Tight Lower Bounds for Binary First-Order Oracles for Convex Optimization

Amitabh Basu; Marco Molinaro; Phillip Kerger

REVIEW 2 major objections 4 minor 1 cited by

Tight Lower Bounds for Binary First-Order Oracles for Convex Optimization

T0 review · 2 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper proves that solving convex feasibility with bit-wise first-order oracles requires a number of queries quadratic in the number of continuous variables, establishing that the standard discretization of cutting-plane methods is esse

desk verdict A real Ω(d²) fixed-scale lower bound for bit and inner-product oracles, but the log(R/ρ) amplification is under-proved and the mixed-integer tightness claim overreaches. read the letter →

arxiv 2511.02082 v2 pith:6PMX2C77 submitted 2025-11-03 math.OC

Amitabh Basu , Phillip Kerger , Marco Molinaro This is my paper · ORCID

classification math.OC MSC 90C2590C1168Q17

keywords informationcomplexityoracleconvexfeasibilitymixed-integeroptimizationseparationbitinnerproductlowerbounds

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks how many bits of first-order information are needed to solve convex feasibility problems, a special case of mixed-integer convex optimization. It proves that under two 'bit-wise' oracles—one returning individual bits or whole coordinates of a separating hyperplane normal, the other returning only the sign of an inner product with that normal—any algorithm must make Ω(2^n d² log(R/ρ)) queries in the bit model and Ω(2^n d²(1 + log(R/ρ)/log d)) queries in the inner-product model. The quadratic dependence on the number of continuous variables d was previously unknown even for purely continuous convex optimization. These lower bounds match the natural strategy of approximating separating hyperplanes to enough bits and running standard cutting-plane methods, so they settle the bit complexity of first-order convex feasibility up to logarithmic factors. The result matters because it says that any algorithm using only bit-wise first-order information cannot beat the straightforward discretized cutting-plane approach.

What carries the argument

The key objects are two recursive adversarial-consistency strategies. For the bit oracle, orthant labeling and a 'commit to a standard basis vector' rule: after d/4 zero-answers in an orthant, all further queries there are answered using a fixed coordinate normal; an orthant-counting argument proves one orthant survives d²/16 queries. For the inner-product oracle, the split-set construction: the adversary answers zero for batches of queries, producing unit normals a_1,...,a_k orthogonal to all queried directions and to each other, so that all convex sets inside the intersection of the strips -R/√d < <a_i,y> < 0 are consistent; a geometric lemma shows this intersection contains a cube of side

What would settle it

For d=4, run the bit-oracle adversary for exactly d²/16 = 1 query in each of two adjacent orthants and then query a point on the boundary between them; check whether the adversary's commitment rule can produce a separating normal that is valid for both orthants' surviving instances. If no such normal exists, the consistency claim for off-orthant queries fails. Alternatively, for the inner-product oracle, verify the claimed ball containment: take a_1,...,a_k from the construction and a point y with ||y-u||_2 < R/(2√d); if any such point violates some split-set condition, the recursion over cube

Watch

Extended reading notes

Core claim

The central discovery is an adversarial oracle argument showing that for the class of convex sets in [-R,R]^{n+d} with ρ-fat feasible regions, the bit and inner-product oracles have information complexity quadratic in d. For the bit oracle (even when queries return an entire coordinate of the separating normal), the adversary answers '0' for the first d/4 queries in each orthant, then commits to a standard basis vector as the normal; a counting argument over orthants shows that after d²/16 queries there remains at least one orthant in which every instance is consistent with all replies. Repeating this inside the surviving orthant shrinks the search box by a constant factor each time, yieldin

Load-bearing premise

The recursive amplification assumes that whenever the algorithm queries a point outside the currently surviving orthant or cube, the oracle can answer with a separating hyperplane that remains consistent with all previous answers and all instances in the surviving region; the proof asserts this is possible but does not write down the formal consistency invariant.

Editorial extensions

If this is right

Any algorithm using bit-wise first-order oracles for constrained convex optimization must make at least quadratically many queries in the number of continuous variables, matching the d² bits used by standard discretized cutting-plane methods.
The lower bound holds even for the stronger oracles that return a full coordinate or a full inner product, so it is not an artifact of restricting to single bits.
For mixed-integer problems, the 2^n factor in the bound shows that the exponential dependence on the number of integer variables is inherent for bit-wise oracle models.
The recursive structure shows the log(R/ρ) scaling cannot be avoided by any query strategy that hopes to exploit partial bit information adaptively.
For the inner-product oracle, the amplification factor is only log(R/ρ)/log d, leaving a small log gap to the upper bound, but the quadratic d² baseline is tight.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A direct extension target is unconstrained minimization with M-Lipschitz convex objectives; the paper conjectures the analogous Ω(2^n d² log(MR/(ερ))) bound is attainable via its transfer result, but the feasibility argument alone does not prove it.
The unformalized consistency invariant for off-orthant and off-cube queries is the most natural spot to probe: a reader who can construct a query sequence where the 'repeat the argument' step produces contradictory separating normals would reduce the bound to a single-scale d² statement.
The results suggest that in oracle models where each query inherently reveals one real number or one bit, adaptive coordinate-wise exploration cannot outperform block-coordinate cutting-plane methods; this may inform the design of practical solvers.
The structure of the inner-product adversary—building orthogonal directions—mirrors lower-bound techniques for linear systems, indicating a possible transfer to linear programming feasibility with bit-wise oracles.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: the paper has a real result hiding in it — an Ω(d²) lower bound at fixed scale for both the bit and inner-product oracles — but the stated log(R/ρ) amplification is not established, and the mixed-integer tightness claim is oversold. The stress-test note is on target.

What's new: Previous best for arbitrary binary first-order queries was Ω(d^{8/7}) from the authors' earlier paper. Getting quadratic in d closes the gap to the natural d² upper bound up to logs, at least for continuous feasibility. The orthant-counting adversary (Thm 3) and the orthogonal split-set construction (Thm 4) are genuinely new and convincing at the single-scale level. The counting in Thm 3 — after d²/16 queries, at least one orthant stays consistent — checks out. The split-set argument in Thm 4, building orthogonal vectors and a surviving cube, is also plausible. So the fixed-scale part deserves serious attention.

The soft spot is the recursion. Both theorems claim a log(R/ρ) factor by repeating the argument inside a shrunken cube or orthant. But the oracle's responses to queries outside the surviving region are only hand-waved: 'simply repeat the argument' in Thm 3, and 'simply report inner products with canonical vectors' in Thm 4. That is not enough. In Thm 3, the first phase commits to a single normal s_i e_i for all future queries in the surviving orthant. Once you shrink to a subcube Q inside that orthant, points outside Q but still in the orthant (e.g., closer to the origin than Q) are not separated by s_i e_i. You'd need a different normal, but then the earlier commitment and any 0-answers at that same point constrain your choices. The paper never states the consistency invariant that would let the adversary answer new queries without contradicting earlier ones. Without it, you only get the single-scale d² bound, not the R/ρ dependence.

Same issue in Thm 4: if a point outside Q was previously answered with inner product 0 along some direction v, then a later canonical-vector normal must be orthogonal to v; that is not guaranteed by the separation-from-Q argument. The stress-test note's example is real.

Also, the abstract and intro claim tightness 'up to a logarithmic term' for mixed-integer convex optimization. That only holds for n=0 or fixed n; the upper bound for general n has an (n+d)² factor, so the lower bound in d and upper in (n+d)² are not tight when n is comparable to d. They should state the continuous case as tight and describe the n>0 setting as a gap.

Who is this for? People working on oracle complexity and communication complexity of convex optimization. The fixed-scale d² lower bound is a true advance. The paper deserves peer review — a serious referee might help repair the recursion or find a different way to get the R/ρ factor. But as it stands, the main theorems overclaim. My recommendation: send to review, flag the recursive consistency invariant as the key issue, and ask the authors to either prove it or soften the theorems.

Referee Report

2 major / 4 minor

Summary. The paper studies the information complexity of the feasibility problem for mixed-integer convex sets under two binary first-order oracles: the bit oracle, which returns a bit of a coordinate of a separating hyperplane normal, and the inner-product oracle, which returns the sign of an inner product of a queried direction with the normal. The main results, Theorems 3 and 4, claim lower bounds of Ω(2^n d² log(R/ρ)) for the bit/coordinate oracle and Ω(2^n d²(1 + log(R/ρ)/log d)) for the inner-product oracle on the class I_{n,d,R,ρ}, matching—up to logarithmic factors—the upper bounds previously given in [1]. The proofs are adversarial: in a first phase, an oracle strategy forces a surviving orthant (Theorem 3) or a surviving cube (Theorem 4) that is consistent with all answers; this phase is then iterated to obtain the dependence on R/ρ. Integer variables are handled by a fiber-wise argument. The paper also observes that the lower bounds hold even if the oracle returns a full coordinate of the normal or the full inner product, which strengthens the results.

Significance. If the main theorems are correct, they close a genuine gap in the oracle-complexity literature: for the natural bit-wise and inner-product first-order oracles, the dependence on the number of continuous variables is quadratic, not merely superlinear. This matches the naive bit-approximation strategy for cutting-plane methods and thus resolves the previously open question for constrained convex feasibility. The single-scale adversarial constructions in §2.1 and §2.2 are elegant and appear sound in their core counting arguments. The paper also usefully proves the lower bounds for the stronger full-coordinate and full-inner-product oracles, making the negative result more robust. The upper bounds from [1] show that the new lower bounds are tight up to logarithmic factors, so the overall contribution is significant. However, the recursive amplification step—which is essential for the R/ρ dependence—is not yet written at the level of rigor required for a journal proof.

major comments (2)

[§2.1, Theorem 3 and §2.2, Theorem 4] The recursive amplification to obtain the log(R/ρ) factors is asserted rather than proved. In §2.1 the text says 'one can simply repeat the argument (with appropriate translations)', and in §2.2 it says 'the oracle can simply report inner products with one of the canonical vectors ±e_1,...,±e_d' for queries outside the surviving cube. No formal invariant is stated that would guarantee these out-of-region responses remain consistent with all earlier answers and with every candidate instance in the surviving region. Because the stated lower bounds depend on the log(R/ρ) term, this is a load-bearing gap. The gap appears repairable: for an axis-aligned cube, any outside query can be separated from the cube by some coordinate normal, and since G(C,·) may depend on the query point, one can fix a normal per point. But this lemma and its interaction with the recursive phase need to be written do
[§2.2, Theorem 4] The split-set construction for the inner-product oracle assumes that each batch of queries can be assigned a fresh normal a_j. The proof does not address what happens when the algorithm re-queries a point that already has a fixed normal from an earlier batch. In that case the oracle must answer the new query using the previously fixed normal, not the current batch's a_j. For example, if a point queried in the first batch is later queried again with a new direction, the answer is v^T a_1, which need not be the 'report 0' answer required by the second-batch construction. The proof's statement 'For any subsequent queries, if they are made outside P_1 ∩ [−R,R]^d, the oracle simply reports inner products with a_1 or −a_1' handles some repeats, but the general invariant for points that move between the 'inside' and 'outside' categories is not established. A per-point invariant is needed to mak

minor comments (4)

[§2.1, Theorem 3] The proof uses quantities such as d/4 and d²/16 without ceilings or integrality assumptions. For d < 4, the threshold 'd/4 − 1' is negative and the adversarial strategy is undefined, even though the theorem states d ≥ 1. Use ceiling conventions and treat the finitely many small d separately, or restrict the theorem to d ≥ 4 with a separate small-d argument.
[§2.1, integer variables] The sentence 'we have C_1 ∩ C_2 ∩ (Z^n × R^d), i.e., C_1 and C_2 have no common mixed-integer point' is missing the equality symbol; it should be 'C_1 ∩ C_2 ∩ (Z^n × R^d) = ∅'. Also, the text uses log_3(R/(2ρ)) in one place and log_2(R/(2ρ)) in the next; the bases should be consistent.
[§2.2] In the description of the oracle responses for queries outside P_1 ∩ [−R,R]^d, the text says 'report inner products with a_1 or −a_1, depending on which side of P_1 the queried point is.' If the queried point lies inside P_1 but outside the box [−R,R]^d, neither a_1 nor −a_1 separates it from P_1 ∩ [−R,R]^d; a coordinate normal is needed. This is a special case of the missing invariant in the first major comment.
[General] There are minor typos, e.g., 'Similarly, let M_2 > 0 be such that such that' in §2.1. The title in the full text differs slightly from the arXiv title; please harmonize.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the new d^2 lower bounds are proved by explicit adversarial constructions; the recursive amplification gaps are proof omissions, not self-referential reductions.

full rationale

The derivation chain for Theorems 3 and 4 does not reduce to its inputs. The lower bounds are established by direct adversary arguments: §2.1 counts orthants after coordinate-oracle answers (0 answers, then committing to a canonical normal s_i e_i), and §2.2 constructs orthogonal split sets P_i and counts queries until a surviving cube is found. These constructions do not invoke the target lower bound as an assumption, do not fit parameters to data and relabel them as predictions, and do not rename an existing result. The paper does cite the authors' prior work [1] for the framework, upper bounds, and Theorem 1, but those citations are background/baseline; no load-bearing step in the new proofs is justified only by a self-citation containing the claimed result. The recursive log(R/rho) amplification in both proofs is asserted by 'simply repeat the argument' (§2.1) and by reporting canonical-vector inner products outside Q (§2.2), without a fully written invariant that all earlier answers remain realizable by one first-order map. That is a completeness/correctness gap in the proof, not circularity: the missing invariant would strengthen the analysis, but the analysis does not define the target quantity in terms of itself or treat a fitted input as a prediction. Therefore no circular step is exhibited, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No numbers are fitted and no new entities are postulated. Parameters R, ρ, d, n in the theorem statements are inputs to the problem class, not fitted constants. The main content is adversarial construction; the audit therefore records the domain model and standard facts it relies on, plus the under-formalized recursive-consistency step covered in weakest_assumption.

assumptions (5)

domain assumption Feasible instances are restricted to the box [-R,R]^{n+d} and feasible points are fat: an ℓ∞ ball of radius ρ is contained in C.
Needed: without these restrictions, even 1D continuous feasibility has infinite oracle complexity (Section 1, half-line example).
standard math Existence of separating hyperplanes with normal G for closed convex sets, as formalized by the first-order map.
Definition 1 and all queries assume the oracle can choose a separating normal whenever the query point is outside C.
domain assumption Adversarial consistency criterion: if two disjoint instances match all oracle replies for some first-order map, then no algorithm can be correct on both.
Section 2 restates this from [1]; it is the logical engine of both lower-bound proofs.
standard math In R^d, a nonzero vector can be orthogonal to at most d−1 prescribed vectors; m points with m−1 difference vectors and m query directions fit in a d−1 dimensional orthogonal complement.
Used in Theorem 4 to construct a_j orthogonal to query directions and point differences at each phase.
standard math Every query outside a surviving axis-aligned cube can be separated from the whole cube by some ±e_i.
Used in Theorem 4's recursion for queries not in Q; a cube is an intersection of axis-parallel halfspaces.

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Cite this review

Pith. "Pith review of Tight Lower Bounds for Binary First-Order Oracles for Convex Optimization." pith.science (2026). https://pith.science/paper/6PMX2C77

@misc{pith2026251102082,
  author       = {Pith},
  title        = {Pith review of: Tight Lower Bounds for Binary First-Order Oracles for Convex Optimization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6PMX2C77}},
  note         = {Machine review of arXiv:2511.02082}
}

read the original abstract

We establish new lower-bounds for the information complexity of mixed-integer convex optimization under two "bit-wise" oracles. The first oracle provides bits of first-order information in the standard coordinate model, and the second oracle answers whether the inner product of a specified vector with the gradient of the function at a point or the normal vector of a separating hyperplane for the feasible region is positive or non-positive, thus also providing one bit of first-order information. The new contribution is that under such oracles, the complexity is quadratic in the number of continuous decision variables, which was not known before even for continuous convex optimization. These new lower-bounds are tight (up to a logarithmic term), matched by a natural discretization of standard cutting-plane methods for convex optimization. These reveal that using a standard bit-representation of the first-order information is, in general, the best one can do with respect to the number of bits of information needed to solve constrained convex optimization problems.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Closing the Oracle-Complexity Gap in Derivative-Free Convex Optimization: A Near-Quadratic Lower Bound from Exact Function Values
math.OC 2026-07 conditional novelty 8.0 of 10 partial

Deterministic derivative-free convex optimization needs Ω(d²/log d) exact-value queries at accuracy Θ(d^{-1/2}), matching Protasov's 1996 upper bound up to log factors.

Reference graph

Works this paper leans on

8 extracted references · 2 linked inside Pith · cited by 1 Pith paper

[1]

Basu, A., Jiang, H., Kerger, P.A., Molinaro, M.S.: Information complexity of mixed-integer convex optimization. Math. Program. 210, 3--45 (2023)

2023
[2]

Proceedings of the 56th Annual ACM Symposium on Theory of Computing (2024)

Ghadiri, M., Lee, Y.T., Padmanabhan, S., Swartworth, W.J., Woodruff, D.P., Ye, G.: Improving the bit complexity of communication for distributed convex optimization. Proceedings of the 56th Annual ACM Symposium on Theory of Computing (2024)

2024
[3]

arXiv preprint arXiv:2203.15260 (2022)

Marsden, A., Sharan, V., Sidford, A., Valiant, G.: Efficient convex optimization requires superlinear memory. arXiv preprint arXiv:2203.15260 (2022)

arXiv 2022
[4]

Ekonomika i matem

Nemirovski, A.S., Yudin, D.B.: E fficient methods of solving convex-programming problems of high dimensionality. Ekonomika i matem. metody (in Russian) (1979)

1979
[5]

1986 25th IEEE Conference on Decision and Control pp

Tsitsiklis, J.N., Luo, Z.Q.T.: Communication complexity of convex optimization. 1986 25th IEEE Conference on Decision and Control pp. 608--611 (1986)

1986
[6]

ArXiv abs/1906.05832 (2019)

Vempala, S.S., Wang, R., Woodruff, D.P.: The communication complexity of optimization. ArXiv abs/1906.05832 (2019)

arXiv 1906
[7]

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ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'after.sentence := #3 '...
[8]

write newline

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Basu, A., Jiang, H., Kerger, P.A., Molinaro, M.S.: Information complexity of mixed-integer convex optimization. Math. Program. 210, 3--45 (2023)

2023

[2] [2]

Proceedings of the 56th Annual ACM Symposium on Theory of Computing (2024)

Ghadiri, M., Lee, Y.T., Padmanabhan, S., Swartworth, W.J., Woodruff, D.P., Ye, G.: Improving the bit complexity of communication for distributed convex optimization. Proceedings of the 56th Annual ACM Symposium on Theory of Computing (2024)

2024

[3] [3]

arXiv preprint arXiv:2203.15260 (2022)

Marsden, A., Sharan, V., Sidford, A., Valiant, G.: Efficient convex optimization requires superlinear memory. arXiv preprint arXiv:2203.15260 (2022)

arXiv 2022

[4] [4]

Ekonomika i matem

Nemirovski, A.S., Yudin, D.B.: E fficient methods of solving convex-programming problems of high dimensionality. Ekonomika i matem. metody (in Russian) (1979)

1979

[5] [5]

1986 25th IEEE Conference on Decision and Control pp

Tsitsiklis, J.N., Luo, Z.Q.T.: Communication complexity of convex optimization. 1986 25th IEEE Conference on Decision and Control pp. 608--611 (1986)

1986

[6] [6]

ArXiv abs/1906.05832 (2019)

Vempala, S.S., Wang, R., Woodruff, D.P.: The communication complexity of optimization. ArXiv abs/1906.05832 (2019)

arXiv 1906

[7] [7]

, " * write output.state after.block = add.period write

ENTRY address author booktitle chapter doi edition editor eid howpublished institution journal key month note number organization pages publisher school series title type url volume year label INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts #0 'before.all := #1 'mid.sentence := #2 'after.sentence := #3 '...

[8] [8]

write newline

" write newline "" before.all 'output.state := FUNCTION n.dashify 't := "" t empty not t #1 #1 substring "-" = t #1 #2 substring "--" = not "--" * t #2 global.max substring 't := t #1 #1 substring "-" = "-" * t #2 global.max substring 't := while if t #1 #1 substring * t #2 global.max substring 't := if while FUNCTION word.in bbl.in capitalize ":" * " " *...