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The Optimal Smoothings of Sublinear Functions and Convex Cones

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Pith's one-line read The paper characterizes all optimal smoothings of sublinear functions and convex cones as exactly the beta-smooth convex objects between two explicit extremal smoothings, with the tradeoff governed by a single width invariant.

desk verdict Complete characterization of optimal smoothings for sublinear functions and convex cones; proofs hold up, and the only real soft spot is the numerical exponential-cone example. read the letter →

arxiv 2508.06681 v2 pith:6ACLG6ON submitted 2025-08-08 math.OC

Thabo Samakhoana , Benjamin Grimmer This is my paper
classification math.OC MSC 90C2552A4149J52
keywords optimalsmoothingsublinearfunctionsconvexconesParetofrontierMoreauenvelopesmoothablesetsconicprogrammingnonsmoothoptimization
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 proves a complete answer to a basic question in nonsmooth optimization: when smoothing a convex function or set, which smooth convex approximation is closest? For sublinear functions and closed convex cones, the answer is an interval: every optimal $\beta$-smoothing lies between two explicit extremal smoothings, and no smooth convex object outside that interval can tie or beat their distance. The whole smoothness-versus-distance tradeoff is controlled by one number, the width of a 'core' assembled from the original object. This matters because standard smoothings such as Moreau envelopes and log-sum-exp are usually ad hoc rather than nearest; the paper shows its optimal max-function smoothing improves the accelerated-gradient iteration count by a factor of $1/\sqrt{\log n}$.

What carries the argument

The carrying mechanism is the pair of 'cores': $C_\sigma := \{(x,r) \mid (x,r)+\operatorname{epi}(\tfrac12\|\cdot\|^2) \subseteq \operatorname{epi}\sigma\}$ for sublinear functions and $C_K := \{x \mid x+B(0,1) \subseteq K\}$ for cones. From each core the paper defines a center and a width, $w_\sigma = r_\sigma + \tfrac12\|x_\sigma\|^2$ and $w_K = \|x_K\|-1$. The extremal smoothings are then infimal convolutions with $\tfrac12\|\cdot\|^2$ for functions, and Minkowski sums with balls followed by rescaling for sets; the rescaling $[f]_\eta(x)=\eta f(x/\eta)$, $[C]_\eta=\eta C$ transfers every $\beta=1$ result to all $\beta>0$. The interval theorems hold because Lemma 4.9 forces every finite-di

What would settle it

Take $\sigma=\max\{x_1,x_2\}$ on $\mathbb{R}^2$. The theorem says every $1$-smooth convex $f$ has $\sup_x |f(x)-\sigma(x)| \ge w_\sigma/2 = 1/8$. A numerical search over $1$-smooth convex functions that obtains a strictly smaller sup-distance would refute Theorem 4.1. For the conic version, compute the minimal Hausdorff distance from a $1$-smooth convex body to the second-order cone $K$ and compare it with $w_K/(2+w_K)=(\sqrt{2}-1)/(\sqrt{2}+1)$; finding a smaller distance falsifies Theorem 4.2.

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Extended reading notes

Core claim

The paper's central claim is that for any sublinear function $\sigma$ and any closed convex cone $K$ with nonempty interior, the set of all optimal smoothings has a complete, explicit description. Theorem 4.1 states that $\sigma$ is $\lambda$-smoothable exactly when $\lambda \ge w_\sigma/2$, and that for any $\beta>0$, a $\beta$-smooth convex $f$ is an optimal $\beta$-smoothing of $\sigma$ exactly when $[F^{\mathrm{gen}}_\sigma]_{1/\beta} \le f \le [f^{\mathrm{gen}}_\sigma]_{1/\beta}$, where $w_\sigma$ is the functional width and $F^{\mathrm{gen}}_\sigma, f^{\mathrm{gen}}_\sigma$ are the lower and upper extremal smoothings built from the core. Theorem 4.2 gives the conic analogue: $K$ is $\l

Load-bearing premise

The proof assumes every candidate smoothing is closed and at finite distance from the original, because Lemma 4.9 then forces the candidate to have the same directions at infinity (its horizon cone) as the original; if a smoothing were non-closed or infinitely far, the bracketing interval would not be forced.

Editorial extensions

If this is right

  • For any sublinear function, an optimal smoothing can be certified just by checking that it sits between two known functions; no minimax distance calculation is needed.
  • For any norm, the Moreau envelope is optimal among outer smoothings; the $\ell_1$ norm has a unique optimal smoothing, while some weighted-$\ell_\infty$ norms have a whole interval of equally good ones.
  • The max function and the maximum eigenvalue function each have a unique optimal smoothing with an explicit piecewise-quadratic (respectively spectral) formula and approximation error $(1-1/d)/(4\beta)$.
  • Composing that optimal max smoothing through a Lipschitz map with Lipschitz Jacobian yields an accelerated-gradient guarantee for finite maxima whose constant improves log-sum-exp smoothing by a factor $1/\sqrt{\log n}$ (Theorem 5.1).
  • Every closed convex set containing a ball is a section of a smoothable cone, so the conic smoothing theory applies to general convex constraint sets (Theorem 5.2).

Reading between the lines

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

  • We infer that the if-and-only-if interval statement makes optimal-smoothing search a convex feasibility problem: any $1$-smooth convex $f$ in $[F^{\mathrm{gen}}_\sigma, f^{\mathrm{gen}}_\sigma]$ is automatically Pareto-optimal, so solvers can select smoothings with extra structural properties for free.
  • The authors explicitly leave open optimal smoothings under general norms; we infer that rebuilding the core around a non-Euclidean unit ball would change $w_\sigma$ and $w_K$, and could decide whether the $\sqrt{\log n}$ gain over log-sum-exp survives in the $\ell_\infty$-Lipschitz regime.
  • We infer that the uniqueness criterion (core equals a translated copy of the original) is a practical diagnostic: cones like the nonnegative, second-order, and semidefinite cones have forced smoothings, whereas the exponential cone does not, so the exponential cone's interval freedom could be exploited to preserve sparsity or decomposability in conic algorithms.
  • We infer that the numerical core computation for the exponential cone is a template for applying the theory to arbitrary convex bodies via the conic section in Theorem 5.2, though the paper stops short of testing that pipeline.
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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

0 major / 5 minor

Summary. This paper studies the problem of optimally smoothing sublinear functions and closed convex cones. It defines the functional/conic core, center, and width, and proves that for every target smoothness level the set of all optimal smoothings is an explicit interval between two extremal smoothings: Theorems 4.1 and 4.2 give the general characterizations, and Theorems 4.3--4.6 give the analogous characterizations for inner and outer smoothings. The theory is applied to norms, the ReLU and max functions, maximum eigenvalue functions, nonnegative/SOC/SDP/exponential cones, amenable composite functions, and general convex sets via conic lifting. Proofs are given in Section 4.3.

Significance. The main results, if correct, provide a complete, essentially parameter-free description of the Pareto frontier between smoothness and approximation error for two natural classes of nonsmooth objects. The paper contains explicit formulas for several important examples, validates them against known cases such as the Moreau envelope and the second-order cone, and derives a concrete improvement over log-sum-exp smoothing for finite maxima. The development is self-contained and nicely exhibits the function/set symmetry. I found the central argument sound; the issues that remain are local proof details and presentation points.

minor comments (5)
  1. [Section 4.3.2, Lemma 4.13, Eq. (4.19)] In the proof of Lemma 4.13, the unit normal ζ is introduced without specifying which element of N_S(P_S(0)) is chosen. The equality ||P_S(0)-ζ|| = ||P_S(0)||+1 requires ζ = -P_S(0)/||P_S(0)|| (with the P_S(0)=0 case handled separately). As written, the equality is not justified. This is a local repair: choose ζ accordingly or add a sentence justifying the existence of such a unit normal.
  2. [Section 4.3.2, Lemma 4.12] There is a sign error in the projection computation. Equation (2.1) gives P_{Sin_K}(0) = P_{CK}(0) + P_{B1}(-P_{CK}(0)), not P_{CK}(0) - P_{B1}(-P_{CK}(0)). The final displayed bound ||P_{Sin_K}(0)|| = ||x_K|| - 1 is correct once the sign is fixed, since P_{B1}(-x_K) = -x_K/||x_K||.
  3. [Section 3.1] The claim that a (λ,Δ)-smoothable sublinear function or cone must be (λ,0)-smoothable 'from Lemma 3.3' is not immediate and, under the stated definition, the scaling argument does not by itself prove it. Since the rest of the paper is developed for Δ=0 and the main theorems are proved there, please reword this reduction, prove it, or explicitly restrict the definition to Δ=0.
  4. [Section 4.2.5, Table 1] The exponential cone entries are computed numerically by dense sampling of the normal cone and numerical projection. The text should state more explicitly that the core, center, width, and the 'No' in the uniqueness column are numerical observations rather than rigorous analytic conclusions, so that the example is not over-interpreted.
  5. [Section 5.1, Theorem 5.1 proof] The proof uses the assertion that ∇fσ(z) ∈ ∂σ(0) for all z, saying this follows from finite distance and Lemma 4.9. This is true, but it deserves a short justification: finite distance implies epiσ is the horizon cone of epifσ, and the subgradient inequality combined with σ(y) = max_{ζ∈∂σ(0)}<ζ,y> gives the inclusion. Please add this one-line argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: optimal-smoothing characterization is derived from explicit core/center/width definitions and independent convex-analytic lemmas.

full rationale

The central theorems (4.1/4.2) characterize all optimal smoothings as the interval between explicit extremal smoothings built from the functional/conic core, center, and width of the input sigma or K. These objects are defined directly from sigma/K, not from the family of optimal smoothings; Lemma 4.3 derives the core independently as an epigraph via Fenchel conjugation, and Lemma 4.9 — the horizon-cone identity for finite-distance smoothings — is proved from Rockafellar's recession-cone calculus without presupposing the interval characterization. The lower/upper bounds in Lemmas 4.10–4.13 follow by direct convex-analytic arguments (Moreau envelope formulas, smooth-set decompositions, and projection identities). The set-side proof relies on [20, Prop. 3] (representation of 1-smooth sets as C+B(0,1)); although this is a same-group citation, it is a parameter-free structural lemma with stated assumptions that do not include the target result, so under the review rules it counts as independent support and does not create circularity. No fitted parameter is renamed as a prediction, and no uniqueness conclusion is imported from prior work. The only issue found is a small, repairable exposition gap in Lemma 4.13: an arbitrary unit normal zeta in N_S(P_S(0)) need not be parallel to -P_S(0), so the equality ||P_S(0)-zeta|| = ||P_S(0)||+1 requires choosing zeta = -P_S(0)/||P_S(0)|| (with the P_S(0)=0 case handled separately). This is a correctness nuance, not a circularity, and does not affect the conclusions. Overall, the derivation is self-contained relative to standard convex analysis.

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

No free parameters are fitted: the core, center, and width are computed from the input sigma or K. The central claim rests on standard convex-analysis representation theorems plus prior smooth-set decomposition lemmas, some from the authors' own earlier work but with independent proofs. No invented entities are introduced.

assumptions (5)
  • standard math Finite sublinear functions on a Euclidean space are support functions of their subdifferential at 0: sigma(y)=sup_{zeta in partial sigma(0)} <zeta,y>.
    Used in Lemma 4.3 to derive C_sigma = epi rho_sigma, the functional core.
  • standard math The Moreau envelope of a closed convex function with 1/2||.||^2 is 1-smooth, and any 1-smooth convex function can be represented as a Moreau envelope of a convex function (Rockafellar-Wets Proposition 12.60).
    Used throughout Section 4.3 to identify smoothings and their extremal envelopes.
  • domain assumption For closed convex sets at finite Hausdorff distance from a cone K, their horizon cones coincide (Lemma 4.9).
    Used in Lemma 4.11 and Lemma 4.13 to replace epi sigma or K with the horizon of an arbitrary optimal smoothing.
  • domain assumption Any 1-smooth closed convex set S decomposes as C+B(0,1), and Minkowski sums with balls preserve smoothness (Liu-Grimmer [20, Proposition 3, Lemma 9]).
    Basis for the cone smoothing characterization in Theorem 4.4; prior paper by a co-author but with independent proofs.
  • domain assumption K is a closed convex cone with nonempty interior and K is not equal to E.
    Needed for the existence of the conic center x_K and for the boundary-normal argument in Lemma 4.13.

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Cite this review

Pith. "Pith review of The Optimal Smoothings of Sublinear Functions and Convex Cones." pith.science (2026). https://pith.science/paper/6ACLG6ON

@misc{pith2026250806681,
  author       = {Pith},
  title        = {Pith review of: The Optimal Smoothings of Sublinear Functions and Convex Cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ACLG6ON}},
  note         = {Machine review of arXiv:2508.06681}
}
read the original abstract

This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal smoothings is given. These provide if and only if characterizations of the set of optimal smoothings for any target level of smoothness. Optimal smoothings restricting to either inner or outer approximations also follow from our theory. Finally, we apply our theory to provide insights into smoothing amenable functions given by compositions with sublinear functions and generic convex sets by expressing them as conic sections.

Figures

Figures reproduced from arXiv: 2508.06681 by the authors.

Figure 1
Figure 1. Left: Five candidate smoothings of ∥x∥2 in 1D. Right: Smoothness and distance bounds, and the implied smoothability bound for the two-norm and second-order cone from each candidate. with distance from σ of ηDi . As a result, each fi proves σ is λi = βiDi-smoothable. Among these choices, f2 provides the best smoothability constant in general and f1 provides the best constant among the outer smoothings. Our Theorems 4… view at source ↗
Figure 2
Figure 2. The exponential cone, its translation xK + K, its conic core, its minimal optimal 1- smoothing, and its maximal optimal 1-smoothing. as Lemma 4.4 dictates. The supremum is numerically computed using a dense sample from the set  ζ ∈ R 3 | ζ ∈ NK(0), ∥ζ∥2 = 1 . By computing a projection onto this computed conic core, one can numerically estimate the exponential cone’s conic center and width. These numerically compute… view at source ↗

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