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First-order methods for stochastic and finite-sum convex optimization with deterministic constraints

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Stochastic solutions can meet hard constraints surely at no extra cost

desk verdict A genuinely useful strengthening of the feasibility guarantee, with careful proofs; the infinite-sample result is conditional on a strong uniform gradient bound, and the numerical section is light. read the letter →

arxiv 2506.20630 v1 pith:Y5OLNMCN submitted 2025-06-25 math.OC cs.LGcs.NAmath.NA

Zhaosong Lu , Yifeng Xiao This is my paper
classification math.OCcs.LGcs.NAmath.NA MSC 90C1590C2590C3065K05
keywords stochasticconvexoptimizationfinite-sumdeterministicconstraintssurelyfeasibleoptimalsolutionquadraticpenaltyacceleratedgradientvariancereductionsampleaverageapproximation
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

Stochastic convex optimization with deterministic hard constraints usually settles for expected constraint violation. This paper asks for a certificate that the violation is at most epsilon with certainty, and shows it can be obtained by running an accelerated stochastic gradient scheme just once on a sequence of quadratic penalty subproblems. For infinite sample spaces the first-order oracle complexity is $O(\epsilon^{-2})$ gradient evaluations, the same order as for unconstrained stochastic optimization. For finite sums with a bounded number of constraints the complexity is $O(s\log s + \sqrt{s}\,\epsilon^{-3/2})$ for a surely feasible solution, or $O(s\log s + \sqrt{s}\,\epsilon^{-1})$ for the weaker expected-feasibility notion. The analysis also yields sample average approximation complexities under the same guarantees.

What carries the argument

The load-bearing object is the quadratic penalty subproblem $F_\rho(x)=F(x)+\tfrac{\rho}{2}\|[c(x)]_+\|^2$ together with Lemma 1's bound $\|[c(x)]_+\| \le 2\Lambda/\rho + \sqrt{2(F_\rho(x)-F^*_\rho)/\rho}$. This inequality shows that controlling the penalty suboptimality $F_\rho(x)-F^*_\rho$ and choosing a large enough $\rho$ yields a deterministic bound on the original constraint violation, and the paper selects $\rho$ as a power of the horizon $K$ so that the two terms balance. Algorithm 1 applies an accelerated stochastic gradient scheme once to a sequence of these penalty problems, while Algorithm 2 runs a modified variance-reduced accelerated scheme in which the stochastic estimator approximates only the finite-sum objective and the penalty gradient is computed exactly, keeping the noise confined to the objective component.

What would settle it

Run Algorithm 1 on a constrained stochastic convex problem satisfying all assumptions except the uniform bound $\max_{\xi\in\Xi}\|\nabla\tilde f(x,\xi)\|\le G$, for example $\tilde f(x,\xi)=\xi^\top x$ with $\xi$ standard normal so the stochastic gradient is unbounded, and check whether $\|[c(x_K)]_+\|\le\epsilon$ holds with probability one at the claimed $O(\epsilon^{-2})$ gradient budget; a single output with positive-probability violation above $\epsilon$ falsifies the SFSO claim.

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Extended reading notes

Core claim

The paper establishes that an $\epsilon$-surely feasible stochastic optimal ($\epsilon$-SFSO) solution, meaning a point $x$ with $\|[c(x)]_+\| \le \epsilon$ and $E[|F(x)-F^*|] \le \epsilon$, can be computed by a single loop of accelerated stochastic gradient steps applied to the quadratic penalty problems $F_\rho(x)=E[\tilde f(x,\xi)]+\psi(x)+\tfrac{\rho}{2}\|[c(x)]_+\|^2$ with suitably chosen penalty parameters $\rho$. For infinite sample spaces, Algorithm 1 achieves first-order oracle complexity $O(\epsilon^{-2})$ with constant penalty $\rho=K^{3/2}$ and $\tilde O(\epsilon^{-2})$ with dynamic penalties, where the oracle counts gradient evaluations during a single pass. For finite sums of size $s$, Algorithm 2 uses a variance-reduced estimator for the average of the $f_i$ and an exact gradient for the penalty term, and under $m=O(1)$ finds an $\epsilon$-SFSO solution in $O(s\log s + \sqrt{s}\,\epsilon^{-3/2})$ gradient evaluations, with the weaker $\epsilon$-expectedly feasible variant at $O(s\log s + \sqrt{s}\,\epsilon^{-1})$. The proof transfers stochastic suboptimality of the penalty problem into a deterministic feasibility bound through Lemma 1, so the noise only enters the optimality gap while the constraint violation stays deterministic.

Load-bearing premise

The deterministic constraint-violation guarantee depends on a uniform almost-sure bound on the norm of the stochastic gradient over the whole feasible domain; without that bound the method's output can only be certified in expectation or with high probability, not surely.

Editorial extensions

If this is right

  • Infinite-sample stochastic convex problems with deterministic constraints admit $\epsilon$-SFSO solutions at $O(\epsilon^{-2})$ gradient evaluations, the same order as the unconstrained case, so the constraints cost nothing in the epsilon-dependence.
  • For finite sums with $m=O(1)$, Algorithm 2 finds $\epsilon$-SFSO solutions in $O(s\log s + \sqrt{s}\,\epsilon^{-3/2})$ evaluations and $\epsilon$-EFSO solutions in $O(s\log s + \sqrt{s}\,\epsilon^{-1})$.
  • Combined with sample average approximation, the proposed solvers yield $O(\epsilon^{-2})$ for $\epsilon$-EFSO and either $O(\epsilon^{-3})$ or $O(\epsilon^{-5/2})$ for $\epsilon$-SFSO solutions of the original infinite-sample problem.
  • In the deterministic special case $s=1$, Algorithm 2 becomes a single-loop first-order penalty method with optimal $O(\epsilon^{-1})$ or nearly optimal $\tilde O(\epsilon^{-1})$ complexity.
  • The framework extends directly to affine equality constraints and to convex conic constraints $c(x)\in-K$ with an exactly computable projection, covering conic programs beyond ordinary inequalities.

Reading between the lines

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

  • One can infer that the deterministic certificate depends on the uniform almost-sure gradient bound in Assumption 2; under heavy-tailed or unbounded noise, the randomness would leak into the constraint violation, and recovering a usable certificate would require a probabilistic analogue of that assumption.
  • The penalty parameter recipes $\rho=K^{3/2}$ and $\rho=s^{2/3}K^{4/3}$ instantiate a general balance between constraint violation and objective gap, and the same trade-off could govern penalty-based stochastic and augmented Lagrangian methods beyond the two algorithms analyzed here.
  • A direct stress test is to replace the uniform bound with a bounded-variance or sub-Gaussian assumption and measure how the deterministic feasibility certificate degrades, which would show when 'surely feasible' claims are genuinely available.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper proposes first-order methods for convex stochastic or finite-sum optimization with deterministic convex constraints, targeting ε-surely feasible stochastic optimal (ε-SFSO) solutions, i.e., deterministic constraint violation at most ε and expected optimality gap at most ε. For infinite sample spaces, Algorithm 1 applies an accelerated stochastic gradient scheme once to a sequence of quadratic penalty problems; under Assumptions 1–2 it is claimed to compute an ε-SFSO solution with FO complexity O(ε^{-2}) for constant penalty parameters and Õ(ε^{-2}) for dynamic ones. For finite-sum problems, Algorithm 2 uses a modified variance-reduced accelerated scheme with exact penalty gradients, obtaining O(s log s + √s ε^{-3/2}) for ε-SFSO and O(s log s + √s ε^{-1}) for ε-EFSO when m=O(1). Section 4 analyzes SAA, yielding O(ε^{-2}) for ε-EFSO and O(ε^{-5/2}) for ε-SFSO. The proofs are detailed, with explicit constants and lemmas leading to Theorems 1–4.

Significance. The deterministic feasibility guarantee is stronger than the commonly studied expected-feasibility notion, and the claimed infinite-sample complexity O(ε^{-2}) matches the unconstrained stochastic oracle bound. The finite-sum analysis with separation of stochastic and deterministic gradient contributions is a meaningful technical contribution, and the SAA byproduct gives concrete complexity comparisons. The paper is self-contained: it proves its own Lemmas 3–10 rather than importing the central recursion, and the main rates are internally consistent. The principal caveats are the strong uniform almost-sure gradient bound required for the infinite-sample deterministic feasibility claim, and the unsupported deterministic s=1 optimal-complexity remarks.

major comments (2)
  1. [Assumption 2; Lemma 3; Theorem 1] The ε-SFSO guarantee of Algorithm 1 for infinite sample spaces rests essentially on the uniform almost-sure bound in Assumption 2. In Lemma 3, inequality (56) bounds the noise pathwise as ∥δ_i∥ ≤ 2G, which is the only mechanism converting the recursion for F_ρ(x_K) − F*_ρ into a deterministic bound; Lemma 1(46) then yields the deterministic constraint violation in Theorem 1. If Assumption 2 is weakened to a variance bound only, the pathwise terms γ_i∥δ_i∥ are uncontrolled, and only an in-expectation or high-probability feasibility statement follows. The abstract and Remark 1 state the O(ε^{-2}) 'surely feasible' result without flagging this restriction. I recommend adding an explicit scope-limitation statement and, if possible, a companion in-expectation feasibility result under weaker noise assumptions.
  2. [Remarks 3(iii), 4(iii); Theorem 3(ii); Theorem 4(ii)] The special-case deterministic O(ε^{-1}) claims are not consequences of the stated theorems. For s=1, Theorem 3(ii) with ρ = √s K = K gives the bound ∥[c(˜x_K)]_+∥ ≤ 8 L_{∇ḟ}^{1/2} D_X K^{-1/2} + ... (first displayed bound after Theorem 3(ii)), so the theorem yields O(ε^{-2}) rather than O(ε^{-1}); the K^{-1/2} term originates from the K-independent term 16 L_{∇ḟ} D_X² in Lemma 8 (80), which decays only through division by ρ. Similarly, Theorem 4(ii) bound (35) has first term 60 L_{∇ḟ}^{1/2} D_X s^{-1/4}k^{-1/2}, giving O(k^{-1/2}) for s=1. If the intended argument is that δ_t=0 in the deterministic case and the noise term vanishes, that analysis must be supplied; otherwise Remarks 3(iii) and 4(iii) should be corrected.
minor comments (4)
  1. [Section 6.2, after eq. (110)] There is a typo: 'togther' should be 'together'.
  2. [Notation, Section 1.1] In Assumption 2, the variance bound E∥∇f~(x,ξ)−∇f(x)∥² ≤ σ² is implied by the uniform almost-sure bound G via σ ≤ 2G; stating both is fine, but the redundancy could be noted.
  3. [Section 4, Approach 2] The equality O(s log s + √s ε^{-3/2}) = O(ε^{-5/2}) suppresses the logarithmic factor in s; writing O(ε^{-5/2} log(1/ε)) would be more precise.
  4. [Algorithm 2, line 11] The notation ˜x_{k+1} = Σ_{t=1}^{T_k} θ_t x_t / Σ_{t=1}^{T_k} θ_t is clear, but the dependence of θ_t on k is implicit; a subscript θ_{k,t} would aid readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the epsilon-SFSO bounds are derived from the stated assumptions via in-paper penalty and ASG/SVRG lemmas; the only self-citation is a non-load-bearing external benchmark.

full rationale

The derivation chain is self-contained. The deterministic feasibility guarantee is not assumed; it is produced by Lemma 1's penalty/Lagrange multiplier inequality and, in the infinite-sample case, by Lemmas 3 and 4, where Assumption 2's uniform bound max_{ξ∈Ξ} ||∇f˜(x,ξ)|| ≤ G is used explicitly (e.g., equation (56)) to control the noise pathwise. This is a stated assumption rather than a hidden restatement of the conclusion, so Theorem 1's “surely feasible” claim is conditional on that assumption but not circular. For the finite-sum case, Algorithm 2's SVRG estimator is analyzed in Lemmas 5–7 with the deterministic component gradient computed exactly, and the penalty parameter choices in Theorem 3 are chosen after the rates are derived, not fit to the target guarantee. The only overlap with the authors' prior work is reference [23] in Section 4, which is cited alongside [17] and [37] as one of several deterministic solvers used to convert a sample-average solution into an ε-SFSO solution. That citation is a comparison benchmark, not an input to the main derivation; removing it would not change Algorithms 1–2 or Theorems 1–4. Hence there is no circular step that reduces a prediction to its inputs by construction.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central claims rest on four domain assumptions: bounded domain with bounded variation, smoothness and bounded stochastic gradients, componentwise smoothness for variance reduction, and uniform bounded variation for SAA. Standard convex analysis inequalities (Young's inequality, Hoeffding's inequality) are used as background. No numbers are fitted to data: all constants are explicit functions of the problem parameters D_X, D_F, L_f, L_c, L_i, sigma, G, and Lambda. The penalty parameters rho are set by theory to balance terms, not calibrated to data. No new physical or mathematical entities are introduced.

assumptions (4)
  • domain assumption Assumption 1: X is bounded (diameter D_X), F has bounded variation D_F on X, each c_i is Lipschitz smooth with bounded magnitude, and the set of optimal Lagrange multipliers Lambda* is nonempty.
    Used in Lemma 1 to relate the penalty problem to the original problem via (45)-(46), and in all subsequent bounds involving D_X, D_F, and Lambda. The boundedness of X is essential for the deterministic estimates.
  • domain assumption Assumption 2: f(x)=E[f~(x,xi)] is L_nabla_f-smooth, E[grad f~(x,xi)]=grad f(x), E[||grad f~(x,xi)-grad f(x)||^2] <= sigma^2, and max_{xi in Xi} ||grad f~(x,xi)|| <= G for all x in X.
    The uniform bound G is used in Lemmas 3 and 4 to bound the noise delta_k almost surely by 2G, which is what makes the constraint-violation bound in Theorems 1 and 2 deterministic. Without this boundedness, the SFSO guarantee cannot hold with certainty.
  • domain assumption Assumption 3: each f_i is L_i-smooth on X.
    Used to build the SVRG estimator in Algorithm 2 and to set the sampling distribution q_i proportional to L_i (Section 3, Lemma 5). This is the standard smoothness assumption for finite-sum variance reduction.
  • domain assumption Assumption 4: there exists V > 0 such that |f~(x,xi)-f~(x',xi')| <= V for all (x,xi), (x',xi') in X x Xi.
    Used in Section 4 to apply Hoeffding's inequality and derive the sample-average optimality gap bound in (41). It is needed for the SAA complexity results but not for the main Algorithms 1 and 2.

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Cite this review

Pith. "Pith review of First-order methods for stochastic and finite-sum convex optimization with deterministic constraints." pith.science (2026). https://pith.science/paper/Y5OLNMCN

@misc{pith2026250620630,
  author       = {Pith},
  title        = {Pith review of: First-order methods for stochastic and finite-sum convex optimization with deterministic constraints},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Y5OLNMCN}},
  note         = {Machine review of arXiv:2506.20630}
}
abstract

In this paper, we study a class of stochastic and finite-sum convex optimization problems with deterministic constraints. Existing methods typically aim to find an $\epsilon$-$expectedly\ feasible\ stochastic\ optimal$ solution, in which the expected constraint violation and expected optimality gap are both within a prescribed tolerance $\epsilon$. However, in many practical applications, constraints must be nearly satisfied with certainty, rendering such solutions potentially unsuitable due to the risk of substantial violations. To address this issue, we propose stochastic first-order methods for finding an $\epsilon$-$surely\ feasible\ stochastic\ optimal$ ($\epsilon$-SFSO) solution, where the constraint violation is deterministically bounded by $\epsilon$ and the expected optimality gap is at most $\epsilon$. Our methods apply an accelerated stochastic gradient (ASG) scheme or a modified variance-reduced ASG scheme $only\ once$ to a sequence of quadratic penalty subproblems with appropriately chosen penalty parameters. We establish first-order oracle complexity bounds for the proposed methods in computing an $\epsilon$-SFSO solution. As a byproduct, we also derive first-order oracle complexity results for sample average approximation method in computing an $\epsilon$-SFSO solution of the stochastic optimization problem using our proposed methods to solve the sample average problem.

Figures

Figures reproduced from arXiv: 2506.20630 by the authors.

Figure 1
Figure 1. Performance of Algorithm 1 with constant and dynamic penalty parameters. [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗
Figure 2
Figure 2. Performance of Algorithm 2 with constant and dynamic penalty parameters. [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Comparison of Algorithm 2 with EAG 6 Proof of the main results In this section, we provide a proof of our main results presented in Sections 2 and 3, which are particularly Theorems 1, 2, 3, and 4. To proceed, we establish a lemma regarding a relationship between problems (1) and (8), which will be used in our subsequent analysis. Lemma 1. Consider the penalty problem associated with (1): F ∗ ρ = min x n Fρ(x) := F(… view at source ↗

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