arxiv: 2604.26735 · v1 · submitted 2026-04-29 · 🧮 math.OC

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Quasar-Convex Optimization: Fundamental Properties and High-Order Proximal-Point Methods

Masoud Ahookhosh , Jose M.M. de Brito , Alireza Kabgani , Felipe Lara , Jinyun Yuan

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Pith reviewed 2026-05-07 11:19 UTC · model grok-4.3

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keywords quasar-convex functionsproximal-point methodshigh-order regularizationconvergence ratesmachine learning optimizationglobal geometry

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The pith

Quasar-convex problems admit high-order proximal-point methods whose convergence jumps from linear to superlinear when the regularization order exceeds 2.

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

The paper first characterizes quasar-convex functions by proving they are stable under standard calculus operations, obey growth conditions, and contain no spurious critical points, yielding a saddle-free global geometry. It then introduces high-order proximal-point algorithms and supplies a unified convergence theory with explicit rates that change sharply with the regularization order p. For p between 1 and 2 the convergence is locally linear with complexity O(log(1/ε)), at p equals 2 it becomes globally linear with the same complexity, and for p greater than 2 it becomes superlinear with complexity O(log log(1/ε)). These results target machine learning objectives that satisfy quasar-convexity but lack strong convexity. Preliminary numerical tests on selected problems are consistent with the predicted rates.

Core claim

For quasar-convex functions the high-order proximal-point methods converge globally to minimizers. The rate is locally linear with complexity O(log(ε^{-1})) when the regularization order p lies in (1,2), globally linear with the same complexity when p equals 2, and superlinear with complexity O(log log(ε^{-1})) when p exceeds 2.

What carries the argument

The HiPPA family of proximal-point algorithms that regularize the objective with a term of order p greater than 1 and generate the next iterate by minimizing the resulting model.

Where Pith is reading between the lines

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

These methods could be applied to non-convex machine learning models if quasar-convexity can be established or promoted during training.
An adaptive strategy that increases the regularization order p during the run might combine fast early progress with eventual superlinear speedup.
The same high-order proximal construction may extend to other function classes that share the saddle-free geometry but lack quasar-convexity.
Stochastic or distributed versions of HiPPA would be a natural next step for large-scale data science problems.

Load-bearing premise

The functions satisfy regularity assumptions sufficient to support the unified convergence analysis and explicit rate derivations.

What would settle it

Numerical computation of the iterates of the method with p equals 3 on a simple quasar-convex test function that shows only linear error reduction rather than superlinear reduction would refute the superlinear convergence claim.

Figures

Figures reproduced from arXiv: 2604.26735 by Alireza Kabgani, Felipe Lara, Jinyun Yuan, Jose M.M. de Brito, Masoud Ahookhosh.

**Figure 1.** Figure 1: Relationships between convexity, star-convexity, and quasar-convexity. Here, “ view at source ↗

**Figure 2.** Figure 2: A nonsmooth, nonconvex function that is κ-quasar-convex with respect to some minimizers; see Example 17. (c) For every θ ∈ (0, 1], the function h is (θκ, γ θ )-strongly quasar-convex with respect to x. Proof. (a): It is straightforward. (b) We have dom(g) = dom(h)− x. Hence, dom(g) is convex and 0 ∈ dom(g). Set z = 0. Since x ∈ arg minRn h, for every z ∈ R n, it holds that g(z) = h(x) ≤ h(x + z) = g(z). T… view at source ↗

**Figure 3.** Figure 3: A nonconvex quasar-convex function whose minimizer set is star-convex but not convex; see Example 41. view at source ↗

**Figure 4.** Figure 4: Numerical comparison on the population ReLU-GLM problem. view at source ↗

**Figure 5.** Figure 5: Numerical comparison on the nonconvex multi-task regression problem. view at source ↗

read the original abstract

We study the optimization of (strongly) quasar-convex functions, a class that arises naturally in many machine learning and data science applications due to its favorable properties. The fundamental properties of this class are first developed, including its stability under standard calculus operations, growth conditions, and the absence of spurious critical points, which together imply a benign global geometry with no saddle points. Motivated by these properties, a class of proximal-point algorithms (HiPPA) with high-order regularization of order $p>1$ is introduced. Conditions are identified under which the iterates converge globally to minimizers, and a unified convergence analysis is provided with explicit rates and iteration complexity bounds under appropriate regularity assumptions. The results reveal a sharp transition in behavior with respect to the order $p$: for $p\in(1,2)$, the method achieves local linear convergence with complexity $\mathcal{O}(\log(\varepsilon^{-1}))$ when initialized sufficiently close to a minimizer; for $p=2$, it attains global linear convergence with the same complexity; and for $p>2$, it exhibits superlinear convergence with complexity $\mathcal{O}(\log\log(\varepsilon^{-1}))$, where $\varepsilon>0$ denotes the target accuracy. The theory is complemented with preliminary numerical experiments on selected machine learning problems, which illustrate the effectiveness of the proposed methods and are consistent with the theoretical findings.

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

The paper gives high-order proximal methods for quasar-convex functions and claims a clean p-dependent shift from local linear to global linear to superlinear rates, but those rates rest on regularity conditions that quasar-convexity alone does not obviously supply.

read the letter

The main point is a new family of high-order proximal-point algorithms (HiPPA) for quasar-convex functions, together with a claimed sharp change in convergence behavior as the regularization order p grows. For p between 1 and 2 the method is only locally linear, at p=2 it becomes globally linear, and above 2 it turns superlinear with double-log complexity. The paper first works out some basic facts about the function class: stability under standard operations, growth bounds, and the lack of spurious critical points. Those properties are useful on their own and support the benign geometry story that makes quasar-convexity attractive for machine-learning problems.

Referee Report

1 major / 2 minor

Summary. The paper develops fundamental properties of (strongly) quasar-convex functions, including stability under standard operations, growth conditions, and absence of spurious critical points implying benign global geometry. It introduces a class of high-order proximal-point algorithms (HiPPA) with regularization order p>1. Under identified conditions and appropriate regularity assumptions, it provides a unified convergence analysis yielding explicit rates and complexity bounds: local linear convergence with O(log(ε^{-1})) for p∈(1,2) when initialized near a minimizer; global linear convergence with the same complexity for p=2; and superlinear convergence with O(log log(ε^{-1})) for p>2. Preliminary numerical experiments on machine learning problems are included to illustrate the results.

Significance. If the rates are shown to apply under conditions that follow from quasar-convexity (or clearly delineated additional assumptions), the work offers a useful unified framework for proximal methods on this class of functions relevant to ML applications. The development of basic properties and the sharp transition in rates with respect to p are strengths; the explicit complexity expressions and numerical support add value.

major comments (1)

[Abstract and convergence analysis section] Abstract and the unified convergence analysis (around the main theorems on rates): The claimed sharp transition in convergence behavior (local linear for p∈(1,2), global linear for p=2, superlinear O(log log(ε^{-1})) for p>2) is stated to hold under 'appropriate regularity assumptions.' Quasar-convexity ensures no spurious critical points and benign geometry via the growth inequality, but does not automatically imply the higher-order smoothness, Lipschitz continuity of derivatives, or error-bound conditions typically required for high-order proximal-point analyses. If these regularity conditions are additional rather than derived from strong quasar-convexity, the explicit rates apply only to a subclass and the claim that the behavior is characteristic of the quasar-convex class is not fully supported. Please clarify in the theorem statements whether the assumptions follow from the quasar

minor comments (2)

Ensure consistent use of notation for the target accuracy ε and the order p across theorems and complexity statements.
The numerical experiments section could benefit from more details on how the test problems satisfy the quasar-convexity and regularity assumptions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We address the major comment point by point below and will revise the paper to improve clarity on the assumptions.

read point-by-point responses

Referee: [Abstract and convergence analysis section] Abstract and the unified convergence analysis (around the main theorems on rates): The claimed sharp transition in convergence behavior (local linear for p∈(1,2), global linear for p=2, superlinear O(log log(ε^{-1})) for p>2) is stated to hold under 'appropriate regularity assumptions.' Quasar-convexity ensures no spurious critical points and benign geometry via the growth inequality, but does not automatically imply the higher-order smoothness, Lipschitz continuity of derivatives, or error-bound conditions typically required for high-order proximal-point analyses. If these regularity conditions are additional rather than derived from strong quasar-convexity, the explicit rates apply only to a subclass and the claim that the behavior is characteristic of the quasar-convex class is not fully supported. Please clarify in the theorem statements是否

Authors: We agree that the higher-order smoothness, Lipschitz continuity of derivatives, and related error-bound conditions do not follow from (strong) quasar-convexity. Quasar-convexity provides the key geometric properties (growth inequality, absence of spurious critical points, and benign global landscape), but the explicit convergence rates for HiPPA require additional regularity to bound the proximal mapping error and establish the stated rates. In the revised manuscript we will: (i) update the abstract to state that the rates hold for quasar-convex functions satisfying the listed regularity assumptions; (ii) revise the statements of the main theorems to list the assumptions explicitly and note that they are independent of quasar-convexity; and (iii) add a short remark after the theorems clarifying that the sharp transition in rates is characteristic of the subclass obeying both quasar-convexity and the regularity conditions. These changes will remove any ambiguity that the rates apply to the entire quasar-convex class without qualification. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper first establishes fundamental properties of quasar-convex functions directly from the given growth inequality definition, including stability under operations and absence of spurious critical points. It then introduces the HiPPA class of proximal-point methods with p-order regularization and derives convergence rates explicitly from the stated regularity assumptions and the quasar-convex geometry. No load-bearing step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the explicit complexity bounds (local linear for p in (1,2), global linear for p=2, superlinear for p>2) follow from the unified analysis under the listed conditions rather than tautological renaming or imported uniqueness. The derivation remains self-contained against the paper's own assumptions and external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the definition and properties of (strongly) quasar-convex functions plus standard regularity assumptions in optimization; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Quasar-convex functions have no spurious critical points and benign global geometry
Stated as fundamental property developed in the paper.
domain assumption Appropriate regularity assumptions hold to obtain explicit convergence rates
Invoked for the unified analysis and complexity bounds.

pith-pipeline@v0.9.0 · 5565 in / 1108 out tokens · 51434 ms · 2026-05-07T11:19:26.397415+00:00 · methodology

discussion (0)

Reference graph

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