Title: Fast Rates in Stochastic Online Convex Optimization by Exploiting the Curvature of Feasible Sets
Venue: NeurIPS 2024
Accepted: yes
Keywords: online convex optimization

online linear optimization

curvature of feasible sets

strongly convex sets

uniformly convex sets

universal online learning
Abstract: In this work, we explore online convex optimization (OCO) and introduce a new condition and analysis that provides fast rates by exploiting the curvature of feasible sets. In online linear optimization, it is known that if the average gradient of loss functions exceeds a certain threshold, the curvature of feasible sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a logarithmic regret. This study reveals that algorithms adaptive to the curvature of loss functions can also leverage the curvature of feasible sets. In particular, we first prove that if an optimal decision is on the boundary of a feasible set and the gradient of an underlying loss function is non-zero, then the algorithm achieves a regret bound of $O(\rho \log T)$ in stochastic environments. Here, $\rho > 0$ is the radius of the smallest sphere that includes the optimal decision and encloses the feasible set. Our approach, unlike existing ones, can work directly with convex loss functions, exploiting the curvature of loss functions simultaneously, and can achieve the logarithmic regret only with a local property of feasible sets. Additionally, the algorithm achieves an $O(\sqrt{T})$ regret even in adversarial environments, in which FTL suffers an $\Omega(T)$ regret, and achieves an $O(\rho \log T + \sqrt{C \rho \log T})$ regret in corrupted stochastic environments with corruption level $C$. Furthermore, by extending our analysis, we establish a matching regret upper bound of $O\Big(T^{\frac{q-2}{2(q-1)}} (\log T)^{\frac{q}{2(q-1)}}\Big)$ for $q$-uniformly convex feasible sets, where uniformly convex sets include strongly convex sets and $\ell_p$-balls for $p \in [2,\infty)$. This bound bridges the gap between the $O(\log T)$ bound for strongly convex sets~($q=2$) and the $O(\sqrt{T})$ bound for non-curved sets~($q\to\infty$).


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\n- This paper studies online convex optimisation on feasible sets with curvature conditions. They introduce a new way to exploit the curvature of the feasible set through sphere enclosure. For feasible sets satisfying this condition, they leverage results from universal online learning to show that there exists algorithms whose regret is logarithmic in $T$ (for stochastic environments). \n- Compared to prior work, the conditions of curvature on the feasible set are more general, the losses are not restricted to be linear (and conditions such as strong convexity of the losses can be simultaneously exploited) and a growth condition on the gradients is not needed. These results are also extended to stochastic corrupted environments. \n- For feasible sets that are uniformly convex, they show that there exists algorithm which can achieve regret bounds interpolating between the $\\log T$ rate of strongly-convex sets (q=2) and the $\\sqrt{T}$-rate of sets without curvature ($q \\rightarrow \\infty$) (in stochastic environments) while prior work suffered linear in $T$ regret when $q \\rightarrow \\infty$.\n\nSTRENGTHS:\n- Sphere-enclosure, a new interesting characterisation of the feasible set is introduced, which allows to obtain $\\log T$ regret with improvements with respect to prior work (see Summary) that are clearly explained. Sphere-enclosure captures previously considered sets with curvature as well as some other more general ones. Helpful figures are also provided to help understand and visualise sphere-enclosure.\n- The theoretical contributions are strong with upper and matching lower-bounds. \n- Exploiting the framework of universal online learning allows proofs that are generally simple to follow.\n- For uniformly convex sets, the interpolation of $\\log T$ and $\\sqrt{T}$ rates closes a gap left by prior work (in stochastic environments).\n\nWEAKNESSES:\n- The improvement over FTL is mostly in the stochastic environment. In Theorem 13, if we consider the fully adversarial environment ($C=T$) for example, then the regret is $\\sqrt{T}$ and the $\\log T$ results for FTL are not improved on. Do you believe this is a fault in the analysis or that something beyond the universal online learning conditions are required ?\n- Many of the results work with the $\\ell_2$-norm and it is not clear what the results would be when working with respect to an arbitrary norm $||\\cdot||$. In particular, it would be nice to at least have a discussion of if/how these results could be extended to arbitrary norms. It also seems that considering the euclidean ball in Definition 5 means that $\\ell_p$-balls for $p > 2$ are not sphere-enclosed sets. It would be good to have a discussion of the relationship between uniformly convex sets and sphere-enclosed sets as well as a discussion on how the results from Section 4.1 link to the results in Section 4.4.\n- The generality of the assumption of sphere-enclosure of the feasible set is well motivated (what is the mild condition on $\\nabla f^{\\circ}(x_\\star)$ in line 231 ?). But the assumption that $x^\\star$ is on the boundary of $K$ could be discussed further. In particular, it is only justified for linear losses. Perhaps a slightly more developed discussion of this could help (or/and more pointers to the literature). Moreover, it would be nice to have an explanation of why this assumption is needed - is it just to guarantee $||\\nabla f^{\\circ}(x_\\star)||_2 \\neq 0$?\n\nSome sections are slightly unclear and a bit confusing:\n- In the statement of Theorem 9, it is stated that $T \\geq C/(\\lambda L)^2$ (which is also stated in line 195) but in the proof the steps are: $T \\geq C/(\\lambda L) \\geq 1/(\\lambda L)^2$, which seems to suggest the condition is $T \\geq C/(\\lambda L)$ - is there a typo ? Furthermore, it is not clear why if the assumption does not hold, the lower-bound becomes vacuous ? Is it that the lower-bound of Theorem 8 then dominates the lower-bound of Theorem 9 ? In addition, the key step in line 207 seems to be that $|L-\\hat{L}|/L| \\leq 1$, which is not the step that is explained and seems non-trivial. In line 211 a square root appears on the ln term in the second inequality but is not present in the inequality before and after ? And then the final result in Theorem 9 has a square root on the ln term again making things quite confusion (and this seems to contribute to the confusion around the bound becoming vacuous compared to Theorem 8). Finally, beside the issue around the square-root on the ln term, it seems that Theorem 9 follows from line 211 by taking the average of the final two inequalities (of line 211) and if so this could be explicitly stated for clarity. Overall, this part is quite confusing and could be made clearer.\n- In line 224-225, you use the notation $x_t$ to discuss the definition of $B_\\gamma^K$ but $x_t$ is the notation used for the point played by the algorithm at time step $t$. It is unclear why this notation is used here (similarly for $\\nabla f^{\\circ}(x_t)$ in the statements of Theorem 8 and 9).\n- In line 249, the assumption on the loss vectors refers to the growth condition assumptions from line 42 ? If so, this could be made clearer, for example just adding “growth” in front of assumptions. This is discussed 2 paragraphs down but it is not necessarily clear at this point.\n- Line 251-254: am I correct in understanding that your approach guarantees a $O(\\sqrt{T})$ regret even in the adversarial setting because it satisfies the universal online learning condition ? If so, it would be nice to make this more explicit.\n\nTypos:\n- Line 222 missing a “the” in front of inequality. \n- Line 219, missing a “the” in front of ball.\n- Line 245: typo “$x_\\star$ in on corners\" ?\n- Line 30, the definition of the regret $R_T$ is written with $f_t(x)$ and not $f_t(x_\\star)$. A similar point for line 102 - perhaps defining the regret directly w.r.t. $x_\\star$ would make the notation clearer instead of with respect to an arbitrary point $x$ without making this dependence explicit in the notation $R_T$.\n- line 147: missing \"a\" in front of \"convex body K\".\n- Line 287: missing “the” in front of ball. \n- Line 286: \"for this $\\tilde{B}^K_\\gamma$\" is at the start of a sentence but not capitalised and $\\tilde{B}^K_\\gamma$ is not defined yet.\n- Line 20: you state that uniformly convex sets include $\\ell_p$-balls for $p \\in [1, \\infty)$ but this should be for $p \\in [2, \\infty)$ I think. (same issue in line 78).\n\nQUESTIONS:\nSome questions are asked in the weaknesses above.\n- In Corollary 12, do you think the dependence on $GD$ can be removed? Would it be possible to adjust the lower-bound so that these also appear ?\n- The upper bound in Theorem 10 is shown to be tight in the case of $K = W_\\lambda$ with $d=2$. Is there any dimension dependence that is hidden in some of the terms in the bound of Theorem 10 ? And could these be shown to be necessary by considering the lower-bound on $W_\\lambda$ in higher dimensions ?\n- For the result on uniformly convex sets, it is mentioned that a similar analysis is possible for corrupted stochastic environments. Without necessarily giving the details of the analysis (which I realise might be quite similar to the one already given), it would be nice to include a statement of the result in this case or at least what the dependence on the corruption level $C$ is in the bound and a bit of a discussion of this. In particular so that the reader can get an idea of the result in the fully adversarial setting when $C = T$ (which is the setting [10] consider it seems) and whether the $\\log T$-$\\sqrt{T}$ interpolation still holds in this case.\n\nLIMITATIONS:\nIn general yes, though it is not necessarily apparent that in the fully adversarial setting, the results do not improve upon FTL.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn this work, the authors study stochastic online convex optimization (SOCO), and establish a tighter regret bound of $O(\\log T)$ for sphere-enclosed sets. Specifically, such a regret bound is derived by exploiting an immediate result of several existing universal online learning algorithms. Moreover, the authors also demonstrate that the immediate result can be utilized to recover the existing regret bound for SOCO over strongly convex sets, and could improve the existing regret bound over uniformly convex sets. In addition, a corrupted setting is also considered, and lower bounds for non-corrupted and corrupted are also provided.\n\nSTRENGTHS:\nThis paper studies stochastic online convex optimization (SOCO), and mainly focuses on feasible sets with special curvature. Although this paper is not the first to study the problem, it has the following strengths compared with previous studies.\n1) Different from previous studies that need the strong convexity of sets to achieve an $O(\\log T)$ regret bound for SOCO, this paper introduces a new condition on the set to obtain the $O(\\log T)$ regret bound, i.e.,  sphere-enclosed sets.\n2) For SOCO over the uniformly convex sets, this paper has established a better regret bound than existing results.\n3) The authors also consider a corrupted setting that interpolates the fully adversarial and stochastic settings.\n4) Matching Lower bounds for a special case of ellipsoid sets are provided for the non-corrupted and corrupted settings.\n\nWEAKNESSES:\nAlthough this paper provides some new results for stochastic online convex optimization (SOCO), I have some concerns. First, compared with stochastic (offline) optimization, the main challenge of online convex optimization (OCO) is how to deal with a fully adaptive adversary. However, this paper mainly focuses on SOCO, where the adversary is even weaker than the oblivious one, which is not very interesting to me.\n\nSecond, although it seems that this paper has made many contributions, some results have already been achieved, and some results can be simply derived. \n1) The authors argue that previous studies [8,16] only consider online linear optimization (OLO) that may prevent us from leveraging the curvature of loss functions, and the existing FTL algorithm [8] will suffer an $\\Omega(T)$ regret if the ideal condition is not satisfied. However, in Section 4 of the existing work [8], Huang et al. have already proposed an adaptive algorithm to avoid the $\\Omega(T)$ regret of FTL, whose regret is at most $O(\\sqrt{T\\log T})$. The main idea is to utilize an expert tracking algorithm to combine FTL with an optimal algorithm for the general OCO. I also refer the authors to a missing related work [*], which provides an online lazy gradient descent algorithm, which enjoys the same regret bound as the algorithm of Huang et al.\n2) Note that universal online learning algorithms utilized in this work follow a very similar idea of using the expert tracking algorithm. Moreover, it is easy to combine FTL with universal online learning algorithms. Without the analysis provided in this paper, one can simply prove that such a combination can achieve the $O(log T)$ regret of FTL for SOCO under ideal conditions, and leverage the curvature of loss functions (and the convexity) to recover the optimal regret bounds for general OCO.\n3) The derivation of the lower bound in Theorem 8 seems to be very simple, i.e., just computing and replacing a constant of an existing lower bound with the gradient norm in the fixed optimal solution. Moreover, the extensions (about both the lower and upper bounds) to the corrupted setting do not bring noteworthy challenges.\n4) The sphere-enclosed sets seem to be a bit complicated. Moreover, as discussed by the authors it is very similar to the Bernstein condition, which is more common and has been used to achieve a tighter regret bound for SOCO. So, it is not surprising that the sphere-enclosed sets can also be used to derive similar results.\n\nThird, the writing and organization of this paper are not very sound.\n1) The context of Section 2.1 is almost the same as the first paragraph in Section 1.\n2) Many short proofs are interluded in the main text, which reduces the readability of this work.\n3) In line 340, \"a similar analysis ideas\" seems to be a typo.\n\n[*] Anderson and Leith. Online Lazy Gradient Descent is Universal on Strongly Convex Domains. In NeurIPS 2021.\n\n----After Rebuttal----\nAfter reading the authors' response, I realize that it is very hard (and even impossible) to combine FTL with an existing universal online learning algorithm. So, the analysis of universal online learning algorithms provided in this paper is valuable, and I increase my score to 5.\n\nQUESTIONS:\nBesides the weaknesses discussed above, I also have the following questions.\n1) Can the authors summarize the fully new contributions compared with existing studies?\n2) In Theorem 14, the authors exploit the curvature of loss functions and feasible sets simultaneously. Are those regret bounds better than the existing regret bounds derived by only exploiting the curvature of loss functions?\n\nLIMITATIONS:\nThe authors have discussed some limitations of this work.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-05"}

comments: SUMMARY:
The paper relates the regrets of (stochastic) online convex optimization to the geometry of the feasible set. Precisely, the paper proposes the sphere-enclosed-feasible-set property which measures the local curvature condition of the feasible set at the offline optimal solution, and shows that many existing online convex optimization algorithms, under this property, achieves regret O(\rho*\log(T)), the fast rate. In addition, similar regret upper bounds are shown to be valid in various settings, including the stochastic with corruption, convex (nonlinear) objectives, etc.

STRENGTHS:
The introduced sphere-enclosed-feasible-set property appears innovative, and provide insights into conditions under which fast rate is achievable.

WEAKNESSES:
The paper shows that many existing algorithms can converge at the fast rate under the proposed sphere-enclosed-feasible-set condition. Thus, one might argue that there is not too much improvement over previous works on the algorithmic side. In addition, the lower and upper bounds provided appear to hold under slightly different conditions (see the question below).

QUESTIONS:
I’m wondering if there is any connection between the \lambda-strongly convex feasible set and the sphere-enclosed feasible set? The lower bounds are provided for the case when the feasible sets are ellipsoid or \lambda-strongly convex, but the upper bounds are provided under the sphere-enclosed condition. It would be great to compare the lower and upper bound under the same set of conditions.

LIMITATIONS:
Yes, the paper points out that one limitation is that the main results hold under stochastic environment.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-13

comments: SUMMARY:
This paper considers the influence of feasible set geometry on the OCO regret. For a class of algorithms satisfying certain regret upper bounds (including MetaGrad and Maler), the authors provide new fast-rate results in the stochastic convex loss setting and the corrupted convex loss setting. Their results also match existing lower bound results.

STRENGTHS:
Most contributions of this paper are new in the OCO area, including:

1.A novel generalization of previous curvature-dependent regret for linear optimization to general stochastic convex loss.

2.New regret results in the corrupted setting.

3.A new result that simultaneously utilizes the curvature of the loss and the feasible set.

WEAKNESSES:
One possible limitation of the results is that, unlike the OLO results, the action set curvature-dependent rates for the convex losses in this paper don't hold for the adversarial setting.

QUESTIONS:
What's the main technical challenge in extending existing results for OCO from the stochastic setting to the stochastically extended adversary (SEA) setting or even the adversarial OCO setting? Could the authors explain more about the gap in simply applying the linearization technique in this setting?

LIMITATIONS:
N.A.
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 3
DATE: 2024-07-13