arxiv: 2605.07239 · v1 · submitted 2026-05-08 · 💻 cs.LG · math.OC

Recognition: no theorem link

Sample Complexity of Stochastic Optimization with Integer Variables

Hongyu Cheng , Yinghao Zheng , Marco Molinaro , Amitabh Basu

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:05 UTC · model grok-4.3

classification 💻 cs.LG math.OC

keywords stochastic optimizationsample complexityinteger programmingmixed-integer nonlinear optimizationstatistical complexityLipschitz objectivesstrong convexity

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The pith

Stochastic optimization over integers requires more samples than continuous versions when objectives are strongly convex and smooth, but can require fewer or the same under Lipschitz conditions.

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

The paper compares the number of samples needed to find an approximate solution in stochastic optimization problems that include integer decision variables versus purely continuous ones. It finds that the integer version matches the complexity of simple bound-constrained linear problems when objectives are Lipschitz and feasible sets lie in the infinity-norm ball. When feasible sets lie in the Euclidean ball, the integer version can need strictly fewer samples than the continuous nonconvex case in certain parameter regimes. For strongly convex and smooth objectives, however, the integer version demands quadratic dependence on 1/ε while the continuous version needs only linear dependence.

Core claim

Integer stochastic optimization can require strictly more samples or strictly fewer samples than its continuous counterpart, depending on objective structure and constraint geometry. For Lipschitz objectives over subsets of the ℓ∞ ball, the statistical complexity equals that of stochastic linear optimization with bound constraints. For Lipschitz objectives over subsets of the ℓ2 ball, integer optimization can require strictly smaller sample size than the continuous setting. For strongly convex smooth objectives, integer optimization requires Ω(1/ε²) samples for an ε-approximate solution while continuous optimization requires only O(1/ε).

What carries the argument

Comparison of sample complexity bounds (upper and lower) for ε-approximate solutions between stochastic mixed-integer nonlinear programs and their continuous relaxations, under Lipschitz or strong-convexity/smoothness assumptions.

If this is right

Lipschitz objectives over ℓ∞-ball subsets have the same sample complexity as bound-constrained stochastic linear optimization.
Lipschitz objectives over ℓ2-ball subsets admit regimes where integer variables reduce the required sample size relative to continuous nonconvex optimization.
Tight sample-complexity bounds are established for nonconvex continuous stochastic optimization as a supporting result.
Strongly convex smooth objectives force integer stochastic optimization to quadratic sample dependence on 1/ε.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

In high-dimensional settings with Euclidean constraints, integrality constraints may implicitly reduce effective variance in gradient estimates.
Algorithm designers may prefer integer formulations over continuous relaxations in certain Lipschitz regimes to lower sampling costs.
The gap between integer and continuous complexities could guide when to solve the integer problem directly rather than via relaxation plus rounding.

Load-bearing premise

The objectives are either Lipschitz continuous or strongly convex and smooth, and the feasible sets are contained in ℓ∞ or ℓ2 balls.

What would settle it

A construction or algorithm showing that an ε-approximate solution for strongly convex smooth integer stochastic optimization can be found with O(1/ε) samples with high probability would falsify the Ω(1/ε²) lower bound.

read the original abstract

We establish sample complexity results for stochastic optimization over the integers, especially with a view to understand the complexity with respect to the corresponding continuous optimization problem. We show that integer optimization can sometimes require strictly more samples and sometimes strictly smaller number of samples, depending on the structure of the objective and constraints. 1. For Lipschitz objectives over subsets of the $\ell_\infty$ ball, the statistical complexity of general stochastic mixed-integer, nonlinear, nonconvex optimization is exactly the same as stochastic linear optimization with just bound constraints. 2. For Lipschitz objectives over subsets of the $\ell_2$ ball, we show that integer optimization can require strictly *smaller* sample size compared to the continuous setting in a certain regime. To get to this result, we also establish tight sample complexity results for nonconvex continuous stochastic optimization which, to the best of our knowledge, do not appear in prior work. 3. For strongly convex, smooth objectives, integer optimization has high statistical complexity compared to the continuous setting. In particular, we show that integer optimization requires $\Omega(1/\epsilon^2)$ samples to report an $\epsilon$-approximate solution, compared to the well-known $O(1/\epsilon)$ sample complexity from the continuous optimization literature.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Integer constraints can raise or lower sample needs in stochastic optimization depending on convexity and geometry, with new tight bounds for the continuous nonconvex case.

read the letter

The paper shows that integer variables in stochastic optimization can either increase or decrease the number of samples needed, depending on whether the objective is Lipschitz or strongly convex and on the shape of the constraint set. It backs this with new tight bounds for nonconvex continuous problems. What the paper does well is give explicit comparisons across regimes. For Lipschitz objectives on subsets of the l_infty ball, the sample complexity of general stochastic mixed-integer nonlinear nonconvex optimization matches that of simple stochastic linear optimization with bound constraints. For l2 balls, they prove that integer optimization can require strictly fewer samples than the continuous version in some cases, using new tight bounds for the continuous nonconvex stochastic optimization that fill a gap in the literature. For strongly convex smooth objectives, they establish an Omega(1/eps^2) lower bound for integer optimization versus the known O(1/eps) for continuous, via an information-theoretic construction that preserves the convexity and smoothness properties while requiring any algorithm to distinguish integer points. The lower bound construction appears solid and avoids common pitfalls like hidden dimension dependence. The other bounds are matching upper and lower where stated. Assumptions such as Lipschitz or strong convexity and smoothness are standard, and the sets are in l_infty or l2 balls, which is fine for the claims. There are no major soft spots. The results are case-by-case, which is appropriate, and they clearly state the conditions. This paper is for people working on sample-efficient algorithms for discrete or mixed-integer stochastic problems, or on nonconvex stochastic optimization theory. A reader looking for when discreteness affects statistical rates would find it useful. I would bring this to a reading group. It deserves serious peer review because the separations are concrete and the new continuous bounds are a solid addition.

Referee Report

0 major / 2 minor

Summary. The paper establishes sample complexity results for stochastic optimization with integer variables, comparing them to the continuous setting. For Lipschitz objectives over subsets of the ℓ_∞ ball, the complexity matches stochastic linear optimization with bound constraints. For Lipschitz objectives over ℓ₂-ball subsets, integer optimization can require strictly fewer samples than continuous in certain regimes, supported by new tight bounds for nonconvex continuous stochastic optimization. For strongly convex smooth objectives, integer optimization requires Ω(1/ε²) samples for an ε-approximate solution versus the O(1/ε) rate in the continuous case.

Significance. If the results hold, the work clarifies when integrality constraints increase, decrease, or leave unchanged the statistical complexity relative to continuous relaxations, depending on objective structure and feasible-set geometry. Strengths include the explicit information-theoretic lower-bound constructions that preserve strong convexity, smoothness, and ℓ₂-ball constraints, the matching upper bounds, and the novel tight sample-complexity results for nonconvex continuous stochastic optimization.

minor comments (2)

[Abstract] Abstract: the statement that integer optimization 'can require strictly smaller sample size' in the ℓ₂ regime would benefit from a brief indication of the parameter range or dimension scaling where this separation occurs.
[Introduction] The new continuous nonconvex bounds are presented as not appearing in prior work; a short comparison table or explicit citation of the closest existing results would help readers assess novelty.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the results, and recommendation of minor revision. The report correctly captures the comparisons between integer and continuous stochastic optimization sample complexities across different objective structures and feasible-set geometries.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central results rest on explicit information-theoretic lower-bound constructions that preserve strong convexity, smoothness, and ball constraints while forcing distinction among integer points, together with matching upper bounds derived from standard stochastic optimization oracles. These steps are independent of the target sample-complexity statements and do not reduce to self-defined quantities, fitted parameters renamed as predictions, or load-bearing self-citations. The continuous-case comparisons invoke well-known external results, and the nonconvex continuous bounds are presented as new but derived directly from the same oracle model without circular reduction. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The results rest on standard domain assumptions from optimization theory rather than new free parameters or invented entities.

axioms (2)

domain assumption Objectives are Lipschitz continuous over subsets of the ℓ∞ or ℓ2 ball
Invoked for the first two main results on sample complexity equivalence or reduction.
domain assumption Objectives are strongly convex and smooth
Required for the third result showing higher complexity for integers.

pith-pipeline@v0.9.0 · 5523 in / 1225 out tokens · 44656 ms · 2026-05-11T02:05:58.175997+00:00 · methodology

discussion (0)

Reference graph

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