Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints

arXiv:2610.13787 · math.OC, cs.LG · Submitted 2026-10-09 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints".

Jane: The gist: New algorithms are presented for stochastic optimization problems where the objective is neither smooth nor convex, subject to a convex constraint set,

Tom: First, who's behind it and why it matters.

Paper summary: Jane: So, wrapping up the discussion on "Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints", we’ve seen that these researchers are tackling problems where the objective function is neither smooth nor convex, but they are constrained within a nice convex set.

Tom: The authors are claiming they have achieved optimal convergence rates for this task in several settings, specifically showing that projection-based and projection-free algorithms hit an Oe(delta minus one epsilon minus three) rate with stochastic gradients.

Lu: And the zeroth-order methods also have a rate dependent on the dimension d, which is good because it shows how they handle the complexity of higher dimensions without losing efficiency.

Meng: What this means for production is that if you’re dealing with these kinds of complex, non-smooth tasks under constraints, you can use these algorithms knowing that the convergence rate isn't unnecessarily slowed down by having those difficult mathematical properties.

Lalam: The paper proves that constrained nonsmooth optimization is as easy as unconstrained smooth optimization in terms of query complexity for nonconvex stochastic problems because their lower bound matches their upper bounds >

Jane: So, to put it plainly, the authors have provided first-order and zeroth-order algorithms with optimal performance bounds for these challenging types of AI problems. This work, "Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints", suggests that even when things are messy and constrained, we can find solutions efficiently if we use the right query strategy.

Conclusion: Tom: So, we've been looking at these new algorithms for solving optimization problems that are super messy—where you don't have smooth or convex functions—but you still have those tough constraints on your problem.

Jane: Right, so the paper is called "Optimal Algorithms for Nonsmooth Nonconvex Stochastic Optimization with Constraints." It seems to be about finding the best way to solve these kind of tricky math problems using stochastic methods.

Lu: Yeah, they're showing that you can get optimal convergence rates in several different settings when you have these kinds of constraints. It’s not just a theoretical thing; they’ve actually built concrete algorithms for it.

Meng: So what does this actually mean for someone trying to deploy an AI model? Does it make running those complex models faster or cheaper?

Lalam: Lalam sees that the paper establishes lower bounds that match the upper bounds for these constrained, nonsmooth problems, showing they are as easy to solve as simpler smooth ones in terms of query complexity. It means the difficulty isn't just in the function shape but in how you query it.

Tom: So, if I’m running a model and it’s not smooth or convex, I don't have to worry that my method is inherently too slow just because the math is ugly. The complexity stays manageable with these new methods.

Jane: Exactly. It shows that even when the objective function is bumpy and you have boundaries to stay within, there are efficient ways to find a good solution using those stochastic queries we talked about before.

Lu: The authors are presenting both first-order and zeroth-order algorithms here, which means they’re giving us tools for both gradient-based approaches and methods that only use function value queries. It’s a comprehensive toolkit.

Meng: A toolkit is good, but I need to know if these optimal rates hold up when we move from theory to real traffic data where the noise level is actually high and unpredictable.

Lalam: The paper covers the stochastic setting, which accounts for that noise you mentioned in production. It shows that the convergence rate doesn't degrade because of those random fluctuations in a way that surprises us.

Tom: So it’s about building algorithms that are robust to the randomness inherent in real-world data, even when the underlying math is complicated. That sounds like something we need for reliable AI systems.

Jane: It really puts things into perspective on what these constraints mean for learning complex patterns in data. It’s about finding a sweet spot where you can't afford to be too slow but also don't want a method that breaks down easily with every tiny change in the data landscape.

Gon Buzaglo, Guy Kornowski

Princeton University · Institute for Advanced Study

math.OC, cs.LG

Submitted: 2026-10-09

Updated: 2026-10-09

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 91/100

The gist: The gist: New algorithms are presented for stochastic optimization problems where the objective is neither smooth nor convex, subject to a convex constraint set, achieving optimal convergence rates

Key concepts

Reduction from Constrained Optimization to Online Learning: This technique transforms a constrained optimization problem into an online learning framework. It uses existing methods from online learning theory to establish convergence guarantees for the original constrained problem, allowing researchers to apply established online algorithms to complex optimization tasks.
Stochastic First-Order Algorithms: These are algorithms designed for problems where the objective function is not smooth and involves randomness. The paper designs these algorithms using optimal gradient oracle complexity, meaning they use the minimum necessary number of stochastic gradient queries to converge to a constrained approximate-stationary point.
Zeroth-Order Algorithms: These methods avoid calculating gradients directly by instead querying function values. They use smoothed versions of the objective function and specific estimators to find an approximate stationary point. This approach provides convergence rates that improve upon previous work in this area.

Summary

The gist: New algorithms are presented for stochastic optimization problems where the objective is neither smooth nor convex, subject to a convex constraint set, achieving optimal convergence rates for constrained problems in several settings

Contributions

- Reduction from Constrained Optimization to Online Learning (Theorem 3.1): The main proof technique extends the online-to-nonconvex framework of Cutkosky et al. [2023] to optimization over a closed convex constraint set.

- First-Order Algorithms (Theorem 4.1 and Theorem 5.1): Stochastic first-order algorithms are designed with optimal gradient oracle complexity for problem (1), achieving Oe(δ−1/ε−3) stochastic gradients for convergence to a constrained approximate-stationary point.

- Zeroth-Order Algorithms (Theorems 6.1 and 6.2): Projection-based and projection-free algorithms are discussed, requiring Oe(dδ−1/ε−3) function-value queries to converge to a constrained approximate-stationary point, which improves over prior work by a sqrt(d) factor.

- Matching Lower Bounds (Section 7): Lower bounds are adapted to prove optimality, establishing that constrained nonsmooth optimization is as easy as unconstrained smooth optimization in terms of query complexity for nonconvex stochastic problems.

How it works

The reduction from Constrained Optimization to Online Learning (Theorem 3.1) extends the online-to-nonconvex framework of Cutkosky et al. [2023] to optimization over a closed convex constraint set, converting any local online learning algorithm into a constrained optimization algorithm and translating the online learner’s regret into a local constrained-stationarity guarantee
The first-order algorithms utilize this reduction to design stochastic first-order algorithms with optimal gradient oracle complexity for problem (1), yielding projection-based and projection-free methods that require Oe(δ−1/ε−3) stochastic gradients to converge to a constrained approximate-stationary point
Zeroth-order algorithms apply Algorithm 1 to the smoothed objective fρ, replacing its stochastic-gradient query with a call to the gradient estimator of Algorithm 5, resulting in convergence rates dependent on function-value queries that improve over Liu et al. [2024] by a sqrt(d) factor

Analysis and Lower Bounds

- The analysis is based on a reduction to online linear optimization, extending a related technique from unconstrained optimization.

- Lemma 8.2 shows that the map η 7→∥Gη,X (x, g)∥ is non-increasing and its supremum over η>0 equals dist−g, NX (x), which serves as a stationarity measure.

- Theorem 4.1 establishes the complexity for projection-based algorithms using Oe(∆G2δe3) stochastic-gradient queries to return x out in X satisfying E[R f,X δ (x out)] ≤ ε.

- Algorithm 5 provides a two-point gradient estimator that queries F(z + ρu; ξ) and F(z - ρu; ξ) to return d 2ρ F(z + ρu; ξ) − F(z - ρu; ξ), which is used in the zeroth-order optimization setting.

- Theorem 8.20 provides a lower bound for the Frank-Wolfe gap even for smooth objectives, showing that achieving expected FWf,X δ (x out) ≤ ε requires T = omega(∆HG2D4ε4) stochastic-gradient queries.

- Corollary 8.23 establishes the lower bound for the Goldstein residual, requiring omega(∆G2δe3) stochastic-gradient queries, matching the upper bounds obtained in Theorem 6.1.

Zeroth-Order Optimization

The zeroth-order results rely on applying Algorithm 1 to fρ, where fρ is a smoothed version of f, and using Algorithm 5 for gradient estimation

Theorem 6.1 shows that by replacing the stochastic-gradient query with a call to Algorithm 5, applying Theorem 4.1 to fρ with specific parameters yields ERf,X δ/2 (x out) ≤ ε

Theorem 6.2 shows that for a compact set X with diameter D, applying Theorem 5.1 to fρ yields E FWf,X δ/2 (x out) ≤ ε

Discussion

The paper provides the first dimension-free algorithms for first-order constrained optimization of arbitrary stochastic Lipschitz functions without smoothness or convexity assumptions, demonstrating that their rates are optimal for (possibly-)randomized zero-respecting algorithms The approach extends to the zeroth-order setting and closes the upper-bound gap left open by Liu et al. [2024] The results show that in nonconvex stochastic optimization, the convergence rate is degraded neither by constraints nor by nonsmoothness

The paper was developed through a highly interactive process involving AI models, specifically, GPT Sol 5.6 and GPT Astra Every mathematical argument appearing in the paper was independently re-derived, rewritten and simplified by the authors The authors take full responsibility for the correctness and presentation of all material in this manuscript.

Improvements for AI systems

Bold header: Optimal First-Order Constrained Optimization Algorithms

This system can solve stochastic optimization problems with neither smooth nor convex objectives subject to a convex constraint set, returning a (δ, ε)-Goldstein stationary point with optimal complexity rates for both projection-based and projection-free methods.

Bold header: Improved Zeroth-Order Query Efficiency

The AI can perform zeroth-order optimization by querying only function values, achieving an optimal convergence rate of Oe(dδ−1/ε−3) using function-value queries, which improves over prior work by a factor of omega(√d).

Bold header: Robustness to Constraints and Nonsmoothness

The algorithms are shown to achieve the optimal convergence rates for nonconvex stochastic optimization in settings where neither constraints nor nonsmoothness degrade the convergence rates, demonstrating that constrained nonsmooth nonconvex optimization is as easy as unconstrained smooth optimization in terms of query complexity.