Improved Regret for Bandit Convex Optimization with Delayed Feedback

Authors: yuanyu  wan
chang  yao
mingli  song
lijun  zhang
Venue: NeurIPS_cc_2024_Conference
Published: 2024-05-13 13:32:09

Abstract: We investigate bandit convex optimization (BCO) with delayed feedback, where only the loss value of the action is revealed under an arbitrary delay. Let $n,T,\bar{d}$ denote the dimensionality, time horizon, and average delay, respectively. Previous studies have achieved an $O(\sqrt{n}T^{3/4}+(n\bar{d})^{1/3}T^{2/3})$ regret bound for this problem, whose delay-independent part matches the regret of the classical non-delayed bandit gradient descent algorithm. However, there is a large gap between its delay-dependent part, i.e., $O((n\bar{d})^{1/3}T^{2/3})$, and an existing $\Omega(\sqrt{\bar{d}T})$ lower bound. In this paper, we illustrate that this gap can be filled in the worst case, where $\bar{d}$ is very close to the maximum delay $d$. Specifically, we first develop a novel algorithm, and prove that it enjoys a regret bound of $O(\sqrt{n}T^{3/4}+\sqrt{dT})$ in general. Compared with the previous result, our regret bound is better for $d=O((n\bar{d})^{2/3}T^{1/3})$, and the delay-dependent part is tight in the worst case. The primary idea is to decouple the joint effect of the delays and the bandit feedback on the regret by carefully incorporating the delayed bandit feedback with a blocking update mechanism. Furthermore, we show that the proposed algorithm can improve the regret bound to $O((nT)^{2/3}\log^{1/3}T+d\log T)$ for strongly convex functions. Finally, if the action sets are unconstrained, we demonstrate that it can be simply extended to achieve an $O(n\sqrt{T\log T}+d\log T)$ regret bound for strongly convex and smooth functions.

Main paper: 
 introduction: 
 Online convex optimization (OCO) with delayed feedback [Joulani et al., 2013, Quanrud and Khashabi, 2015] has become a popular paradigm for modeling streaming applications without immediate reactions to actions, such as online advertisement [McMahan et al., 2013] and online routing [Awerbuch and Kleinberg, 2008] . Formally, it is defined as a repeated game between a player and an adversary. At each round t, the player first selects an action x t from a convex set K ⊆ R n . Then, the adversary chooses a convex function f t (•) : R n → R, which causes the player a loss f t (x t ) but is revealed at the end of round t + d t -1, where d t ≥ 1 denotes an arbitrary delay. The goal of the player is to minimize the regret Reg(T ) = T t=1 f t (x t ) -min x∈K T t=1 f t (x), i.e., the gap between the cumulative loss of the player and that of an optimal fixed action, where T is the number of total rounds. Over the past decades, plenty of algorithms and theoretical guarantees have been proposed for this problem [Weinberger and Ordentlich, 2002 , Langford et al., 2009 , Joulani et al., 2013 , Quanrud and Khashabi, 2015 , Joulani et al., 2016 , Héliou et al., 2020 , Flaspohler et al., 2021 , Wan et al., 2022a ,b, Bistritz et al., 2022] . However, the vast majority of them assume that the full information or gradients of delayed functions are available for updating the action, which is not necessarily satisfied in reality.For example, in online routing [Awerbuch and Kleinberg, 2008] , the player selects a path through a given network for some packet, and its loss is measured by the time length of the path. Although this loss value can be observed after the packet arrives at the destination, the player rarely has access to the congestion pattern of the entire network [Hazan, 2016] . To address this limitation, it is natural to investigate a more challenging setting, namely bandit convex optimization (BCO) with delayed feedback, where only the loss value f t (x t ) is revealed at the end of round t + d t -1.It is well known that in the non-delayed BCO, bandit gradient descent (BGD), which performs the gradient descent step based on a one-point estimator of the gradient, enjoys a regret bound of O( √ nT 3/4 ) [Flaxman et al., 2005] . Despite its simplicity, without additional assumptions on functions, there does not exist any practical algorithm that can improve the regret of BGD. Therefore, a few studies have proposed to extend BGD and its regret bound into the delayed setting [Héliou et al., 2020 , Bistritz et al., 2022] . Specifically, Héliou et al. [2020] first propose an algorithm called gradient-free online learning with delayed feedback (GOLD), which utilizes the oldest received but not utilized loss value to perform an update similar to BGD at each round. Let d = max{d 1 , . . . , d T } denote the maximum delay. According to the analysis of Héliou et al. [2020] , GOLD can achieve a regret bound of O( √ nT 3/4 + (nd)1/3 T 2/3 ), which matches the O( √ nT 3/4 ) regret of BGD in the non-delayed setting for d = O( √ nT 1/4 ). Very recently, Bistritz et al. [2022] develop an improved variant of GOLD by utilizing all received but not utilized loss values one by one at each round, and reduce the regret bound to O( √ nT 3/4 + (n d) 1/3 T 2/3 ), 1 where d = (1/T ) T t=1 d t is the average delay. However, there still exists a large gap between the delay-dependent part in the improved bound and an existing Ω( √ dT ) lower bound [Bistritz et al., 2022] . It remains unclear whether this gap can be filled, especially by improving the existing upper bound.In this paper, we provide an affirmative answer to this question in the worst case, where d is very close to d. Specifically, we first develop a new algorithm, namely delayed follow-the-bandit-leader (D-FTBL), and show that it enjoys a regret bound of O( √ nT 3/4 + √ dT ) in general. Notice that both the O((nd) 1/3 T 2/3 ) and O((n d) 1/3 T 2/3 ) terms in previous regret bounds [Héliou et al., 2020 , Bistritz et al., 2022] can be attributed to the joint effect of the delays, and the one-point gradient estimator, especially its large variance depending on the exploration radius. To improve the regret, besides the one-point gradient estimator, we further incorporate the delayed bandit feedback with a blocking update mechanism, i.e., dividing total T rounds into several equally-sized blocks and only updating the action at the end of each block. Despite its simplicity, there exist two nice properties about the cumulative estimated gradients at each block.• First, with an appropriate block size, its variance becomes proportional to only the block size without extra dependence on the exploration radius. • Second, the block-level delay, i.e., the number of blocks waiting for computing the cumulative estimated gradients at each block, is in reverse proportion to the block size.Surprisingly, by combining these properties, the previous joint effect of the delays and the one-point gradient estimator can be decoupled, which is critical for deriving our regret bound. Compared with the existing results, in the worst case, our regret bound matches the O( √ nT 3/4 ) regret of the non-delayed BGD for a larger amount of delays, i.e., d = O(n √ T ), and the delay-dependent part, i.e., O( √ dT ), matches the lower bound. Moreover, it is worth noting that our regret bound actually is better than that of Bistritz et al. [2022] as long as d is not larger than O((n d) 2/3 T 1/3 ), which even covers the case with d = O(1) partially. To the best of our knowledge, this is the first work that shows the benefit of the blocking update mechanism in delayed BCO, though it is commonly utilized to develop projection-free algorithms for efficiently dealing with complicated action sets [Zhang et al., 2019 , Garber and Kretzu, 2020 , Hazan and Minasyan, 2020 , Wan et al., 2020 , 2022c , Wang et al., 2023 , 2024b] . Furthermore, we consider the special case of delayed BCO with strongly convex functions. In the non-delayed setting, Agarwal et al. [2010] have shown that BGD can improve the regret from O( √ nT 3/4 ) to O((nT ) 2/3 log 1/3 T ) by exploiting the strong convexity. If functions are also smooth and the action set is unconstrained, BGD has been extended to achieve an O(n √ T log T ) regret bound [Agarwal et al., 2010] . Analogous to these improvements, we prove that our D-FTBL can achieve a regret bound of O((nT ) 2/3 log 1/3 T + d log T ) for strongly convex functions, and its simple extension enjoys a regret bound of O(n √ T log T + d log T ) for strongly convex and smooth functions over unconstrained action sets. These regret bounds also match those of BGD in the non-delayed setting for a relatively large amount of delay. Moreover, the O(d log T ) part in these two bounds matches an Ω(d log T ) lower bound adapted from the easier full-information setting with strongly convex and smooth functions [Weinberger and Ordentlich, 2002] .Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?Answer: [Yes] Justification: The abstract and introduction have clearly stated the contributions of this paper.Guidelines:• The answer NA means that the abstract and introduction do not include the claims made in the paper. • The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or NA answer to this question will not be perceived well by the reviewers. • The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings. • It is fine to include aspirational goals as motivation as long as it is clear that these goals are not attained by the paper. 
related work: 
 In this section, we briefly review the related work on online convex optimization (OCO) and bandit convex optimization (BCO), as well as delayed feedback.If d t = 1 for all t ∈ [T ],OCO with delayed feedback reduces to the standard OCO [Zinkevich, 2003] . Online gradient descent (OGD) [Zinkevich, 2003 , Hazan et al., 2007] is one of the most popular algorithm for this problem, which simply updates the action x t via a gradient descent step based on ∇f t (x t ). By using appropriate step sizes, OGD can achieve O( √ T ) and O(log T ) regret bounds for convex and strongly convex functions, respectively. Follow-the-regularized-leader (FTRL) [Hazan et al., 2007 , Shalev-Shwartz, 2011 , Hazan, 2016 ] is an alternative algorithm, which chooses the new action by minimizing the linear approximation of cumulative loss functions under some regularization. With appropriate regularization, FTRL achieves the same O( √ T ) and O(log T ) regret bounds as OGD. Moreover, Abernethy et al. [2008] have presented a lower bound of Ω( √ T ) for convex functions, and a refined lower bound of Ω(log T ) for strongly convex functions, which implies that both OGD and FTRL are optimal.BCO is a special yet more challenging case of OCO, where the player can only receive the loss value f t (x t ) at each round t. The first algorithm for BCO is bandit gradient descent (BGD) [Flaxman et al., 2005] , which replaces the exact gradient used in OGD with an estimated gradient based on the single loss value (known as the classical one-point gradient estimator). By incorporating the approximation error of gradients into the regret analysis of OGD, Flaxman et al. [2005] establish an O( √ nT 3/4 ) regret bound for BGD with convex functions. Later, Agarwal et al. [2010] show that BGD enjoys an O((nT ) 2/3 log 1/3 T ) regret bound for strongly convex functions, and can be extended to achieve an O(n √ T log T ) regret bound in the special case of unconstrained BCO with strongly convex and smooth functions. Saha and Tewari [2011] develop a new algorithm for BCO with smooth functions, and establish the O((nT ) 2/3 log 1/3 T ) regret bound without the strongly convex assumption. van der Hoeven et al. [2020] propose novel BCO algorithms, which adaptively improve the previous regret bounds for convex and smooth functions if the norm of the comparator is small. By revisiting the case with strongly convex and smooth functions, several algorithms [Hazan and Levy, 2014, Ito, 2020] have been developed to achieve the O(n √ T log T ) regret bound in the constrained setting.Moreover, a series of studies [Bubeck and Eldan, 2016 , Hazan and Li, 2016 , Bubeck et al., 2017 , Lattimore, 2020 , Bubeck et al., 2021] have been devoted to designing nearly optimal algorithms, which almost match the Ω(n √ T ) lower bound for the general BCO [Shamir, 2013] without any additional assumption. However, the running time of their algorithms are either exponential in n and T , or polynomial with a high degree on n and T , which is not suitable for practical large-scale applications. We refer the interested reader to Lattimore [2024] for a comprehensive survey on BCO. Additionally, we notice that BCO is closely related to the zero-order stochastic optimization (ZOSO) problem [Duchi et al., 2015 , Bach and Perchet, 2016 , Shamir, 2017] , where the stochastic values are available for minimizing a fixed loss function. However, ZOSO is less challenging than BCO in the sense that it does not need to deal with time-varying functions and is usually allowed to query the loss value at two points per iteration.The seminal work of Weinberger and Ordentlich [2002] first considers the case with a fixed delay, i.e., d t = d for all t ∈ [T ], and proposes a black-box technique that can covert any traditional OCO algorithm into the delayed setting. The main idea is to maintain d instances of the traditional algorithm, and alternately utilize these instances to generate the new action. If the regret of the traditional algorithm is bounded by Reg(T ), this technique can achieve an d Reg(T /d) regret bound. Moreover, there exist Ω( √ dT ) and Ω(d log T ) lower bounds for convex functions, and strongly convex and smooth functions, respectively [Weinberger and Ordentlich, 2002] . However, the delays are not always fixed in practice, and its space complexity is d times as much as that of the traditional algorithm, which could be prohibitively resource-intensive. Although Joulani et al. [2013] have generalized this technique to deal with arbitrary delays, the space complexity remains high. Besides these black-box techniques, there exists a surge of interest in developing and analyzing specialized algorithms for delayed OCO [Langford et al., 2009 , McMahan and Streeter, 2014 , Quanrud and Khashabi, 2015 , Joulani et al., 2016 , Li et al., 2019 , Flaspohler et al., 2021 , Wan et al., 2022a ,b, 2023 , 2024] , which do not require additional computational resources.Despite the great flourish of research on OCO with delays and BCO, delayed BCO has rarely been investigated. GOLD [Héliou et al., 2020] is the first algorithm for this problem, which originally has the O( √ nT 3/4 + (nd) 1/3 T 2/3 ) regret, and is further refined to enjoy the O( √ nT 3/4 + (n d) 1/3 T 2/3 ) regret [Bistritz et al., 2022] . However, Bistritz et al. [2022] also present an unmatched lower bound of Ω( √ dT ). Although two recent advances in a more complicated bandit non-stochastic control problem [Gradu et al., 2020 , Sun et al., 2023] provide some intermediate results about OGD and FTRL with the delayed bandit feedback, they focus on the case with a fixed delay and can only recover the O( √ nT 3/4 + (nd) 1/3 T 2/3 ) regret in general. In this paper, we take one further step toward understanding the effect of arbitrary delays on BCO by establishing improved regret bounds such that the delay-independent part is equal to the regret of BGD, and the delay-dependent part matches the lower bound in the worst case. Moreover, we notice that although the block-box technique of Joulani et al. [2013] can also convert BGD into the delayed setting, it only achieves an O( √ nd 1/4 T 3/4 ) regret bound for convex functions, which is much worse than that of GOLD and our algorithm.Question: Does the paper discuss the limitations of the work performed by the authors?Answer: [Yes] Justification: In Appendix A, the authors discuss one limitation of the results, and introduce a related open problem.Guidelines:• The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper. • The authors are encouraged to create a separate "Limitations" section in their paper.• The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect on how these assumptions might be violated in practice and what the implications would be. • The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on a few datasets or with a few runs. In general, empirical results often depend on implicit assumptions, which should be articulated. • The authors should reflect on the factors that influence the performance of the approach.For example, a facial recognition algorithm may perform poorly when image resolution is low or images are taken in low lighting. Or a speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon. • The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size. • If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness. • While the authors might fear that complete honesty about limitations might be used by reviewers as grounds for rejection, a worse outcome might be that reviewers discover limitations that aren't acknowledged in the paper. The authors should use their best judgment and recognize that individual actions in favor of transparency play an important role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations. 
main results: 
 In this section, we first introduce the necessary preliminaries including definitions, assumptions, and an algorithmic ingredient. Then, we present our improved algorithm for BCO with delayed feedback, as well as the corresponding theoretical guarantees.We first recall two standard definitions about the smoothness and strong convexity of functions [Boyd and Vandenberghe, 2004] .Definition 1. A function f (x) : R n → R is called β-smooth over K if for all x, y ∈ K, it holds that f (y) ≤ f (x) + ⟨∇f (x), y -x⟩ + β 2 ∥y -x∥ 2 2 . Definition 2. A function f (x) : R n → R is called α-strongly convex over K if for all x, y ∈ K, it holds that f (y) ≥ f (x) + ⟨∇f (x), y -x⟩ + α 2 ∥y -x∥ 2 2 .Note that as proved by Hazan and Kale [2012] , any α-strongly convex function f (x) :EQUATIONfor any x ∈ K, where x * = argmin x∈K f (x).Then, following previous studies on BCO [Flaxman et al., 2005 , Héliou et al., 2020 , Garber and Kretzu, 2020 , 2021] , we introduce some common assumptions. Finally, we introduce the one-point gradient estimator [Flaxman et al., 2005] , which is a standard technique for exploiting the bandit feedback. Given a function f (x) : R n → R, we can define the δ-smoothed version of f (x) asEQUATIONwhere the parameter δ ∈ (0, 1) is the so-called exploration radius. As proved by Flaxman et al. [2005] , the δ-smoothed version satisfies the following lemma. Lemma 1. (Lemma 1 in Flaxman et al. [2005] ) Given a function f (x) : R n → R and a constant δ ∈ (0, 1), its δ-smoothed version fδ (x) defined in (2) ensures∇ fδ (x) = E u∼S n n δ f (x + δu)uwhere S n denotes the unit Euclidean sphere centered at the origin in R n .From Lemma 1, the randomized vector n δ f (x + δu)u, which can be computed by only utilizing a single loss value, is an unbiased estimator of ∇ fδ (x). Moreover, Flaxman et al. [2005] have also shown that fδ (x) is close to the original function f (x) over a shrunk setK δ = (1 -δ/r)K = {(1 -δ/r)x|x ∈ K}.(3)Therefore, this one-point estimator can be utilized as a good substitute for the gradient ∇f (x) in the bandit setting. For example, we notice that at each round t, BGD [Flaxman et al., 2005] first plays an action x t = y t + u t , where y t ∈ K δ and u t ∼ S n , and then updates y t asEQUATIONwhere Π K δ (y) = argmin x∈K δ ∥x -y∥ 2 2 denotes the projection onto the set K δ , and η t is the step size.Before introducing our algorithm, we first briefly discuss the joint effect of the delays and the bandit feedback in GOLD [Héliou et al., 2020] , which will provide insights for our improvements. Recall that in the delayed setting, the loss value f t (x t ) will be delayed to the end of round t+d t -1, and thus the player can only receive {f k (x k )|k ∈ F t } at the end of round t, whereF t = {k|k + d k -1 = t}.Since the set F t may not contain the round t, the vanilla BGD in (4) is no longer valid. To address this issue, GOLD [Héliou et al., 2020 ] replaces f t (x t ) in (4) with the oldest received but not utilized loss value at the end of round t. Intuitively, the update of this approach is O(d) rounds slower than that of the vanilla BGD, which is analogous to those delayed OCO algorithms. However, due to the use of the one-point gradient estimator, the slower update causes a difference of O(ηdn/δ) between its action and that of BGD, and the cumulative difference will bring additional regret of O(T ηdn/δ), where a constant step size η t = η is discussed for brevity. Note that from the standard analysis of BGD, to control the total exploration cost, the value of 1/δ should be sublinear in T . Therefore, it will amplify the effect of delays, and finally results in the O( √ nT 3/4 + (nd) 1/3 T 2/3 ) regret [Héliou et al., 2020] .To reduce the effect of delays, we propose to incorporate the delayed bandit feedback with a blocking update mechanism [Zhang et al., 2019, Garber and Kretzu, 2020] . Specifically, we divide the total T rounds into T /K blocks, each with K rounds, where T /K is assumed to be an integer without loss of generality. For each block m ∈ [T /K], we only maintain a preparatory action y m ∈ K δ , and play x t = y m + δu t with u t ∼ S n at each round t in the block. Due to the randomness of u t and the independence of x t in the same block, it is not hard to verify that for each block m ∈ [T /K], the sum of randomized gradients generated by the one-point estimator, i.e., ∇ m = mK t=(m-1)K+1 n δ f t (x t )u t , satisfies (see Lemma 5 presented in Section 3.4 for details)E[∥∇ m ∥ 2 ] = O( Kn 2 /δ 2 + K).By using an appropriate block size of K = O(n2 /δ 2 ), this upper bound will be E[∥∇ m ∥ 2 ] = O(K). By contrast, without the blocking update mechanism, one can only achieve E[∥∇ m ∥ 2 ] = O(Kn/δ). Moreover, we notice that the cumulative estimated gradients ∇ m will be delayed at most O(d/K) Algorithm 1 Delayed Follow-The-Bandit-Leader 1: Input: δ, K, α, and η > 0 if α = 0 2: Initialization: set ḡ0 = 0 and choose y 1 ∈ K δ arbitrarily 3: for m = 1, 2, . . . , T /K do 4:for t = (m -1)K + 1, . . . , mK do 5:Play x t = y m + δu t , where u t ∼ S n 6:Query f t (x t ), and receive{f k (x k )|k ∈ F t } 7: Update ḡt = ḡt-1 + k∈Ft n δ f k (x k )u k 8:end for9: Set R m (x) = 1 η ∥x -y 1 ∥ 2 2 if α = 0 m i=1 Kα 2 ∥x -y i ∥ 2 2 otherwise10:y m+1 = argmin x∈K δ {⟨ḡ mK , x⟩ + R m (x)} 11: end for blocks, because even the last component n δ f mK (x mK )u mK is available at the end of round mK + d -1.As a result, one possible approach to determine y m for each block is to extend the update rule of GOLD [Héliou et al., 2020] into the block level with K = O(n 2 /δ 2 ). Combining with previous discussions, it will reduce the effect of delays on the regret from O(T ηdn/δ) toO T η d K Kn 2 δ 2 + K =O (ηdT )which is good enough for deriving our desired regret bounds. However, it requires a bit complicated procedure to maintain the cumulative estimated gradients for any block that has not been utilized to update the action. For this reason, instead of utilizing this approach, we incorporate FTRL [Hazan et al., 2007 , Hazan, 2016] with the delayed bandit feedback and blocking update mechanism, which provides a more elegant way to utilize the delayed information.Specifically, we initialize y 1 ∈ K δ arbitrarily, and use a variable ḡt to record the sum of gradients estimated from all received loss values, i.e., ḡt =t i=1 k∈Fi n δ f k (x k )u k .Then, according to FTRL, an ideal action should be selected by minimizing the linear approximation of cumulative loss functions under some regularization, i.e.,EQUATIONwhere the regularization is set as R m (x) = 1 η ∥x -y 1 ∥ 2 2 for convex functions [Hazan, 2016] andR m (x) = m i=1 Kα 2 ∥x -y i ∥ 22 for α-strongly convex functions [Hazan et al., 2007] . Unfortunately, due to the effect of delays, the value of m i=1 ∇ i required by (5) may not be available. To address this limitation, we generate y m+1 by replacing this term with the sum of all available estimated gradients, i.e., ḡmK . 2The detailed procedures are outlined in Algorithm 1, where the input α is the modules of the strong convexity of functions, and it is called delayed follow-the-bandit-leader (D-FTBL).We first present the regret bound of our D-FTBL for convex functions. Theorem 1. Under Assumptions 1, 2, and 3, Algorithm 1 with α = 0 ensuresE [Reg(T )] ≤ 4R 2 η + ηT γ 2K :=A + ηT G 2 2 d 2 K 2 + 4 γ :=B + 3δGT + δGRT r :=C (6) where γ = K nM δ 2 + K 2 G 2 .Remark. To help understanding the regret bound in (6), we notice that the term A actually is derived from the expected regret of the ideal action y * m on a sequence of surrogate losses, and the term B is caused by the cumulative distance between our preparatory action y m and the ideal one. Additionally, the term C is caused by the exploration error of the one-point gradient estimator. At first glance, it seems that term B suffers a multiplicative joint effect of the maximum delay d and the exploration radius δ due to the existence of γ. However, as discussed before, this joint effect can be decoupled by setting an appropriate block size of K = O(n 2 /δ 2 ), which allows us to derive an improved regret bound. Specifically, by substituting α = 0, K = n√ T , η = 1/ max{ √ T d, √ nT 3/4 }, and δ = c √ nT -1/4 into (6), where c is a constant such that δ < r, our D-FTBL can enjoyEQUATIONfor convex functions. 3 It is tighter than the O( √ nT 3/4 + (nd) 1/3 T 2/3 ) regret of GOLD [Héliou et al., 2020] , and matches the O( √ nT 3/4 ) regret bound of BGD in the non-delayed setting as long as d is not larger than O(n √ T ). Even for d = Ω(n √ T ), our regret bound is dominated by the O( √ dT ) part, which matches the Ω( √ dT ) lower bound [Bistritz et al., 2022] in the worst case. Moreover, Bistritz et al. [2022] could benefit from a small average delay, it is also worse than our regret bound when d is not larger than O((n d) 2/3 T 1/3 ).although the O( √ nT 3/4 + (n d) 1/3 T 2/3 ) regret bound ofThen, we establish an improved regret bound for α-strongly convex functions. Theorem 2. Under Assumptions 1, 2, and 3, if all functions are α-strongly convex, Algorithm 1 with α > 0 ensuresEQUATIONwhereγ = K nM δ 2 + K 2 G 2 , C T = 1 + ln T , and C ′ T = 6 + 4 ln T .Remark. By comparing Theorem 2 with Theorem 1, we find that the strongly convexity can be exploited to reduce the expected regret of the ideal action y * m on the surrogate losses, and the cumulative distance between our preparatory action y m and the ideal one, i.e., improving terms A and B in (6) to terms A ′ and B ′ in (8). By further substituting α > 0, K = (nT ) 2/3 ln -2/3 T , and δ = cn 2/3 T -1/3 ln 1/3 T into (8), where c is a constant such that δ < r, our D-FTBL can enjoyE [Reg(T )] ≤O (nT ) 2/3 log 1/3 T + d log T (9)for strongly convex functions. This regret bound is tighter than the above O( √ nT 3/4 + √ dT ) regret bound achieved by only utilizing the convexity condition, and can match the O((nT ) 2/3 log 1/3 T ) regret bound of BGD in the non-delayed setting as long as d is not larger than O((nT / log T ) 2/3 ). Even if d = Ω((nT / log T ) 2/3 ), it is dominated by the O(d log T ) part, which matches the Ω(d log T ) lower bound [Weinberger and Ordentlich, 2002] , and thus cannot be improved. Moreover, different from the case with convex functions, the parameters for achieving the bound in ( 9) do not require the information of delays. Furthermore, we consider the unconstrained case, i.e., K = R n , with α-strongly convex and βsmooth functions, and extend our D-FTBL to achieve a better regret bound. Specifically, without the boundedness of K, Assumptions 2 and 3 may no longer hold over the entire space [Agarwal et al., 2010] . Therefore, we first introduce a weaker assumption on the Lipschitz continuity, i.e, all loss functions are G-Lipschitz at 0. Combining with (1), it is not hard to verify that the fixed optimal actionEQUATION)As a result, the player only needs to select actions from the following setK ′ = x ∈ R n ∥x∥ 2 ≤ 2G α (11)which satisfies Assumption 1 with r = R = 2G/α, and it is natural to further assume that all loss functions satisfy Assumptions 2 and 3 over the set K ′ . Now, we can apply our D-FTBL over the shrink set of K ′ , i.e.,K ′ δ = (1 -δ/r)K ′ = 1 - αδ 2G K ′ (12)instead of the original K δ , and establish the following regret bound. Theorem 3. Let K = R n . If all loss functions are α-strongly convex and β-smooth over K, and Assumptions 2 and 3 hold over K ′ defined in (11), applying Algorithm 1 with α > 0 over  and C ′ T = 6 + 4 ln T . Remark. By comparing Theorem 3 with Theorem 2, we find that the exploration error of the one-point gradient estimator is reduced, i.e., improving the last two terms in (8) to the term C ′ in (13). Then, by substituting α > 0, K = n T / ln T , and δ = cn 1/2 T -1/4 ln 1/4 T into (13), where c is a constant such that δ < 2G/α, we can achieve an O n √ T log T + d log T regret bound for strongly convex and smooth functions in the unconstrained case. It is better than the O((nT ) 2/3 log 1/3 T + d log T ) regret bound achieved by only utilizing the strong convexity. Moreover, this bound matches the O(n √ T log T ) regret bound achieved by using BGD in the non-delayed setting as long as d is not larger than O(n T / log T ). Otherwise, it is dominated by the O(d log T ) part, which cannot be improved as discussed before.EQUATION)where γ = K nM δ 2 + K 2 G 2 , C T = 1 + ln T ,Due to the limitation of space, here we only prove Theorem 1, and the omitted proofs can be found in the appendix. Specifically, let x * = (1 -δ/r)x * where x * ∈ argmin x∈K T t=1 f t (x), and recall the ideal action defined in (5). As in Lemma 2, we first notice that the expected regret of Algorithm 1 can be bounded by the sum of three parts including the expected regret of ideal actions on some surrogate losses, the cumulative distance between y m and the ideal one, and the exploration error of the one-point gradient estimator. Lemma 2. Under Assumptions 1 and 2, Algorithm 1 with α = 0 ensuresEQUATIONNote that the part regarding the exploration error in ( 14) is exactly the same as the term C in (6). So, we only need to analyze the first two parts in ( 14). For the first part, we define surrogate losses as ℓ 1 (x) = ⟨∇ 1 , x⟩ + 1 η ∥x -y 1 ∥ 2 and ℓ m (x) = ⟨∇ m , x⟩ for any m = 2, . . . , T /K. Combining with (5) for convex functions, it is easy to verify that y * m+1 = argmin x∈K δ m i=1 ℓ i (x). Then, we introduce the following lemma to bound the regret of y * 2 , . . . , y * T /K+1 on ℓ 1 (•), . . . , ℓ T /K (•). Lemma 3. (Lemma 6.6 in Garber and Hazan [2016] ) Let {ℓ t (x)} T t=1 be a sequence of functions over a set K, and letx * t ∈ argmin x∈K t i=1 ℓ i (x) for any t ∈ [T ]. Then, it holds that T t=1 ℓ t (x * t ) - min x∈K T t=1 ℓ t (x) ≤ 0.Specifically, by applying Lemma 3, we haveT /K m=1 ℓ m (y * m+1 ) - T /K m=1 ℓ m (x * ) ≤ 0.Combining this inequality with Assumption 1, we haveEQUATIONMoreover, to replace y * m+1 in the left side of ( 15) with y * m , we introduce the following lemma.Lemma 4. (Lemma 5 in Duchi et al. [2011] )Let Π K (u, η) = argmin x∈K ⟨u, x⟩ + 1 η ∥x∥ 2 2 . We have ∥Π K (u, η) -Π K (v, η)∥ 2 ≤ η 2 ∥u -v∥ 2 .Combining Lemma 4 with (5) for convex functions, we haveEQUATION)Then, combining ( 15) with ( 16), we haveEQUATION)We notice that the term ∥∇ m ∥ 2 2 in (17) can directly benefit from the blocking update mechanism, as shown by the upper bound in the following lemma. Lemma 5. Under Assumptions 2 and 3, for any m ∈[T /K], Algorithm 1 ensures E[∥∇ m ∥ 2 2 ] ≤ K nM δ 2 + K 2 G 2 .However, to completely bound the right side of ( 14), we still need to analyze ∥y m -y * m ∥ 2 , which is more complicated due to the effect of delays. Specifically, letEQUATIONbe the set consisting of the time stamp of loss values that are queried but still not arrive at the end of round (m -1)K. By using Lemma 4 again, we haveEQUATIONMoreover, we establish the following lemma regarding the right side of (19). Lemma 6. Under Assumptions 2 and 3, for any m ∈ [T /K], Algorithm 1 ensuresE   t∈Um n δ f t (x t )u t 2 2   ≤ 2 d 2 K 2 + 4 K nM δ 2 + K 2 G 2 .Combining ( 14), ( 17), ( 19), Lemma 5, Lemma 6, andγ = K nM δ 2 + K 2 G 2 , we have E [Reg(T )] ≤ 4R 2 η + E   η 2 T /K m=1 ∥∇ m ∥ 2 2   + KG T /K m=1 E [∥y m -y * m ∥ 2 ] + 3δGT + δGRT r ≤ 4R 2 η + ηT γ 2K + ηT G 2 2 d 2 K 2 + 4 γ + 3δGT + δGRT r .Question: For each theoretical result, does the paper provide the full set of assumptions and a complete (and correct) proof?Question: Does the paper provide open access to the data and code, with sufficient instructions to faithfully reproduce the main experimental results, as described in supplemental material? Answer: [No] Justification: Any interested people can send the authors an email to query the source code. Guidelines:• The answer NA means that paper does not include experiments requiring code. 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experiments: 
 In this section, we compare our D-FTBL against GOLD [Héliou et al., 2020] and improved GOLD [Bistritz et al., 2022] by conducting simulation experiments on two publicly available data sets-ijcnn1 and SUSY from the LIBSVM repository [Chang and Lin, 2011] . All algorithms are implemented with Python, and tested on a laptop with 2.4GHz CPU and 16GB memory. Specifically, we randomly select T = 40000 examples from the original data sets, and consider online binary classification over a convex set K = {x ∈ R n |∥x∥ 2 ≤ 50}. The dimensionality of ijcnn1 and SUSY are n = 22 and n = 18, respectively. In each round t ∈ [T ], the adversary chooses the hinge loss f t (x) = max 1 -y t w ⊤ t x, 0 where w t and y t ∈ {-1, 1} are the feature vector and class label of the t-th example, respectively. Different values of the maximum delay d in the set {200, 600, 1000, . . . , 5000} have been tried in our experiments. For each specific d, to simulate arbitrary delays, d t is independently and uniformly sampled from [d] . In this way, the average delay d is equal to (d + 1)/2 in expectation, and thus is close to the maximum delay.According to the previous discussions about Theorem 1, we set α = 0, K = ⌊n √ T ⌋, δ = c √ nT -1/4 , and η = c ′ / max{ √ T d, √ nT 3/4 } for our D-FTBL by tuning these two constants c and c ′ . For those two baselines, we only need to set parameters δ and η. In addition to the theoretically suggested value of δ and η, we also introduce c and c ′ as the scale factor, respectively. For all algorithms, c and c ′ are respectively selected from {0.1, 1.0, 10} and {0.01, 0.1, . . . , 100} simply according to their performance for d = 200. Moreover, due to the randomness of these algorithms, we repeat them 20 times and report the average of their total loss. Fig. 1 shows the results of all algorithms on both data sets. We first find that when d increases from 200 to 5000, the total loss of our D-FTBL grows slowly, which is consistent with the dependence of our regret bound on d. It is worth noting that d = 5000 is larger than n √ T in our experiments. Second, from d = 600 to d = 5000, the total loss of our D-FTBL is better than both GOLD and improved GOLD, which verifies the advantage of our algorithm in the delayed setting. By contrast, due to d ≈ d, the performance of improved GOLD is very close to that of GOLD. Finally, we also notice that D-FTBL is slightly worse than baselines for d = 200. However, it is reasonable because the block update mechanism enlarges each delay to be at least the block size, which could result in a slightly larger constant factor in the regret. 
conclusion and future work: 
 In this paper, we investigate BCO with delayed feedback, and propose a novel algorithm called D-FTBL by exploiting the blocking update mechanism. Our analysis first reveals that it can achieve a regret bound of O( √ nT 3/4 + √ dT ) in general, which improves the delay-dependent part of the existing O( √ nT 3/4 + (n d) 1/3 T 2/3 ) regret bound as long as d is not larger than O((n d) 2/3 T 1/3 ). Furthermore, we consider the special case with strongly convex functions, and prove that the regret of D-FTBL can be reduced to O((nT ) 2/3 log 1/3 T + d log T ). Finally, if the action sets are unconstrained, we show that D-FTBL can be simply extended to enjoy the O(n √ T log T + d log T ) regret for strongly convex and smooth functions. Nonetheless, there still exist several open problems, which are discussed in the appendix due to the limitation of space. 
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 Question: Does the paper describe safeguards that have been put in place for responsible release of data or models that have a high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?Answer: [NA] Justification: This paper is mostly theoretical and poses no such risks.Guidelines:• The answer NA means that the paper poses no such risks.• Released models that have a high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere to usage guidelines or restrictions to access the model or implementing safety filters. • Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images. • We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make a best faith effort.12. Licenses for existing assets Question: Are the creators or original owners of assets (e.g., code, data, models), used in the paper, properly credited and are the license and terms of use explicitly mentioned and properly respected?Answer: [Yes] Justification: In Section 4, we have indicated that data sets are selected from the LIBSVM repository [Chang and Lin, 2011] .Guidelines:• The answer NA means that the paper does not use existing assets.• The authors should cite the original paper that produced the code package or dataset.• The authors should state which version of the asset is used and, if possible, include a URL. • The name of the license (e.g., CC-BY 4.0) should be included for each asset.• For scraped data from a particular source (e.g., website), the copyright and terms of service of that source should be provided.• If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswithcode.com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of a dataset. • For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided. • If this information is not available online, the authors are encouraged to reach out to the asset's creators. 
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 Appendix: First, we notice that all our regret bounds depend on the maximum delay, and thus it is natural to investigate whether these bounds can be further improved to be depending on the average delay. It seems highly non-trivial to obtain such results with our D-FTBL because the blocking update mechanism actually enlarges each delay to be at least the block size.Second, it is appealing to apply our algorithm to an emerging task-memory-efficient fine-tuning of large language models [Malladi et al., 2023 , Zhang et al., 2024] . The key insight is that our algorithm only requires the delayed loss value to update the model, and thus could achieve an asynchronous acceleration while avoiding memory costs of the backpropagation.Moreover, in the above task, we actually only need to handle a sequence of stochastic loss functions. Note that Agarwal and Duchi [2011] have shown that the delay only increases the regret of OCO with stochastic and smooth loss functions in an additive way, i.e., an O( √ T + d 2 ) regret bound. Thus, it is interesting to investigate whether the stochastic setting or an intermediate setting [Chen et al., 2024 , Wang et al., 2024a] can make delayed BCO easier in a similar way.Finally, it is also worth extending other BCO algorithms into the delayed setting, e.g., generalizing the algorithm of Saha and Tewari [2011] to improve the regret for smooth functions. However, a more complicated analysis is required because they utilize additional techniques, e.g., the self-concordant barrier [Nemirovski, 2004] .Recall the definition of x * and x * in the proof of Theorem 1. First, it is easy to verify thatEQUATIONwhere the first inequality is due to Assumption 2, and the second inequality is due to Assumption 1 and u t ∼ S n . Note that y 1 , . . . , y T /K in Algorithm 1 are computed according to approximate gradients of the δ-smoothed version of original functions, i.e., ft,δ (x) = E u∼B n [f t (x + δu)], ∀t ∈ [T ].Therefore, before utilizing the unbiasedness of these approximate gradients in Lemma 1, we introduce the following lemma regarding the connection between the original function and its δ-smoothed version over K δ defined in (3). Lemma 7. (Lemma 2.6 in Hazan [2016] ) Let f (x) : R n → R be α-strongly convex and G-Lipschitz over a set K satisfying Assumption 1. Its δ-smoothed version fδ (x) defined in (2) has the following properties:• fδ (x) is α-strongly convex over K δ ; • | fδ (x) -f (x)| ≤ δG for any x ∈ K δ ; • fδ (x) is G-Lipschitz over K δ .Combining (20) with the second property in Lemma 7, it is easy to verify that Reg(T ) ≤ ⟨∇ ft,δ (y m ), y m -x * ⟩ + 3δGT + δGRT r .From the above inequality, it is easy to verify thatReg(T ) ≤ T /K m=1 mK t=(m-1)K+1 ⟨∇ ft,δ (y m ), y * m -x * + y m -y * m ⟩ + 3δGT + δGRT r ≤ T /K m=1 mK t=(m-1)K+1 ⟨∇ ft,δ (y m ), y * m -x * ⟩ + KG T /K m=1 ∥y m -y * m ∥ 2 + 3δGT + δGRT r (22)where the last inequality is due to the last property in Lemma 7.Moreover, according to Lemma 1, we havemK t=(m-1)K+1 E ⟨∇ ft,δ (y m ), y * m -x * ⟩ = mK t=(m-1)K+1 E n δ f t (y m + δu t )u t , y * m -x * =E [⟨∇ m , y * m -x * ⟩]. Finally, we can complete this proof by first taking expectations of both sides in ( 22) and then substituting the above equality into the right side.Lemma 5 can be proved by simply following the proof of Lemma 5 in Garber and Kretzu [2020] . In the following, we first prove Lemma 6, and then include a simple proof of Lemma 5 for completeness.For brevity, let g t = n δ f t (x t )u t for any t ∈ [T ]. Since g 1 , . . . , g (m-1)K-d+1 must be available at the end of round (m -1)K, it is not hard to verify thatEQUATION)whereA k = {(k -1)K + 1, . . . , kK} ∩ U m . Because of |A k | ≤ K, for any k = m -1 -⌈d/K⌉, . . . , m -1, we have E   t∈A k g t 2 2   =E   t∈A k ∥g t ∥ 2 2 + i,j∈A k ,i̸ =j ⟨g i , g j ⟩   ≤|A k | nM δ 2 + E   i,j∈A k ,i̸ =j ⟨E[g i |y k ], E[g j |y k ]⟩   ≤K nM δ 2 + E   i,j∈A k ,i̸ =j ∥E[g i |y k ]∥ 2 ∥E[g j |y k ]∥ 2   ≤K nM δ 2 + (|A k | 2 -|A k |)G 2 ≤ K nM δ 2 + K 2 G 2 (24)where the first inequality is due to Assumption 3, and the third inequality is due to Assumption 2, Lemma 1, and the last property in Lemma 7.Combining ( 23) with ( 24), we haveE   t∈Um g t 2 2   ≤ 2 d 2 K 2 + 4 K nM δ 2 + K 2 G 2which completes the proof of Lemma 6.Additionally, let T m = {(m -1)K + 1, . . . , mK}. Following (24), it is easy to verify thatE ∥∇ m ∥ 2 2 = E   t∈Tm ∥g t ∥ 2 2 + i,j∈Tm,i̸ =j ⟨g i , g j ⟩   ≤ K nM δ 2 + K 2 G 2 (25)which completes the proof of Lemma 5.This proof is similar to that of Theorem 1, but requires some specific extensions to utilize the strong convexity. Note that by combining ( 21) in the proof of Lemma 2 with the strong convexity, we haveEQUATION)where T m = {(m -1)K + 1, . . . , mK}, x * = (1 -δ/r)x * , and x * ∈ argmin x∈K T t=1 f t (x). Moreover, by reorganizing ( 26) and inserting the ideal action y * m ,4 we haveReg(T ) -3δGT + δGRT r ≤ T /K m=1 t∈Tm ⟨∇ ft,δ (y m ), y * m -x * ⟩ + ⟨∇ ft,δ (y m ), y m -y * m ⟩ - α 2 ∥y m -x * ∥ 2 2 ≤ T /K m=1 t∈Tm ⟨∇ ft,δ (y m ), y * m -x * ⟩ - α 2 ∥y m -x * ∥ 2 2 + T /K m=1 KG∥y m -y * m ∥ 2 (27)where the last equality is due to Assumption 2 and the last property in Lemma 7.For brevity, we define surrogate losses as ℓ m (x) = ⟨∇ m , x⟩ + αK 2 ∥y m -x∥ 2 2 for m ∈ [T /K], and notice thatEQUATIONin the strongly convex case. Then, it is not hard to verify thatE   T /K m=1 t∈Tm ⟨∇ ft,δ (y m ), y * m -x * ⟩ - α 2 ∥y m -x * ∥ 2 2   =E   T /K m=1 t∈Tm n δ f t (y m + δu t )u t , y * m -x * - α 2 ∥y m -x * ∥ 2 2   =E   T /K m=1 ∇ m , y * m+1 -x * + y * m -y * m+1 - αK 2 ∥y m -x * ∥ 2 2   ≤E   T /K m=1 ℓ m (y * m+1 ) -ℓ m (x * )   + E   T /K m=1 ∥∇ m ∥ 2 ∥y * m -y * m+1 ∥ 2   ≤E   T /K m=1 ∥∇ m ∥ 2 ∥y * m -y * m+1 ∥ 2   (29)where the first equality is due to Lemma 1, and the last inequality is due to (28) and Lemma 3. From ( 27) and ( 29), we still need to bound ∥y m -y * m ∥ 2 and ∥y * m -y * m+1 ∥ 2 .To this end, we notice that y * m for any m = 2, . . . , T /K is equal toEQUATION)Similarly, for any m = 2, . . . , T /K, the action y m of Algorithm 1 with α > 0 is equal toEQUATION)Combining ( 30) and ( 31) with Lemma 4, for any m = 2, . . . , T /K, we have∥y m -y * m ∥ 2 ≤ 1 α(m -1)K ḡ(m-1)K - m-1 i=1 ∇ i 2 = 1 α(m -1)K t∈Um n δ f t (x t )u t 2 (32)where U m is defined in (18).Moreover, from (1), for any m = 2, . . . , T /K, we have∥y * m -y * m+1 ∥ 2 2 ≤ 2 αmK m i=1 ℓ i (y * m ) - m i=1 ℓ i (y * m+1 ) ≤ 2 αmK ℓ m (y * m ) -ℓ m (y * m+1 ) ≤ 2 αmK ⟨∇ m + αK(y * m -y m ), y * m -y * m+1 ⟩ ≤ 2 α(m -1)K (∥∇ m ∥ 2 + 2αKR) ∥y * m -y * m+1 ∥ 2where the first inequality is due to the definition of y * m , and the last inequality is due to Assumption 1. The above inequality further implies thatEQUATIONCombining ( 27), ( 29), (32), and (33), we haveE [Reg(T )] ≤E   T /K m=1 ∥∇ m ∥ 2 ∥y * m -y * m+1 ∥ 2   + T /K m=1 KGE [∥y m -y * m ∥ 2 ] + 3δGT + δGRT r ≤E[∥∇ 1 ∥ 2 ∥y 1 -y * 2 ∥ 2 ] + E   T /K m=2 2 ∥∇ m ∥ 2 2 + 2αKR∥∇ m ∥ 2 α(m -1)K   + G α(m -1) T /K m=2 E   t∈Um n δ f t (x t )u t 2   + 3δGT + δGRT r .Combining the above inequality with Assumption 1, Lemma 5, Lemma 6, andγ = K nM δ 2 +K 2 G 2 ,it is easy to verify thatE [Reg(T )] ≤2R √ γ + T /K m=2 1 m -1 2γ αK + 4R √ γ + G α 2 d 2 K 2 + 4 γ + 3δGT + δGRT r ≤2R √ γ + (1 + ln T ) 2γ αK + 4R √ γ + G α 2 d 2 K 2 + 4 γ + 3δGT + δGRT r = 2γC T αK + C ′ T R √ γ + GC T α 2 d 2 K 2 + 4 γ + 3δGT + δGRT rwhere C T = 1 + ln T and C ′ T = 6 + 4 ln T .The main idea of this proof is to combine the proof of Theorem 2 with an improved property of the δ-smoothed version of smooth functions [Agarwal et al., 2010] .Specifically, for any t ∈ [T ] and x, according to the smoothness of functions, we haveft,δ (x) ≤ E u∼B n f t (x) + ⟨∇f t (x), δu⟩ + βδ 2 ∥u∥ 2 2 2 = f t (x) + βδ 2 2 (34)where ft,δ (x) = E u∼B n [f t (x + δu)] and the last equality is due toE u∼B n [u] = 0.Moreover, due to the convexity of functions, for any t ∈ [T ] and x, we haveEQUATIONThen, let x * = argmin x∈R n T t=1 f t (x) and x * = (1 -δ/r)x * , where r = 2G/α. According to (10), we have x * ∈ K ′ and x * ∈ K ′ δ , where K ′ and K ′ δ are defined in ( 11) and ( 12), respectively. By further defining T m = {(m -1)K + 1, . . . , mK}, it is not hard to verify thatEQUATIONwhere the first inequality is due to the smoothness of functions, and the last inequality is due to (34) and ( 35).Then, we follow the definition of y * m in (28), but replace K δ utilized in (28) with K ′ δ . Combining (36) with the strong convexity of functions, we haveEQUATIONwhere we simply set y * 1 = y 1 , and the last inequality is due to Assumption 2 and the last property in Lemma 7.Let R = 2G/α denote the radius of K ′ . It is not hard to verify that (29), (32), and (33) in the proof of Theorem 2 still hold here. Therefore, we haveEQUATIONwhere the first inequality is due to ( 37) and ( 29), and the last one is due to (32) and ( 33).Finally, combining (38) with Lemma 5, Lemma 6, andγ = K nM δ 2 + K 2 G 2 , we have E [Reg(T )] ≤2R √ γ + T /K m=2 1 m -1 2γ αK + 4R √ γ + G α 2 d 2 K 2 + 4 γ + βδ 2 T + βδ 2 GT α ≤2R √ γ + (1 + ln T ) 2γ αK + 4R √ γ + G α 2 d 2 K 2 + 4 γ + βδ 2 T + βδ 2 GT α = 2γC T αK + 2C ′ T G √ γ α + GC T α 2 d 2 K 2 + 4 γ + βδ 2 T + βδ 2 GT αwhere the last equality is due to R = 2G/α, C T = 1 + ln T , and C ′ T = 6 + 4 ln T .F A refined regret bound for Bistritz et al. [2022] From Theorem 4 of Bistritz et al. [2022] , their algorithm can achieve the following regret boundEQUATIONfor BCO with delayed feedback, where δ > 0 and η > 0 denote the exploration radius and the step size, respectively. Then, by further substituting Bistritz et al. [2022] have established the O(nT 3/4 + √ n d1/3 T 2/3 ) regret bound. However, we notice thatδ = max T -1/4 , T -1/3 d1/3 and η = min n -1 T -3/4 , n -1/2 T -2/3 d-1/3 into (39),EQUATION)where the equality holds withEQUATION) From (40), if n 2 T δ -2 ≥ n dT δ -1 , we have min δ>0,η>0 O δT + ηn 2 T δ 2 + 1 η + nη dT δ = min δ>0 O δT + n √ T δ = O √ nT 3/4 (42)where the last equality holds with δ = √ nT -1/4 .Note thatBistritz et al. [2022] actually only argue a regret bound of O(nT 3/4 + √ n d1/3 T2/3 ). However, as discussed in our Appendix F, it is not hard to derive this refined bound by tuning parameters more carefully.From the above discussions, one may replace m i=1 ∇i with the sum of all available ∇i. However, we find that simply utilizing ḡmK can attain the same regret, though they have a slight difference.One may notice that the step size for achieving this result depends on the maximum delay d, which may be unknown beforehand. Fortunately, as discussed in previous studies[Quanrud and Khashabi, 2015, Wan et al., 2024], there exists a standard solution-utilizing the "doubling trick"[Cesa-Bianchi et al., 1997] to adaptively estimate the maximum delay d and adjust the step size, which can attain the same bound as in (7).Note that the definition of y * m in (5) for strongly convex functions is only valid for m ≥ 2. For m = 1, we simply set y * 1 = y1. 


Decision: This work studies the BCO problem with delayed feedback and presents a novel FTRL algorithm that uses a blocking update mechanism to decouple the effect of the delayed feedback on the regret of the algorithm. Most reviewers find the papers presentation to be very good, that the proofs are sound and that the authors have done a good job with literature review and comparison to prior work. Most reviewers also find the blocking update idea to be novel and interesting and point out that it could have application in other problems. One reviewer has suggested that the authors include an empirical evaluation of their algorithm which the authors do in the rebuttal. The only raised concern during the discussion phase is that of how impactful the algorithm or technical contribution of this paper are, given that the analysis builds on the standard FTRL framework. I find that the blocking update is interesting enough by itself, as other reviewers have also pointed out, and that the technical contributions are good enough for NeurIPS. As such I would recommend this paper for acceptance. I encourage the authors to incorporate the empirical evaluation in the final version of the paper.

Reviews: review: paper_summary: paper_summary: The authors consider the problem of bandit convex optimization in the adversarial setting under delays. In each round, an adversary selects a convex function $f_t$, the optimizer selects an input $x_t$ and observes $f_t(x_t)$ with a delay of $d_t$ timesteps. The goal is to minimize regret with respect to the best action in hindsight.

Without any delay, the best regret is obtained by the zeroth order one-point flaxman updates. Prior work in the delayed setting adapts this algorithm to use the oldest available feedback to make the update. However, the observation that the authors make is that according to this scheme, information might become available much before it is used -- if there is older information that has not been used yet.

So instead, to minimize the gap between availability and utilization of information, the authors propose a blocking algorithm. Where at the end of each block, all available information so far is used to make the update. This enables them to get improved regret bounds.
strength_weakness: strength_weakness: **Significance** The practical impact of this theoretical toy seems limited, and the problem was of more interest a couple of years ago. In order to increase the reception, it would be instructive to connect the implications of these findings to other topics of recent interest or include some simulations
questions: questions: 1. Could the authors comment on what the problem looks like for the regular stochastic case with all $f_t = f$ for some fixed $f$, and how this might make the problem easier under delays?
2. Why do the authors conjecture that the lower bound depends on $\bar{d}$, whereas the upper bound depends on $d$? What improvements would be required to get these to match?
limitations: limitations: Yes
paper_summary: paper_summary: This paper studies the bandit convex optimization problem with delayed feedback, where the loss value of the selected action is revealed under an arbitrary delay. 

Previous work achieves $\mathcal{O}( \sqrt{n} T^{3/4} + (n\bar{d})^{1/3} T^{2/3})$ regret bound of this problem. The authors develop a novel algorithm and show that it improves the delay-related part to $\mathcal{O}( sqrt{dT})$ when $d$ the maximum delay, is close to $\bar{d}$, the average delay (specifically, strictly better when d = $\mathcal{O}( (n\bar{d})^{2/3} T^{1/3}  )$. The authors claim that the primary idea is to decouple the joint effect of the delays by incorporating the delayed bandit feedback with a blocking update mechanism, which reduces the correlation between recent delayed updates (otherwise there could be a $d^2$ term).
strength_weakness: strength_weakness: 1. The contribution of this work is quite concerning. It would be better for the authors to emphasis the contribution (either algorithmic or analytic) on improving the delayed feedback result for BCO problem in certain conditions that the max delay $d$ is close to $\bar{d}$? Some parts of the proof are rather straightforward and standard.
questions: questions: The questions are raised in the weakness section. I am willing to re-evaluate the scores if these questions are properly answered.
limitations: limitations: This paper is pure theoretical and does not have any limitations.
paper_summary: paper_summary: This paper studies the problem of bandit convex optimization with delayed feedback (where the feedback for round $t$ is delayed by an arbitrary number of rounds $d_t$).
For this problem, they show $O(\sqrt{n} T^{3/4} + \sqrt{d T})$ regret in general, $O((n T)^{2/3} \log^{1/3} (T) + d \log(T))$ regret for strongly-convex functions, and $O(n \sqrt{T \log(T)} + d \log(T))$ regret for smooth and strongly-convex functions in an unconstrained settting.
In these bounds, $n$ is the dimension, $T$ is the horizon and $d$ is the maximum delay.
strength_weakness: strength_weakness: - The paper only makes an improvement on state-of-the-art for certain delay sequences, i.e. when $d = O((n\bar{d})^{2/3} T^{1/3})$.
questions: questions: No questions at this point.
limitations: limitations: There is some discussion of the limitation due to the fact that the regret is given in terms of maximum delay rather than average delay.
paper_summary: paper_summary: The paper investigates the problem of bandit convex optimization (BCO) with delayed feedback, where the value of the action is revealed after some delay. The authors proposed an algorithm D-FTBL, and proved that it enjoys a regret bound of $O\left(\sqrt{n} T^{3/4}+\sqrt{d T}\right)$, closing the gap between the previous result and the lower bound on delay-dependent part. Furthermore, the proposed algorithm can improve the regret bound to $O\left((n T)^{2 / 3} \log ^{1 / 3} T+d \log T\right)$ for strongly convex functions, and if the action sets are unconstrained, the proposed algorithm can achieve an $O(n \sqrt{T \log T}+d \log T)$ regret bound for strongly convex and smooth functions.
strength_weakness: strength_weakness: The paper lacks numerical experiments. Besides this, I did not spot any significant weaknesses.
questions: questions: I would be curious about how much improvement can be made in practice. In bandit setting, people often consider large time horizon, in which cases the regret is dominated by the delay-independent term.
limitations: limitations: The authors addressed the limitations and consider it as future work.
paper_summary: paper_summary: The paper investigates the online convex optimization problem with delayed bandit feedback.

The main contribution is introducing a new algorithm that uses a block updating mechanism with FTRL, proving the algorithm achieves a delay-dependent regret of $O(\sqrt{dT})$, which is known to be optimal when the average delay is close to the maximal delay: $\bar{d} \approx d$.
strength_weakness: strength_weakness: * The paper could be written more clearly. I found myself understanding parts of the introduction only after going through the entire paper (e.g. line 69). 
* The proofs are hard to follow. Some explanations before each lemma to understand how those are used would be helpful.
* It seems hard to use the blocking updates to further improve the regret, so the value of the presented algorithm could be temporary.
questions: questions: Can you explain each term in the theorems and where does it come from? As is, the theorems are packed with different terms and besides substituting the optimal values I don't know how to interpret them
limitations: limitations: The authors properly acknowledge the limitations of their work.

Comments: title: Thanks for the clear response
Thank you for the response
comment: Thank the authors for their response. I would like to keep the current score.
Thank you, I am keeping my score.
Thank you for the response. I will keep my score positive.
Thank you for the response, which I find quite helpful. I will stick with my positive score.
Thanks for the clear response. I'm sticking with my positive score.