Uniform Wrappers: Bridging Concave to Quadratizable Functions in Online Optimization
Authors: Mohammad Pedramfar
Christopher John Quinn
Vaneet Aggarwal
Venue: NeurIPS 2025 poster ()
Keywords: DR-submodular Optimization
Convex Optimization
Primary area: optimization

Abstract: This paper presents novel contributions to the field of online optimization, particularly focusing on the adaptation of algorithms from concave optimization to more challenging classes of functions.
Key contributions include the introduction of uniform wrappers, a class of meta-algorithms that could be used for algorithmic conversions such as converting algorithms for convex optimization into those for quadratizable optimization.
Moreover, we propose a guideline that, given a base algorithm $\mathcal{A}$ for concave optimization and a uniform wrapper $\mathcal{W}$, describes how to convert a proof of the regret bound of $\mathcal{A}$ in the concave setting into a proof of the regret bound of $\mathcal{W}(\mathcal{A})$ for quadratizable setting.
Through this framework, the paper demonstrates improved regret guarantees for various classes of DR-submodular functions under zeroth-order feedback. Furthermore, the paper extends zeroth-order online algorithms to bandit feedback and offline counterparts, achieving notable improvements in regret/sample complexity compared to existing approaches.

Uniform Wrappers: Bridging Concave to Quadratizable Functions in Online Optimization
Mohammad Pedramfar
Christopher J. Quinn Iowa State University
Mila Quebec AI Institute/McGill University mohammad, pedramfar@@ila. quebec
ciguinn@iastate.ddi
Vaneet Aggarwal
Purdue University
vaneet@purdue. edu
Abstract
This paper presents novel contributions to the field of online optimization, par- ticularly focusing the adaptation of algorithms from concave optimization more challenging classes functions. Key contributions include the introduction of uniform wrappers, class of meta-algorithms that could be used for algorith- mic conversions such as converting algorithms for convex optimization into those for guadratizable optimization. Moreover, propose guideline that, given base algorithm. for concave optimization and a uniform wrapper W, describes how to convert proof of the regret bound of AA in the concave setting into proof of the regret bound ((A) for quadratizable setting. Through this frame- work, the paper demonstrates improved regret guarantees for various classes of DR- -R-submodular functions under zeroth-order feedback. Furthermore, the paper extends zeroth-order online algorithms to bandit feedback and offline counter- parts, achieving notable improvements regret/sample complexity compared to existing approaches.
Introduction
The optimization of continuous DR-submodular functions has become increasingly prominent in recent years. This form of optimization represents an important subset of non-convex optimization problems the forefront of machine learning and statistics. These challenges have numerous real- world applications like revenue maximization, mean-fiet inference, and recommendation systems,
among others [Bian al. 2019 Hassani al. 2017 Mitra et al. 2021 Djolonga and Krause 2014
Ito and Fujimaki 2016 Gu et al. 2023 Li et al. 2023].
A natural starting point for DR subuuuoou maximization is to begin with convex optimization algorithm and adapt it to the setting of DR-submodular functions. Online Convex Optimization (OCO) is extensively utilized across various fields due to its numerous practical applications and robust theoretical underpinnings. The tools from the area of online convex optimization have been applied to many online non-concave optimization algorithms, e.g., to converge to stationary points in online non-concave optimization (Yang et al.] 2018), or algorithms with approximation guaran- tees for Dusuubuou optimization [Chen al. 2018 Niazadeh et al. 2020 Zhang al. 2022 Pedramfar et al. 2023).
In this paper, we focus on large class of functions, namely the class of quadratizable functions, first introduced in (Pedramfar and Aggarwal (2224]] Quadratizable functions includes special sub- classes of non-comvex/non-cooccww functions where the offline constrained optimization problem NP-hard to solve but we can find an a-approximation the optimal value in polynomial time. Indeed, it is shown that this class of online upper quadratizable optimization includes up-concave
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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optimization (a generalization of DR- Susbmuuou and concave optimization) in the following cases: (i) monotone y-weaklyy p-strongly -usuuboau functions with curvature e over general convex sets, (ii) monotone 7-weakly DR- DR-subbodular functions over convex sets containing the origin, and (iii) non-monotone DR-submodular optimization over general convex sets.
Even though the tools from OCO have proven effective in more challenging classes, much of the past work along these lines take inspiration from OCO and manually designing new algorithms and analyzing them specific to each problem setting. This raises the following question
When and how can we adapt algorithms from the (simpler) setup of online convex optimization into algorithms for online optimization over more general classes of functions?
In this paper, we try provide partial solutions to this question for adapting OCO algorithms to algorithms for online quadratizable optimization. The notion of quadratizability is built upon a gen- eralization of the defining condition f(x) f(y)= 2 (V (y)))
y) of convex functions. This sim- ilarity with convex functions is a starting point which allows us to define a class of meta-algorithms called "uniform wrappers". Uniform wrappers provide a straightforward way convert OCO al- gorithms into algorithms that can handle quadratizable functions. We also develop a guideline to convert the existing proofs for regret bounds of the base algorithms in the convex setting into regret bounds of the new algorithms over quadratizable functions.
We note that for specific class of algorithms, this question was partly addressed in [Pedramfar and Agggaraal 2024a]. Specifically, as we will discuss in Appendix their result can be formulated as special case of ours, where they assume that the starting algorithm is deterministic first order online algorithm with semi-bandit feedhack. This condition is too restrictive to facilitate adapting many of the ideas in the OCO literature. For example, their result could not be applied to Follow- The-Perturbed-Leade type algorithms, or more generally any algorithm that is not deterministic. Another important setting where their result could not be applied is the setting with zeroth-order feedback. Without using zeroth-order algorithm as the base algorithm, we can obtain zeroth-order results by converting first-order algorithms to zeroth-order ones. However, such conversion comes at cost. For example, in the convex case, taking an optimal first-order algorithm such as Online Gradient Descent and converting it into zeroth-order algorithm will result in sub-optimal OTT3/4) regret Flaxman al. 2005). In contrast, using our framework, starting from a zeroth-order base algorithm, obtain (T2)33 regret for zeroth-orde feedback. In particular, our approach recovers the result of [Wan et al.) 2023 while their approach could not. (See Table
In this paper, we take step further and can handle broader classes of algorithms, including the more challenging setting of zeroth order feedback. Note that, without limiting ourselves to classes of algorithms such as the one discussed in (Pedramfar and Aggarwal, 2024a],. there is not much hope of obtaining a single theorem that captures conversions of any OCO algorithm (and proof) into one for online quadratizable optimization. Instead, we provide guideline that captures the process of o
converting algorithms and proofs for convex optimization into ones for quadratizable optimization.
As an application of our framework, we propose a variant of a bandit convex optimization algorithm that was introduced in (Saha and Tewari 201T] as the base algorithm, namely Zeroth Order Fol- low the Regularized Leader (ZO-FTRL) and demonstrate how it can be converted using uniform wrappers (denoted by W) to obtain algorithms for function classes (i)-(iii) mentioned above. See Tables [ and 2 for details. Note that ZO-FTRI. and W(ZO-FTRL) are zeroth order, but they are not bandit feedback algorithms. We also extend the results to those with bandit feedback, as well as derive sample complexity guarantees for the offline algorithm.
We consider 3 classes of weakly DR-submodular functions, specifically (i) monotone 7-weakly j--troongy DR-submodular functions with curvature over general convex sets, (ii) monotone 7- weakly subsouuoa functions over convex sets containing the origin, and (iii) non-monotone DR- DR-submodular optimization over general convex sets. The main contributions in this work include:
1. We develop a general framework for converting algorithms and their regret guarantees from on- line convex concave optimization to online quadratizable optimization. Conversion of the algo- rithm could be applied to any online optimization algorithm, and the conversion of the proof described using a general guideline.
2

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Table 1: Online up-coneave maximization
36
Feedbank
Referemee
Apps
Fel queries
log, (eeeeeee
al 2023
Fall Information
stoch.
Pedramtar et 3004a
10 10 1/20
2/3 0/3
VA
Pedranta
Semi bundit
stoch.
Kolramiar A/44
374
Pedramtar and Arr KM
2/3
Pedrantar
DEK
Information
socc.
Therru
2/3
stoch.
Pedramtar
Pedramiar el 2044
1 1/4)
475 1 075
3/4
Mototone
det.
Zhang el al. 2041
al
(*)
3/4
4/5
Bandit
stoch.
Teeceem
3/4
Moch.
Pedrant of and Accrrell 27/
Pedramfar
1-07
5/6
4/5
Information
Pelrantar
1
4/3
Lich sl 201
7 7/1-7"
172
gonsal
Semi-bandit
stoch.
Pedramrar and
Pearaniri E dell P324a
+17
1/2
3/4
1/2
Varaml and Agcirell
Full Information
sooh.
[edraamlar or ml M749
1/2
710E 1/40
475 #75
Theveem
2/3
570
Pedramiar and Aggaraal 2024
+
3/4
3/4
1/2))
2/3 0/3
Full Information
stoch.
Zhang dhill 2lel
1/2
VP
Pedranirar and 24 Aarara 20240
1/2
Non Monotone
Semi-bandir
stoch.
Pedramiarr
Pearamrir di 2054
200la
2/3
3/4
ganeal
and Aggirall AlMR
2/3
Full Information
stoch.
Pedramirr
Pedramiar et al 2024a
puriaa
(1 4324
6324
10
1/4))
4/5 0/5
3/4
stoch.
Pedralirar and Aggireml p0zza (*)
Iedramiar et all AMda
R3/4
4/5
5/8
This table compares different static regret results for the online up up-concave maximization. The logarithmic
terms in regret are ignored. Here h := minzek |z] Rows marked with (*) are results in the literature that are special cases of the results stated here and therefore fit within the framework described in this paper. The rows describing results with stochastic feedback that marked with assume that the random query oracle is contained with a cone, as detailed Theorem Note that such oracles cover deterministic oracles as a special case.
To the best our knowledge, in all cases considered, our framework recovers all algorithms in the literature for continuous DR-submodular maximization that are not Frank- Wolfe type. (See Appendix B for more details.) This includes all results marked with (*) in Tables and
In all online optimization settings considered, our framework either beats the SOTA success- fully recovers it (see Table [}. Note that our framework also recovers all known results for non-stationary DR-submodular maximization. (See Remark 11 and Table 3 in (Pedramfar and
Aggarwal, 2024a).)
In all offline optimization settings considered-excep for deterministic first-order feedback and the special case of 7-weakly non-monotone functions with 7 < 1-our framework either beats the SOTA or successfully recovers it. (See Table(2)
When considering either deterministic or fully stochastic query oracles, we obtain superior regret guarantees for 3 settings. Specifically, for online or offline zeroth order feedback, we improve the SOTA for (ii) and for bandit feedback, we improve the SOTA for (iii). (See Tables and 2
We define a new notion of query oracles that are stochastic, but contained within cone. (See Section When considering these types of oracles (together with deterministic and fully stochas- tic query oracles), we improve SOTA for 6 more settings. Specifically, for online zeroth order, offline zeroth order and bandit feedback settings for (i) and (iii). (See Tables I and
To simplify the notation and statements, we define regret for maximization problems and focus concave maximization and DR-submodular maximization.
2 Background and Notation
For set D C Rdd we define its affine hull (D) to be the set ax (1 (1-a)y for all x, D and E R. The relative interior of 'D is defined as relint(D): == (x E D 3r >> 0, (x) all (D) C D). All convex functions are continuous on any point in the relative interior of their domains. this work, will only focus (everywhere) continuous functions. x relint(KC) and f is convex
3

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and is differentiable at x. then we have f(y) f(x) 2 (Vf(x), - x)), for all y E K. More
generally, given lr 2 0, we say a vector o E TRd is a p-subgradient of f at x if f(y) - f(x) >
(o.y - x) + Elly -xl for all y E K. Given a convex set K, function f K + R is p-strongly convex if and only if it has a y-subgradient at all points x K. A function F: : + - is G- Lipschitz continuous if for all x, E DD [[ F(y)ll < G|| yll. A differentiable function F: R$ L-smooth if for all x, E D, [ VVF(y) < L|x-y||. Given a continuous
monotone function/ f K + R. its curvature is defined as the smallest number E [0. 1] such that fly+ f(y) 2
f(x)), , for all x, E K and 2 such that x x y + z E K. We define the curvature of a function class F as the supremum of the curvature of functions in F.
(f-c) (f
We say Vf :7: - Rd is μ ptstrongly 7-weakly up-super-gradien of f if for all x S y in IC, we have ((F((y), y -x- + f - x x1²) < f(y) f(x) S ((V - x) - 51ly-113 Then we say f is p-strongly 7-weakly up-concave if it is continuous and it has a j--strongly 7-weakly up- super-gradient. When 7 1 and the above inequality holds for all x, K. we say f is p-strongly concave. A differentiable function K + I is called continuous DR-submodular if for all x S y. we have [x( 2 V sly). More generally, we say f is 7-weakly continuous DR-submodular if for all x S y, we have V fxx 2 /Vy( It follows that any 7-weakly continuous Dubsubaadular functions is y-weaklly up-concave.
3 Problem setup
Online optimization problems can be formalized as a repeated game between an agent and an adver- sary. The game lasts for I rounds on a convex domain K. where T' and K. are known to both players. In the t-th round, the agent chooses an action x, from an action set IC S RR,, then the adversary chooses loss function fi F and a query oracle for the function fa. Then, for some k2 2 0, and 1 S i < ks, the agent chooses a points yt, and receives the outputs of the query oracle.
To be more precise, an agent consists of a tuple (IA , AAuerry, where Q is probability space that captures all the randomness of A. We assume that, before the first action, the agent samples wa E Q. The next element in the tuple, A Aaction functions such that A that maps the history Ω2d x K x II (KC x O)k to x, E K. where we use O to denote range of the query oracle. The last element in the uple, Aquery, is the query policy. For each S t S T and S i S ki, Aquerry Q x XC* x II (K O)h, (K Oyi-1 is a function that, given previous actions and observations, either selects point y E K. i.e., query, or signals that the query policy at this time-step is terminated. We may drop w as one of the inputs of the above functions when there is no ambiguity. We say the agent query function is trivial k and Yr.1 = x, for all < t < T. In this case, we simplify the notation and use the notation A Auction (A..... Ar) to denote the agent action functions and assume that the domain of
(A ... .Apccioo) is a sequence of
A is Ω- x (AC x O)1-1
A query oracle is function that provides the observation to the agent. Formally, query oracle for function f is a map Q defined on XC such that for each x E X., the Q(x) is a random variable taking value in the observation space O. The query oracle is called a stochastic value oracle stochastic zeroth order oracle o = R and f(x) = [x Similarly, it is called a stochastic up-super- gradient oracle or stochastic first order oracle O Rd and E E([(x)) is a up-super suupr-gradient of f at x. In all cases, if the random variable takes single value with probability one, we refer to it as a deterministic oracle. Note that, given function, there is at most a single deterministic gradient oracle, but there may be many deterministic up-super-gradicn deterministic gradient oracle. We say an oracle is bounded by B if its output is always within the Euclidean ball of radius B centered at the origin. We say the agent takes semi-bandit feedback if the oracle is first-order and the agent query function is trivial. Similarly, it takes bandit feedback if the oracle is zeroth-order and the agent query function is trivial. If the agent query function is non-trivial, then we say the agent requires tull-information feedback.
oracles. We will use V to denote the
An adversary Adv is a set such that each element B E Adv, referred to as realized adversary, a sequence (B),... .... Br) of functions where each B, maps a tuple (xi...., x,) E K to a tuple (Se, 2.) where /1 E F and Q, is a query oracle for ft. We say an adversary Adv is oblivious if for any realization B- = (B,,..., Br), all functions B, are constant, i.e., they are independent of (x1,"". xx). In this case, realized adversary may be simply represented by sequence of func-

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tions .f) frr) E F" and sequence of query oracles work we also consider adversaries that are fully adaptive, ie.. adversaries with no restriction. Given a function class F and i E {0, 1), we use Adv to denote the set of all possible realized adver- saries with deterministic i-th order oracles. If the oracle is instead stochastic and bounded by B, we use Adv (F, B) to denote such an adversary. Finally, we use Adv; (F) and AF,, B) to denote all oblivious realized adversaries with i-th order deterministic and stochastic oracles, respectively.
Qr) for these functions. In this
In order to handle different notions of regret with the same approach, for an agent A, adversary Adv, compact set U C K.",, approximation coefficient 0 < S and I a S b S T, we define regret
as
6
RA. Aay (a))aa bj == sup
E
a
max u=(u EET
E.(w) - Σ f((x))
BEAdv
t=o
where the expectation the definition of the regret is over the randomness of the algorithm and the query oracle. We use the notation R2) [a, == (2 bj when Adv {B) is singleton. We may drop a when it is equal to 1. When a < we often assume that the functions are non-negative. Static adversarial regret or simply adversarial regret corresponds to a = 1, b = T x E 1C). When a = 1, b = T and 21 contains only a single element
and U K.- == {(x, .x) | then it referred to as the dynamic regret [Zinkevich] 2003 Zhang et al. 2018). Adaptive regret, is defined as max SASETT Rg,Adv (KC. |a, bj [Hazan and Seshadhri 2009 We drop a, b and U when the statement is independent of their value or their value is clear from the context.
4 Uniform wrappers
We next introduce class of meta-algorithms that will be a central element of our proposed frame- work for adapting algorithms. At a high level, the meta-algorithms we consider wrap around the base algorithm and translate each action and feedback signal between the base algorithm and the adversary. The qualifier "uniform" highlights that the translations are one-to-one and independent of time.
Definition 1. Given function class F and family of query oracles Q over F, we say uniform wrapper W- (M Wflumttoon Wyuuery is a tuple of maps where W KC K, Wifunction
H for function class H and for any f E F and any query oracle 2, E Q Wiquery (2₂) is a query oracle for Wilunction())) E H. [ Given an oblivious adversary Adv choosing functions in F and query oracles in Q, we define W A(dvv) to be the adversary over H where the selected function and query by the adversary are transformed according to Wifunction and Wipeery all the maps in W are identity.
F +
We say W = Id if
Meta-algorithm 1:
Application of uniform wrapper to the base algorithm - W(A) Input: horizon 7. algorithm A, uniform wrapper W for = 1,2...., , do
In Section C we will discuss several exam- ples of uniform wrappers for up-concave op- timization. We drop the superscripts and use W()), W(J) and W(O,) to denote Waccion (x), Wifunetion()) (f) and (a re-
Play Masciion (x+) where is the action chosen by A
spectively, when there is no ambiguity.
The adversary selects fo and query oracle Q for ft
Meta-algorithm [ details the pseudo-code for ((A) for a uniform wrapper W and an on- line optimization algorithm A. Note that, when W = Id,, the meta-algorithm also reduces to the identity meta-algorithm and we see that ((A)= = A. Note that in the special case where A is an online algorithm with semi-bandit feed- back, Meta-algorithm I reduces to Algorithm end
for starting from while A is not
terminated for this time-step do
yt. be the query chosen by Amwery Return Or (Q) (yr,)) as the output of the query oracle to Amway
end
In general Wastion may map point to a random variable taking values in K.. In this work, we only consider the cases where Wultion is deterministic.
'Recall that, in the case of non-oblivious adversary, each realized adversary B Adv is function mapping the history to the next choice the adversary. In this case, given a history hy, we define W(B) (h.) be (Milinction (fi), Willey (Q.)) where (fe, ②.) Ba (h,)). When the adversary oblivious, the output of any realized adversary B does not depend on the history actions h. and we may define W(Adv) only using W Aifanction and Waueey
5

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in Pedramfar and Aggarwal 2024a]. See Ap-
pendix A for more details.
In this paper, we will design uniform wrappers that could convert algorithms for concave opti- mization into algorithms for more general class of functions that contains many DR-subbodular functions. Specifically, we consider aopee-quudratzablllnnnaaaaa functions which we will dis- cuss in the following section.
5 Linearizable and quadratizable functions classes
We next define an important function class significantly generalizes concavity but preserves enough structure that will enable to obtain improved regret bounds for various problems.
Definition 2 (Peedramfar and Aggarwal 2024a])- Let K. S Ra be convex set, F be function class over K. We say the function class is upper quadratizable there are maps g Fx Rd and h K K. and constants 2 O, 0 < <. 1 and B > 0 such that
1 af(y) f((()) 3 (a(f)x), - x) 1 Ily x1()
(1)
As a special case, when μ 0, say F is upper linearizable. By setting g(f.x) Vf(x), = 1dx and a = 3 = 1, we see that the notion of upper linearizability generalizes concavity and upper quadratizability generalizes strong concavity. It was shown in [Pedramfar and Aggarwal, 2024a that several classes of Dubbubbod (and up-concave) functions are upper quadratizable. (see Lemmas [ and 3 A similar notion of wttr--aaaaaaaaeeleeeeracaa defined for minimization problems such as convex minimization
may be similarly
Definition 3. We say F upper quadratizable with uniform wrapper W W(F) defined and differentiable over K and, for all f E F, we have
of(y)) - f(W(x)) 3 (1VWU)(K),) - x) 2 x112)
-
Note that uniform wrapper is not uniquely determined by h and g in the definition of upper quadra- tizable functions as it also needs to describe transformations of query oracles. The special case with 3 = 1, W = Id reduces to the definition of (strong) concavity. In Appendix C we will construct uniform wrappers for several classes of upper quadratizable functions, specifically, mono- tone j-sttroggly 7-weakly up-concave functions with bounded curvature (FM), monotone y-weakly up-concave functions over convex sets containing the origin (FMO), and non-monotone up-concave functions over general convex sets (FNM).
6 When Is Concave Optimization Enough?
As can be seen in Meta-algorithm] [ we may apply a uniform wrapper W to any online optimization algorithm A. However, even if the original algorithm has sublinear regret over concave functions and F is a function class that upper quadratizable with W, this does not guarantee that the resulting algorithm ((A) has sublinear regret over F. In this section proofs of the regret bound for A over concave functions into proof of a similar regret bound for ((A) over F. That starting point is to notice that game played between ((A) and Adv is mathematically equivalent to one played between and W(Avv). Therefore we need to convert the regret guarantees of A against concave adversary into guarantees for A against W(Avv). See Appendix A for more details.
diseuss how might convert the
The guideline for converting proof for concave optimization into proof' for upper-quadratizzale optimization can be informally summarized in few steps:
(0) Sometimes, if the algorithm A is the result of application of a meta-algorithm to another algo- rithm B, e.g. A = (FTT(B) (the meta-algorithm SFTT converts algorithms that require full- information feedback to ones that work with
feedback; see Appendix M). we may
We say F is lower quadratizable if af(x) f(h(x)) 2 3 (lo()xx) x) x)+ fll -013) x1(²). eralizes the notion of convexity and strong convexity.
This gen-

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need to consider the base algorithm instead. For example, in the example of SFTT, we might want consider (FTT(W(B)) instead of W(SFTT(B)) = W(A).
(1) Rewrite the parts of proof (after possibly adapting the algorithm) of the original regret bound without assuming that the function class in concave, in order to isolate the use on concavity in the proof. In this step, we hope obtain a result that would only require a single use of an inequality
of the type f(y) f(x) I y - x) lly - xII to complete the proof for the concave
case, See Theorems I (as an example of family of zeroth order results), and 11 (as an example of family of first order results) for examples of this step.
(2) Verify that the results of the previous step could be adapted to upper-quadratizable setting. See the proof of Theorems
and 12 for examples of this step.
In the following section, we discuss a version of Follow The Regularized Leader (FTRL) algorithm for concave optimization and adapt it to fit the guidelines discussed above. As another application of the guideline, we refer to Appendix E for discussion of applying this guideline to recover some previous results in the literature, including all the results in Tables and that are marked with (*).
Examples
In this section, we provide examples of applying uniform wrappers and the guideline described earlier. The first example is illustrative and recovers some existing results, while the second example provides new results, as we will discuss in Section g
7.1 Online Gradient Ascent
Here clarify the guideline in Section by applying it to simple base algorithm, namely Online Gradient Ascent (OGA).
Note that this is only an illustrative example and a much more general result is proven in Appendix E
In Algorithm 2 stochastic first order query oracle for ft which returns unbiased estimates of Vi The ef- fect of Meta-algorithm [ is that transforms the action of the base algorithm using Waction and query oracle of the base algorithm using Wqquer, If we apply the first-order first-order uniform wrapper WAN (for non-monotone up- concave functions over general convex sets; see Appendix C.3 for additional details), we get first order algorithm for online optimization of non-monotone DR-submodular function over general convex sets. The resulting algorithm detailed Algorithm
Qs to denote the
Algorithm 2: Online Gradient Ascent (OGA)
Input: horizon T', convex set K. X1 E K.,
step-sizes (ve )=
for 1,2,..., do Play x
The adversary selects fo and query oracle Q for ft
Observe sample of (x.), unbiased estimate of V (x.. X1+1 Px(x + n...)
end
By plugging in the definitions of (WWN Jaction and (WFMMyyuurr as per Appendix C.3 get the Algorithmm4 4 which is Algorithm in Zhang
Algorithm 3: Application of W/NM Online Gradient Ascent (WNN (OGA))
al. 2024 for the case of non-monotone func- tions.
Input: horizon convex set K. x1, K. step-sizes )==
for = 1,2,.... do
The conversion of the regret bound is the fo- cus Section 6, in the form of three-step guideline. Since that is guideline that depends on the structure and the details of the original proof, any explanation of the application of the guideline requires an understanding of the orig- inal proof. For OGA specifically, we may sim- plify the proofs and the guideline in the follow- ing manner:
Play (WANM (x:).
The adversary selects fo and query oracle Q for ft
Observe sample of (WNNM yuuery (2.)(x;)
X++1 + Px(x; 1/,,,)
end
7

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As per Step of the guideline, we first rewrite a part of the proof the OGA, in a way that does not require concavity, but the proof of the regret bound for the concave case can be re- covered using the inequality f'y) - f(x) I (Vf((x), x).
Algorithm 4: Application of WNM Online Gradient Ascent (WNM (OGA))
Input: horizon convex set K. X1, E..xKK step-sizes (17 )
for f = 1,2,...
Play (x+ + x)/2
Theorem For any differentiable fince- tion class F, with stochastic first-order
Sample according to ZNM (as defined in
Appendix C.3)
query oracles bounded by and step-sizes
The adversary selects f. and query oracle 2 for ft
= GVP the algorithm OGA guarantees that maxxeK Et B((V((xx),x -xxi] I GGVT.
Observe os, a sample of ( x) + x),
an unbiased estimate of
V (x: x x)
Xx+1 Px(xi THOO.)
The proof of Theorem D is simple rewriting end
of the classical results. We include a proof in Appendix D for completeness. Note that the boundedness of the stochastic first-order query oracle by G implies that the gradients functions in F are bounded by G, i.e., functions in are G- Lipschitz continuous.
In Step of guideline, we adapt the result of Theorem L the upper-linearizable setting.
Theorem 2. F F is a function class over K. that spper-linearizable with a 1 and B 2 and first-order uniform wrapper W, then with stochastic first-order query oracles bounded by G and step-sizes The fe (WWection
D the algorithm W(OGA) guarantees that maxxeK Σt=1 B(f((x)
GVC
(xx))] I GDVT.
We refer to Appendix D for proof.
7.2 Zeroth Order Follow The Regularized Leader
Follow The Regularized Leader is popular online optimization algorithm. When applied to sequence of vectors {gr ) in IC, FTRL outputs a sequence of points {x.) where
argmin (x), X²-1 xEK
argmin xEX
ge x) + ((x).
Here (xx) is an arbitrary regularizer and parameter. In this paper, we use self- concordant barrier of IC as the regularizer of FTRL. Self-concordant barriers were first pro- posed in the convex optimization literature, with [Abernethy et al., 2008) the first use in bandit feedback setting. We refer Ap- pendix H for an overview of the main ideas present FTRL, including the definition of self-concordant barrier & the Minkowski set K,,,, and E-smoothing of function to obtain f=
Algorithm 5: Zeroth Order Follow The Reg-
ularized Leader ZO-FTRL
Input: Horizon T, smoothing radius 8, learning rate 7, p-self-concordant barrier db
X1 + argmin xEK: ((x)
for = 1,2,.... do
Play X;
The adversary selects f. and reveals zeroth-order query oracle O. for ft -1/2
E, < (v2
Draw ve uniformly from sd-1
+ sample of Q. at x, + 55.V.
Here we propose FTRL variant for zeroth- order feedback, based [Saha and Tewari 2011], which will be a key base algorithm for our framework. See Algorithm 5 for pseudo- code.
O, + 1022 1.21 Lve
X f- argmni E
-7/0,, x) ((x)
end
The following theorems demonstrate how to apply the guideline described in the beginning of Sec- tion KlonBooo the results of (Saha and Tewari 2011]. The first step is to analyze the proof and modify the base algorithm so that we could obtain a result that is valid for non-convex functions and would only require single use of an inequality similar to f(y) f(x) < (Vf(x),y x) to obtain a regret bound for concave case. By small modification in the original algorithm, we get FO-FTRL which

---

differs from the original in that it is no longer a bandit algorithm. While the agent plays x, it queries the oracle at X, + V,,V, * x,. This modification allows us to obtain the following result.
Theorem 3. Let F be an ---Lisschitz M2 M2-smooth function class that is bounded by Mo and let Bo 2 Mo- Also let B E (F, Bo) be realized adversary that returns f1,-.. - let u, arK Ei=1 fe(u) and ú, E argminxeK.... ||u - x|| where y= TT Then, when running Algorithm S against B, we have
T
E [f((u)) fe(xx]] -(a)))))(((()))) - 0((2T) -
E
Fo ( - -5e20 -s12)(xx))
1=1
T
and Σ
E
V bZ (x.). ú. -x,x x1) S - (11 (n8022 2T
+1 + "log0 T)
See Appendix D for the proof. Note that if f is concave, then right hand side of the first inequality is bounded by the left hand side of the second inequality and obtain the regret bound for the concave case. See Appendix for the proof.
use Lemma | to that the
Theorem 4. Let F be an M1 MI-Lipschitz Mz-smooth function class over K. that is upper- upper-linearizabe linearizable with 0 < S 1, 3 2 O and zeroth-order uniform wrapper W. Also assume that Wastion is M Lipschitz and M3- Moosshhth. If Adv is a zeroth order oblivious adversary over such that for for any I E F and any query oracle 2, returned by Adv, W(2r) is stochastic zeroth order query oracle for W() that bounded by By, then
R FTRL FTRL) =0(10 ( 17 T + a.Adv
In particular, by setting T-3 and ô = 1-16 we that WIZO FTRL) = O(T2/33. R,,Adv
We emphasize that Theorems and 7 correspond to steps 1 and 2 in the guideline discussed in Section&rrspecttin Specifically, for step 1, we note that Theorem is NOT concave functions. Moreover, if we assume that the functions are concave, then we may use Theorem and the the inequality
S (V 651 (xx), ax)
to immediately obtain a regret bound. For step 2, we note that Theorem 3 adapts the result of previous step uniform wrappers obtain regret bound for linearizable functtioobleunncionn....
8 Applications
We next discuss some specific online and offline non-convexynon-coocaax optimization problems for which we can use our new framework to derive improved regret and sample complexity bounds respectively by applying uniform wrappers proposed in Section the zeroth order feedback OCO base algorithm ZO-FTRL (Algorithm 5). We note that we can also apply our framework to other base algorithms to recover many existing results the literature. (See Section E for more details).
We start with a definition. For x E I. and C > O. we say a zeroth order query oracle 2, is contained in (x, C.) cone if we have (2)(z) fxx) I C'Iz- for all z E A. In other words, the randomness of the query oracle approaches to zero at least linearly as we approach the point x. We use the notation & (F, Cone(x, C)) denote the oblivious adversary over F with query oracles that are contained within this cone. Note that 2, < AF, Cone(x, C)) is equivalent to WNM( 22) being bounded. See condition (iii) of Definition 5 details. If 2, does not belong to a cone as described above, we can see that the term
($) causes WNM Q,) to blow up. Similarly, the special case when æ = and f(0) = 0, it is also equivalent to (C,) being bounded.
We begin by showing ( a-regret bounds for online optimization problems for the three func- tion classes discussed in Section C Junder zeroth order feedback. See Section for for the proof.
Theorem 5. Let FM FM0 and FNM denote the function classes described in Lemmas 7 and respectively and let a MD and a NM be the values of a. If the function classes are M, Lipschitz and Ma-smooth, then,fo for any C' > 0 and Bo 2 Mo = maxxek. fxx), the following are (
WW (ZO-FTRL)
WMM WMM0 (ZO-FTRL)
R RWO (20-FTRL)
aM Advg ,B))" _Bo)'
NM Adyg (x,)

---

Remark 1. For each function class, the SOTA for noisy zeroth order feedback achieved Õ(T3/4) a-regret bounds while we achieve ((T2/3). For the special case of exact zeroth order feedback, the SOTA is Ö(VT). All the SOTA algorithms mentioned are special cases of our framework.
We next show O(T9/4) a-regret bounds for online optimization problems for the three function classes discussed in Section under bandit feedback. For full information zeroth order algorithms, the query location may differ from the action taken. Here we convert them into bandit algorithms using the meta- veta-algorithm Stochastic Full-in mull-information To Trivial query (SFTT) from [Pedramfar and Aggarwal 2024a]
Cunder
(see Section for details). The proof is in Section N
Theorem 6. Under the assumptions of Theorem
the following are O(T3/A):
R SFTT (zO (TRI)) RMM (Fo. Bo)
SETT(W (ZO-FTRL)) R
SPTT(W (ZO FTRL)) NN g (FF) Con
MO ,Advi o Cono(O,C]))
where SFTT Algorithm 4 in Pedramfar and Aggarwal 2024a with L T
Remark Note that Algorithm 3 in [Wan et al. 2023) is in fact SFTT (20- (ZO-FTRL)) How- ever, our analysis simplifies the proof and generalizes the result to allow for stochastic feedback. Remark 3. For the class FNM of non-monotone up-concave functions over general convex sets, our Ö(T3/4) bound beats the SOTA O(T4/5) bounds for exact and for noisy bandit feedback. For the class FMO of monotone w-eekkly up-concave functions over convex sets containing the origin, our O(T3/4) bound beats the SOTA ((T4/5) bound for noisy bandit feedback and matches the bound for exact bandit feedback. For the third class FM of monotone ju-strongly 7-weakly up- concave functions with bounded curvature, our results match the SOTA. All of the SOTA algorithms mentioned here are special cases of our framework.
Conversions of online algorithms to offline are referred to online-to- to-batch techniques and are well- known in the literature (See Shalev-Shwarre Shwartz|2012], A simple approach is to simply run the online algorithm and if the actions chosen by the algorithm X1, Xy, return x; for 1 t S T with probability 1/T. We use OTB to denote the meta-algorithm that uses this approach to convert online algorithms to offline algorithms.
We next show that using OTB conversion (on top of W(ZO-FTRI.)) we obtain ((1/e²) sample complexity for finding an a-approximate solution in each function class under noisy value oracle model, beating the SOTA Ó(1/d) sample complexity. The proof is Section O
Theorem 7. Under the assumptions of Theorem the following is true.
(i) If the stochastic query oracle is bounded by Bo, then the sample complexity of the offline algorithm OTB(W) (ZO-FTRL)) over F. is Ö(E-)).
(ii) If the stochastic query oracle is contained in the cone Cone(O, C), then the sample com- plexity of the offline algorithm (TB(WA
over FMO is ((-=))
(iii) If the stochastic query oracle is contained in the cone Cone(x, C)2 then the sample com- plexity of the offline algorithm OTB (ZO-FTRL)) over NM is Ō(c
Conclusion
We proposed a class of meta-algorithms that can be used to convert convex optimization algorithms (with various types of feedback) into algorithms for quadratizable optimization. Using this frame- work, we improved regret guarantees for various classes of DR-submodular functions under zeroth- order feedback. Furthermore, we extended zeroth-order online algorithms to bandit feedback and offline counterparts, achieving notable improvements in regret/sample complexity compared to ex- isting approaches.
Acknowledgments
This work was in part supported by IVADO, CIFAR, and the National Science Foundation under grants CCF-2149588 and CCF-2149617.
10

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Table of Contents
Introduction
2
Background and Notation
3 Problem setup
Uniform wrappers
Linearizable and quadratizable functions classes
When Concave Optimization Enough?
Examples
7.1
Online Gradient Ascent
7.2
Zeroth Order Follow The Regularized Leader
8
Applications
9
Conclusion
10
Appendices
13
How uniform wrappers work
15
Additional Related Works
15
Uniform wrappers for up-concave optimization
17
Proof of Theorems 2
19
Recovering previous results in the literature
20
Proof of Theorem
21
Proof of Theorem 12
23
H Follow The Regularized Leader
23
Technical Lemmas
24
J Proof of Theorem 3
26
Proof of Theorem
27
Proof of Theorem
29
M Stochastic Full-information To Trivial query SFTT
29
Proof of T'hoorem 6
30
Proof of Theorem 7
30
14

---

W
W(zs)
Agent
Adversary Adv
((2.)
Qt
Figure 1: Effect of uniform wrapper the interaction between the agent and the adversary.
A How uniform wrappers work
Figure demonstrates the effect of uniform wrapper on the interaction of the agent and the ad- versary. The vertical line depicts the uniform wrapper, acting as translation layer, where any information passing through is changed accordingly. The action of the agent is changed from x; to Waetion (xt) and the query oracle selected by the adversary is changed from Q₂ to Wileery (Qt).
Remark 4. Note that neither Figure A nor Meta-Algorithm 1 require Wylunctionn Thus, we define an abstract uniform wrapper as a tuple of maps W = (Wmcttion Wauery) where Vyaction defined as before. On the other hand, Wiqeery is defined as a function that, for any query oracle 2, E Q, the value of W(Q,) behaves similar to a query oracle, but is not necessarily one. Specifically, if i, 2 and W is an i-order to j-order abstract uniform wrapper, then for any i-th order query oracle Q,, w(2,) is just function from K. to Rd In particular, there may not exist function F K - IR such that D F(x) = ([W(O))(x)) where D denotes the j-th order derivative. The generality of abstract uniform wrappers allow for considering quadratizable function classes where g. as described in Definition 2 might not satisfy g(f. x) = FF(x) for any function F K. + RR. Note that, unlike uniform wrappers, we may longer be able to define W(Adv) for abstract uniform wrappers.
The figure may be interpreted in two equivalent ways. If we think of W as a meta-algorithm that wraps around the agent, then we get W(A), described in Meta-Algorithm [ playing against the adversary Adv. Alternatively, we may think of W as wrapping around the adversary. In this interpretation, the agent A is playing against W(Adv) Since these two interpretations are equivalent, we may use one to better understand the other. Specifically, we may modify the proof of the regret bounds of the game between A and a concave adversary bound the regret of against W(Adv) and therefore use that to obtain regret bounds for W(A) against Adv. The guideline in Section describes how this might be achieved.
B Additional Related Works
DR-submodular maximization Two of the main methods for continuous DR- Susuubmad maxi- mization are Frank- Wolfe rype methods and Boosting based methods. This division is based on how the approximation coefficient appears in the proof. To the best of knowledge, the only other algo- rithms in the literature are those that are designed for specific choices of domain and use techniques such discretization and rely heavily on the detailed structures on the constraint set.
In Frank-Wolf type algorithms, the approximation coefficient appears by specific choices of the Frank-Woll update rules. (See Lemma 8 in [Pedramfar et al. (2024a] The specific choices of the update rules for different settings have been proposed [Bian al.) 2017b Mualemm and Feldman
2023 Pedramfar et al.) 2023, Chen et al. 2023). The momentum technique of [Mokhtari et al.
2020 has been used to convert algorithms designed for deterministic feedback stochastic feedback setting. (Hassani et al. 2020) proposed Frank- Wolfe variant with access to stochastic gradient oracle with known distribution. Frank-Wolds type algorithms been adapted to the online setting using Meta-Frank-Wail -olfe [Chen et al.] 2018 2019) using Blackwell approachablity NNiazdeh et al.) 2023). Later [Zhang et al. 2019] used a Meta-Frank-Wo Wolfe with random permutation technique to obtain full-information results that only require single query per function and also bandit results.
15

---

Table 2: Offline up-coneave maximization
F
Set
Feedhack
Reference
Appx.
Complexity
Mookhtari et al.
2020
0(1/e))
V F
stoch.
Hassani et al 2020
-
0(1/2)
Zhang 2022
6
0(1/22)
Pedramlar and Aggarwal
2024a
e
0(1/2)
0 K
Pedramfar ct al.
2024b
1 - 6
0
det.
Pedramlar Aggarwal,
20)24a
0(142)
Monotone
F
stoch. f
Theorem
1-
0(1/e))
Pedramfar al.
2024b
1-
0(1/e))
stoch.
Peddramaar and Aggarwal
2024a
(*)
1-6
0(1/e)
Hassam et al 2017
(*)
7/11+7)
0(1/2)
VF
stoch.
PPddammar et al.
2024b
+/1++9
Ó(1e2)
general
Pedramlar and Aggarwal
2024a
(4)
0(122)
Pedramffr al. 2023
2/11+79)
O(1/2))
det.
Pedramiarr and Aggarwal, 2024a
+/(1+
01122)
F
Pedramfar et al.
2023
+/11+79
0(1/25)
stoch.
Pedramlar and Aggarwal,
20124a
7/(1+
0(1/e)
Theorem
0(1e2)
Pedramfar et al.
2024b
X(1-A
0(1/e)
Non- Monotone
F
stoch.
Zhang ot
2024
(*)
(1
h)/4
0(1e²)
general
Pedramlar and Aggarwal
2024a
(*)
(1
h)/4
0(122)
det.
Pedramfar et al.
2024b
-11
2
0(1/e)
F
Pedramlar Aggarwal
2024a
(1
h)/4
0(1/e)
stoch.
Theorem
(1
h)/4
0(1/²)
Podramfar al.
2024b
O(1/e)
stoch.
-1
2 2
Pedramlar and Aggarwal,
2024a
(1
h)/4
0(1/e)
This table compares the different results for the number of oracle calls (complexity) within the constraint for up-concave maximization. We refer to [Pedramfar et al.) [2024b] for a more comprehensive table that includes results for deterministic first order feedback. Here h min [|z]]>> and r 7+1/7. Rows marked with (+) are results in the literature that fit within the framework described in this paper. The rows describing results with stochastic feedback that marked with assume that the random query oracle contained with a cone, as detailed Theorem 5
This was extended another settings by [Zhang al., 2023 and generalized to many different settings with improved regret bounds by [Peraatarr et al. 2024a
Some techniques construct an alternative function such that maximization of this function results in approximate maximization of the original function. Given this definition, we may consider the result of (Hassani al. 2017 Chen al. 2018 Fazel and Sadeghi 2023] the first boosting based results. However, in the case of monotone DR-submodular functions over general convex sets, the alternative function is identical the original function. The term boosting in this context was first used in [Zhang et al.] 2022), based on ideas presented in [Filmus and Ward 2012 MMtra al. [202T) for monotone functions over convex sets containing the origin. This idea has been used later in |Wan et al. 2023 |iao et al. 2023 in bandit and projection-free full-information settings. Finally, in [Zhang et al. [024 boosting based method was introduced for non-monotone functions over general convex sets.
Up-concave maximization Not all continuous Dusuuuuou functions are concave and not all concave functions are continuous DR-submodular. [Mitra et al.) [001]] considers functions that are sum of a concave and a continuous DR-submodular function. well-known that continuous DR-submodular functions are concave along positive directions [Calinescu et al.) 2011 Bian et al.) 2017b). Based on this idea, [Wilder] 2018) defined an up-coneave function as function that concave along positive directions. Up-concave maximization has been considered in the offline
16

---

setting before, e.g. in [Lee et al.) 2023 and [Pedramfar and Aggarwal 2024a]. In this work, we focus on up-coneave maximization which is generalization DR-submodular maximization.
C Uniform wrappers for up-concave optimization
In this section, we study three classes of up-concave functions and show that they are upper- quadratizable with appropriate uniform wrappers.
Remark 5. As we will see in the following, all of these uniform wrappers fit into the following pattern. There is random variable Z taking values in [0, 1] and parametrized family of functions he : K + K for z E [0, 1], such that, for some a E (0, 1] and B, f 2 O, we have
x) af(y) - f(h (x)( S ((
H f f1xxx(l)) 2
xI()).
where F(x) = 2f ((a0))-((hh(())))) [(ho(x))]. If such he and Z exist, we define Wuccion = h1, and Wfunction (f) = F. We also need to define Wapeey (Qy) such that it converts a query oracle for f into a query oracle for F using the following equality.
VF(x) B, (f(h (x) ()(ho(x))Jho [x)
Note that the identity wrapper, where F f. can be recovered by choosing (x) = * x and Z 81, the Dirac delta distribution with mass at 1.
C.1 Monotone p-strongly 7-weakly up-concave functions with bounded curvature (FM)
For differentiable DR- -R-suumodular functions, the following lemma is proved for the case Y in [Fazel and Sadeghi 2023] and for the case μ = 0 in [Hassami et al. 2017). The general form we use here is proved in Lemma in [Pedramfar and Aggarwal 2024aa.
Lemma 1. Let f 1] I. be non-negative monotone p-strongly 7-weakly up-concave function with curvature bounded by c. Then, for all x,y E [0, 1] we have
1+ cg 2
2 [(y)
" [(x) S I+ 1 (1VI(x), - x) 5 fly x113)
?
where f an up-super-gradient for f.
Lemma D together with Definition of uniform wrappers, immediately imply the following.
Theorem 8. Let FM be the class functions over K. where every f E FM may be extended to a non- negative differentiable monotone justtrongyy 7-weakly up-concave function with curvature bounded by defined over [0, 1]d Then FM is upper-quadratee euarraazaaa with uniform wrapper WM = Idd
If A is one of the algorithm described in Theorem 12 using the above theorem recovers Theorem 2 in [Pedramfar and Aggarwal| [2024a] which itself generalization of Theorem 2 in [Chen al.,
2018 and Theorem 3 in Fazel and Sadeghi 2023
C.2 Monotone 7-weakly up-concave functions over convex sets containing the origin (FMU)
For differentiable monotone OR--ubmmodula functions, the following lemma is proved in (Zhang et al. 2022). The general form here is proved in Lemma in [Pedramfar and Aggarwal 2024a
Lemma 2. Let f [0, 1]d + Th be a non-negative monotone 7-weakly up-concave differentiable
function and let F: [0, 1]d R. be the function defined by F(x) -1 ze 1)
(1-
Then F is differentiable and (1 - ->>) - f(x) < 1- (V FVF(x), x).
Let the random variable MO E [0, be defined by the law V2 E [0, 1]. P(ZMO < 2) =
So >=21==" du. Then we have Er-zmo [2 (f(z x) [()) = F(x). Moreover, for 2 1, if f is times differentiable then also have Ea / f(2*x)] F'F(x).
17

---

Definition 4. Let xC S [0, 1] be convex set containing the origin and, for any i 2 0, let F FMO be the class of functions over K. that are max 1} times differentiable and where every f E FMO may be extended to non-negative monotone w-aekkl up-coneave function defined over [0, 1]4 We also as- sume that f(0) = 0 for all E F We define WMMo := ((WMM))action ( (Murr))
to be the uniform wrapper with (i) (W)20) yationn == Idg: (ii) for any J'EF. FM
( (f) := x + ZM (f(= + x) [(0))] K. -> IR; and
(iii) for any f E FN and any i-th order query oracle 2, for f. we have (WMO) youry (e)))) ==
2' 12(2 * x), where z is sampled according to ( - 2).
Remark 6, This wrapper fits into the pattern described in Remark 5 by defining h hx(x) == 2 x and Z = ZMO.
Theorem 9. For any i 2 0, the function class F defined above is upper-linearizable with the uniform wrapper WM
Remark 7. The meta-algorithm A -> OMBQ(A, BQMO, Id), described in Pedramfar and Aggar- wal. 2024a] is identical to A + (M (A). In other words, the results of Theorem 3 in [Pedramfar and Aggarwal 2024a] are about the first order uniform wrapper WMO Here consider a more general case where we are not necessarily limited to first order.
C.3 Non-monotone up-concave functions over general convex sets (FNM)
For differentiable monotone DR-submodular functions, the following lemma is proved in [Zhang et al. 2024]. The general form we use is proven in Lemma 3 in [Pedramfar and Aggarwal, 2024a].
Lemma 3. Let f : [0, 1j - R be a non-negative continuous up-coneave differentiable function and let x E K. Define F [o, 11d - R as the function F(x) == =Jo 3=(1- gya (f(2 (x x) +x)
2
(z) then F is differentiable and we have
1 ||2| f(y) - s(y)
5 (0x0)
x x
3 V x).
Let the random variable NM E [0, be defined by the law Vz E [0. 1]. TP S 2) So 31- du. Then we have EenzNN((()) x (5 (x x) +x) 13)-f(x f(x)) F(x). Moreover, if 2 1 and]] is i times differentiable, then Ez-zaM ((2) vif(; *(x (x-x)
x) + x)] F(F)))
Definition 5. Let K. S 10, be convex set and, for any i 2 0, let F.N be the class of fune- tions over IC where every f E FNM may be extended to non-negative up-concave function defined over [0, 1]a We also assume that FNM is max 1} times differentiable for all 2 0 and, for some known constant e > O and all f E F f(x) = For i 2 O, we define be the uniform wrapper with
WNM J (WNM ytunction (W
() (WNNAaaaction = x+x K K: (ii) for any f E FNM
(WNM yuuection (f) == x x BZ-gNM (G) -1 * ( (x - x) +x) ((x)) : K - IR; and
(iii) for any f E FNM and any i-th order query oracle 2 for f.
(i) i-1 2, (i + (x-xx x) + x)
if 1
(W,N (2)((x) =
(5)
Q1 ( * (x- x) + +x) -c) i
where: = sampled according to PM <²).
Remark 8. This wrapper fits into the pattern described in Remark 5 by defining h=(x) == ] (x x) + x and Z = 2NM
Theorem 10. For any i 2 0, the function class FMM defined above is upper-linearizable with the uniform wrapper WNN.
Remark 9, The meta-algorithm A + OMBQ(4, BQN, X+* ), described in Pedramfar and
Aggarwal 2024a). is identical to A + WNN (A). In other words, the results of Theorem 4 in [Pe- dramfar and Aggarwal 2024a] are about the first order uniform wrapper WNN, Here we consider a more general case where we are not necessarily limited to first order.
18

---

D Proof of Theorems aand 2
The proof of Theorem ii simple rewriting of the classical results. We start with lemma which contains the main idea commonly used in the proof of the regret bound of Online Gradient Descent.
Lemma 4. Let X, X, E KC let G >> 0 and lei (g. (g:){= be sequence of vectors such that l|g: || S G. Define the sequence (xx)?=1 such that X++1 < Px(xx 1/.B:),, where D GVT
Then we have
T
Σ
g. x) & GDVT.
3
2
The following proof is copied from the proof of Theorem 3.1 in (Hazan al. 2016
Proof. Using Pythagorean theorem, we have
|x++1 x|| || n)ge)) xx12 S ||x: 1/8rr x12
Hence,
|2e11 - x|| < ||x, - x|| 11.1B: 112 21/ (gr, X x).
27: X1 xx) v 1/t (1xx, x11² - ||x+11 - x1(²) + G
Summing this from t = 1 to T, and setting =
D we see that
Gvi
T
T
2
Th gr. X - -x)
Xx - x12 - |xx+1 - -x((²) + +mc
[=1
t=1
T
1
1
_11
|X]
x||
+
| - x|2
+
G²
n
1
11t
T/-1
1=2
1=2
T
< <
1
+
1
1
+
G2
Σ
111
Th-1
1=2
1=2
D2
+
G22
1/8
t=2
< 3DGVT,
where the second inequality follows from the facts that diam(K)) D and mh is decreasing and therefore 1 is always positive; and the last inequality follows from the facts that n) GVD
D
and ΣT q1 71 1
S 2VT. VT
Using this lemma, we can easily prove Theorem I
Proof. By applying Lemma 4 to gr, (to account for the fact, unlike OGD, that we update
according to X++1 <- Px(x, + n),,)) we see that
Σ
or,, - x1) 5 2
GDVT.
1=1
If we take the expectation, we see that
E (V)((x) x;)l
EE [x xi)
X[E x1|
T
El x - x 3
GDDV
19

---

Next we prove Theorem 2
Proof. At first glance, may seem that moving from Theorem T to Theorem E might be non-trivial. However, as elaborated in Section A whether we apply W(OGA) to an adversary Adv that selects functions from F apply OGA to the adversary W(Adv), the sequence x, remains the same. Thus, by using Theorem we immediately see that
T
Σ (VW(())()). xi) XEK.
GDVT. CDVT.
max
Another way to arrive at this inequality is to follow the proof of Theorem and adapt is here. Specifically, for all x K., we have
T
B((VW())(x)) xx)
E [ x1)
[[ x1||X111
&
- x 3
where the last inequality follows from Lemma Now we may use the definition of uniform wrap- pers to complete the proof of Theorem
E Recovering previous results in the literature
Algorithm 3 in [Wan et al. 2023) is in fact SFTT and therefore their result fits within our framework. The way the remaining results the tables that are marked with (*) is discussed in the following.
As mentioned in Remark 2
(20 (ZO-FTRL))
We demonstrate how to apply the guideline described in the beginning of Section to Theorem in [Pedramfar and Aggarwal 2024b). This allows us to obtain a generalized version of Theorems in [Pedramfar and Aggarwal 2024a] As we will diseuss below, this will allow to recover all the remaining results in Tables I and 2 that are marked with (*) and all the results of (Pedramfar and Aggarwal 2024a). Note that the results of [Pedramfar and Aggarwal 2024a] in non-stationary setting are not discussed in this paper, but they are also recovered.
definitions. Given function class F, class of functions g(y) = (g(J,)), x) l - : IR, for all f E F and E K. This the class of quadratic (or linear, when μ = 0) functions that form the upper bound Equation [ the notation Q [,]],,]
We start with
the notation F F,.g denote the
Similarly, for any B1 > O, we
denote the class of functions g (o.y - x) / lly x||2 IC -> R, for all x E K and E BB, (0). In the following theorems, we will obtain results that allow us to reduce the problem of online optimization over to the problem of online optimization over the quadratic (or linear) function class F F..g.
Theorem 11. Let A be algorithm for online optimization with semi-bandit feedback. Also let be differentiable function class over K. and μ. 2 0.Then the following true.
If query oracles Adv are deterministic gradient oracles, then we have
(VVL,))))) u. - xx) xx) - -enixxxx)) xiI)
sup
E max
Σ tm
< R.a Ad
BEAdy
uELI
On the other hand, Ff is M Lipischhitz and query oracles in Adv are stochastic gradient oracles that are bounded by B1 Z M2, then we have
sup max
Σ
11 (x)())) - -xx) ~ "luu xxll
$ R- (Q,
BEAdy uell
20

---

See Appendix F for proof. Note that if fr are ju-strongly concave, then this result reduces to Theo- rem 2 in (Pedramfar and Aggarwal 2024b]. Next, we follow step (2) in the guideline obtain the following result.
Theorem 12. Let F be function class over that is upper- roper-quadratizable with Z 0.0<a51 and B 2 0 and first-order uniform wrapper W.
IfW(V) = V, i.e., it maps deterministic gradient oracles into deterministic gradient ora- cles, then have R ((A)
a.AdvV( S SBRA 1,Adv' (,Adv,(F...)) (Fn.v)
If. for any f E F and any query oracle 2 bounded by B1, W(2y) is a stochastic query oracle for W(J) that is bounded by B', then have R (())
a.Adv, Bi) S BRA (A.vV(
See Appendix G for proof. In this theorem, by using the uniform wrappers described in Section C in the special case of 1, we recover Theorems and in [Pedramfar and Aggarwal 2024a] (See Remarks 7 and 9 In other words, we recover all meta-algorithms in [Pedramfar and Aggar-
wal 2024a] that are used to convert concave optimization algorithms into up-concave optimization algorithms.
Remark 10. Online Gradient Ascent requires semi-bandit feedback. Thus, Theorems and special cases of Theorems 11 and 12 respectively.
Remark I.. By applying these uniform wrappers to base algorithms SO-OGA (Garber and Kretzu 2 or IA ([Zaang et al.) 2018)). ). we recover all the results of [Pedramfar and Aggarwal 22024a] In particular, we also recover the results for non non-stationary regret described in Table 3 in PPdrammar and Aggarwal 2024a].
F Proof of Theorem
Proof.
Deterministic oracle:
For any realization = (BI,...,,B, E Adv S (F) we define (((x,, "(,x, to be the tuple (g,, V) where
xt)
(((x1, x1) == 0: == + (V)(())) x1)
2 11y - xel1?
and B' = (Bi. ... Br), Note that each Bi is deterministic function of x1 and therefore B'' Adv (F,,,) v). Since the algorithm uses semi-bande feedback, the sequence of random vectors (x), xX) chosen by A is identical between the game with and B'. Hence
H [u xx(?²)
sup E
max DElf
(xxx)) ur x1)
BE
sup E BEAdv
Σ (ue)
- Ea(x)))
ueld
t=a
-RA (F
sup B'EAdv, (F..v)(FF.v))
Stochastic oracle:
Let Ω of x ... o2 capture all sources of randomness in the query oracles of Advī(F, BI) i.e. for any choice of 0 E Ω2 the query oracle is deterministic. Hence for any 0 Ω and realized adversary B E Adv C Advi (I), we may consider By as an object similar to an adversary with a deterministic oracle. However, note that B, does not satisfy the unbiasedness condition of the oracle, i.e., the returned value of the oracle is not necessarily the gradient of the function at that point. Recall that B, maps tuple (xi,... xx) to a tuple of Se and a stochastic query oracle for fe. We will use Ege to denote the expectation with respect to the randomness of query oracle and
21

---

Enol] == xx) to denote the expectation conditioned on the action of the agent and the adversary. Similarly, let En denote the expectation with respect to the randomness of the agent. Let o, be the random variable denoting the output of Q at time-step t and let
O2 = Elo, ft. x
a [o.] (x((
Similar to the deterministic case, for any realization B (B Br) Adv and any 0 Ωº we define (X.."... X²) to be the pair (gt, V) where
qt y (or, y x1) マ y xrl1?.
We also define By == 1B,,,'''.. )r Note that specific choice of necessary to make sure that the function returned by B, is a deterministic function X1. " X2 and not a random variable and therefore B3 belongs Adv (F.,..)).
Since the algorithm uses (semi-)bandi, feedback, given a specific value of 0, the sequence random vectors (x1,- .x)) chosen by A is identical between the game with Bg and Bg- Therefore, for any u E U, we have
b
(Vi((x)): u, x.) -11, 5111-xx11?) )
E
E
Se, Xe] u1 xx)
11 IIlu, xxl13)
E
(0 Ui - -xx) b Iluu xil1? 2 fer Xxc E
E [u. (x(x)r) fe, f(ix2i))
E
t=a
E
(gc(ur) di (xx))
Hence we have
max E uell
Σ
(V)e(x)), -x.) - ur xill)
max E usu
Σ (dc(ux) -x (x()))
Σ (gc( qe (x())))
< E
max
u=(u.... urJEu
R (2)
where the inequality follows from Jensen's inequality. Therefore
b
11 (V us - x.) 1 I11 - x:ll
sup max E BEAdy uell
sup
Rd
BERd
<
sup
R4
FF
RA
(F,
22

---

G Proof of Theorem 12
Proof.
(i):
We have
R ((4A)
(FF BEAdv,(F)
sup E
u= .. uuyJe Jeu
max
(afelu.)- -
t=a
b
Σ 3 (VW(F))(X))) ur -xx) -x)) - x t=
sup BEAdv u= Jr)
E
1, (H,.v)"
(ii):
Since Adv is oblivious, the sequence of functions (f1, fr) is not random and we have
R
(F,d))(F.
((A)
sup E BFAdv
max
Σ (af((u)) f((W(x))))
u=(u:
uy EEU
t=
b
sup BeAdv, um( ur)
E
Σ (afi(u.)) -
(cf((w))s((((x))))) S((((x))))
tmo
Σ 8 (VW(F:)(x:)) -x.) - -1 xx1)
S sup
max
E
BEAdv, (F, u=(u u= Jel
t=
BR L
H Follow The Regularized Leader
We start by defining the notion of self-concordant barrier.
Definition 6 (Hazan et al. 2016). ). Let K. IRd be convex set with non empty interior int R u-self-concoodamt barrier of K. if:
We call function Φ int
& three-times continuously differentiable, convex, and tends infinity along any quence of points approaching the boundary of K.;
(ii) For every h E Rdd and x E int((C), we have:
(') h, hj] 2 h))3/2,
[h]x S 1 <V hh)
where the third-order differential is defined as V* h. h] 62h + =
St, 0t. 4 thh
Next we define the notion of local norm and dual norm with respect to self-concordant barrier. Definition 7. For every int (K), the Hessian of the self-concordant barrier induces local norm, denoted as I|, and dual norm, denoted ||. x,,,,,, where for any Rd
|vllo = V (VV2
- (V
important result for FTRL is the following theorem which was proved [Abernethy et al., 2008). It shows that if we set the regularizer to be self-concordant barrier of K. and the algorithm can access the unbiased estimator of gr, then the regret of the generated solution sequence {xt)]= can be bounded in terms of the local norm of the estimator.
23

---

Theorem 13 ([Abernethy et al. 2008)), Let K S R be a convex set, ((x) be self-coneordant barrier on K. (ge)i= be a sequence of random vectors in Rd. Then running FTRL (described
17" in Rd with (xx) as the regularizer will produce a in Equation 3) on a vector sequence {gil== sequence of point {xx)rm in KC where
X + p(y) ((x1)
Σ g x1)
S
8: ll
"
for any y E K.
The ellipsoid gradient estimator was proposed in [Abernethy et al. 2008 where the authors use it along with Theorem 13 to design an Ö(vT) regret algorithm for bandit linear optimization. For continuous function but possibly non-smooth f R IR and invertible matrix E gddd define the E-smoothed version of f.
Definition 8. For function f(x) Rd RR and invertible matrix E E Rdxd we call f=((x E-smoothed version of fx), where (x) EvaRs [f(x + [v)] Here Bd means that is sampled from the unit ball Bd uniformly random.
There is surprising fact that there is unbiased estimator of V (x) for any x, and the estimator uses only a single query to the value oracle of) f.
Lemma ([Abernethy et al. 2008 ). Let Σ E Raxd be an invertible matrix, f(x)) : TRd R be an arbitrary function. Then (x) dlEv-ga-1 ( Here v Sd-1 means that
Ev)y-
sampled from the (d 1) dimdeension unit sphere $d-1 uniformly at random.
If is a linear function, (x) fx). so Lemma provides one-sample unbiased estimator of the gradient of the linear function. The ellipsoid gradient estimator is usually used along with FTRL. with a self-concordant regularizer & of IC. When the invertible matrix is set to be (V2((x)) (x)) and x E int(K), the sampled action Vv is located in the surface of so-called Dikin ellipsoid centered at x, i.e. {x' ||x' x||0,x & 1}. The fact that Dikin ellipsoid entirely contained in IC allows us to define f at x.
We finish this section with quick overview of the concept of the Minkowski function, the Minkowski set and some of their useful properties.
Definition 9. Let K. be compact convex set, the Minkowski function xx K. R parameterized by polc x E int (nC) is defined as T == inf 2 0 + - (y x) E KC). Given 8 e R+ and X1 E inC( we define the Minkowski
K-y,x = Ex Txi (x) S (1+
Lemma ([Abernethy et al. 2008 ). Let K. be compact convex set, in) with diameter D, u, E K. and u, == argmin u. be the projection of onto the Minkowski set K.,, then
I. ú. S yD
The following lemma provides upper bound of the difference between the function value of self-concordant barrier at two different points.
Lemma (Nesterov and Nemirovskij 1994]). Let be a v-self-concoorlant barrier over a compact convex set K, then for all x, y E int(K)):
( - 4(x) & vlog
1 TT()))
I Technical Lemmas
This section provides some technical lemmas that will be used in the proofs later.
Lemma 8. Let K. be compact set and let f : K. - Rd be M2-smooth function. Then f may be extended to Ma-smooth function f pd - IR
24

---

Proof. The function VF is an Mæ-Lipschitz function defined on K. Therefore, according to Kirszbraun theorem Kirszbraun 1934] may be extended function g R R that is M2-
Lipschitz. Now the result follows directly from Whitney's s extension theorem [Whitney] 1934 17
Parts (iiiiii) of the following lemma are well-known in the literature. (See Lemma A.5 in Wan et al. 2023 for a proof). Here we provide a proof for part (iv).
Lemma 9. Following properties hold for E-smoothed version of a function [xx for an invertible
matrix Σ.
(i) If f(x) is monotone function, then so is f (x).
(ii) If f(x) is Mi-Lisschizz then so is (x).
(ii) If fxx is Mz-smooth, then so fx)
(iv) If f is upper-quudraaizable with uniform wrapper W and a, and μs, then have
())))))))))) sB ((TWL))))))))) x)
1 f x11²)
Proof. We have
- (f - (x)
Eyare [af(y + f(W)-
Bywad 8 ((VWL)6X+ Ev),y x) 1 ly x1² x
2
(Ev-Ea (N(ff(*+Ev))) --x- -x)
/ 2 ly x11²
((VENz) [W())(+ Ev) .y - x) 11 ll
1 ly x
((TWWE))()))) - x1(²
Lemma 10. If K. + R is Mj Lipschitz and My-smooth and K. - KC is Mi -MpLiicchitz and M2- Mi-smooth, then f g is M" Lipschitz and Mí smooth where M" = Mj M, and MI
M2 + MzM(?.
Proof. We have
|D[ o (x)]l = [(D(((()))))())))))))))))))))))) -((o(x))))((x]111)) S M2M
and therefore for all x,y E K. we have
|| e g)(x) - D(f o g) = [Df(((())) Dg(x) Df(gly)) Dg(y)l
< [D](((x)) Dg(x) DS(((x)) Do(y)l
+ +IDf(g(x)) Dg(y) - -Df(g(y)) -Dgl))
=[DI(g()))]I(Dg()}> - D9(x))l
+|(((((x) -D(G()))))DD(()I
S M + (.Milll(x)-g(1))
-||l-
25

---

J Proof of Theorem 3
Proof. We have
T
E
[f((u.) - [x(x))]
T
T
1P
(ū.)) -5e 1xx)
R
Se (u.) - -5000 (ū.)
(4)
(4)
T
r
+
f((u,) ft (u.)
+
E
(xx) - f((x,)
(B)
(C)
Note that, for the terms above to be well-defined, we need to be able to define f over K. which requires computing fi over a set that is slightly larger than K.. Using Lemma g functions fi are well-defined and M2-smooth over Rd
we assume that all
Bounding (A): Since fx(x) is M₁ -Liischcii continuous, fe is also M Mi-Lischhitz continuous by Lemma Since ||û. u. I 7: by Lemma
E EE (u.) 50 (ū.)
E
(e,)( f f(,)(
t=1
(5)
M MIyDT.
Bounding (B): Since f((x) is M2-smooth, by Lemma 9 1e is M2-smooth. Thus,
fe(u.) fe (u.) = EywBa [f((u.) - Se(u, + +5,v]]
(VV.((u.) 02ix) + M2
< E8 EBd
|لااا v1
2
M2
Ev
[-(Vs((u.), gE.r + EVERy
|o
112
2
M2
Ey-aaa
2
M2o²10 D2²
2
Note that the last inequality, we used the fact that the Dikin ellipsoid centered at x; is contained in KC which implies that x, + Eev E IC and therefore |اx.|||
s D. It follows that,
22
V E [f((a)))
fo E. f25)((a)) s
(6)
2
Bounding (C): Similarly,
A (xx) f(())) Eynodd ff(xx+0211) - fe(xx)]
- -VVf((x)) + M2
M202
< Evnay
||
2
2
Therefore,
T
E (xx) f((xx.) < M202122T
2
26

---

Putting 7 in 4
we see that
T' 1
E [s(w.)) - [e(x)]]
E
St T (ū.) (
- 11 (xx)
+ +aMiyDT+ + M20 + 2
M282
D2T
which completes the proof of the first claim.
To prove the second claim, we first use Lemma 5 with E = 5Eg, to see that E [o, xx
=
V fft (xx)) On the other hand, since Q, is bounded by Bo, we have
d V
de 52 y D (v2)(xi)) L S d
BBB
1 =
Bollva11
S
82
8²
Xx.
Hence, using Theorem
13
with g: = os and y = ú. we see that
T
T
E
(xx) u. - x1)
E
EE Or xi] "ü, -x,
T
E [(0, ar - xi) xx|]
T
E
(0 û. - xxi)
Or 4xx,, + v(û.) (xx))
< E
11
T'
de B
v(ū.) - (((1))
$ 17
82
"
1=1
nd B
v log [-(++)--
+
8²
where we used Lemma] in the last inequality.
K Proof of Theorem
Proof. Let B E Adv be a realized adversary and let fi, fr be the sequence of functions selected
by B. Also let u, E argmaxuex Σ (u) and ú, E argmin Ilu. x|| where 7 = T We have
T
R
WizO FTRL)
Σ
- - fc((x)))]
E
t=1
T
E
aafe Sie (ū.) - o1) (xi) +
E
St
p (u.) - LT (ū.)
1=1
1=1
(4)
T
T
E [(u...) - -o (u.)
(Se o 1) - s(((xx)))
+ 0
+
(C) As in the proof of Theorem 3 we use Lemma to extend all functions ft to My-smooth functions over d and we bound the terms (A) and (B) by DI and D²T respectively. To bound (C),
(B)
27

---

we first use Lemma 10 to see that ft o W is M2 smooth, where MA MMM3 + M2M2 Hence,
we see that
(fe o (x)) = BU
fi(W(x; + o2iv)) - fe(W(xx))]
-VVS:(W(x.)) + MI
Bunay
2 16
2
Therefore,
B
S 2
Putting the bounds for (A), (B) and (C) together, we see that
R, R,
FTRL)
B [aff(uu.) -f(W(.)))((((())))
aft e (ū.) - (fro () + +aMyDDT + (a + M2 T
Σ
2
(X xi) +aMrDD+ (a:22 + M² 2 D
<
2
t=1
(8)
where the second inequality follows from Lemma g
To bound the remaining term, we use an argument similar the one used the proof of Theorem
again. Using Lemma 5 with Σ 8E, and the fact that yr is an unbiased sample of W(f.) at
X, + dE,v, we see that E [; | xx] v(W(f))) (xx). On the other hand, since W(O,) is bounded by Bp, we have Iyel < Bu. which implies that
d 2
V2
82
Me
T 2,'
(V2O(xi))
Dx vv Di'v,
d 5z BBllv
dB 82
xi.
Hence, using Theorem 13
with g: O2 and y ú, we that
T
R
((V u. xi)
3
E
E [or
xx] 0. xz)
1=1
E
E [fo, u. -xx) xxo]
3E
O1 û. xii
T
4(ū.) - ((xx)
< 3E
17
|o 11 Pxx.+
7)
t=1
& CB
((ū.)
(((1)
8m Σ
B
82
17
Bnd BBT
By log(
(-
82
17
28

---

where we used Lemma
7 in the last inequality. Plugging this into Equation 8 and using M that
MIM; Mr M2Ml1² and
T-1
Bnd B
Bu log( 1-
FTRL)
R o.B
82
a.2 + MiM2 + M2M{²) 8
aMiyDD
=0 2T + +1 + log T+ TTPT).
L Proof of Theorem 5
Proof. Note that in all three cases, Wascion is I-Lipschitz and O-smooth. Now the result for the first case follows immediately from the fact that WM Id. Also note that for any zeroth order query oracle Q for function f E PMO and any y E K
|MAilo = === y)1 -=-
yll -
|y <
Thus the query oracle W(2,) is bounded by D and the assumptions of Theorem arr satisfied. The proof of boundedness of (N)((2) for any f E FNM is similar.
M Stochastic Full-information To Trivial query - SFTT
In this section, we discuss the SFTT meta-algorithm (Algorithm 4 in [Pedramfar and Aggarwal, 2024a]) which converts algorithms that require full-information feedback into algorithms that have a trivial query oracle. In particular, it converts algorithms require zeroth-order hull-information feedback into bandit algorithms.
We say function class F closed under convex combination if for any f1, ff F and any 51,-.. 6x 2 O with E, 6, = 1, we have Σ, 8 E F.
Theorem 14 (Theorem 7 and Remark and Corollary in [Pedramfar and Aggarwal 2024a) Let A be an online optimization algorithm with iall-iffrmation feedback and with K queries at each time-step where Aquery does not depend on the observations in the current round and A' (ATT(A). Then, for any M1 Mı-Lipschitz function class F that is closed under convex combination and any Bi M1, < a S and S a S b S T, let a' === =L 1)) + b' = [b/L] L}, D diam(K) and let (T} and [T/L} denote the horizon of the adversary. If we also have R.A' (F (IC) b] = O(BT), = 0(1) and = O(T then
R. Adv (K4) 0 (BT
More generally, the above result holds even if the query oracles are not bounded. Specifically, what we require is that the set of query oracles to be closed under corvex combinations.
Algorithm 6: Stochastic Full-information To Trivial query SFTT(A)
Input: base algorithm A, horizon I', block size L > K.
for 1,2,.... 1,2.,.,,T/L T/LL do
Let &,, be the action chosen by _4
Let (y v 1
queries selected by Aquery
Let (Iq. .tg....)........ be a random permutation of {(g - 1)L + I....,gL}}
for = (q 1)L ... 9L do
if Lq.3 for
< i K then
Play the action x = 8 Return the observation to the query oracle as the response to the i-th query
Play the action x; = &g
end
end
end
29

---

N Proof of Theorem 6
Proof. All three class of functions considered are closed under convex combination. Therefore may directly apply Theorems 5 and 14 to obtain this result for the first case.
For any sequence of functions f1, ... fi and query oracles Q1. Qs for these functions that contained within cone COne(O, C*) and non-negative numbers .1. Òд such that >; 6, = 0, the query oracle Q that uses 2, with probability ó, is trivially a query oracle for Σ; ó,ff that is also contained within this cone. Therefore, we may apply Theorem 14 to obtain this result for the second case as well. The proof of the last case is similar.
O Proof of Theorem 7
First we state the following simple result about OTB.
Theorem 15 (Theorem 8 in (Pedramfar and Aggarwal 2024a If A is an online algorithm that queries no more than K = 1 times per time-step and obtains an c-regret bound of O(T"), ), then the sample complexity of OTB(A) is 12((6 ---))
Proof of Theorem 7 This is an immediate corollary of Theorem 5 and the guarantees for the OTB meta-algorithm stated Theorem [3]
30

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Reviews:
[{"review_id": "0vzmOdV312", "reviewer": "Reviewer_ww6m", "summary": "This paper introduces a general technique for reducing problems involving (online) optimization of quadratizable losses to the much more well-studied problem of online optimization of concave losses (and in the course of doing so, recovering or improving many state-of-the-art results on quadratizable optimization). Quadratizability is a generalization of concavity that allows for proving approximate optimization guarantees (e.g. that some form of gradient descent will reach an alpha-approximation to the optimal); the condition that f is (upper) quadratizable can be ~written as f satisfying\n\nalpha * f(y) - f(h(x)) <= beta * (<g(f, x), y - x> - ||y-x||^2).\n\nThe key observation of this paper is that many quadratizable functions actually satisfy a slightly stronger constraint, where the functional g(f,x) is equal to $\\nabla G(f)(x)$ for some simple transformation G(f). In particular, they can construct such transformations (which they call “uniform wrappers”) for fairly broad classes of quadratizable functions, including monotone gamma-weakly concave functions with bounded curvature, monotone gamma-weakly concave functions over convex sets containing the origin, and non-monotone concave functions over general convex sets. \n\nThis transformation is fairly powerful and allows for transferring a wide variety of results from the concave optimization regime to the quadratizable optimization regime (although it does seem like there is no completely generic claim along these lines, and every application of this framework does require a little bit of custom work). One example the authors give is the generalization of a classic zeroth-order FTRL algorithm (due to Saha and Tewari) to the quadratizable setting, where it immediately gives an O(T^{2/3}) alpha-regret algorithm for the class of alpha-upper-linearizable losses.", "questions": "See section above. Feel free to reply to any part of this review.", "limitations": "Yes.", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Optimization of (online) quadratizable losses is strongly connected to several problems in online submodular optimization, an area of key importance in machine learning and algorithm design. This paper introduces what seems to me (a non-expert in this area) a fairly powerful technique for translating between this problem and (the very well-studied problem of) online convex optimization.\n\nEarlier work by Pedramfar and Aggarwal (who also introduced the class of quadratizable functions) provided a method to automatically translate between these two regimes, but their method is somewhat more lossy than the method presented in this paper (in the zeroth-order FTRL application mentioned above, the algorithm of Pedramfar and Aggarwal results in suboptimal T^{3/4} rates). The downside is that this new method seems to be somewhat less automatic (e.g., one step is to “rewrite the steps of the analysis so that the application of concavity is limited to a single line”). But the authors do successfully apply it to a fairly large class of problems, and so the generality of this method is fairly convincing.\n\nThe paper was in general well-written and easy to understand, although as someone who is not an expert in the area, it would have been helpful to have a bit more exposition explaining which of the new results in Table 1 arethe most significant improvements over the existing state-of-the-art algorithms. (Right now my impression is that the improvements largely are limited to previously not-well-studied domains, e.g. stochastic-within-a-cone queries, but there are many results and I could have missed some important ones).", "was_revised": true, "final_justification": "I have read the author's response and the other reviews and my score remains the same. The authors introduce a general framework for studying these generalizations of online convex optimization, and I expect this to be of interest to the optimization / online learning communities at NeurIPS.", "rebuttal": "We thank the reviewer for their review and positive assessment of our work.\n\n> ... Earlier work by Pedramfar and Aggarwal (who also introduced the class of quadratizable functions) provided a method to automatically translate between these two regimes, but their method is somewhat more lossy than the method presented in this paper (in the zeroth-order FTRL application mentioned above, the algorithm of Pedramfar and Aggarwal results in suboptimal $T^{3/4}$ rates)\n\n\nWe note that the result of [Pedramfar and Aggarwal, 2024a] for the zeroth order case is obtained by transforming a first-order algorithm and then converting the resulting first-order algorithm into a zeroth-order algorithm using one-point gradient estimators (a la [Flaxman et al., 2005]).\nSpecifically, they require the base algorithm to be first-order and their result cannot be directly applied to a zeroth-order base algorithm.\n\n> ... it would have been helpful to have a bit more exposition explaining which of the new results in Table 1 arethe most significant improvements over the existing state-of-the-art algorithms. (Right now my impression is that the improvements largely are limited to previously not-well-studied domains, e.g. stochastic-within-a-cone queries, but there are many results and I could have missed some important ones).\n\nThe most significant improvements are for previously studied oracle types.\nNote that a deterministic oracle is a special case of stochastic-within-a-cone oracle.\nThus, when considering either deterministic or fully stochastic query oracles, we obtain superior regret guarantees for 3 settings. Specifically, for online or offline zeroth order feedback, we improve the SOTA for monotone up-concave functions over general convex sets and for bandit feedback, we improve the SOTA for non-monotone up-concave functions over general convex sets. (See Tables 1 and 2)\n\nHowever, we note that the results in Tables are by no means the only applications of our work.\nWe have only considered the zeroth-order algorithm of [Saha and Tewari, 2011] due to the limited space in the paper.\nWe refer to our response to reviewer ALb7 for more details on other possible applications."}, {"review_id": "LoOwG1Qvgh", "reviewer": "Reviewer_ALb7", "summary": "The main contribution of the paper is providing a procedure named the uniform wrapper that converts algorithms and their regret guarantees from online concave optimization to online quadratizable optimization, which includes online DR-submodular maximization as a special case. By applying the proposed framework, the paper can recover almost all the known results in the literature for online/offline DR-submodular maximization (and indeed its generalization, online/offline up-concave optimization).", "questions": "It is recommended to leave more room for introducing the uniform wrapper. A good manner is to describe the whole procedure in detail for a concrete scenario.", "limitations": "yes", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Quality. The paper presents a systematic method for solving online quadratizable optimization. It is, in general, technically sound. All the claims are well supported by theoretical analysis. \n\nClarity. Perhaps due to the paper's extensive coverage of scenarios, substantial space is devoted to presenting results, leaving limited room for presenting and explaining the approach. The description of conversion steps in Section 6 is overly simplistic, making it difficult to grasp the essence without consulting the appendix. Although Section 7 provides an example of zeroth-order FTRL and proposes a variant algorithm, the authors still fail to explain how this algorithm successfully integrates with the uniform wrapper. In a word, the main body alone provides an inadequate explanation for the algorithm's technical success.\n\nSignificance. Seeking a unified algorithmic framework for online quadratizable optimization represents a fascinating research attempt. The proposed method is, in general, successful since it either beats the SOTA or at least recovers it. However, it is worth noting that the approach only beats the SOTA in a limited number of cases. This means that the approach does not contain too many substantial insights that can break the barrier for online up-concave optimization. This paper may be regarded as an interim summary in some sense.\n\nOriginality. The paper proposes a unified framework for adapting online concave optimization algorithms to algorithms for online quadratizable optimization, improving the previous results such as [Pedramfar and Aggarwal, 2024a]. This validates its originality.", "was_revised": true, "final_justification": "I went through the part about the contributions again. Combining with the reply from the authors, I find it unnecessary to be so demanding about the paper’s findings.​ So I decided to raise my score. And due to the writing, I will increase it only by 1.", "rebuttal": "We thank the reviewer for their review and suggestions.\n\n> Clarity: ... substantial space is devoted to presenting results, leaving limited room for presenting and explaining the approach.\n\nThe main concepts and ideas are explained in Sections 4, 5, and 6 in the main text and Appendices A and C.\nThe core ideas are mathematically complete, but somewhat abstract, which makes them difficult to explain and understand without examples.\nThis is why we have devoted Sections 7 and 8 to examples and applications of our method, which takes a total of 2.5 pages.\nHowever, there is a tradeoff between the simplicity of the example and the significance of its application.\nIn Section 7, we have chosen an example that would allow us to obtain novel results.\n\nWe believe that, given the abstract nature of the ideas, the best way to increase clarity might be to add a simpler example.\nIn the response to reviewer ESFd, we have detailed how our guideline may be applied to a simpler base algorithm, namely Online Gradient Ascent.\nThe resulting algorithms are not novel, but the procedure will increase the clarity of our presentation.\nWe will include this example in the final version in the main text to further clarify the main ideas.\n\nIn the final version, we will expand Section 6 further to clarify the ideas.\nWe will also add examples of pseudo-codes for application of the specific uniform wrappers we consider to the base algorithm mentioned in Section 7.\n\n\n\n> Significance: ... However, it is worth noting that the approach only beats the SOTA in a limited number of cases. This means that the approach does not contain too many substantial insights that can break the barrier for online up-concave optimization. This paper may be regarded as an interim summary in some sense.\n\nAs discussed in the paper, and in the Online Gradient Ascent example in our response to reviewer ESFd, even though the changes to the original algorithm and proof could be very minor, one can not simply claim that the original proof could be adapted to the format we require without justification.\nThus, applying our guideline requires an almost complete rewriting of the proof of the original regret bound.\n\nIn this work, we provide an example for zeroth-order case.\nHowever, this is merely because of limited space.\nOur technique is general enough that, a priori, it is not possible to rule out *any* algorithm for online concave optimization as a base algorithm.\nThe result of [Pedramfar and Aggarwal, 2024a] covers base algorithms that are deterministic and require first order feedback.\nHowever, there are first order algorithms that are not deterministic.\nFor example, all Follow-The-Perturbed-Leader type algorithms are by definition stochastic as they require perturbation.\nSimilarly, the SOTA projection-free online concave optimization algorithm using membership oracles is also stochastic (See [1]). \nOn the other hand, the results of [Pedramfar and Aggarwal, 2024a] do not cover any base algorithm that is not first order.\nThus none of the zeroth-order or second-order (e.g. Newton method) feedback algorithms are directly covered by [Pedramfar and Aggarwal, 2024a].\nWhile it is possible to add more results in this paper, by considering more base algorithms, we would need to introduce the notation, ideas, and proofs of those results in entirety in order to do so.\nThe paper [1] is 77 pages. Even if adapting this result using our framework increases our page count by only 30 pages, it would still result in an overly lengthy paper and reduce the clarity of the presentation.\nMoreover, none of the ideas used in [1], or in second-order algorithms, or even in the zeroth-order algorithm that we have discussed in Section 7, are directly relevant to the main contribution of our work.\n\n\n\n[1] Efficient Projection-Free Online Convex Optimization with Membership Oracle - Mhammedi - 35th Annual Conference on Learning Theory - 2022"}, {"review_id": "9Goga7JG5k", "reviewer": "Reviewer_yWTu", "summary": "This paper introduces a meta-algorithmic framework called \"uniform wrappers\" that transforms algorithms designed for concave/convex optimization into ones applicable to non-convex, quadratizable optimization problems, specifically targeting classes of DR-submodular functions. The main results are: development of a conversion guideline that adapts both algorithms and their proofs; application to zeroth-order feedback, bandit feedback, and offline optimization;", "questions": "1. For many practical functions, how should we construct a suitable wrapper W? Are there heuristics or automatable steps, especially for less structured non-convex functions?", "limitations": "n/a", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths: 1) the proposed uniform wrappers is a novel development. It enables systematic translation of actions and oracles while maintaining regret guarantees. 2) The detailed proof adaptation framework leads to new regret bounds under zeroth-order and bandit settings. These theoretical results either match or improve existing regret/sample complexity bounds. 3) The framework handles multiple function classes, including monotone, non-monotone, weakly DR-submodular functions. This demonstrates the framework’s generality by recovering and unifying prior SOTA results.\n\nWeakness: 1) The paper is purely theoretical. No simulations is provided to demonstrate practical performance of the meta-algorithms (e.g., on DR-submodular test functions).", "was_revised": false, "final_justification": "", "rebuttal": "> The paper is purely theoretical. No simulations is provided to demonstrate practical performance of the meta-algorithms\n\nThe reason we have not added simulations is two-fold.\n\nFirst, given the theoretical nature of our work and the extensive coverage of difference scenarios, it is not clear what experiments could support our main contributions.\nAn experiment for the zeroth-order algorithm in one of the settings will only demonstrate that, in that setting, that algorithm is performant.\nSuch an experiment only marginally supports our main contribution, which is Meta-algorithm 1 and the guideline described in Section 6.\n\nSecond, it is common for papers that are theoretical like this to not have experiments. \n\n> For many practical functions, how should we construct a suitable wrapper W? Are there heuristics or automatable steps, especially for less structured non-convex functions?\n\nGiven the difficulty of non-convex optimization, finding a uniform wrapper for a single function class is a powerful result.\nIn general, even for offline optimization, there are few families of non-convex functions where global convergence results are currently known.\nThus, any heuristic that could be applied to a general family of function classes would be an immensly powerful breakthrough, as it would result in global convergence guarantees for online optimization of the all the function classes in that family.\n\n\n\nIn order to construct a uniform wrapper $\\mathcal{W}$, we need to specify $\\mathcal{W}^{\\operatorname{action}}$, $\\mathcal{W}^{\\operatorname{function}}$, and $\\mathcal{W}^{\\operatorname{query}}$.\n\nSo far, except for the identity wrapper which applies to both concave functions and monotone up-concave functions over general convex sets, the examples of uniform wrappers fit into the following framework:\n\nThere is a random variable $\\mathcal{Z}$ taking values in $[0, 1]$ and a parametrized family of functions $h_z : \\mathcal{K} \\to \\mathcal{K}$ for $z \\in [0, 1]$, such that, for some $\\alpha \\in (0, 1]$ and $\\beta, \\mu \\geq 0$, we have\n\n$\\alpha f(y) - f(x) \\leq \\beta \\left( \\langle \\nabla F(x), y-x \\rangle - \\frac{\\mu}{2}\\| y - x \\|^2 \\right)$,\n\nwhere $F(x) = \\mathbb{E}_{z \\sim \\mathcal{Z}} [f(h_z(x)) - f(h_0(x))]$.\n\nIn this case we define\n$\\mathcal{W}^{\\operatorname{action}} = h_1$,\nand \n$\\mathcal{W}^{\\operatorname{function}}(f) = F$.\nWe also define\n$\\mathcal{W}^{\\operatorname{query}}(Q_f)$\nsuch that it converts a query oracle for $f$ into a query oracle for $F$ using the following equality:\n\n$\\\\nabla F = \\mathbb{E}\\_{z \\sim \\mathcal{Z}} [\\\\nabla f(h_z(x)) J_{h_{z}}(x) - \\\\nabla f(h_z(x)) J_{h_{0}}(x)]$,\n\n\nWe will add a discussion on these lines in the final version."}, {"review_id": "SzT85JzokZ", "reviewer": "Reviewer_ESFd", "summary": "This paper introduces uniform wrappers, a meta-algorithmic framework that systematically converts online convex optimization (OCO) algorithms into algorithms for quadratizable function classes (a generalization of concave/DR-submodular functions). The core innovation lies in providing provable regret guarantees through automated transformation of both algorithms and their regret proofs. By applying this framework to zeroth-order base algorithms (e.g., ZO-FTRL), the work achieves improved regret bounds for zeroth-order feedback and bandit feedback—surpassing prior state-of-the-art results. The framework can recover or obtain superior regret for DR-submodular maximizations in different setting.", "questions": "1. Could the authors provide a concise user guide for converting an OCO algorithm into an online quadratizable optimization algorithm, along with an explanation of how the regret bound proof is transformed? This should include specifying which technical parts of the paper should be applied and in what order. If the authors have already provided such part somewhere in the paper, it will be appreciate that the authors could remind me the location.\n\n2. Could the authors provide examples of potential applications for quadratizable function classes, or provide references that demonstrate the significance of this function class?\n\n3. For the future work, could the regret lower bound proof of some OCO setting be convert to that of the online quadratizable optimization?\n\ntypos:\n58-59: \"fully adaptive Another...\"", "limitations": "yes", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 1, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The article's content is very dense, leading to a somewhat cluttered presentation. I believe the core contribution is that any OCO algorithm can be converted into an algorithm for quadratizable function classes via the proposed uniform wrappers, and that the regret bound proof for the original OCO algorithm can be transformed into a regret bound proof for the new algorithm. If I understand correctly, this would be a very nice result. The remaining parts essentially serve as examples demonstrating the generality of this core conclusion. Therefore, I recommend that the authors significantly enhance the introduction by adding a practical guide outlining the conversion process and proof transformation. Specifically: Assuming a researcher in a related field proposes a new OCO algorithm, how should they use the authors' framework to derive a online quadratizable optimization algorithm?\n\nI'd be happy to reevaluate if the authors could alleviate my confusions as shown in the question part.", "was_revised": true, "final_justification": "I believe this work possesses significant scholarly merit despite needing writing improvements. It provides meaningful guidance for future research, though my main reservation remains the lack of a general selection methodology for the wrapper. I increase my rating accordingly.", "rebuttal": "Q1.\n\n> 1. Could the authors provide a concise user guide for converting an OCO algorithm into an online quadratizable optimization algorithm, along with an explanation of how the regret bound proof is transformed? This should include specifying which technical parts of the paper should be applied and in what order. If the authors have already provided such part somewhere in the paper, it will be appreciate that the authors could remind me the location.\n\n\nWe first discuss \n> a concise user guide for converting an OCO algorithm into an online quadratizable optimization algorithm\n\nThe conversion of the algorithm is detailed in Meta-algorithm 1. Basically, this meta-algorithm transforms the base algorithm  action using $\\mathcal{W}^{\\operatorname{action}}$ and the base algorithm  query oracle using $\\mathcal{W}^{\\operatorname{query}}$.\n\nFor example, consider the  Online Gradient Ascent as a base algorithm:\n\n---\n\nInput: convex set $\\mathcal{K}$, $x_1 \\in \\mathcal{K}$, step-sizes $\\eta_n$  \n\nFor $t=1$ to $T$\n\nPlay $x_t$\n\nObserve $o_t$, a sample of $\\mathcal{Q}_{f_t}(x_t)$, an unbiased estimate of $\\nabla f_t(x_t)$\n\n$x_{t+1} \\gets P_{\\mathcal{K}}(x_t + \\eta_t o_t)$  \n\n---\n\nWhere we use $\\mathcal{Q}_{f_t}$ to denote the stochastic first order query oracle for $f_t$ which returns unbiased estimates of $\\nabla f_t$.\n\nNow, if we apply the first-order to first-order uniform wrapper $\\mathcal{W}_1^{NM}$ (for non-monotone up-concave\nfunctions over general convex sets; see Appendix C.3 for additional details)  according to Meta-algorithm 1, we get a first order algorithm for online optimization of non-monotone DR-submodular function over general convex sets:\n\n---\n\nInput: convex set $\\mathcal{K}$, $x_1 \\in \\mathcal{K}$, $\\underline{x} \\in \\mathcal{K}$, step-sizes $\\eta_n$\n\nFor $t=1$ to $T$\n\nPlay $(\\mathcal{W}_1^{NM})^{\\operatorname{action}}(x_t) = (x_t + \\underline{x})/2$\n\nObserve $o_t$, a sample of $(\\mathcal{W}_1^{NM})^{\\operatorname{query}}(\\mathcal{Q}_{f_t})(x_t)$\n\n$x_{t+1} \\gets P_{\\mathcal{K}}(x_t + \\eta_t o_t)$\n  \n---\n\nBy plugging in the definitions of $(\\mathcal{W}_1^{NM})^{\\operatorname{action}}$ and $(\\mathcal{W}_1^{NM})^{\\operatorname{query}}$ as per Appendix C.3, we get the following algorithm:\n\n---\n\nInput: convex set $\\mathcal{K}$, $x_1 \\in \\mathcal{K}$, $\\underline{x} \\in \\mathcal{K}$, step-sizes $\\eta_n$\n\nFor $t=1$ to $T$\n\nPlay $(x_t + \\underline{x})/2$\n\nSample $z$ according to $\\mathcal{Z}^{NM}$ (as defined in Appendix C.3)\n\nObserve $o_t$, a sample of $\\mathcal{Q}_{f_t}(\\frac{z}{2}(x_t - \\underline{x}) + \\underline{x})$, an unbiased estimate of $\\nabla f_t(\\frac{z}{2}(x_t - \\underline{x}) + \\underline{x})$\n\n$x_{t+1} \\gets P_{\\mathcal{K}}(x_t + \\eta_t o_t)$\n  \n---\n\nThis becomes Algorithm 1 in [Zhang et al., 2024] in the case of non-monotone functions.\n\n\n\nWe will add this example with more explanation in the Appendix to demonstrate exactly how Meta-algorithm 1 works.\n\n\n\n> an explanation of how the regret bound proof is transformed\n\n\nThe conversion of the regret bound is the focus of Section 6, in the form of a three-step guideline. Section 7 and Appendix D are examples of how to apply such a guideline.\nSince this is a guideline that depends on the structure and the details of the original proof, any explanation of the application of the guideline requires an understanding of the original proof.\nThis is why Section 7 and Appendix D are as long as they are, we needed to go through the proof of the original regret bound, explain all the ideas there, and rewrite the proofs in order to match the guideline.\n\n\n\nIn Section 7, lines 285-310 describe exactly how Theorems 1 and 2 provide an example of how to apply the guideline in this case.\nIn Appendix D, lines 984-997, specifically the statement of Theorems 9 and 10 and the paragraph between the theorems describe how the guideline applies in this case.\n\n\n\nOnline Gradient Ascent (OGA) mentioned above is a special case of what is covered in Appendix D.  For OGA specifically, we can simplify the proofs and guidelines as follows.\nHowever, in this specific case, we may simplify the proofs and the guideline in the following manner:\n\nAs per step 1 of the guideline, we first rewrite a part of the proof of the OGA, in a way that does not require concavity, but the proof of the regret bound for the concave case can be recovered using the inequality $f(y) - f(x) \\leq \\langle \\nabla f_t(x), y - x \\rangle$.\n\n\n\n\n\n\n\n\n> Theorem A. For any differentiable function class $\\mathbf{F}$, with stochastic first-order query oracles bounded by $G$ and step-sizes $\\eta_t=\\frac{D}{G \\sqrt{t}}$, the algorithm OGA guarantees that\n$\\max_{x \\in \\mathcal{K}} \\sum_{t=1}^T \\mathbb{E}[ \\langle \\nabla f_t(x_t), x - x_t \\rangle ] \\leq \\frac{3}{2} G D \\sqrt{T}$.\n\nThe proof of this theorem is a simple rewriting of the classical results.\nNote that the boundedness of the stochastic first-order query oracle by $G$ implies that the gradients of functions in $\\mathbf{F}$ are bounded by $G$, i.e., functions in $\\mathbf{F}$ are $G$-Lipschitz continuous.\n\n\nNext we want to adapt the result of Theorem A to upper-quadratizable setting.\n\n\n> Theorem B. If $\\mathbf{F}$ is a function class over $\\mathcal{K}$ that is upper-linearizable with $0 < \\alpha \\leq 1$ and $\\beta \\geq 0$ and a first-order uniform wrapper $\\mathcal{W}$, then with stochastic first-order query oracles bounded by $G$ and step-sizes $\\eta_t=\\frac{D}{G \\sqrt{t}}$, the algorithm $\\mathcal{W}(OGA)$ guarantees that\n$\\max_{x \\in \\mathcal{K}} \\sum_{t=1}^T \\mathbb{E}[ f_t(x) - f_t(\\mathcal{W}^{\\operatorname{action}}(x_t)) ] \\leq \\frac{3}{2} G D \\sqrt{T}$.\n\n\nAt first glance, it may seem that moving from Theorem A to Theorem B might be non-trivial.\nHowever, as elaborated in Appendix A, whether we apply $\\mathcal{W}(OGA)$ to an adversary $\\operatorname{Adv}$ that selects functions from $\\mathbf{F}$ or applying OGA to the adversary $\\mathcal{W}(\\operatorname{Adv})$, the sequence $x_t$ remains the same.\nThus, by using Theorem A, we immediately see that\n$\\max_{x \\in \\mathcal{K}} \\sum_{t=1}^T \\mathbb{E}[ \\langle \\nabla \\mathcal{W}(f_t)(x_t), x - x_t \\rangle ] \\leq \\frac{3}{2} G D \\sqrt{T}$.\nNow we may use the definition of uniform wrappers to complete the proof of Theorem B.\n\n\nWe will include the discussion above in the appendix to show how our guideline may be applied to a setting that is simpler than the zeroth-order algorithm mentioned in Section 7.\n\n> 2. Could the authors provide examples of potential applications for quadratizable function classes, or provide references that demonstrate the significance of this function class?\n\nQ2. So far, the main classes of functions that are covered by quadratizable functions are concave functions and continuous DR-submodular functions (as described in the paper). However, we would like to point out that this concept has been introduced in the literature in [Pedramfar and Aggarwal, 2024a] which was published in December 2024. Thus, given more time, we expect that this concept will be better understood and more examples of quadratizable/linearizable functions will be found.\n\n\n> 3. For the future work, could the regret lower bound proof of some OCO setting be convert to that of the online quadratizable optimization?\n\nQ3. That is a great question.\nUnfortunately, we do not think our framework could be used for converting lower-bound proofs.  In order to prove a lower bound, it is essential to already know the optimal approximation coefficient. (To see why, given a function class $\\mathbf{F}$ with a uniform wrapper with $0 < \\alpha \\leq 1$, it may be possible to find another uniform wrapper with a worse approximation coefficient $0 < \\alpha' < \\alpha$.  If an algorithm for $\\mathbf{F}$ has sublinear $\\alpha$-regret, then it will have negative $\\alpha'$-regret.)\n\nFinding the optimal approximation coefficient is highly non-trivial.  For example, for non-monotone DR-submodular functions over downward-closed convex sets, the optimal approximation coefficient is still unknown."}]

Author rebuttal:

Meta-review: Overall all reviewers were positive about this work and I support their decision.
Decision: Accept (poster)