<B
Moreover, by taking into account that the gradient decent scheme in Definition dK(:: Pn)e. (Nn+1 v'n) 20., || Vnlle(x)11 S O, we can obtain that
5.5
with v thn implies
||
viallcc)
s -dK((, uny pn)en (Wn+1 vx)
2an
$ || - vallo(x).11
which is equivalent to
|| villocx),11 S 2Ba
(46)
By the similar argument, we obtain
||o+1 PallM(x) 2Ban, || - - 2Ba..
(47)
By combining
42 with
43
47
we conclude
14
K(ŌN. GN) - XC((. Mnx, (2)) S
an Σ
C. 6B2
n=1
n=U
26

---

Published in Transactions on Machine Learning Research (06/2024)
B Proof of Theorem 5.14
Before the proof of the main result, we will show two lemmas used in the proof of Theorem 5.14
and 5. 11 hold. Then, we have the following: Lemma B.1. Let Assumptions 5.8 5.9 5. 10
(1) (1 1202, sl - 11 walls, for 1n. t/2 E S1,0 f S"
(1) [ fi) f2)lss, S fl - f2lls for vp St,e f1, f2 E S.
(1i) + G(b, L ( + +1 -mmooth with respect to || lls, over Sre for each S
(it) G(b, is 4 + -smooth with respect to || I| over S for each ab E St,e-
Proof. The proof is generalization of Lin al. 2020aa Lemma 4.3) to infinite dimensional function spaces with two variable. By the optimality, we have
dg( So)o (2(02)) f) f)) 0, (g f, (002))))) (0(11, 01122, f]) O,
which implics that
(1)tee) Dete +1.1, s 0.
(48)
With Assumption
5.9 and
48
we estimate that
3 1 (bbi, f) (vh,, l)ss
- [dG(vi, f:) (1b,,, (n f) - ( f))
[dg(1/2 (a)wa.)) dg(wi, ( f) (V), (1 f, (e(2))) Vg(V1, ) lls l1 (1h1,) (112,f)
< L dells. 10(1,, (1/2, fl
where last inequality results from Assumption 5.11 Hence, we obtain (i). (ii) is given by the same arguments of (i).
By the envelop theorem Milgrom and Segal 2002). the Gâteaux differential dG(- f)s of G(: f) at ab S is represented as dG(,, f)v dG (et, f) Using this, we estimate that
IVG(- f),, - Voll
|V f. f)) V. f, 2V2, f))
< [VV[ (. f)) VG(. ((V,,)
+IVV9( ( f)) Vg(, t. (1/2,, f))
< T. (1(
1 (/2 M) + 11 )2ll))))
L
< T
+1
|/
rb
27

---

Published in Transactions on Machine Learning Research (06/2024)
where last inequality follows from (i). Using the above estimate, we further estimate that
G(Di, 1) - G(V2, f) - dG(- Des - -b2)
d G(v22 c(1) 122), dG(;, (1/1 4/2dde
de
! (C((::))tttte-vsaal
1/22 dz2 - s/2)de
(49)
$
L
+1) <l|v> - -2113,
+
0
3
+1
$ 2
| v
Hence, we obtain (ii)). (iv) is given by the same arguments.
Lemma B.2. Let Assumptions 5.8
5.9 5. 5. and 5.11 hold. Let n > and yp E S2,es and we denote by
f+ Pso.o (s + (()))
Then, it holds that for to E S1,0, f E Sh and o E S2.c.
-g(0, ((v.S.0) s (p+ 2, -) + (5 - (
3
11 110 - 4113,
2 211o 3bls.
Proof. The proof is generalized from the finite dimensional case (Bubeck et al. 2015 Lemma 3.6).
By property of the projection Psa.. (see Nesterov (2003 Lemma 3.1.4)), we have
(s+ (s + (VV, ))) 8+ x)) S 0.
(50)
By Assumption 5.11 and same argument
49
we can show that
- (( - f. ()
- -p) 2 0110+ =pll22.
which implies that with Assumption 5.9 and (5
+
S +(0):,:)+++ g(v. G(b)
I
- 8 2111 $11
dg(i, )pp0--0p0000000000000000 - +)+ _ || - -11ss3 + dg(w, )) x))
L
< (VG(1, )p: 2+ -dss, マ/2
11 $113.
L
B
(e+ -,, -,)s
|1+0 =
77
2
2
(L 1 ||ا
< (pt 9, Φ 4)s, +
=1133
11 21.3.
2 n
2
Proof of Theorem 5.14 The proof is generalization Lin et al. 2020a Theorem 4.4) to infinite dimensional function spaces with three variables.
We denote by Lg (+ + +1). First, we estimate the upper bound of G(Un+1, fa+1). By Lemnaa B.1 we
have
Lg Ilfnt1 fral33...
G(yn+1: fn+1) S G(Nn41 + dG(pn41. )), (fat1 - fa) +
(51)
2
28

---

Published in Transactions on Machine Learning Research (06/2024)
By property of the projection Ps" (see Nesterov 2003
Lemma 3.1.5) and Young's inequality, have
1l f.
5 /. Vg( Pn) 13
(52)
< 20 VVg(n, (()n; ( + 20 ,, (V B(vn,
$ 2L a ( (n) pullss + +2af.n [G((nn -) 11.s..
We estimate that
dG(Un+1, Is (JutI
(V9(1n+1s'' (1 fn)) Jn+1 fn)ss
- (V @(Nn+1 0(0 fn)) afnn Vg(wn: ; Φ(vn, fn))sa)ss
+ (Vg(vn+1, 1. fn fnt1
fn + ay, Vg((n. ( fa))s
(53)
and by Lemma B.1 and Assumption
5.11
V (1n+i, fn))
< (9(+n+1: :, ((v++1, fa))s,
V9(vn) P(Un+1, fa))s, lls VG(Wn. (V fn))s= V9(Vn. (( fn)) lls"
Vg(+,, ((pns fa)) Ilst,
< T11 + [ - thllls. |[V )),, lls",
(54)
and by property of the projection Pso.o (see Nesterov 2003
Lemma 3.1.5))
|| - fn + ayn (g D(vn, fn)),,lls" $ a ( [V p Vg(vn: $(v fn)) 11s"
Lag ( fnllss..
(55)
Combining 53
with
54
and 55
we futhre estimate that
fn)
$ T.(1 + (1)) whllls: VG(Wn 1ss"
- VG(Wn, 15,11.
+ (1 Lg)aa,,nllll............ whllss ll.a - T(vin fn)sssss
+ Laya VG(ln, I), 11sl:>llp - (1bn, fn)llss
L(1 + LB)
|Wa+1 th 1n + L(1 + Lg)a fm |IV 11.
(56)
2
-ffm
VG(Wns
+ T.2(1 + Lg)
( + Lg)a
|uat1 Un1
|| ((U'nfa)l3
La f.u
I,
+
VV(Wn, 113. 7/2 1110, - 4(v., fl))
where we have employed Young's incquality for last incquality. By the same way with 52 we have
| - vn113. -
< s VVl. Sn, Pn) 113.
(57)
< 20 [V(. fnr 4 ((vn, fa)) Vn. Pn).. v.. Il3, 2a VV P(vn, fa)) 113,
< 21 2l (1 - pn 113, + +2a VG(, fn Jn).
29

---

Published in Transactions on Machine Learning Research (06/2024)
With
51
52
56
and
57
obtain that
((tn11;5411)
G(
T1(1 + L.g
Lays Lgassm } IIVG((n: -sn I1s.
+ tafm
-1+
2
2
L(1 + L) + LB)
|1 vinlls.
[ L
(1 + LB)
} 110(v/n spnll3s
1.2 L.g.a
2
(58)
$ G(vn+1.f
-1+ L(1 + Lo)ası
Laja + Loaasm } |IVG(()): 1s.
+ +asm
2
2
+ L(1 + L) + LB)a
2
[VG::
Min
L (1 + Lg)a
+ L2g = L L3 D)L+ Lo)ag.. } 10(2nn)))
+ 2
Second, estimate the upper bound of G(va+I, By Lemma B.1 we have
G(vn+1, S G( dG(, (Wh+1 - vn)) + Lg || 1.ullss
(59)
2
By the same way with 58 we estimate that
dG( Jnlv. (Vn+1 thn))
(Vg(, Snr (1Vnn (n)) Un+1 vnn)s
= ap..lllV(( ((Vn: In)) 11,
+ (VS(., fnr 2 ((vn Wwn Un+1 - thn + aww V9(: P(pn, fn))o.)s.
(60)
S V. fn) vnl. 13
+ Lll VG, 1ls ll (b fnllls=
+ La
/ Dn 113, + Z 11 - D(Wnn fn)l
2
Thus, by combining
57
58
59
and 60
we get
G( 1nt1) Glynn In)
Lawi
I(1 + I.)(1 + L3)acp, + Lagaoiin }
ap,m VVG(s
Lafn
T(1 L.)as. + LBaFn ag } (VG(mn ))n
(61)
L² + Lg)a
1.2 Lg3
(1 L) + Lo)a L. +
2
Assumption 5.13 (1))CC
(1 pa 12..
30

---

Published in Transactions on Machine Learning Research (06/2024)
Third, estimate (V fu) 1p 8n. Using Lemma B.2 and Assumption 5.13 (1), evaluate that
Inn Φ(fn 4n 1)-
< || - Φ 4n- 133, + 2(p - (f 2 10,11,,,0 Pn Pn-1)ss + 11 - Pn-1ll3,
< (1 - 8 11 (f 2n-, 1n-1)1 + (-11+ + La pn- 113
< (1 90ppn-1)6n
which implies that by using Young's inequality, Lemma 3.1
and
we have
8, (1 fn) - &plll3s
$ (1 + pap.a-11 ()) - 4n 11:
1
+ 2
1+
(110(0n-1, fni - 10 (WWn: full33. - -112(0n-1,,)1) - (1 fn fn-1))1))))
Bag.n-1
2L
< (1
3² p,n- )6n
1
(1o/a1 - 2/x ll's + -inni-Snll'
/32
/30
(62)
fa
+
21.
1+
30 )
(a + +af/n-1) 6n-1
82
2L3
(ap.
[V 1) 113s, + a n- VV((n/ .l3.))
32
300/
212
$ n6n-1
1
(a
(op.n-1/IVGi (n- 13, + a VG 11-1,, 15-1111)))
B²
BC,
where we have employed Assumption 5.13 (iii) for last inequality. Then, have
21.²
n n- VG(. I1', + + (VV((a)))))
8, 000000 +
(1+
Σ
(a o
32
30%
1=0
By this and
61
we have
Lawa L(11+ L) Lg)ag,n Locrem } aw VVI
Lafn
L(1 L(1+ LoagA 1 a.. (t.
2
< G(wn. fn))
G(un+11 + +07"00
2T2C
+
|+
Σ
" + (VC()
82
3Co
1=0
31

---

Published in Transactions on Machine Learning Research (06/2024)
and taking the summation over the interval n 0 to N - 1,
N
Lote,
T(1 + L)(1 + Lz)a - Iga assn CosallVVG (VG:
2
Larf
L(1 + Lg)ayn
Lgars } asa (..aaVV((onn))) VV(()n)) 11
2
n=0
N-1
-G(UN, fo(
+ +op
Σ
?
212C
N-1
1+
Σ
(1
(VG(
8
3Co/
==0 i=0
8
$ {G(bb.fo)
inf G( f) + Co V.
7"
212C
1+ )
N-1
+
Σ 7d
Σ
(
Fn(, 12', I ))s I12sr)
8²
1==0
n=0
which is equivalent to
N-1
Logg
2T,20m (1 (1+ 3Co 3Co) L(1 + L) + Lg)ag,n Lgagsn
1
}
2
B² (1)
0
Assumption
5.13
(iv) 20.20
aighin VVG,
N
L(1 + Lg)afm L3a5n
213Caff..
+
1
2
2
(2(1-2)
3Co
n=0
Assumption
5.13 (v) 20,50
a.f.n VG((n) (s, 11,
C5D inf G(b,f)
I G(yo, fo)
+
v.f
1-7
Finally, we estimate that
VGW.N/
/|s
ΣN=1 aw, VG(fn: )00
s.
EN-O ashi
N-1 En asnn
VG -ev llss
NN awpn || VG
) 13,,
1/2
<
Σn agin
N-1 015)
1/2
By the same way, we estimate that
EN'r fffn
VG(nnn )), 123
1/2
J 50
S
1/2
Σ Σ as,n
Therefore, we conclude Theorem
5.14
32

---

Published in Transactions on Machine Learning Research (06/2024)
C Proofs in Section 6.1
C.1 Proof of Proposition 6.1
Proof. For a E [0, 11, 11, 122 ((X), and M1, E M(X),
R(a4/1 a)22 au1 + (1 a))2))
S s²² / ((0ri woldu1 + (1 a)² /
e(v2. vo)du2
+ +a(1-o0 aa) (f(ttss ((ubi. vo)d + vo)
(vV( +(1 (1- - aVV a(1 a)2 || 02l12(x1,2
+aW((n) + (1 a) a(1 - a)y
|1 - 122 (2)1c(x)111
2
=a (fe (1b51.ooddus + (( + W((1) +(1 a) (/ e(v2,, Ho)d + ((22) + ((22)
=1((22112)
+ +a(1 +a(1-a) a) (-/(e(ti - L(12, -(122, (-(()(())))))(((b)))))))) -M2)
1 02llexx
7 2
|1ui -
and (+) is non-positive because we have
(+) [[((v), L(v22, Vollo, | 12lle(x
Sellei Valllcox, vi Valle(x))2
7 |/41 wallecx)22 ? 311m-12lllce -2llc(x)
2
(1 vbellcxx122 1|mi -eellcx)).) S 0.
C.2 Proof of Lemma 6.2
Proof. For v. p Sc.a.b,
Dia (1b = (1b) - Inp(() - d( d(Th.p)) - sp)
= =f (k((w) -h(pp= - 11(() - -)) du
< L / I/ "p?2 du |1 - -
C.3 Proof of Lemma 6.3
Proof. By the definition of f-divergence 24 we have
/
sup / pdpu aiiimj edp. J,(11)
( dup
(du))
du
J7(1)) sup HEM(X)
duo.
We solve a concave maximization problem for + I sdpp If de dvo. We consider
d (f + +xx) 1:(((utt) din
(du+txx) dvo)
= 0,
de
33

---

Published in Transactions on Machine Learning Research (06/2024)
which is equivalent to By the assumption, I' is invertible, and &p E Se,. Substituting
/vdx - -f
dyc dx dx duo = / (x- S
d4i dup
))
(des duro
dx 0,
for all Then, the optimal a satisfies
8 f
dv
du
= (())
into
ava
J7(4) sup /¹
/
4 dup
du dul - / (
duo
(1(t)) duo
dui
dvo.
then, we obtain that
Jj(()) = / 11-00) dus.
C.4
Proof of Lemma 6.6
Proof. By the definition of IPM
28
we have
/ pdp -
/ pdvo X(pEF).
Jo (4)
sup vEC(X)
By the Fenchel-Morean theorem, we obtain that
/ pdvo + xlx FJ.
(imM).(()) -
C.5 Proof of Proposition 6.9
Proof. (2) holds due to the convexity of k(-). (4) follows from the linearity of ~ K. (vh, u4, s)) and the norm 1 I1-11%, induced by inner products.
For (3), it holds that
1 / (15) - 222 + 3 11 w22l Havza" 2
and by S', have
1
f(( (11
22) S 1/2 / (wi (111-)2) -/)) = 2 1 r222 -
1
2
2
For (5), we estimate by using the Le-smoothness of : (a, b) + R and vo S"
D D_x, = D k()don (121
= =/ [M((p)))) - -k((2) k' ((((2)((1) - (2)) dub
Ls / 1:21 1p22 12p
2
2 11 -Pelii((x.n.""
34

---

Published in Transactions on Machine Learning Research (06/2024)
Finally, will prove (1). To apply Proposition 6.1 2 2b) = (- 0o)), V( ||
11 W(H)
2 I|ll| we verify assumptions (i), (ii,, and (iii) Proposition 6.1 (i) holds due to the convexity of
t (t (5-s) s)?. (ii) holds because norms || Il and Il-Il are induced by inner products. We prove (ii) as
followings:
By Chu et al.
2020
Proposition 14), the RKSH norm ll, is represented as
Ifl
-5 k=O la|=k
Σ
1 al
|00l ffll
(63)
for ʃ Ho. Here, we employ the multi-index notation with d-dimensional multi-index a (a) (o,, aa) Ng where the sum of its components denotes the |o|| on ...... + Ota- Additionally, define the factorial of the multi-index a! = aj! aal, and the partial derivative as 0 8m ... -.3
We estimate that
10 ((1/1 + 4/2 - 21/0 - (2)))l(2(()))
Σ
a
-a- 9 (t1 + 4/2 - 200 (1b 1b2)
a
(12(8))
10
(vi + t/ - 220 ( 110 - +22) -22)ll2((3))
3
(64)
Σ1C,
2
2
< (4C))²
Σ
1 4/2 4/2) (
Asa
B5a
=(24)²
<2" Σ ssa
105 - li
S $16C88
Σ
10 +222 ( (((R)))
85a
where the first equality employs the Leibniz formula, the second incquality utilizes the Canchy-Schware inequality and the result of 11, 122, tu E Sr, and the third inequality makes use of the Cauchy-Schward inequality and multi-binomial theorem.
35

---

Published in Transactions on Machine Learning Research (06/2024)
By using 63 and 64). we further estimate that
1 20² - /D)2
2 41 v0)²
1o
11(121 + 1/2 - 20/2 - 2/2))
4
2
Σ
(402)k
Σ
10 4
k=
|a|=k
B
(402)18) 1
|1a8 (1/1 rdse)l (2()))
!!!
|a|
$2a
< 408
(402g
Σ (4o2) 14
1102 +22))12233
a!!
k=
la/=k
82a
< -C
(2V20)
(2v 20
}31
Σ
Σ
(402)31-aa ...
100 +22 22)
a!
k=0
|
B420a
91201
=[Enza =C
=
< <40,0z ad Σ
- (2v (2V 20)
Σ
1
(0)
+2))
a!
k=0
la/=k
= =C0f0z 11 2
where Co = EjeNo (402)) < oo, which implies that [ 4h)2 is with respect to Il.1lz, and [1-11xewzi Thus, by the assumption of 7 2 40%0 and applying Proposition 6.1 to our setting, we conclude that (1),1)) (1,,,, p. is convex.
D Proofs in Section 6.2
D.1 Proof of Lemma 6.10
Proof. Since I: : M(X) IR is (1/8)-smooll with respect to || [lM(x), the convex conjugale I C(() - R is B-strongly convex with respect to || [)x By this and
(Jro + IJ" J +I"
then (Jon 4I* is f-strongly convex with respect to ||. 1111x due the fact that strong convexity is preserved by adding convex functions.
D.2 Proof of Lemma 6.11
Proof. As Lipschitz contimuus,, absolutely continuous. Thus, the derivative Vh of defined a.e. in X with respect to the Lebesgue measure m. By the assumption μ << m, the derivative Vh is also defined a.e. in X with respect to probability measure &, which implies that
d Jn + tco) c= =1 /V((f())) ((2))(d2)) de
36

---

Published in Transactions on Machine Learning Research (06/2024)
D.3 Proof of Proposition 6.13
Proof. (1) is given by Lemma 6.10 (2) holds from Lemma 6.11 and Gâleaux differentials are given by
(9((-5,)))(1) = / (10 o - 40 Dn fdCo:
/ (10 f 2o f)((v= - Vuo) o f} gdEo + / /VV ff gdEo.
dGi (Qi(v),)))(())
60 fd5o / (((p) odvs - Blo, p)H..
dg (l((v.f)))o))
We will confirm Assumption 5.11 as follows:
(a): for 1i, 2/2 E S1,e2 J E S1,es sp E S2,0, and 11 E Si with (x)l s
dC - (. = 2 / (w' (010f- of f- o fin fdCo
= / (1) (22)n((((5))
S 20al/vi V2lle((0)l1ll
$ 20al|01 v2lla4(x),
[IdGi(;f. deils (.1)) 2C31l/11 2 (2))
(b): for t S1,r f1 f2 E Sh v E S2,s and n E S₁ with Inll $
dg1l. f1, dG, p)
2 / (w - vn o fn) o Sio - 2 / [1b 12- o f2)n o f2dCo
2 / po do o fi fi 170 f2ldEo
+ 1 - v o fallm o f2ldEo + 2 o 1w00 o fi to felln o fald5o
< 4C2Lip(n) / 1/1 f2 + 40, / 1/1 falln feldEo
I 4C21VV/l1== fell: + 4C Inlle - 1M
fallx
- 11000xx221/+44C00 Ilfi -fllle(((((xx6))
+ +1Id0((-ff1.4) (2
() S (4 + 4//c//2) llfi - fall
where the last inequality results from Lemma D.1 where Ča,x 0 is some constant depending d and X. (c): for vb E S1,c, f E S1,e, P1, E S2,e, and 77 E S1 with [x)]l 1
d(1(:.f.p:))()) - =0
[IdG)(sf.111) - d. sclpp1 P
for any 0.
37

---

Published in Transactions on Machine Learning Research (06/2024)
(d): for 3b E S1,0 fi.f2 E Sy 4 e S2,e, and g E S" with (10l) S
dG (1))))))) (0) doi (66,,,),)333
-2 / (wo vo (1) - V o fil gdEu + / (Vio fit gdCo
2 / (wo 12 40 o [2) Vxbo f2] gdEo / IVpo f2] gdEo
s 443 / 1V8- Vubo o lo - (V - Veb) f2l lgl
+ + / fo10 v f2l + 100 fi 40 f2l3 lgldEo / /ipofi - vp o fallg/dEo.
s (160102+ + C1) / 1f1 ful Ig|dEo
s (160102+ Cill f2ll))(2xxx)))
= (V: - -gg 10s S (16C)C2+C))ll - S2lle:(2xx.60)
(e): for wh e S1,0 f Sh pe S2.e and g E s" with (gl $
dg i (g)(())))))))))) dgi
= 2 (01 of w fV Vvoo) f} gd5o
- / (12 (1205- - ~ (V Vebo 0 ʃ} gdLo
2
1 0 vn V44
V4h2 0l ggldEn
+2 / 111 oo - 222 o -2
V Vun o fl Ig|dEo
s 400, / 1441 of ol + 4C2/ / 111 o - 222 o fl fglddo
s 1/ //11 (2ll
+ IId dG 0000::/000000000000000000000a0a0 $4C20//1105 1|1 - 42)l
(f): for v E S1,e, E 8h 11, 122 S2,: and g E S" with | $
dV dG (2))))
= = o ---off/-----ve (Vp2 - gdEo
$ / V
-//2 1|p1 p2l1H.
||d9 (w) - (2)) C./2 -22ll,,
where the last inequality follows from
63 where C2 > 0 is some constant depending on
38

---

Published in Transactions on Machine Learning Research (06/2024)
(g): for vb E S1,es f E Si,er P1, P2 E S2,c, and S2 with l - 1,
d5i (v), S ) (6) dgi (v) S )22 (6)
= 1 (p(2)) ((11) } Φdvr + :(0; 92 +11H..
duo
1/2
S Lx sup zex
dm (x)
|1 p2ll22(x) + +B1181 822lHH
deo
1/2
+0) 11 8211210
2 La sup
(x)
zeX dm
dug dm
1/2
La sup =EX
+ 11 s221m4.
(h): for si, 1/2 S1,es I E S1.0 & E S2 and S2 with 11011 1,
dGi ( (v/2 Jz (0)) 0,
+
- plli S clbbi - w2ll
for any >
(i): for vb Si,es f2 E S1 10 S2,,, and & S2 with |101 $1,
(v. (0)) de, fo. / 6o fi -00 f2d50
Lip(ø) / 1ifi feldEo
(2) fallx2(2xx66))
+ Idd ( -
)lline S (caxx56(2))2 - fAlLE((.XX66))
where the last inequality results from Lemma D.1 where Co,,x > 0 is some constant depending d, and X.
We have employed the following fundamental Lemma in the proof of Proposition 6.13
Lemma D.1. Let Q c: Idd be compact set, and let f f Ω + R be Lipschitz continuous. Then, we have
| I max { (d+ 1) ||/ll ( (d + 1112/1111111 के
|
2d
2d
Proof. Assume that f: Q + R be L-Lipschitz with L 2 We denote by
argmax
and
M = ||] max ((x DEft
Without loss of generality, we can assume that M > As f: Q R is L-Lipschitz, that is,
If(x) f(a)) s L|x a/, x E Ω
39

---

Published in Transactions on Machine Learning Research (06/2024)
estimate that
L [
MIL
al
M S
l -a-al
L
$ L|x al + M = f(x),
which implies that
J [5(x)do 2T, / MIL
1 1(1o-o/50)
Ja
= 1 J M/L
J. Sup
M/L d
Rta,
L
(lm=al/s//aa,dze drdv
J0
-Rto;
L /. M/T.
QR.alo)du,
where
d
Vd R
Qu,((u)=
Vd (2R)*
2 R.
Here, R. > 1 is chosen large enough such that
aC
II
R + 0, R R+ a:).
By direct computation, can show that
[ ldx
0
2 2d ( M - RLVd
I. > VdR
2d Rd v/ I.Re+1
1/4 VdR
a+1
/ || (|)))1))
7, VaR
( || [|
2d
I VdR.
M
Therefore, we conclude that
[[flzz(())
d+1 1
(d 1)dd/2
atr
< max
|
|| || (1
2d
2
(d + 1)/2//22
(d 1244)///210 1/2
24.
< max
|
||
(1
T
2d
2
40


Reviews:
[{"review_id": "RRe8AACIpr", "reviewer": "Reviewer_tDMF", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "### Strengths\n\n1. The paper is well-written and easy to follow.\n2. The technical part is solid. \n3. Different conditions have been verified for certain GAN and UDA settings. \n\n### Weaknesses\n1. Unclear problem formulation\n    * The optimization problem for UDA was defined as $\\min_{(\\psi, \\mu) \\in \\mathcal{C}(X)\\times \\mathcal{P}(X)} R(\\psi, \\mu) + J_{\\nu_0}(\\mu)$, where $\\psi$ is the predictor, $\\mu$  and $\\nu_0$ are respectively the source and target distribution on $X$. \n    * However, in UDA, $\\mu$ and $\\nu_0$ are both fixed distributions, which cannot be optimized. In particular, $\\mu$ is the source marginal data distribution on $X$. The distribution matching in UDA was realized by either conducting matching on the marginal representation space [1,2,3,4] or on the joint probability space [5,6].\n    * So, I am not convinced by this part. Later on, the authors have defined $\\mu=f\\_{\\\\#}\\xi_0$ as a push-forward measure in eq(10). \n    * However, in the context of UDA, it should be claimed that $\\xi_0$ is the source distribution and $\\mu$ is some induced measure defined on the representation space.  \n    * A *very close formulation in [6]* using Wasserstein distance lacks a discussion, which is a minimax problem formulated to maximize some discriminator and minimize w.r.t both the predictor and feature extractor. \n\n2. Related works \n\nThe related works in UDA  have not been sufficiently discussed, as mentioned above. In addition, some work has studied when the equilibrium exists for GAN, e.g., [7], which lacks a discussion.\n\n3. Novelty\n\nThe paper is an extension of [8] to the infinite functional space with the same order of $N$ for convergence. Hence, the novelty is not that impressive to me. In addition, an improved minimax algorithm has been studied in [9], so it's better to involve a discussion. \n\n[1] Ben-David, Shai, John Blitzer, Koby Crammer, Alex Kulesza, Fernando Pereira, and Jennifer Wortman Vaughan. “A theory of learning from different domains.”  In: Machine learning 79.1 (2010).\n\n[2] Ben-David, Shai, John Blitzer, Koby Crammer, and Fernando Pereira. “Analysis of representations for domain adaptation”. In: Advances in neural information processing systems 19(2006).\n\n[3]Ganin, Yaroslav, Evgeniya Ustinova, Hana Ajakan, Pascal Germain, Hugo Larochelle, François Laviolette, Mario Marchand, and Victor Lempitsky. “Domain-adversarial training of neural networks”. In: The journal of machine learning research 17.1 (2016).\n\n[4]Zhao, Han, Remi Tachet Des Combes, Kun Zhang, and Geoffrey Gordon. “On learning invariant representations for domain adaptation”. In: International Conference on Machine Learning. PMLR. 2019.\n\n[5] Acuna, David, Guojun Zhang, Marc T Law, and Sanja Fidler. “f-domain adversarial learning: Theory and algorithms”. In: International Conference on Machine Learning. PMLR. 2021.\n\n[6]Chen, Qi and Mario Marchand. “Algorithm-Dependent Bounds for Representation Learning of Multi-Source Domain Adaptation”. In: arXiv preprint arXiv:2304.02064 (2023) \n\n[7] Arora, Sanjeev, et al. \"Generalization and equilibrium in generative adversarial nets (gans).\" International conference on machine learning. PMLR, 2017.\n\n[8] Tianyi Lin, Chi Jin, and Michael Jordan. On gradient descent ascent for nonconvex-concave minimax problems. In International Conference on Machine Learning, pages 6083–6093. PMLR, 2020.\n\n[9] Lin, Tianyi, Chi Jin, and Michael I. Jordan. \"Near-optimal algorithms for minimax optimization.\" Conference on Learning Theory. PMLR, 2020.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper formulates GAN and UDA as the minimax problem in convex-concave and nonconvex-concave settings. Specifically, the study considers infinite-dimensional functional spaces and mainly focuses on the convergence under small optimization errors.\n1. Convex-concave:\n$$\\min_{\\psi, \\mu \\in S_1\\times S^\\prime}\\max_{\\phi\\in S_2}  \\mathcal{K}(\\psi, \\mu, \\phi),$$\nwhere $S_1, S_2 \\subset \\mathcal{C}(X)$ are compact convex subset of some functional space and $S^\\prime \\subset \\mathcal{P}(X)$ is a compact convex subset of all the measures over $X$.  This is the general form of eq(8).\n\n2. Nonconvex-concave:\n$$\\min_{\\psi, f \\in S\\_{1,c}\\times S_{c}^{\\prime\\prime} } \\max_{\\phi\\in S\\_{2,c}}  \\mathcal{G}(\\psi, f, \\phi),$$\nwhere $S_{1,c}\\subset S_1, S_{2,c} \\subset S_2$ and $S^\\prime \\subset \\mathcal{C}(Z, X)$ are compact convex subset of some functional spaces. This is the general form of eq(10).\n\nThey further proved that,  under proper assumptions, the convex-concave setting exists convergence to the true minimax solution, and the nonconvex-concave setting shows convergence to the stationary point.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please fix the problems mentioned in the above section.", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "OEnSvak3Tm", "reviewer": "Reviewer_cgJA", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n1. This work presents a thorough proof and analysis of the convergence of the minimax optimization problem in the scheme of gradient descent. Specifically, under appropriate assumptions, the authors analyze the convex-concave problem of continuous functions and the probability measures and also consider the nonconvex-concave problem over spaces of continuous functions.\n2. Most of the assumptions and definitions are clear and well-presented.\n3. In the reviewer's view, there are no fatal issues in the analysis and the corresponding proof. The overall framework is rigorous for the convergence analysis of the minimax problem in the infinite-dimensional spaces of continuous functions and the probability measures.\n4. The analysis framework draws theoretical insights into understanding the conditions for the convergence properties in certain GANs and UDAs settings.\n\nWeaknesses:\n1. The organization and the presentation of the paper can be further improved by better structuring the analysis framework. \n2. It would be better if the authors could explicitly point out the distinguishable differences on the analysis framework compared with the previous related work (e.g.,  the unique assumption this work has made and the derivation conducted different than previous ones), or further discuss the major technical contribution for the optimization problem under the infinite-dimensional spaces.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper focuses on the convergence of the minimax optimization problems over the infinite-dimensional spaces towards stability in adversarial training. Specifically, this work tackle the instability problem of adversarial optimization through a functional analysis, where the results show that the stabilization techniques (e.g., spectral normalization and gradient penalty) can serve as the necessary condition for a good convergence.\n\nTo be more concise, the paper's contributions are,\n- present an analysis of the convergence to the minimax solution over the infinite-dimensional spaces;\n- verify the fulfillment of sufficient conditions for the convergence properties in certain GANs and UDA settings, and provide the theoretical interpretation of existing techniques like spectral normalization and gradient penalty.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Overall, the reviewer thinks this work presents a rigorous framework for analyzing the convergence for minimax optimization problems over infinite-dimensional spaces towards the stability in adversarial training. The following lists some specific comments,\n\n1. Please summarize all the notations in a table to allow a easy reading.\n2. Beyond the problem setting of the GAN and UDA, please also introduce and discuss further on the limitation of previous analysis for the finite-dimensional spaces of probability measures or continuous functions. \n3. Could the authors further discuss the future problems of the existing theoretical frameworks except for the empirical verification, like the validity of the assumptions in more practical scenarios?", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "bR7vXkC0pG", "reviewer": "Reviewer_qvrN", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths** \n1. This paper extends previous analyses of minimax optimization from finite-dimensional to infinite-dimensional spaces, focusing on both convex-concave and nonconvex-concave settings.\n\n2. In addition, the analysis framework presented here applies to both GANs and UDA. While GANs have received attention in previous literature, this work supplements the understanding of convergence properties in adversarial training for UDA, an area yet to be extensively explored.\n\n\n**Weaknesses** \n1. From a technical standpoint, it is unclear how much new contribution is made by extending previous theoretical analyses in Chu et al.(2019, 2020) and Lin et al.(2020) from finite-dimensional to infinite-dimensional cases. Specifically, it seems Chu et al. (2020) are already aware that certain smoothness conditions can be extended to the infinite-dimensional case, as indicated in the first paragraph in their Page 9 (following Proposition 13), where the last sentence states that \"this regularization results in a function with Lipschitz gradients, so it is unsurprising that this property carries over to the infinite-dimensional case.\"\n\n2. In terms of conclusions, the effectiveness of spectral normalization and gradient penalty in adversarial training for GANs has been previously justified by Chu et al. (2020). Therefore, this work may not provide additional insights into adversarial training, limiting its significance.\n\n3. Two additional minor weaknesses include the limitation of the current analysis to convex-Lipschitz continuous loss functions in UDA (e.g., squared loss), and the difficulty in empirically verifying the theoretical results, a challenge acknowledged by the authors.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper uses functional analysis tools to study the convergence behavior of minimax optimization problems, focusing specifically on GD applied in infinite-dimensional spaces. Under appropriate assumptions for their minimax problem, the authors demonstrate that GD converges to the minimax solution in convex-concave settings, while in nonconvex-concave scenarios, GD has a vanishing gradient norm and converge to a stationary point. Furthermore, they illustrate how these assumptions can be met in certain GANs and UDA algorithms, and they justify the effectiveness of techniques like spectral normalization and gradient penalty in stabilizing adversarial training.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Below are some suggestions for adjustments, with the first two points being my primary concerns, while the remaining points can be used to strengthen the work.\n\n1. *Improve the clarity of the motivation behind analyzing the infinite-dimensional space.* Given that the effectiveness of spectral normalization and gradient penalty has already been justified in finite-dimensional spaces by Chu et al. (2020), what advantages does extending the analysis to infinite-dimensional spaces provide? Is the motivation to better characterize the case of UDA by analyzing the minimax problem over infinite-dimensional space? It seems that with additional assumptions for the source risk term (e.g., convexity and smoothness), the framework of Chu et al. (2019, 2020) could also be applied to study UDA.\n\n2. *Provide more details about the adaptations in the analysis compared to previous works.* If studying the minimax problem over infinite-dimensional space has advantages, could the authors highlight the main differences between their analysis and previous works, such as Lin et al. (2020) and Chu et al. (2019, 2020) in finite-dimensional settings? Although the related work section and other parts of this work briefly discuss similarities and differences, readers may not grasp the key modifications without digging into all of these works.\n\n3. *Verify whether the $\\mathcal{H}\\Delta\\mathcal{H}$-divergence satisfies the assumptions.* In Section 6.13, concrete examples of $f$-divergence and IPMs stratify the smoothness conditions. Can you also elaborate on the $\\mathcal{H}\\Delta\\mathcal{H}$-divergence in [R1]? Note that $\\mathcal{H}$-divergence and $\\mathcal{H}\\Delta\\mathcal{H}$-divergence are studied in seminal works [R1,R2] of UDA theory, thus taking $\\mathcal{H}\\Delta\\mathcal{H}$-divergence into discussion is meaningful.\n\n[R1] Shai Ben-David, et al. \"A theory of learning from different domains.\" Machine learning 79 (2010): 151-175.\n\n[R2] Shai Ben-David, et al. \"Analysis of representations for domain adaptation.\" Advances in neural information processing systems 19 (2006).\n\n4. *Comment on Assumption 5.13.* Assumption 5.13 is hard to understand and may not be clear enough for why it can be satisfied immediately in Section 6.2.3. This assumption, which seems to suggest that step sizes should be sufficiently small for convergence, also implies that $L$ and $\\beta$ should not be too large. For instance, if $L\\gg \\beta$, and $C_0$ is very small such that condition (i) is satisfied, then it seems possible that the LHS in (iii) can be very large such that $\\gamma\\in(0,1)$ does not exist. Can you clarify why Assumption 5.13 can be immediately satisfied in Section 6.2.3?\n\n5. *Discuss the empirical distribution case and SGD.* In practical adversarial training, since the true distribution $\\mu$ and $\\nu_0$ are unknown, we usually use the training sample as a \"proxy\". If $\\mu$ is replaced by its empirical version $\\hat{\\mu}$, are any modifications required in the analysis? Additionally, SGD is often used in practical algorithms. Is it possible to formulate SGD rather than GD in your framework? If such a formulation is feasible, considering that SGD introduces some randomness or noise, will it affect convergence to a stationary point in the nonconvex-concave setting?\n\n6. *Minor point: Add a table summarizing the notations used in this paper for the convenience of readers.*", "broader_impact_concerns": "I do not see any ethical concerns with this paper, as it is purely theoretical in nature.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: **Paper Summary**  

This paper examines the convergence of minimax optimization problems in infinite-dimensional spaces within the context of adversarial training, including GANs and unsupervised domain adaptation.

**Claims and Evidence**  

This is a theoretical paper. In the initial review, reviewers tDMF and cgJA expressed concerns about the clarity of the problem formulation. After the rebuttal, none of the reviewers raised further significant concerns regarding the theoretical results. Reviewers also noted the omission of several related works in the domain adaptation field, which the authors addressed in the revised version. Therefore, I find this part satisfactory.

**Audience**

In the initial review, some reviewers questioned the significance of the theoretical results. I acknowledge the relatively modest significance of these results. However, the acceptance criteria for TMLR emphasize interesting insights rather than the significance of the results (i.e., this part should be considered for the featured certificate rather than acceptance). Based on the official recommendations from the reviewers, all of them agree that this paper could offer certain valuable insights for the sub-community. Thus, I find this part satisfactory as well.


**Decision**

Based on the aforementioned discussions, I think this paper meets the criteria for acceptance. The reviewers did not oppose the acceptance decision in their official recommendation. Therefore, I recommend acceptance.
Decision: Accept (poster)