Fundamental Convergence Analysis of Sharpness-Aware Minimization
Authors: Pham Duy Khanh
Hoang-Chau Luong
Boris Mordukhovich
Dat Ba Tran
Venue: NeurIPS 2024 poster (2024)
Keywords: Convergence Analysis
Deep Learning
Inexact Gradient Descent Methods
Neural Network
Sharpness-Aware Minimization
Primary area: optimization
TL;DR: A convergence analysis for Sharpness-Aware Minimization and its variants.

Abstract: The paper investigates the fundamental convergence properties of Sharpness-Aware Minimization (SAM), a recently proposed gradient-based optimization method (Foret et al., 2021) that significantly improves the generalization of deep neural networks. The convergence properties including the stationarity of accumulation points, the convergence of the sequence of gradients to the origin, the sequence of function values to the optimal value, and the sequence of iterates to the optimal solution are established for the method. The universality of the provided convergence analysis based on inexact gradient descent frameworks (Khanh et al., 2023b) allows its extensions to the normalized versions of SAM such as F-SAM (Li et al. 2024), VaSSO (Li & Giannakis, 2023), RSAM (Liu et al., 2022), and to the unnormalized versions of SAM such as USAM (Andriushchenko & Flammarion, 2022). Numerical experiments are conducted on classification tasks using deep learning models to confirm the practical aspects of our analysis.

Fundamental Convergence Analysis of Sharpness-Aware Minimization
Pham Duy Khanh
Hoang-Chau Luong
Ho Chi Minh City University of Education khanhpd@hcmue edu. vn
VNU-HCM University of Science
1hchau208apes. fitus. edu, vn
Boris S. Mordukhovich
Dat Ba Tran
Wayne State University boris@math. wayne. edu
Wayne State University
tranbadat@wayne..c edu
Abstract
The paper investigates the fundamental convergence properties of Sharpness-Awar Minimization (SAM), recently proposed gradient- optimization method [Foret et al. 2021] that significantly improves the generalization of deep neural networks. The convergence properties including the stationarity of accumulation points, the convergence of the sequence of gradients to the origin, the sequence of function values the optimal value, and the sequence of iterates to the optimal solution established for the method. The universality of the provided conver- gence analysis based on incxact gradient descent frameworks [Khanh et al. 2023b allows its extensions to efficient normalized versions of SAM such as F-SAM LL et al. 2024), VaSSO [Li and Giannakis, 2023), RSAM [Liu et al., 2022]. and to the unnormalized versions of SAM such USAM (Andriushchenko and Flammarion 2022]. Numerical experiments are conducted on classification tasks using deep learning models to confirm the practical aspects of our analysis.
Introduction
This paper concentrates on optimization methods for solving the standard optimization problem
minimize f(x) subject to E IR".
where f IR" + IR is continuously differentiable (C) -moooth) function. We study the conver- optimization algorithm Sharpness- Aware Minimization (Foret
gence behavior of the gradient- et al. 2021] together with its efficient practical variants (Liu et al., 2022 Li and Giannakis) 2023 Andriushchenko and Flammarion, 2022]. Given an initial point x² IR" the original iterative procedure of SAM is designed as follows
Vf(a*)
x h+1 = -
for all l IN where > and p > 0 are respectively the stepsize other words, the learning rate) and perturbation radius. The main motivation for the construction algorithm is that by making the
((=*)
backward step a + P [[ it avoids minimizers with large sharpness, which is usually poor for the generalization of deep neural networks as shown in Keskar et al. (2017).
1.1 Lack of convergence properties for SAM due constant stepsize
The consistently high efficiency of SAM has driven a recent surge of interest in the analysis of the method. The convergence analysis of SAM is now primary focus on its theoretical understanding
38th Conference on Neural Information Processing Systems (NcurIPS 2024).

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with several works being developed recently (e.g., Ahn et al. [2024]. Andriushchenko and Flammarion 2022], Dai al. 2023 and Yun 2023 However, none of the aforementioned studies have addressed the fundamental convergence properties of SAM, which are outlined below where the stationary accumulation point (2) means that every accumulation point of the iterative sequence
(x*) satisfies the condition V f() 0.
lim Vf(2*)
k-+00
stationary accumulation point
lim [[] =0
k=+00
(4)
lim [xx) = ((x) with Vf(x) = 0
(5) converges to some 2 with Vz =0
Table 1: Fundamental convergence properties of smooth optimization methods
The relationship between the properties above summarized as follows:
(1)
(2) (3) (4).
The aforementioned convergence properties standard and are analyzed by various smooth optimiza- tion methods including gradient descent-type methods, Newton-type methods, and their accelerated versions together with nonsmooth optimization methods under the usage of subgradients. The readers are referred to [Bertsekas) (016)) Nesterov| (2018), PPlyyk] [1987] and the references therein for those results. The following recent publications have considered various types convergence rates for the sequences generated by SAM as outlined below:
(i) Dai et al.
2023
Theorem 1]
tL?p²
(1 +
2p(2 Lt)
where I. the Lipschitz constant of VJ, and where is the constant of strong convexity constant for f.
(ii) Si and Yun
2023
Theorems 3.3, 3.4]
|/((>)|| =
1 k
vi('')| -
+ + L2'
VK) and
1=1
where L is the Lipschitz constant of f.. We emphasize that none of the results mentioned above achieve any fundamental convergence properties listed in Table The estimation in (i) only gives us the convergence of the function value sequence to a value close to the optimal one, not the convergence to exactly the optimal value. Additionally, it is evident that the results in (ii) do not imply the convergence of [V((*)) to O. To the best of our knowledge, the only work concerning the fundamental convergence properties listed in Table Tis /ndriisshhhhenk and Flammarion [2022]. However, the method analyzed in that paper is unnormalized SAM (USAM), variant of SAM with norm being removed in the iterative procedure (2c). Recently, Dai et [2023) suggested that USAM has different effects in comparison with SAM in both practical and theoretical situations, and thus, they should be addressed separately. This observation once again highlights the necessity for a fundamental convergence analysis of SAM and its normalized variants.
Note that, using exactly the iterative procedure 2), SAM does not achieve the convergence for either { 1=*t }. or {f(=*)} or {v*(=*)] to the optimal solution, the optimal value, and the origin, respectively. It is illustrated by Example below dealing with quadratic functions. This calls for the necessity of employing an alternative stepsize rule for SAM.. Scrutinizing the numerical experiments conducted for SAM and its variants (e.g.. Foret et al. 2202T, Subsection C1], Li and Giannakis 2023 Subsection 4.1.), we can observe that in fact the constant stepsize rule is not preferred strategy. Instead, the cosine stepsize scheduler from Loshchilov and Hutter (016). designed to decay to zero and then restart after each fixed cycle, emerges more favorable approach. This observation

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motivates to analyze the method under diminishing stepsize, which is standard and employed in many optimization methods including the classical gradient descent methods together with its incremental and stochastic counterparts [Bertsekas and Tiitsiklis 2000]. Diminishing step sizes also converge to zero as the number of iterations increases, a condition satisfied by the practical cosine step size scheduler in each cycle.
1.2 Our Contributions
Convergence of SAM and normalized variants
We establish fundamental convergence properties of SAM in various settings. In the convex case, we consider the perturbation radii to be variable and bounded. This analysis encompasses the practical implementation of SAM with constant perturbation radius. The results in this case are summarized in Table
Classes of function
Results
General setting
lim inf f((=") =
Bounded minimizer set
stationary accumulation point
Unique minimizer
{x*)
is convergent
Table 2: Convergence properties of SAM for convex functions in Theorem
In the nonconvex case, we present a unified convergence analysis framework that can be applied to most variants of SAM, particularly recent efficient developments such VaSSO [Li and Giannakis 2023]. F-SAM [Li et al., 2024], and RSAM (Liu et al. 2022]. We observe that all these methods can be viewed as inexact gradient descent (IGD) methods with absolute error. This version of IGD) has not been previously considered, and its convergence analysis is significantly more complex than the one in Khanh et al. [203b 2024a 2023a 2024b), the function value sequence generated by the new algorithm may not be decreasing. This disrupts the convergence framework for monotone function value sequences used in the aforementioned works. To address this challenge, we adapt the analysis for algorithms with nonmonotone function value sequences from Li et al. [0023), which was originally developed for random reshuffling algorithms, context entirely different from ours.
We establish the convergence of IGD with absolute error when the perturbation radii decrease at an arbitrarily slow rate. Although the analysis of this general framework does not theoretically cover the case of constant perturbation radius, it poses no issues for the practical implementation of these methods, as discussed in Remark 3.6 A summary of our results in the nonconvex case is provided in the first part of Table 3
Convergence of USAM and unnormalized variants
Our last theoretical contribution in this paper involves refined convergence analysis of USAM in Andriushchenko and Flammarion 2022]. In the general setting, we address functions satisfying the
L-descent condition (4). which is even weaker than the Lipschitz continuity of V as considered in
Andriushchenko and Flammarion 22022]. The summary of the convergence analysis for USAM is given the second part of Table
As will be discussed Remark G.4 our convergence properties for USAM use weaker assumptions and cover broader range of applications in comparison with those analyzed in (Andriushchenko and Flammarion] 2022). Furthermore, the universality of the conducted analysis allows us to verify all the convergence properties for the extragradient method [Korpelevich 1976 that has been recently applied in Lin et al. 2020 to large-batch training in deep learning.
SAM and normalized variants
USAM and unnormalized variants
Classes of functions
Results
Classes of functions
Results
General setting
lim vf(x* =0
General setting
stationary accumulation point
General setting
lin [(x =
General setting
lim -(=*)=f'
KL property
xxk is convergent
f is Lipschitz
lim vf(====
KL property
{ 1x is convergent
Table 3: Convergence properties of SAM together with normalized variants (Corollary 3.5 Ap-
pendix D). and USAM together with unnormalized variants (Theorem 4.2)
3

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1.3 Importance of Our Work
Our work develops, for the first time in the literature, a fairly comprchensive analysis of the funda- mental convergence properties of SAM and its variants. The developed approach addresses general frameworks while being based on the analysis of the newly proposed inexact gradient descent meth- ods. Such an approach can be applied in various other cireumstances and provides useful insights into the convergence understanding of not only SAM and related methods but also many other numerical methods in smooth, nonsmooth, and derivative-free optimization.
1.4 Related Works
Variants of SAM. There have been several publications considering some variants to improve
the performance of SAM. Namely, Kwon al. [2021] developed the Adaptive Sharpness- Aware
Minimization (ASAM) method by employing the concept of normalization operator. [Du et al. [2022]
proposed the Efficient Sharpness- -ware Minimization (ESAM) algorithm by combining stochastic weight perturbation and cerrppeess-eessitiv data selection techniques. [Liu et al. [2022] proposed a novel Random Smoothing-Basse SAM method called RSAM that improves the approximation quality in the backward step. Quite recently. Li and Giannakis (2023) proposed another approach called Variance Suppressed Sharpness-aware Optimization (VaSSO), which perturbed the backward step by incorporating information from the previous iterations. ei al. [2024] identified noise in stochastic gradient as a crucial factor in enhancing SAM's performance, they proposed Friendly Sharpness- Aware Minimization (F-SAM) which perturbed the backward step by extracting noise from the difference between the stochastic gradient and the expected gradient at the current step. Two efficient algorithms, AE-SAM and AE-LookSA.., are also proposed in Jiang et al. [2023] by employing adaptive policy based on the loss landscape geometry.
Thcoretical Understanding of SAM. Despite the success of SAM in practice, theoretical under- standing of SAM was absent until several recent works. Barlett et al. [2023] analyzed the convergence behavior of SAM for convex quadratic objectives, showing that for most random initialization, it
converges to a cycle that oscillates between either side of the minimum in the direction with the largest curvature. Ahn et al. [2024] introduces the notion of E-approximate flat minima and investigates the iteration complexity of optimization methods to find such approximate flat minima. As discussed in Subsection 1.1 Dai al. 2023] considers the convergence of SAM with constant stepsize and constant perturbation radius for convex and strongly convex functions, showing that the sequence of iterates stays in neighborhood of the global minimizer while (Si and Yun [2033 considered the properties the gradient sequence generated by SAM in different settings.
Theoretical Understanding of USAM. This method was first proposed by Andriushchenko and Flammarion [022] with fundamental convergence properties being analyzed under different settings of convex and nonconvex and optimization. Analysis of USAM was further conducted in Behdin and Mazumder [2023] for a linear regression model, and in Agarwala and Dauphin [2023] for a quadratic regression model. Detailed comparison between SAM and USAM, which indicates that they exhibit different behaviors, was presented in the two recent studies by [Comaagnoni et al. 2023) and Dai et al.
(2023). During the final preparation of the paper, we observed that the convergence of USA. can also be deduced from Mangasarian and Solodov| (1994), though under some additional assumptions, including the boundedness of the gradient sequence.
2 Preliminaries
First we recall some notions and notations frequently used in the paper. All our considerations are given in the space IR" with the Euclidean norm || ||. As always, IN == 21, .} signifies the
collections of natural numbers. The symbol x² 2 z means that x² + 5 as l + co with & E J c I. Recall that Z is stationary point of a -mooth function f: IR" + IR if Vz)) = 0. A function f: IR" > IR is said to posses a Lipschitz continuous gradient with the uniform constant L > 0, or  equivalently it belongs to the class CL.., if we have the estimate
[V) v f())l $ L vil for all B E IR..
(3)

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This class of function enjoys the following property called the L-descent condition (see, e.g., Izmailov and Solodov 2014 Lemma A. and Bertsekas (2016] Lemma A.24]):
f(v) S [()) + (VI(x) v- x) + 1ly - x|
for all I, y E IR". Conditions and
equivalent to each other when f is convex, while the equivalence fails otherwise. A major class of functions satisfying the L-descent condition but not having the Lipschitz continuous gradient is given by Khanh et al. [2023b] Section 2] as
1 f(z) == 2 (Ax, x) + (b. + c h(z), where A is an n X matrix, b IR", c IR and
h: R + R is a smooth convex function whose gradient is not Lipschitz continuous. There are also circumstances where function has a Lipschitz continuous gradient and satisfies the descent condition the same time, but the Lipschitz constant is larger than the one in the descent condition.
Our convergence analysis of the methods presented in the subsequent sections benefits from the Kurdyka- -ajjsie-t (KT.) property taken from Attouch al. [2010].
Definition 2.1 Kojssiwa-Lossseewiiz property). We say that a smooth function : RR" IR enjoys the KL property at F E dom f if there exist E c0), a neighborhood U of Σ. and desingularizing concave continuous function yp
: [0, 1)) + [0, oo) such that (0()) 0, yp C*-smooth
on (0, m). & > 0 on (0, 17), and for all x E U with 0 < f()) f(z) < we have
('((())
f=))]l 1.
Remark 2.2. It has been realized that the KL property satisfied in broad settings. particular, holds at every nonstationary point of f: see Attoouh et al. (2010 Lemma 2.1 and Remark 3.2(b)). Furthermore, it proved at the seminal paper [Eojasiewicz 1965) that any analytic function f IR" - IR satisfies the KI. property on IR" with (tt)) = Mr- for some q E [0, 1). Typical functions that satisfy the KL property are semi-algebraic functions and in general, functions definable in o-minimal structures; see Attouch al. 2010 2013). Kurdyka [1998.
We utilize the following assumption the desingularizing function in Definition 2.1 which is employed in Li et al. 2023). The satisfaction of this assumption for a general class of desingularizing functions is discussed in Remark G.1
Assumption 2.3. There is some C' such that whenever x, y E (0, with x + < 17, it holds that
( +
((d())))++))
3
SAM and normalized variants
3.1 Convex case
We begin this subsection with an example that illustrates SAM's inability to achieve the convergence of the sequence of iterates to an optimal solution of strongly convex quadratic functions by using constant stepsize. This emphasizes the necessity of avoiding this type of stepsize in our subsequent analysis.
Example 3.1 (SAM with constant stepsize and constant perturbation radius does not converge). Let the sequence {2*) be generated by SAM in (2) applied to the strongly convex quadratic function f(x) = 2 (Ax, x (b, æ),, where A is x symmetric, positive-definite matrix and b E IR". Then for any fixed small constant perturbation radius and for some small constant stepsize together with initial point close to the solution, the sequence {a*) generated by this algorithm does not converge to the optimal solution.
The details of the above example are presented in Appendix A.I illustration for Example 3.1 The graph shows that, while the sequence generated by GD converges to 0, the one generated by SAM gets stuck at a different point.
Figure gives an empirical
As the constant stepsize does not guarantee the convergence of SAM, we consider another well-known stepsize called diminishing (see (7). The following result provides the convergence properties of SAM in the convex case for that type of stepsize.
5

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SAM
Iterations
Figure I: SAM with constant stepsize does not converge to optimal solution
Theorem 3.2. Let f IR" - IR be smooth corvex function whose gradient is Lipschitz continuous with constant L >> Given any initial point x² IR",, let { {a* be generated by the SAM method with the iterative procedure
V [V
x
1x VS
x. + Pk
for all l E with nonnegative stepsizes and perturbation radii satisfying the conditions
t OC, sup Pk
KEIN
Assume that V (x(x)) + for all E IN and that infReI (x* > DO. Then the following assertions hold:
ii) lim inl (x(x*)= 0. k-000
(ii) If f has nonempty bounded level set, then {a*} is bounded, every accumulation point of {at} is a global minimizer of J, and {[(x*)} converges to the optimal value of J. If in addition I has unique minimizer, then the sequence {a*) converges to that minimizer.
Due to the space limit, the proof of the theorem presented in Appendix
3.2 Nonconvex
In this subsection, we study the convergence of several versions of SAM from the perspective of the inexact gradient descent methods.
Algorithm 1 Inexact Gradient Descent (IGD) Methods
Step O. Choose some initial point xº IR", sequence of errors {sx} C (0, oo), and sequence of stepsizes [ta} C (0, 20). For
1, 2, do the following
Step 1. Set k+1 z - tug with || (x* < Ek-
This algorithm is motivated by while being different from the Inexact Gradient Descent methods proposed in [Khanh et al.) 2023a b) [0. The latter constructions consider relative errors in gradient calculation, while Algorithm| | Jsees the absolute ones. This approach is particularly suitable for the constructions of SAM and its normalized variants. The convergence properties of Algorithm [ are presented the next theorem.
Theorem 3.3. Let f: IR" + IR be a smooth function whose gradient is Lipschitz continuous with some constant L > O, and let {a*} be generated by the IGD method in Algorithm with stepsizes and errors satisfying the conditions
l 0o, l 40,
Σ km
LEsk < 00, limsup < 2.
(8)

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Assume that inf f(x²)> -cc. Then the following convergence properties hold:
(i) Vf(=") + O, and thus every accumulation point of {x*) is stationary for f.
(it) If i an accumulation point of the sequence {>*), }, then [(x*) + f(3).
(iii) Suppose that f satisfies the KL. property at some accumulation point & of {x²) with the desingularizing function yp satisfying Assumption 2.3 Assume in addition that
Σ
BKEK
OC;
and that f(x²) > f f()) for sufficiently large N. Then x² as k co. In particular, if ã is global minimizer f. then either f(x²) (() for some k E N, or x² + T.
The proof of the theorem is presented in Appendix C.2 The demonstration that condition
9 is
satisfied when (t) = Mr--9 with some M > O and E 1). and when & = and c
kF with
p 2 0 for all 1: E N. is provided in Remark G.3
The next example discusses the necessity of the last two conditions in (8) in the convergence analysis of IGD while demonstrating that employing constant error leads to the convergence to nonstationary point of the method.
Example 3.4 (IGD with constant error converges nonstationary point). Let f: IR + IR be O and an initial point x. >
defined by f far = x² for æ E IR. Given a perturbation radius consider the iterative sequence
xx for k IN, [
glt1 x² 2tx
(10)
where { 10,1/2,, t and Ex to 00. This algorithm in fact the IGD applied
with g" f x p Then {sr converges to P, which is not stationary point of f.
f((x')
The details of the example are presented in Appendix A.2 We now propose a general framework that encompasses SAM and all of its normalized variants including RSAM [Liu al.) 2022], VaSSO [Li and Giannakis 2023 and F-SAM [Li et al., 2024]. Due the page limit, we refer readers to Appendix ( for the detailed constructions of those methods. Remark D. in Appendix [aaaso shows that all of these methods are special cases of Algorithm la and thus all the convergence properties presented in Thcorem 3.3 follow.
Algorithm la General framework for normalized variants of SAM
Step O. Choose x² E IR", {pk} {4x} C (0, co), and { C IR" {0} For k: 2 1 do the following:
Step 1. Set k+1 x² t f (x + Pk Iid
Corollary 3.5. Let f IR + IR be a CL. function, and let {at) } be generated by Algorithm la with the parameters
2 L
Σ 44 = 00, t 40,
Σ
tek < co, lim Pk <
(11)
k=1
k=1
Assume that inf f(x*) oc. Then all convergence properties presented in Theorem 3.3 hold.
The proof of this result is presented in Appendix C.5
Remark 3.6. Note that the conditions in [I]) do not pose any obstacles to the implementation of constant perturbation radius for SAM practical circumstances. This is due the fact that possible selection of t; and Pk satisfying (T) is t 1 and where > 0. Then the initial perturbation radius is C, while after C million iterations, remains greater than 0.99C'. This phenomenon is also confirmed by numerical experiments in Appendix E on nonconvex functions. The numerical results show that SAM with almost constant radii As has similar convergence behavior to SAM with a constant radius perturbation radius has sufficient empirical evidence for its efficiency in Foret et al. 2021 ] this also supports the practicality of our almost constant perturbation radii.
C 40.001 for all & E N (almost constant),
C. As SAM with constant
7

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USAM and unnormalized variants
In this section, we study the convergence of various versions of USAM from the perspective of the following Inexact Gradient Descent method with relative errors.
Algorithm 2 IGDr
Step O. Choose some x. E IR", v 2 0, and (4x) C [0, 0o). For k 1,2, do the following:
Step 1. Set *+1 x lxg where I|0* (((=") [V((**) ||
This algorithm was initially introduced in Khanh et al. [2023b in a different form. considering different selection of error. The form of IGDr closest Algorithm was established in [Khanh et al.)
[024a] and then further studied in Khanh et al. [2024a] 2023a 2024b). In this paper, we extend the analysis of the method to a general stepsize rule covering both constant and diminishing cases, which was not considered in Khanh et al. [2024a].
Theorem 4.1. Let I: IR" - IR be a smooth function satisfying the descent condition for some constant L > 0, and let {a*) be the sequence generated by Algorithm 2 with the relative error [0, 1), and the stepsizes satisfying
2 2v 0,
E 4h = and tk E
( L(1 + v)²
(12)
for sufficiently large k E IN and for some 8 0. Then either f(x²) -o0, we have the assertions:
(i) Every accumulation point of {a*}
} is stationary point of the cost function f.
(ii) If the sequence {x*} has any accumulation point ã, then f(x*) 1 f(z).
(iii) If f E C.,LL, then V (x**) + 0.
(iv) If satisfies the KI. property al some accumulation point 5. of S, then {a - T.
(v) Assume in addition to (iv) that the stepsizes are bounded away from O, and the KI. property in
(iv) holds with the desingularizing function g(t)= -Mt--9 with M > 0 and q (0, 1). Then either
{ stops finitely stationary point, the following convergence rates achieved:
Ifq -1/2, then {a*}. {V)()))) {f(x*)} converge linearly as l + E, O, and f()).
Ifq E (1/2, 1), then
1 - æ = - ) Vf(x*) 0
(* ( (1
f(x²) - f(i) ( = 0 3
Although the ideas for proving this result is similar to the one given in Khanh et al. 2024 do provide the full proof in the Appendix CC33 for the convenience the readers. We now show that using this approach, derive more complete convergence results for USAM in Andrishchhenko and Flammarion 2022 and also the extragradient method by Korpelevich 1976
Lin et al. 2020).
Algorithm 2a Annrrisshhhenoo and Flammarion, 2022 tion (USAM)
Unnormalized Sharpness-Awee Aware Minimiza-
Step O. Choose d E IR", {px] C 10, oc), and (4x c [0, oo). For k: = 1, 2,.,.., do the following:
Step 1. Set k+1 x² tx fx + V+(x())
Algorithm 2b Korpelevich 1976] Extragradient Method
Step O. Choose d IR" {pk} [0, oc). and (tx} C. [0, oo). For 1,2...., .,, do the following:
Step 1. Set kk1 = x² - 4 fs Pk V ()))).

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We are ready now to derive convergence of the two algorithms above. The proof of the theorem is given Appendix C.4
Theorem 4.2. Let f: IR" + IR be C--soooth function satisfying the descent condition with some constant L > O. Let {x*] } be the sequence generated by either Algorithm 2a or by Algorithm 2b with
Pk: I T, for some v E [0, 1) and with the stepsize satisfying 12). Then all the convergence properties
in Theorem 2.7
5 Numerical Experiments
To validate the practical aspect of our theory, this section compares the performance of SAM employing constant and diminishing stepsizes in image classification tasks. All the experiments are conducted computer with NVIDIA RTX 3090 GPU. The three types of diminishing stepsizes considered in the numerical experiments are ///n (Diminish 1), 1/ 0.500 (Diminish 2). and m / log (Diminish 3), where 77 is the initial stepsize, represents the number of epochs performed, and m = [n/5] + 2. The constant stepsize in SAM is selected through a grid search over {0.1, 0.01, 0.001} } to ensure fair comparison with the diminishing ones. The algorithms are tested 0.01, on two widely used image datasets: CIFAR-10 (Krizhvskyy et al. 2009 and CIFAR-100 (Krizhessky al. 2009].
CIFAR-10. We train well-known deep neural networks including ResNet18 [He et al. 2016), ResNer34 [He et al. 2016], and WideResNee28-10 (Zagoruyko and Komodakis 2016 on this dataset by using 10% of the training set as a validation set. Basic transformations, including random crop, random horizontal flip, normalization, and cutout (DeVries and Taylor, [017], are employed for data augmentation. All the models are trained by using SAM with SGD Momentum as the base optimizer for 200 epochs and batch size of 128. This base optimizer is also used in the original paper Fooret et al.] 2021] and in the recent works on SAM [Ahn et al. 2024 Li and Giannakis 2023 Following the approach by Foret al. [2021], we set the initial stepsize to 0.1, momentum to 0.9, the fz-reuularization parameter to 0.001, and the perturbation radius to 0.05. Setting the perturbation radius to be constant here does not go against our theory, since by Remark SAM with a constant radius and our almost constant radius have the same numerical behavior. We also conduct the numerical experiment with an almost constant radius and got the same results. Therefore, for simplicity of presentation, a constant perturbation radius is chosen. The algorithm with the highest accuracy, corresponding to the best performance. highlighted in bold. The results in Table report the mean and 95% confidence interval across the three independent runs. The training loss in several tests is presented in Figure _
C1F0R-100. The training configurations for this dataset are similar to CIFAR10. The accuracy results are presented in Table while the training loss results are illustrated in Figure 2
e
D cereuct O mm
Dremon
Figure 2: Training loss on CIFAR-10 (first two graphs) and C0F (last two graphs)
CIFAR-10
CIFAR-100
Model
ResNet18
ResNet34
WRN28-10
ResNet18
ResNet34
WRN28-1B 10
Constant
94.10 + 0.27
94.38 + 0.47
95.33 + 0.23
71.77 + 0.26
72.49
0.23
74.63 0.84
Diminish
93.95 + 0.34
93.94 + 0.40
95.18 + 0.03
74.43 1 0.12
73.99 + 0.70
78.59 1 0.03
Diminish
94.60 0.09
95.09 1: 0.16
95.75 + 0.23
73.40
0.24
74.44 0.89
77.04 0.23
Diminish 3
94.75 + 0.20
94.47 + 0.08
95.88 + 0.10
75.65 + 0.44
74.92 + 0.76
79.70 + 0.12
Table 4: Test accuracy CIFAR-I0 and CIFAR-100

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The results on CIFAR-10 and CIFAR-100 indicate that SAM with Diminish stepsize usually achieves the best performance in both accuracy and training loss among all tested stepsizes. In all the architectures used in the experiment, the results consistently show that diminishing stepsizes outperform constant stepsizes in terms of both accuracy and training loss measures. Additional numerical results on a larger data set and without momentum can be found in Appendix
6
Discusison
6.1
Conclusion
In this paper, we provide fundamental convergence analysis of SAM and its normalized variants together with a refined convergence analysis of USAM and its unnormalized variants. Our analysis is conducted in deterministic settings under standard assumptions that cover broad range of applications of the methods in both convex and nonconvex optimization. The conducted analysis is universal and thus can be applied in different contexts other than SAM and its variants. The performed numerical experiments show that our analysis matches the efficient implementations of SAM and its variants that are used in practice.
6.2 Limitations
Our analysis is only conducted in deterministic settings, which leaves the stochastic and random reshuffling developments to our future research. The analysis of SAM coupling with momentum methods is also not considered in this paper. Another limitation pertains to numerical experiments, where only SAM was tested on three different architectures of deep learning.
Acknowledgment
Pham Duy Khanh, Research of this author is funded by the Ministry of Education and Training Research Funding under the project B2024-SPS-07, Boris S. Mordukhovich, Research of this author was partly supported by the US National Science Foundation under grants DMS-1808978 and DMS- 2204519, by the Australian Research Council under grant DP-190100555 and by Project 111 of China under grant D21024. Dat Ba Tran, Research of this author was partly supported by the US National Science Foundation under grants
and DMS- MMS-2204519.
The authors would like thank Professor Mikhail V. Solodov for his fruitful discussions the convergence of variants of SAM.
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12

---

Contents
Introduction
1.1
Lack of convergence properties for SAM due to constant stepsize
Our Contributions/
Importance Our Work
.4
Related Works
Preliminaries
3
SAM and normalized variants
Convex case
Nonconvex case
4
USAM and unnormalized variants
Numerical Experiments
Discusison
10
Conclusion
10
6.2 Limitations
10
Counterexamples illustrating the Insufficiency of Fundamental Convergence Properties
14
Proof of Example
14
A.2
Proof of Example 3.4
14
Auxiliary Results for Convergence Analysis
15
Proof of Convergence Results
17
Proof of Theorem 3.2
17
Proof of Theorem
19
3 Proof of Theorem
22
4
Proof of Theorem 4.2
................................
24
Proof of Corollary 3.5
24
D
Efficient normalized variants of SAM
24
Numerical experiments on SAM constant and SAM almost constant
25
Numerical experiments on SAM with SGD without momentum as base optimizer
27
Additional Remarks
27
13

---

A Counterexamples illustrating the Insufficiency of Fundamental Convergence Properties
A.1 Proof of Example 3.1
(b) the gradient of and the optimal solution are given by Proof. Since f(x (Ax, x)
Vf(z) = Ax - b and x² = -A-bb.
Let Amina Amax > O be the minimum, maximum eigenvalues of A, respectively and assume that
1 Lpxmin
1 Aminn
p>> O, and 0 < ||x² x²
(13)
Amaa
Amnin
Amin
1 tdmia
The iterative procedure of (6) can be written
vf(at)
x² - 1 ((A((*
x² + - 1| 611 )
Ar -
- b)
x+tl 1VS
[V[(2*]||
(14)
Then {x*) satisfies the inequalities
bpdmin
11 t A/
tAmin for all E IN.
(15)
1
It is obvious that (15) holds for k = 1. Assuming that this condition holds for any k E IN, let us show that it holds for k 1. We deduce from the iterative update (14) that
b 1 A (
P A1x - b1 )
Ax *
| k+1 - A -b1
R +
A A- -b
=
A( A(Ax* - b)
I -AA)(** - -A) tp
1|Az b)ll
A(Ax* b)
([ 1x A A>
1| bll
tpxaiin (dmin)
11 A-bl/1 O.
(16)
In addition, we get
1 1+1 - 2b1 < 1p A(Ax* 6)
+ 11((TT- (A)((** - A-
Aaa - bll
< Ipdmax + (1 (imin) 1| - A b1l
Amin
S tpAmax + tpAmin < tp
tdmin
where the last inequality follows from t Amnn Amart
from
13
Thus,
15
is verified. It
follows from (16) that k A x²
A.2 Proof of Example 3.4
Proof. Observe that x² > p for all & E IN. Indeed, this follows from x > p, tt 1/2, and
x k+1 - p == 1² p 2tx - p) = (1 12tx) - p).
In addition, we readily get
p= S p- - -1- (1-2ts)(a (xx) -p) = =.. (x
2t.).
(17)
==1
Furthermore, deduce from Σ 2t, xc that Il -2tx) 0. Indeed, we have
1
2t.) I (1 2tx) 1 >x 2tx
This tells by 17 and the classical squeeze theorem that xx p as k 00.
14

---

B Auxiliary Results for Convergence Analysis
We first establish the new three sequences lemma, which is crucial the analysis of both SAM, USAM, and their variants.
Lemma B.1 (three sequences lemma). Let {a (Bx), {re) be sequences of nonnegative numbers satisfying the conditions
a++1 ak S BROx + 7kk for sufficient large & IN
(a)
(Bxb is bounded, Σ BR 00,
The < 0C, and Brak < 00.
(b)
k=1
1=1
Then have that ag + 0 as k oc.
Proof. First we show that lim in og = 0. Supposing the contrary gives us some > 0 and N IN such that og d for all 2 N. Combining this with the second and the third condition in
(b) yields
o
20
00
8,0
82
BR
00,
which is a contradiction verifying the claim. Let us now show that fact limg- ag = 0. Indeed, by the boundedness of { define 3 == suPKEIN 33 and deduce from (a) that there exists K e IN such that
ak+1 as S 8E0 + 7 for all K.
(18)
Pick 0 and find by lim infa on and the two last conditions in (b) some Kg IN with Kg 2 K, a. S.
0
8
$² 3'
and BBaa S 22
Σ k=K.
7h
Σ k=K,
B
as Ke.
(19)
3
It suffices to show that ag < 2E for all k 2 Kg. Fix k 2 K2 and observe that for og I E the desired
inequality is obviously satisfied. If a4 E, we use ak, I E and find some k' such that ke 2 K₂ and
0g S and 0g > for i i= k, k 1, k' +
Then we deduce from
18 and 19 that
k-1
k-1
ag
o2
(0++1 - as) S
(Beas + 75)
k=
k=b
h-1
Biaq
Σ
Ys + Bw
i=
i=k'
k-1
3a0 +
Σ
7i + v Ber v Braw
(=
i=k
20
3,0 2 + Σ
7 / 1BBw
(=K,
k=
1
+
=E.
8 3
3
As consequence, we arrive at the estimate
2 ag + ag. aw S e 25 for all k 2 K₃,
which verifies that ag + 0 as l + sand thus completes the proof of the lemma.
15

---

Next we recall some auxiliary results from Khanh et al. (002b).
Lemma B.2. Let {x*} and { dd be sequences in IR" satisfying the condition
El
- x*1 || xo.
z is accumulation point of the sequence {-²} and 0 is accumulation points of the sequence dh }. then there exists an infinite set J C. IN such that we have
x 4 x and d 4.
(20)
Proposition B.3. Let f IR" IR be -'-moooth function, and let the sequence {a²} C IR" satisfy the conditions:
(H1) (primary descent condition). There exists O such that for sufficiently large k E we have
f(x²) f(x [ - ==11
|
(H2) (complementary descent condition). For sufficiently large l E IN, we have
[f(xh+))= =(x²)) = [x 1phtl=="" -1 =x"].
E is accumulation point of {a*) and satisfies the KI. property T. then x. as k co.
When the sequence under consideration is generated by a linesearch method and satisfies some conditions stronger than (HI) and (H2) in Proposition B.3 its convergence rates are established in Khanh et al. (2023b, Proposition 2.4] under the KL property with (bt) = Mt as given below.
Proposition B.4. Let f IR" - IR be a -mooth function, and let the sequences {s*} C IR", {Tx) C [0, oo), {at) IR" satisfy the iterative condition x*+ x² + Tyd for all k E IN. Assume that for all sufficiently large IN we have x*+1
+ x² and the estimates
f(x²) f((htt)) BTx [|a" and S
(21)
where a., > 0. Suppose in addition that the sequence {Tk} is bounded away from 0 (i.e., there is some 7 > 0 such that Tk 7 for large &s E IN), that 2 is an accumulation point of {a }. and that satisfies the KL property at z with vt() Mt!-q for some M > O and (0, 1). Then the following convergence rates are guaranteed:
(i) If 10,1/2), then the sequence {x*} converges linearly to T.
(ii) lʃq (1/2,1), then have the estimate
1 - I =
Yet another auxiliary result needed below is as follows.
Proposition B.5. Let f: IR." + IR be a C³ -s-soooth function functiion satisfying the descent condition (4 with some constant O. Let xx } be sequence in IR. that converges and let be such that
a ||(()>)) s f(x²)) f(a for sufficiently large IN.
(22)
Consider the following convergence rates of {
(i) k - T. linearly.
(ii) 1z I = O(m(k)), where m(k) 0 as k + co.
Then (i) ensures the linear convergences of f(**) to f(ā), and vf(x*) to 0, while (ii) yields f(**) f(3) Omm² and ff(x*)) (m((k) as k co.
16

---

Proof. Condition 22 tells us that there exists some N E IN such that f(x²-1) S f(x*) for all k 2 IN. As x* + v. we deduce that f(**) f() with f(*) 2 f()) for k 2 N. Letting k: + oo
in 22 and using the squeeze theorem together with the convergence of {**) f to I and the continuity
of lead us to Vf(() O. It follows from the descent condition of f with constant >> 0 and
from that
L ||
0 0 f<1( f()) < (Vf(a), x -x)
2 || &
- x
=
2
This verifies the desired convergence rates of {f(x*)) EEpllyyig finally 22) and f(x*+1) 2 f()). we also get that
a vf(2>)) & -(=)))((a)) for all 2 N.
This immediately gives us the desired convergence rates for {V](**) and completes the proof.
C Proof of Convergence Results
C.1 Proof of Theorem 3.2
Vi(a") Proof. To verify (i) first, for any & IN define g" = Vf (2t + Pb: [[
and get
v/(ot)
)
110 - V5(aa))
4s
1* + [V
v f((=*)
2t||
$ I
+ Pk
Lpk S Lp,
(23)
[
where p = suPkEN Pk. Using the monotonicity of f due to the convexity of f ensures that
(g vf(xtt)
vf
t Pk
vf(at), vf(at) + |\
(v
vf((*)) x + Pk [Vl(at)])) Vf((*))
[f((+)]|| + +|\ > ||
vf(x**
Pk
(24)
With the definition of g², the iterative procedure (6) can also be rewritten xx+1 = tag for all E IN. The first condition in 7 yields t 4 O, which gives us some K IN such that Lty < 1 for all k > K. Take some such k. Since V is Lipschitz continuous with constant L >> 0, it follows from the descent condition in and the estimates in (3) 24) that
L
S S(x*) f(x+11)
-at)
x01l"
2
Lt 2
-(x*) 4 (vI(x*)) ) +
|1a
Lt
f((²) (1 - Ltt)
V ff(a" ),
+
-118--
-
2
Lt
\f((">] 1'tGp2
< ff(**) t4 Lth) V
11
2
f((*) tx (2 Lt+) | (z >|
2
t
|| L3g
< <(
(25)
2
2
Rearranging the terms above gives us the estimate
t*
Vf(x") )| S f(x²) - f(ax+1)+ +
I3
(26)
2
2
17

---

Select any M K, define S := 1'/2² R==1 -oo that
tk oc, and get by taking into account inf f(a"))
1 M
M
M Σ
T.
t
- Σ (f(a*) - -1 ((x
2 k=
k=K
k=
S f(x²)) - f(x +
S f(x - inf f(x²) S.
Letting M + oo yields Σx=K th [f(*))| ||
co. Let us now show that lim inf V ) = 0.
Supposing the contrary gives us e > 0 and tells us that
N > K such that [V((>*]] 2 for all k 2 N, which
8
2
Σ 4x 00.
This clearly contradicts the second condition in (7) and this justifies (i).
L'p² To verify (ii), define us == 2 Einh 12 for all E and deduce from the first condition in that up 0 as k + co. With the usage of {ux}. estimate (26) is written
t f(ah+1) + ubt1 S f(x") + us 2
[f(*")] for all k
which means that {f(**) + ux J22K is nonincreasing. It follows from İnfREIN f(**)> -o and we O that {f(=*) + ux is convergent, which means that the sequence {[(**)} } is convergent as
well. Assume now / has some nonempty and bounded level set. Then every level set of f bounded by Ruszczynski 2006, Exercise 2.12]. By (26). we get that
f(x*+1) f(**)
p²p² 2
th
LE
|f(a>)]| s f(x*)
L2'2 th for all 2 K.
2
Proceeding by induction leads to
_3p² &
f(a s(xx))
Σ e < f(x + for all k K,
2
S} for all k 2 K and thus justifies the which means that xhtl E E IR" [f()) S f(x boundedness of {
Taking lim inf f(*))|| 0 into account gives us an infinite set J C IN such that
[[((*)]| 2 o. As {x f is bounded, the sequence {a*) key is also bounded, which gives another infinite set I C. and F e IR. such that x² 4 5. By
lim KET
lim KEJ
[ =0
and the continuity of ff, we get that fz()) ensuring that is global minimizer of f with the optimal value f=== f() Since the sequence { [a*)) is convergent and since 7 is an accumulation point of {x*)) } we conclude that f f()) is the limit of (f(a*)) Now take any accumulation point ž: of {a**) and find an infinite set J C IN with xy J F. As {{(**} converges to f", we deduce that
(
lim l(x REJ'
lim f(x))=
kEIN
which implies that 0ेr is also global minimizer of f. Assuming in addition that has unique minimizer and taking any accumulation point of {-*} we get that is a minimizer of f, i.e.
= F. This means that is the unique accumulation point of {x*) }. and therefore at as k + co.
18

---

C.2 Proof of Theorem 3.3
Proof. By (8. we find some e₁ > 0, E (0, 1). and K E IN such that
2/22- Llg Ex L(uEs) 2 ct.
2 [1))
Lln) + LLEcK
S cz and Ll < for all k K.
2
(27)
Let us first verify the estimate
fa S f(x²) city [V((=>||| + c2t2s* whenever k 2 K. To proceed, fix l E IN and deduce from the Cauchy-Sehwarz/ inequality that
(28)
< V((())
(g V)(=", ([(*)) + ||
ス 2 1|0 vf(=") V + [[(())]]
- |} V /
(29)
Since ff is Lipschitz continuous with constant L, it follows from the descent condition in (4) and the estimate (9) that
k+1 -²ah ++++, + f(a S $(a") + (VI(a*),
1 - =011
= -((²)) - 4a (v)(x*), ),9²) + L12 110/1²
2
Li
= -s(²) - (1 T1A) (VI ),1²) +
(11g (5(01)) -|
2
S <((a²) 4(1 T1p) (-ex |
()(()+)))))) +
IIReZ
Lt
2
= f(x²) -
t (2 Ltt)
Vf 11 +x - Lx (x >|
2
2
t 1 (2 Ltx) | (z I1? + gta(1 LLN)EN (1 + ||[
+ LtAE2
S f(*))
2
2
ls (2 Ltk Ek + k
V {|=))| + 1 [RLEE(L Ltt))=
f(ah)
2
= =((²)) (2 Ltk Ek + Ex Vf(("')] )1?
tREk (2(1- Ltt)
LtpEk
2
Combining this with 27 gives us 28
Defining us == Co iink le; for l E IN, we get that Ui - 0
as k: + 2o and un uk+1 t For all k IN. Then 28 can be rewritten
f(a + uk+1 S f(x*) + un caty |f(x")]
k
(30)
To proceed now with the proof of (i), we deduce from 30 combined with inf f(x²) > 0o and Uk + 0 le + 0o that
tu [ < E(fatt) - f(ahtl) + u -u++1)
k=
k=
$ {s(a") - inf f(ark) uu < 00.
KEIN
Next we employ Lemma B.1 with 0g = V ||. Br := Ltv, and 7 = LtyeR for all k E IN to
derive (((h)) + O. Observe first that condition @ab is satisfied due to the the estimates
|| [ (-*1)) >| [[V <
0+++1 or
S T. 1|,/:11 - -+1
Lt ||0*||
5 LCL(( [(((( + I1g (((
3 <Laa( +
= BaaR + 7/ for all l E IN.
19

---

Further, the conditions in b hold by 8 and Σ 00
/|² < co. As all the assumptions (z - aq + 0 as k + 0c.
k= tx |((
(b) are satisfied, Lemma B.1 tells us that /
To verify (ii), deduce from 33)p that {f(a*) us} is nonincreasing. As infRem ((*)) > -o and uk - O, we get that {- + ] } is bounded from below, and thus convergent. Taking into account that uy - 0. it follows that f(**) is convergent as well. Since 2 is an accumulation point of {**}. the continuity of f tells us that f()) is also an accumulation point of (f(x*), which immediately yields f(x*) + f()) due to the convergence of (fz
It remains to verify (iii). By the KL property of f at ã, we find some > 0, neighborhood U of J, and a desingularizing concave continuous function yp: [0, n) - [0, oo) such that (0 = 0, yp is
-m-soooth on (0, n). 4 > 0 on (0, 17). and we have for all x E U with 0 < f(x) - f(x) < 1y that
v'(((x) ((E) 2 1.
(31)
Let K > K be natural number such that f((**)= > (z) for all k: 2 K. Define AA == ((((aa)))
(()) + uz) for all k 2 K, and let R > be such that B(T, R) C U. Taking the number C from
Assumption 2.3 remembering that is an accumulation point of {-*} and using ff fa²*) + uk + f().
) + uu < A& + 0 as k- + oo together with condition (9, we get by choosing larger K that f(*
f() + 17 and
00 Σ
+
Cei
A
K
Σ kesp' k=R
4
L&E < R.
(32)
Let us now show by induction that x² E B(x, R) for all k 2 R. The assertion obviously holds
for l = K due to (32 Take some R 2 K and suppose that x² E U for all h = k..., R. We intend to show that x R+1 E B(T, R) as well. To proceed, fix some h: E R R) and get by
f() < f(**) < f( + < f()) + that
((((xtt) - f())) |f(
(33)
Combining this with un >> 0 and /(x*) - [() > gives us
A Auti 2 ('[(t>t) -[() mn)(((t)))))) un 1(x -ux+1)
(34a)
2 (f((+*) f(z) + un)ctts ||
(34b)
( - f()))) + (()())))) Cits [( |1²
C
(34c)
C
[f((*)|| + (od( cits [f(=t)]
(34d)
where 34a)
follows from the concavity of s. (34b) follows from 30 34c
follows from Assump-
tion 2.3 and (3d) follows from (33.. Taking the square root of both sides in (34d) and employing
the inequality yield
t [f(=*)]| = vtx Vta (V
1 V Cc (As - At + ((((
< 2Cc
(Ax AR+1)
+ Ll ((0((u=))) +
(35)
Using the nonincreasing property of o' due to the concavity of and the choice of o2 E (0. 1) ensures that
(vd(w))) - E4er)
5
48s1)
i=
imk
20

---

Rearranging terms and taking the sum over A = K,... R of inequality 33
gives us
K
R (Ax
R
Σ k=R
t
[[
1
Σ
AR+1)+ +
Σ
Cc1
k=
k=R
(AR
AR) + Σ
C2
Σ
tiei
Cc1
k=
i=
Ce1
AK
k
Σ
K
V==k
The latter estimate together with the triangle inequality and (32 tells us that
R
R+1
R
k+1
x*1
1
k=
R
tk ||g*||
1
+
k= R
R
R
+
+ Σ
4
-
vs(z")]]
km k
b= k
R
R
R
tk
|
+
Σ
tkEt
k=
b=k
R
+
1
A R
+
Σ
x
lap''
Σ
+
x teck
R.
Ca
k=k
k=R
By induction, this means that x² E B(z, R) for all & 2 K. Then a similar device brings us to
K to [f('*)||
8
AR + Σ twp''
Etees imnh
thes
for all K 2 K,
Cc
k=R
k=R
which yields Ex= le [[({t)]||| < co. Therefore,
12++1 - 111
t 110 11 Σ ts Vf(2*))
||| +
+ k=
te 110" vf(**)]
f V V5 ||
<00
k=1
k=1
which justifies the convergence of {a*) and thus completes the proof of the theorem.
21

---

C.3 Proof of Theorem 4.1
Proof. Using (f(=*)-99* S / [[[ || gives us the estimates
||o^1|² = || 1| || -  (V - 112 + +
< 2 [|] || 1 [[]()( + +2(
= -11- (2) |](x>)|] + (
(36)
+
= (VI(a))) - v/(xx)) + V/(xt||
3 || 1100 V/(20) )| | /12
17//4 ( =
3 (1 + 1/) [[
(37)
(V - - |
-
$ 4 [+)()=)] || 111 1100---/(6 -/((=)| - ((+))) /1
2 ((1 () ||]
(38)
|>/((>>)|| - 1|0* - v |[( < ||66 < vf((=)) ||| + 1|9 - vf("")]|
which in turn imply that
(1 (-))| s || < (1 + [V
|| for all k E IN.
(39)
Using condition 12). we find NN N that 2 - 2v Lt 6 for all k 2 N. Select such natural number and use the Lipschitz continuity of fff with constant L to deduce from the descent condition (4). the relationship x² x² - -tag", and the estimates (36) (38) that
f(x S fa*)) + (Vf(a*) ), x -²a) + L
1
-211
2
L1
= f(*h)) - t (V gk)) +
|10/12
2
12)
S s(0²) - 4 (V
+ 1 (V (1 (,²"),,²*)))
||>((>>)|1
2
Lt v²)
S Ss(a²) 4(1 v) |
L + w) ||\
|
= -((²) 4 (2 2v - L1 (1 + +イン ||
8t
f(*²²) 2 |V
I| for all t 2 N.
(40)
It follows from the above that the sequence (f(x²) ) k2N is nonincreasing, and hence the condition
infEEN f(**) xo ensures the convergence of ). This allows us to deduce from 40 that
8
t
||
& Σ (s(xt)) - f(a S s(xk)-
inf f(xt) < co.
(41)
2
KEIN
K=N
K=N
Combining the latter with 39 and x++11 = xat txg gives us
|| a²1l
|10
Σ tx 11o*112 s (1 + x)
Σ tx \f(x*))] Amm
< 00.
(42)
Now we are ready to verify all the assertions of the theorem. Let us start with (i) and show that 0 in an accumulation point of {9*}. Indeed, supposing the contrary gives us > 0 and K E IN such that ||o*| || 2 E for all k 2 K, and therefore
8
4
|10811
>
4
-0,
k=
k
R
K
22

---

which is a contradiction justifying that 0 is an accumulation point of {o*) J. If x is an accumulation point of {a*) }, then by Lemma B.2 and 42 we find an infinite set J C N such that x* 2 I and
gh 2 0. The latter being combined with (39 gives us fz(*) 4 O, which yields the stationary condition V O.
To verity (ii,, let be an accumulation point of {-²} and find an infinite set J IN such that rk 2 E. Combining this with the continuity and the fact that (f(a*)} is convergent, we arrive at the equalities
f(z)
lim f(=²) kEj
lim f(a"),
REIN
which therefore justify assertion (ii).
To proceed with the proof of the next assertion (iii), assume that V is Lipschitz continuous with constant L > 0 and employ Lemma B.1 with ag == f(=*)] ||. 3xx Lt + v),, and * := 0 for all k IN to derive that (x=* to the the estimates
Observe first that condition (a of this lemma satisfied due
ak+1 0k || || [>((*)]|
S || - S L. 11 -1a1|
Ltx ||gh|| S Lt + v) ||(} 3Ras.
The conditions in (b) of the lemma are satisfied since {4x} is bounded, Ex=1 by 12 7kk = 0, and
Σ Baa =
Σ
< co,
where the inequality follows from (41). Thus applying Lemma
B.1 gives us { - 0 as k + co.
To prove (iv), we verify the assumptions of Proposition B.3 for the sequences generated by Algo-
rithm 2 It follows from (40) and x z² 14g that
8t*
f(a S f(1²)
2(1 v) |[
-1g
8
= s(xh))
[[(x*>]] ||a -
(43)
2(1 +
which justify (HI) with a 2[+z] Regarding condition (H2), assume that f(1*+1) = f(a*)
5
and get by (40) that f((x)) ) = O, which implies by ||g vf((x) || S v |V that g = O.
Combining this with kk+1 = z² - leg gives us xh+1 = x², which verifies (H2). Therefore,
Proposition B.3 tells us that {x*} is convergent.
Let verify the final assertion (v) of the theorem. It is nothing to prove if {x*) stops at stationary point after finite number of iterations. Thus we assume that V fx(*)) + for all I. The assumptions in (v) give us & >> 0 and N E IN such that t, 2 for all k 2 N. Let us check that the assumptions
Proposition [B.4 hold for the sequences generated by Algorithm 2 with 7 == 4 and d : -g* for all & E IN. The iterative procedure I**+1 x² t can be rewritten as xk+l = + tgd Using the first condition in (39) and taking into account that ff * 0 for all
k E IN, we get that 9* o 0 for all & E IN. Combining this with -x* =a* 1499 and 4. for all
k 2 N, tells us that ++1 # x* for all & 2 N. It follows from
40 and (39h that
St
f(x < f(**)
g
(44)
2(1 + w)
This estimate together with the second inequality in 39 verifies (1 with
2 2 a -
As all the assumptions are verified, Proposition B.4 gives us the assertions:
If q (0, 1/2), then the sequence {ø*) converges linearly to æ.
23

---

If (1/2,1), then we have the estimate
| - x = =0
The convergence rates of {f(x*)} and {I) (***))) follow now from Proposition B.5 and thus are done.
C.4 Proof of Theorem 4.2
Proof. Let { (x*) be the sequence generated by Algorithm
2a
Defining g² = Vf(x* + Pk: PvV(((*)
and utilizing Pk I y. we obtain
1
[>(>>>||
V5(x* + VVVS(> - v((*")|
< L Pa V//V((())) S [)(>")]
which verifies the inexact condition in Step of Algorithm Therefore, all the convergence properties in Theorem 4.1 for Algorithm 2a be conducted similarly.
The proof for the convergence properties of Algorithm 2b can
C.5 Proof of Corollary 3.5
Proof. Considering Algorithm and defining gh Vs (2+
at we deduce that
ph Taa
de
199 ff(xz))
x + Pk: |d
Lpk-
Therefore, Algorithm Ta is a specialization of Algorithm with & Lpp- Combining this with
11] also gives (8), thereby verifying all the assumptions in Theorem 3.3 convergence properties outlined in Theorem 3.3 hold for Algorithm la
Consequently, all the
D Efficient normalized variants of SAM
In this section, we list several efficient normalized variants of SAM from [Foret al.
2021
Liu
et al. 2022 Li and Giannakis 2023 Li et al. 2024 that are special cases of Algorithm la
As
consequence, all the convergence properties in Theorem 3.3 are satisfied for these methods.
Algorithm 2c (Foret et al. Step O. Choose x² IR", {pk} C [0, oc). and (tt} c o0). For k 1, 2....., do the following: Vile
2021 Sharpness-Aware Minimization (SAM)
Step 1. Set a x² t V (2* + PK (
Algorithm 2d [Lu et al. 2022] Random Sharpness- Awarre Aware Minimization (RSAM)
Step O. Choose x! IR" { [0, oc), and (tx} } C [0, 20). For 1.2...., do the following:
Step 1. Construct random vector A E IR" and set g* Vf(** + A"
Step 2. Set k+1 ² t Vf (at + PK PKTA
Algorithm 2e [Li and Giannakis 2023] Variance suppressed sharpness aware optimization (VaSSO) 0 (0, For 1, do the Step O. Choose x E IR". d' IR". {px} c o0), {tx} [0,
following:
Step 1. Set t = (1 - 0)d -1 +0V[(*)
Step 2. Set h nt 14 VS (rr + Pk |
24

---

Algorithm 2 et al. 2024] Friendly Sharpness-Aware Minimization (F-SAM)
Step 0. Choose x² E IR", IR", m E IR", a E IR, {ou} [0, 00) {4x} C [0, oo), 0 0. For
k 1:
Step 1. Set m² (1 10-00m=+++07
OV (x*h).
Step 2. Set dh = ()(²)) om"
d² Pk ( Step 3. Set s S² V (x
Remark D.1. It is clear that Algorithms 2c 2f are specializations Algorithm la with dt (z
in Algorithm 2c de A
Ag* Algorithm 2d and d constructed inductively for Algorithm 2c
and Algorithm 21
E Numerical experiments on SAM constant and SAM almost constant
In this section, we present numerical results to support our claim in Remark that SAM with an almost constant perturbation radius Pt: = E for close to 0, e.g., results to SAM with constant perturbation radius = C. To do so, we consider the function f()) Ei= log(1 + (Ar: b)?), where A is n X n matrix, and b is vector in R". In the experiment, we construct A and b randomly with n E 2, 20, 50, 100. The methods considered in the experiment are GD with diminishing step size, SAM with diminishing step size and a constant perturbation radius of 0.1, and lastly, SAM with a diminishing step size and variable radius Pk KP for p E 1, 0,1, 0.001. We refer to the case = 0.001 as the "almost constant" case, as numerically similar to C when we consider small number of iterations. The diminishing step size chosen as tj = (0.1/n)/K / at the tt iteration, where n is the dimension of the problem. To make the plots clearer, where 00 is a solution of Ar = b, and 1, is the all-ones vector in IR. All the algorithms are executed for 100n iterations. The results presented in Figure show that SAM with constant perturbation and SAM with an almost constant perturbation have the same behavior regardless of dimension of the problem. This simply because şा almost the same as C. This also tells us that the convergence rate of these two versions of SAM is similar. Since SAM with a constant perturbation radius is always preferable in practice Fooee 2021 Dai et 2023 the practicality of our development.
0.00T, generates similar
C is JFP
choose the initial point _ near the solution, which is x x + (0.1/n²)1.,
this highlights
25

---

harfrmance on 54- Σ up2 +AA
Performncc on Se) Σ loge (Ax-bi))
Perfrmarce on Rx) 2 2. (Ax 58F)
SAN sonserr
S1Mg--0.0 0.003
5A/ conttett
SA1 pm
10/0p 6.1
501 0.002
100
20
132
315
100
25
10
200
130
1/5
200
23
200
12)
395
(a) n =
2
Parformance on fxt- Z.agt1 14x- Aff1
Parformance on Axt Lost (AX dift
Par Parformance ADd - 2 lyt 2073
12
SAM complant
12"
SAMii 0.001
SAMS
10-1
10%1
SMMS
+100p -
10"0
1077
10-2
304
(000) 1200 1963
1735 2000
230
993
7X0
(000)
1252
1998
1752
2000
235
101
130
-000 1259
1590
1792
2000
sanation
(b) 20
Parformance iogg B1))
Parformance
Parformance (5d 2 lugt
12
12'
12P
5 21
12/
1044
12000
Seritoo
evttstion
Senation
(c) 17
Aeformacc Rx( Aogil (Ax A02)
Performance on Rx) f loy'3 (Ax bif)
Perforaanc Rx) Σ Luge1 bif)
StM
10
10-1
10.
10-1
10-3
1077
10
10-2
Eriadon
(d) m
100
Figure SAM with constant perturbation and SAM almost constant perturbation
26

---

Numerical experiments on SAM with SGD without momentum as base optimizer
CIFAR-10, CIFAR-100, and Tiny ImageNet. The training configurations for these datasets follow a similar structure to Section 5 excluding momentum, which we set to zero. The results in Table 5 report test accuracy CIFAR-10 and CIFAR-100. Table T shows the performance of SAM on momentum and without momentum settings. Each experiment is run once, and the highest accuracy for each column is highlighted in bold.
Table 5: Additional Numerical Results CIFAR-10, CIFAR-100 for SAM without momentum
CIFAR-10
CIFAR-
Model
ResNer18
ResNet34
WRN28-10
ResNet18
ResNet34
WRN28-10
Constant
93.64
94.26
93.04
72.07
72.57
71.11
Diminish
88.87
89.79
86.81
65.99
67.04
51.31
Diminish 2
94.56
94.44
93.66
74.24
74.95
74.23
Diminish 3
90.84
91.23
88.70
69.69
70.54
60.64
Table 6: Additional Numerical Results Tiny ImageNet
[e and Yang 2015
for SAM with and
without momentum
Tiny ImageNet
Momentum
Without Momentum
Model
ResNet18
ResNet34
WRN28-10
ResNet18
ResNet34
Constant
48.58
48.34
53.34
54.90
57.36
Diminish
50.36
51.24
58.37
55.82
55.96
Diminish
48.70
49.06
52.74
57.30
60.00
Diminish
51.46
53.98
58.68
57.86
57.82
G Additional Remarks
Remark G.I. Assumption 2.3 is satisfied with constant C for sp(t) -Mt- with M > 0 and q E [0, 1). Indeed, taking any x, with x + y < n, we deduce that (x + y) S x + y and hence
(x' -u++ M(1
9) (x (2+ M(1 q) q) (x
(('(1)))
Remark G.2. Construct an example to demonstrate that the conditions TT do not require that
Pk: converges O. Let I. > 0 be Lipschitz constant of V.. let C' be positive constant such that
C' < 2/L, let P c IN be the set of all perfect squares, let tę & for all k E IN,p > O, and let {ps} be constructed as follows:
[C,
kEP, P.
C
KEP,
which yields t&P: = 1F+I,
F
C
k &
kF
It is clear from the construction of {px) that supk- 2k C> 0, which implies that (pk) does not convergence to O. We also immediately deduce that Σk=1 4 = 00, 44 O. and limsup Pk = C' < L. which verifies the first three conditions in (IT). The last condition in TI follows from the estimates
Σ C KEN
Σ t&Pk +
E S
tkpk
kp+1
KEP
KEP
C
C'
12
kP+1
O0.
KEN
KEIN
27

---

Remark G.3 (on Assumption 9 Supposing that go(t) = Mt - with M > 0,g E (0. 1) and letting
C' (1 q)), we get that (t)(t)) Ct for >> is an mereasing function. If t, and Ei : k with p > O, we have
Σ 4G
30
Iltp dx
1 pkp
1tp
prp
k
imn
which yields the relationships
(w(S
))
p(k 1)p ))
C pakPp
Therefore, arrive the claimed conditions
C
tk
tye
00.
p
Remark G.4. Let us finally compare the results presented in Theorem 4.1 with that in Andrisschhenko and Flammarion 2022). All the convergence properties in Andriushchenko and Flammarion [2022] considered for the class of CI.L functions, which more narrow than the class of Ledeeent
functions examined in Theorem 4.1 i). Under the convexity of the objective function, the convergence
of the sequences of the function values averages of iteration is established in [Andriushchenko and Flammarion 2022, Theorem 11], which does not yield the convergence of either the function values, or the iterates, or the corresponding gradients. In the nonconvex case, we derive the stationarity of accumulation points, the convergence of the function value sequence, and the convergence of the gradient sequence in Theorem 4.1 Under the strong convexity of the objective function, the linear convergence of the sequence of iterate values is established Andriushchenko and Flammarion [2022] Theorem 11]. On the other hand, our Theorem 4.1 derives the convergence rates for the sequence of iterates, sequence of function values, and sequence of gradient under the KL property only, which covers many classes nonconvex functions. Our convergence results address variable stepsizes and bounded radii, which also cover the of constant stepsize and constant radii considered in Andriushchenko and Flammarion [202]].
28

---

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Answer: [Yes]
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Reviews:
[{"review_id": "YZMPYa7XvW", "reviewer": "Reviewer_wdGg", "summary": "### Summary:\n \nThis paper studies various notions of convergence for SAM. The results are summarized in Table 3. Indeed, in Theorem 3.2, they prove the convergence of SAM for smooth convex function for several definitions under diminishing stepsize. Moreover, in Figure 1, they argue that SAM cannot converge with constant stepsize. In Theorem 3.3 they use Inexact Gradient Descent (IGD) in Algorithm 1 to prove convergence results in the non-convex setting, and they generalize it to unnormalized gradient methods like U-SAM in Theorem 4.1. The paper concludes with experiments.", "questions": "### Questions/Comments:\n\nThis is an interesting paper about the theoretical understanding of SAM convergence. However, I think this paper should be majorly revised. The authors introduce a lot of notions and formulae in the first pages and later without accurately explaining them in the paper. I think because this paper could have general audience, it should be understandable to most part of the community. I recommend authors to move a lot of detailed formulae in the main body of the paper and explain necessary definitions/ideas in words through the paper. The writing can be improved for the next version of the paper. \n\nMoreover, the use of IGD seems too general for proving SAM convergence. The authors fail to well motivate using IGDs, as well as how they are enough for the study of SAM. In other words, SAM is way more structured than IGDs and the considered framework seems too general. More explanations are expected for this. \n\n- line 42 -- typo", "limitations": "N/A", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "### Pros:\n\n- study of convergence of SAM is an important problem in optimization\n- well citation to previous related works", "weaknesses": "### Cons:\n\n- the paper is not well-written\n- the method for non-convex convergence (IGD) is not well motivated and it seems too general to be used for the study of SAM", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer wdGg,\n\nWe really appreciate your precious time reviewing our paper. However, we hardly agree with your opinions on the paper.\n\n*1. The paper is not well-written.*\n\nThis comment is going against with opinions of other reviewers, as all of them they view the presentation of our paper in a good way (3/4 points).\n\n*2. IGD is not well motivated and it seems too general to be used for the study of SAM*\n\nFirstly, the motivation for using IGD stems from the construction of SAM, which is already demonstrated by a one-line proof within our results (Appendix C5). Although IGD is more general, to understand the convergence properties of SAM alone, it is sufficient. In addition, we successfully proved the convergence properties of SAM using IGD, which further emphasizes the validity of our approach.\n\nWe would appreciate it if you could reexamine the paper and consider raising your score based on the contributions of our work.\n\nBest regards,\n\nAuthors", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "BqrIU0Koab", "reviewer": "Reviewer_Yksm", "summary": "In this paper, the authors provide a comprehensive understanding of the SAM and its variants. They establish the convergence for SAM under different settings. Unlike the convergence results in the literature, they establish the last-iterate convergence rather than the best-iterate convergence or function value convergence for SAM. By regarding SAM as an inexact gradient descent method (IGD), their analysis can also be applied to other variants.", "questions": "See the Weakness part.", "limitations": "They have mentioned some in their paper.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "* They establish the fundamental convergence properties for SAM and its variants.\n\n* Their analysis is universal and can be applied to other versions of SAM.\n\n* The paper is well-written and easy to follow.", "weaknesses": "* In line 480, you should refer to Proposition 3.9 in Khanh et al.\n\n* Without the KL assumption, would it be possible to derive the convergence rates for SAM based on your analysis?\n\n* The main idea is to regard the SAM as a special case of IGD. Would you please elaborate more on your theoretical contribution compared with IGD?", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer Yksm,\n\nThank you for your precious time reviewing our paper. We answer your question as follows.\n\n*1. In line 480, you should refer to Proposition 3.9 in Khanh et al.*\n\n It is actually Proposition 2.4 in the published version of the paper, as we cited. \n\n*2. Without the KL assumption, would it be possible to derive the convergence rates for SAM based on your analysis?*\n\nWithout the KL assumption, we don't know how to derive the convergence rates for SAM or its variants. This is mainly because of the generality of the KL property, which is satisfied for almost all objective functions in practice. We may consider using other types of regularity properties, such as the metric subregularity of the gradient mapping. However, this property is again equivalent to the KL property in most circumstances.\n\n*3. The main idea is to regard the SAM as a special case of IGD. Would you please elaborate more on your theoretical contribution compared with IGD?*\n\nIn this paper, we develop a version of IGD with absolute error, which can cover variants of SAM. This version of IGD **has not been considered in previous works**, and the convergence analysis of this version is **significantly more complex**. This complexity is because the function value sequence generated by IGD with absolute error may not be decreasing, which makes all the convergence framework for monotone sequence before broken. This is the main contribution of our paper, both in terms of the idea and technical aspects.\n\nBest regards,\n\nAuthors", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "Y5vcmDFkMl", "reviewer": "Reviewer_ZtLt", "summary": "This paper studies the convergence properties of Sharpness-Aware Minimization (SAM) and some of its variants. The authors use the inexact gradient descent to present convergence guarantees for convex and non-convex cases.", "questions": "NA", "limitations": "NA", "rating": 4, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "I think it is interesting to unify different variants of SAM under the inexact gradient framework.", "weaknesses": "- The results do not say much about the quality of the solution in the non-convex case. I think this makes the results less interesting in practice.\n\n- The results seem rather a straightforward application of prior work. I think the contribution of the work is limited.\n\n- The results do not discuss the stochastic setting.\n\nI think based on these points, the results are rather limited and I don't think the paper comes up with a new proof technique.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer ZtLt,\n\nWe really appreciate your precious time reviewing our paper. However, we hardly agree with most of your opinions on the weakness of the paper.\n\n*1. The results seem rather a straightforward application of prior work. I think the contribution of the work is limited.*\n\nIn this paper, we develop a version of IGD with absolute error, which can cover variants of SAM. This version of IGD has **not been considered** in previous works, and the convergence analysis of this version is **significantly more complex**. This complexity is because the function value sequence generated by IGD with absolute error may not be decreasing, which makes all the convergence framework for monotone sequence before broken.This main contribution, in addition to minor ones for unnormalized version of SAM is not straightforward, and is significant.\n\n*2. The results do not say much about the quality of the solution in the non-convex case. I think this makes the results less interesting in practice. The results do not discuss the stochastic setting.*\n\nAs introduced, we only focus on the fundamental convergence properties of SAM and its variants. Therefore, it is reasonable that we do not focus on the quality of the solution, given that the **ideas and techniques** in the convergence analysis are **significantly novel**.\n\n*3. I don't think the paper comes up with a new proof technique.*\n\n**The techniques we used in this paper are new**. This is discussed in the answer 1 above.\n\nWe would appreciate it if you could reexamine the paper and consider raising your score based on the contributions of our work.\n\nBest regards,\n\nAuthors", "claims_and_evidence": "", "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": "", "extra_scores": {}}, {"review_id": "VgjjgkWEhf", "reviewer": "Reviewer_cPcD", "summary": "The authors provide a fundamental convergence analysis of sharpness-aware minimization algorithms,\nincluding the normalized and unnormalized variants.\nThe analysis of normalized variants is based on the theoretical results of inexact gradient descent methods,\nwhile the analysis of unnormalized variants is based on the inexact gradient descent method with relative errors.\nFundamental convergence properties have been established for SAM algorithms in both convex and non-convex cases.", "questions": "- the results reported in Table 4 are much lower than other publications. e.g.,\n  - resnet18, cifar10, 94.10 vs 96.52 (https://arxiv.org/pdf/2110.03141)\n  - resnet18, cifar100, 71.77 vs 80.17 (https://arxiv.org/pdf/2110.03141)", "limitations": "n/a", "rating": 6, "confidence": 3, "soundness": 3, "presentation": 3, "contribution": 3, "strengths": "- SAM is an optimizer widely used in searching for flat minima, thus, studying its convergence is an important problem.\n- The paper establishes theoretical convergence for both normalized and unnormalized SAM algorithms, while most existing convergence results are only for unnormalized variants. As the normalized SAM is more standard than the unnormalized SAM, establishing convergence for normalized SAM is a key contribution.\n- Compared with existing analysis on SAM, many new fundamental properties are established in Sec 3.2 and Sec.4\n- the paper is well-written, easy to follow", "weaknesses": "- I think the analysis of the convex case (Sec 3.1) is meaningless. As for the convex function, usually, we can use some convex programming methods to obtain the global optimal solution. \n- If Theorem 3.3 and Theorem 3.4 are from existing works, some citations for them are necessary (e.g., [Khanh et al., 2023a,b,c, 2024])\n- In Corollary 3.5, the assumption on $\\rho_t$ is not standard. In practice, $\\rho_t$ is a constant, e.g., 0.05\n- The convergence analysis of SAM in this paper cannot provide some theoretical insights on the advantage of SAM over ERM.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Dear Reviewer cPcD,\n\nThank you for your precious time reviewing our paper. We answer your question as follows\n\n*1. I think the analysis of the convex case (Sec 3.1) is meaningless. As for the convex function, usually, we can use some convex programming methods to obtain the global optimal solution.*\n\nWe agree that SAM is not designed for convex functions. However, for the completeness of the analysis, and motivated by the fact that convex functions (even quadratic functions) have been considered in recent studies (Bartlett et al., 2023; Ahn et al., 2023), we presented the analysis for the convex case.\n\n*2. If Theorem 3.3 and Theorem 3.4 are from existing works, some citations for them are necessary (e.g., [Khanh et al., 2023a,b,c, 2024])*\n\nThe proof of Theorem 3.3 is completely new and not deduced from [Khanh et al., 2023a,b,c, 2024]. Could you please check the location of Theorem 3.4 again, as we only have Example 3.4?\n\n*3. In Corollary 3.5, the assumption on rho is not standard. In practice, rho is a constant, e.g., 0.05.*\n\nWe agree that in practice rho is a constant as you mentioned. In our analysis, it is required that when rho is almost constant,  e.g., 0.05/(k^0.00001), with k being the number of iteration. This is, in fact, aligned with the practical use of rho, as  0.05~0.05/(k^0.00001). This phenomenon is also numerically tested in Appendix E, and thus consequently explains why SAM seems to converge to a stationary point in practice.\n\n*4. The results reported in Table 4 are much lower than other publications.*\n\n This is because we use **diminishing stepsize and constant stepsize** to demonstrate our **theory**. In practice, **consine stepsize scheduler** is usually employed to achieve the best performance. We unfortunately could not process the analysis using this type of stepsize and will consider it in the future development.\n\nBest regards,\n\nAuthors", "claims_and_evidence": "", "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": "", "extra_scores": {}}]

Author rebuttal:
Dear Reviewers,

Thank you all for the valuable feedback. We discuss questions and concerns raised in the answers below. If you have any further questions, please do not hesitate to contact us. We are more than happy to answer them!

Best regards,

Authors

Meta-review: Most of the reviewers think the paper is interesting.  There seems to be something new in the proof.  I recommend an acceptance.
Decision: Accept (poster)