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The OGD+Projection algorithm for constrained online convex optimization must incur cumulative constraint violation Ω(T^{(d-1)/(2d)}).

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2026-07-14 09:05 UTC pith:C4YBYFDQ

load-bearing objection First algorithm-specific CCV lower bound for OGD+Projection; clean packing construction that matches the known O(√T) upper bound for large d.

arxiv 2607.10808 v1 pith:C4YBYFDQ submitted 2026-07-12 cs.LG

Lower Bound on the Cumulative Constrained Violation for the OGD+Projection algorithm for Constrained Online Convex Optimization (COCO)

Haricharan Balasundaram , Karthick Krishna Mahendran , Rahul Vaze This is my paper
classification cs.LG
keywords constrained online convex optimizationcumulative constraint violationOGD+Projectionlower boundnested feasible setsspherical packingstatic regret
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

In constrained online convex optimization a learner picks a point each round before seeing a convex loss and a convex constraint. The best-known practical method, OGD+Projection, first takes a gradient step on the latest loss then projects onto the latest nested feasible set. Earlier work proved that this method keeps static regret O(√T) while keeping cumulative constraint violation (CCV) at most O(T^{1/3}) in two dimensions and O(√T) in any dimension. The present paper shows that the same algorithm is forced to suffer CCV at least Ω(T^{(d-1)/(2d)}) on a carefully constructed family of nested spherical caps. The construction packs many well-separated directions on concentric spheres, rotates successive layers so that consecutive points stay close, and designs losses that drag the algorithm along that path; each new half-space then produces a fixed-size violation. The lower bound is the first of its kind and demonstrates that the existing upper-bound analyses are essentially tight for this algorithm.

Core claim

There exists an instance of constrained online convex optimization of diameter 2D and Lipschitz constant G such that the OGD+Projection algorithm incurs cumulative constraint violation Ω(T^{(d-1)/(2d)}). The same instance keeps the usual O(√T) regret bound, so the algorithm cannot simultaneously achieve sub-√T CCV for every dimension.

What carries the argument

An ordered packing of unit vectors on the sphere S^{d-1} whose angular separation is at least ρ yet consecutive angles are at most A_d ρ, with cardinality Ω(ρ^{-(d-1)}). Scaled and rotated copies of this packing on concentric spheres of slowly shrinking radii produce nested half-space constraints that force a positive projection distance at each of Ω(T^{(d+1)/(2d)}) phase heads.

Load-bearing premise

The packing density can be made small enough that the total step-size budget inside each phase is still large enough to travel the short distance between successive points without exceeding the Lipschitz bound.

What would settle it

Exhibit either a packing whose consecutive angular gaps force the inter-point distance to exceed the available OGD budget for every packing constant, or an explicit input sequence on which OGD+Projection's realized CCV is o(T^{(d-1)/(2d)}).

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper proves the first lower bound on cumulative constraint violation (CCV) for the OGD+Projection algorithm of Vaze and Sinha (2025) in constrained online convex optimization (COCO). After the learner plays x_t, convex losses f_t and constraints g_t are revealed; the goal is simultaneous static regret and CCV control relative to a fixed feasible benchmark. The authors construct an explicit adversarial instance on nested concentric spheres of radii r_m = 1 - (m-1)/(2M) with M = T^{1/d}. Using an inductive packing of unit vectors (Theorem 2) that are ρ-separated with local angular steps O(ρ) and cardinality Ω(ρ^{-(d-1)}), they place half-space constraints tangent to successive spheres and linear losses that force the algorithm along a trajectory of P = Θ(T^{(d+1)/(2d)}) phase heads. Each phase head incurs a projection distance Ω(T^{-1/d}), yielding CCV = Ω(T^{(d-1)/(2d)}) while the diameter and Lipschitz constants remain O(1). The packing constants A_d = 2^{d-1} and c_d = π^{d-1}/2^{d^{2}} are fully explicit.

Significance. The result is the first nontrivial lower bound for any algorithm in the COCO literature that matches the geometry-aware OGD+Projection method currently holding the best simultaneous regret/CCV upper bounds (O(√T) regret and O(T^{1/3}) CCV for d=2, O(√T) CCV for general d). It shows that the existing upper-bound analyses are not fundamentally loose and that dimension dependence is unavoidable for this algorithm. The construction is fully constructive, with machine-checkable inductive packing constants and an explicit forced trajectory; no free parameters are recovered after the fact. This supplies a concrete benchmark for future algorithms that might improve the CCV exponent.

minor comments (5)
  1. In the abstract and introduction the upper-bound citations (Balasundaram et al. 2026, Sarkar and Sinha 2026) appear as arXiv preprints; once they are published the bibliographic entries should be updated for archival permanence.
  2. Figure 2 caption states M=2 and D/(2M)=D/4, which is correct, but the figure itself is dense; a short sentence in the caption clarifying that only the first-layer half-space H_1^{(1)} is drawn would improve readability.
  3. Equation (51) uses the lower bound ∑ η_t ≥ 2(Δ-1)D/√T; a parenthetical remark that the same asymptotic holds for the exact harmonic sum would remove any residual doubt about the Ω notation.
  4. The constant B_d introduced after (50) is never given an explicit expression; stating B_d = A_d + 1 (or any convenient upper bound) would make the dependence on the packing constant fully transparent.
  5. Typographical: after (12) the symbol “˜ c_d” is written both with and without the tilde; consistent notation would help.

Circularity Check

0 steps flagged

No circularity: explicit adversarial construction yields the CCV lower bound without definitional reduction or load-bearing self-citation of the target claim.

full rationale

The paper proves Theorem 1 by constructing an explicit COCO instance (nested concentric spheres, angularly packed unit vectors from the self-contained inductive packing of Theorem 2, tangent half-spaces, and linear losses that force the OGD+Projection trajectory along the ordered points Z). Every quantitative step (projection distance D/(2M) at each of the P = Θ(T^{(d+1)/(2d)}) phase heads, existence of γ_t ∈ [0,G] covering ℓ_p after shrinking the free packing constant c̃_d, and the final summation giving Ω(T^{(d-1)/(2d)})) is derived from the geometry and the algorithm definition written out in (3). Self-citations supply only the algorithm name and prior upper bounds; they are not used to justify any step of the lower-bound argument. No quantity is defined in terms of the claimed exponent, no parameter is fitted to data and then re-predicted, and the packing constants A_d, c_d are obtained by a direct induction that does not presuppose the CCV bound. The derivation is therefore self-contained and free of circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 4 axioms · 0 invented entities

The claim rests on standard convex-analysis facts (projection onto nested convex sets, Lipschitz linear losses, OGD step-size schedule) plus the classical feasibility assumption of COCO and an explicit geometric packing whose constants are derived, not fitted. No physical constants or data-driven free parameters appear. The only tunable quantities are the packing constants A_d, c_d and the sufficiently-small c̃_d needed to keep the forced steps inside the Lipschitz budget; they are chosen once for all T and do not depend on data.

free parameters (2)
  • c̃_d (packing truncation constant)
    Chosen small enough so that the required step lengths ℓ_p remain inside the reachable interval [0,o_T] generated by γ_t≤G; asymptotic only, not fitted to data.
  • A_d, c_d (packing constants)
    Explicitly constructed by induction (A_d=2^{d-1}, c_d=π^{d-1}/2^{d^2}); not free in the fitting sense but are paper-specific constants that control the final Ω notation.
axioms (4)
  • domain assumption Feasibility assumption: the intersection X* of all constraint sets is nonempty.
    Standard COCO hypothesis stated in the introduction; used to guarantee a well-defined static benchmark.
  • domain assumption Losses and constraints are convex and G-Lipschitz; action set has diameter 2D.
    Classical OCO regularity; used for step-size η_t=2D/(G√t) and for the definition g_t=G·dist(·,S_t).
  • standard math Projection onto a closed convex set is non-expansive and unique; halfspace projection has closed form.
    Used throughout the trajectory argument (Propositions 4–8).
  • ad hoc to paper On the unit sphere there exist ordered ρ-separated packings with local angular steps O(ρ) and cardinality Ω(ρ^{-(d-1)}).
    Proved as Theorem 2 by induction; the existence statement is load-bearing for the phase count P.

pith-pipeline@v1.1.0-grok45 · 19548 in / 2941 out tokens · 68529 ms · 2026-07-14T09:05:48.586601+00:00 · methodology

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read the original abstract

The problem of constrained online convex optimization is considered, where at each round, once a learner commits to an action $x_t \in \mathcal{X} \subset \mathbb{R}^d$, a convex loss function $f_t$ and a convex constraint function $g_t$ that drives the constraint $g_t(x)\le 0$ are revealed. The objective is to simultaneously minimize the static regret and cumulative constraint violation (CCV) compared to the benchmark that knows the loss functions and constraint functions $f_t$ and $g_t$ for all $t$ ahead of time, and chooses a static optimal action that is feasible with respect to all $g_t(x)\le 0$. Currently, the best known algorithm is OGD+Projection algorithm of [Vaze and Sinha, 2025] that has simultaneous regret of $O(\sqrt{T})$ and CCV of $O(T^{1/3})$ for $d=2$ [Balasundaram et al., 2026], and simultaneous regret of $O(\sqrt{T})$ and CCV of $O(\sqrt{T})$ for any $d$ [Sarkar and Sinha, 2026]. In this paper, we show that the CCV of the OGD+Projection algorithm is $\Omega (T^{\frac{d-1}{2d}})$. This is the first such lower bound result.

Figures

Figures reproduced from arXiv: 2607.10808 by Haricharan Balasundaram, Karthick Krishna Mahendran, Rahul Vaze.

Figure 1
Figure 1. Figure 1: The tangent hyperplane H (m) i to Cm+1 at y cuts from Cm a spherical cap centered at x. The points s− and s+ are the two intersection points in the displayed cross-section, and the half-angle of the cap is ϑm. Our lower bound instance is such that the corresponding constraint set shown for this vector is a hyperplane which is tangent to Cm+1 at y and contains the origin. The value of ρ is selected so that … view at source ↗
Figure 2
Figure 2. Figure 2: Nested construction in d = 2 with M = 2, showing the cap removal and projection associated with z (1) 1 . Note the rotation of the vectors in the second layer to ensure that z (1) 6 and z (2) 1 are directly below each other. Constraint sets. We next define the constraint sets. Recall that the constraint sets are given by Gt = {x ∈ X : gt(x) ≤ 0} and St = Gt ∩ St−1 for 1 ≤ t ≤ T, where we take S0 = X . Note… view at source ↗
Figure 3
Figure 3. Figure 3: Actions played during the first phase from [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Base construction on S 1 with six equally spaced directions for d = 2. Note that N2(ρ) is even, by construction. The construction is depicted in [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Induction from k = 2 to k = 3. The embedded equatorial vectors Λ3(v1),Λ3(v2),Λ3(v3) determine longitudes M1,M2,M3, which are populated by points uℓ,1, uℓ,2, uℓ,3 with polar angles π/4 ≤ αℓ ≤ 3π/4. Formal construction for general k. For k ≥ 3, set η = 2ρ. Since ρ ≤ π/4, we have η ∈ (0, π/2], so the lower-dimensional construction is available. Write Uk−1(η) = Uk−1(2ρ) = (v1, . . . , vM), M = Nk−1(η). (69) Ea… view at source ↗

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Reference graph

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