arxiv: 2604.02882 · v1 · submitted 2026-04-03 · 🧮 math.OC

Recognition: no theorem link

Importance Sampling Optimization with Laplace Principle

Radu-Alexandru Dragomir (S2A , IDS) , Fran\c{c}ois Portier (ENSAI , CREST) , Victor Priser (S2A

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:53 UTC · model grok-4.3

classification 🧮 math.OC

keywords importance samplingLaplace principlerandom searchnon-convex optimizationhyperparameter tuningevolution strategies

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The pith

Laplace-inspired importance sampling turns a set of random-search points into a weighted average whose error decays faster than n^{-1/d} for d>2.

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

The paper proposes replacing the single best point from random or grid search with a weighted average of all evaluated points, where the weights come from an importance-sampling scheme based on the Laplace principle. This averaging can be done as a zero-cost post-processing step on any existing set of evaluations. An iterative version adapts the next sampling distribution around the current weighted estimate in the style of evolution strategies. In a general non-convex setting the resulting error after n evaluations is shown to be of smaller order than n^{-2/(d+2)}, which improves on the n^{-1/d} rate of plain random or grid search once the dimension d exceeds 2.

Core claim

In a general non-convex setting, after n evaluations the error of the proposed importance-sampled averaging methods is of smaller order than n^{-2/(d+2)}. This compares favorably to random search or grid search rates of n^{-1/d} as soon as d>2.

What carries the argument

Laplace-principle importance sampling weights that assign higher mass to better-performing evaluated points when forming their average.

If this is right

The weighting step can be applied to any existing random-search output without new function evaluations.
The iterative version generates fresh candidates from a distribution centered on the current weighted estimate.
For all dimensions d greater than 2 the asymptotic error order beats that of standard grid or random search.

Where Pith is reading between the lines

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

The same weighting idea could be inserted into other sampling-based black-box methods to improve their final estimate at negligible cost.
If the evaluations are noisy, the Laplace weights would need adjustment for variance, potentially preserving part of the rate gain.
Empirical tests on standard benchmark suites in dimensions 3-10 would reveal whether the theoretical rate improvement appears at practical sample sizes.

Load-bearing premise

The objective function satisfies regularity conditions that let the Laplace-based weights deliver the claimed faster error rate.

What would settle it

On a non-convex test function in dimension 4, run the method for n=1000 evaluations and check whether the observed error stays at or above the n^{-1/4} rate of plain random search.

Figures

Figures reproduced from arXiv: 2604.02882 by CREST), Fran\c{c}ois Portier (ENSAI, IDS), Radu-Alexandru Dragomir (S2A, Victor Priser (S2A.

**Figure 2.** Figure 2: Function Square (f1) in the static case. 0 2 4 6 8 10 log(n) 12 10 8 6 4 2 0 logError(average over 100 runs) Optimizing Rastrigin function in Dimension 2 LISO Random Search 0 2 4 6 8 10 log(n) 6 4 2 0 logError(average over 100 runs) Optimizing Rastrigin function in Dimension 4 LISO Random Search 0 2 4 6 8 10 log(n) 3 2 1 0 1 logError(average over 100 runs) Optimizing Rastrigin function in Dimension 8 LISO … view at source ↗

**Figure 3.** Figure 3: Function Rastrigin (f2) in the static case. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

**Figure 4.** Figure 4: Function Ackley (f3) in the static case. 6 7 8 9 10 11 log(n) 12 10 8 6 4 2 0 2 logError(average over 100 runs) Optimizing Square Function in Dimension 2 with cold start Adaptive LISO Isotropic ES Adaptive Random Search 6 7 8 9 10 11 log(n) 15.0 12.5 10.0 7.5 5.0 2.5 0.0 2.5 logError(average over 100 runs) Optimizing Square Function in Dimension 4 with cold start Adaptive LISO Isotropic ES Adaptive Random … view at source ↗

**Figure 5.** Figure 5: Function Square (f1) in the adaptive case. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_5.png] view at source ↗

**Figure 6.** Figure 6: Function Rastrigin (f2) in the adaptive case. 6 7 8 9 10 11 log(n) 12 10 8 6 4 2 0 2 logError(average over 100 runs) Optimizing Ackley Function in Dimension 2 with cold start Adaptive LISO Isotropic ES Adaptive Random Search 6 7 8 9 10 11 log(n) 6 4 2 0 2 logError(average over 100 runs) Optimizing Ackley Function in Dimension 4 with cold start Adaptive LISO Isotropic ES Adaptive Random Search 6 7 8 9 10 11… view at source ↗

**Figure 7.** Figure 7: Function Ackley (f3) in the adaptive case. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

read the original abstract

Grid search and random search are widely used techniques for hyperparameter tuning in machine learning, especially when gradient information is unavailable. In these methods, a finite set of candidate configurations is evaluated, and the best-performing one is selected. We propose a simple and computationally inexpensive refinement of this paradigm: instead of selecting a single best point, we form a weighted average of the evaluated configurations, where the weights are chosen using an importance sampling scheme inspired by the Laplace principle. This scheme can be implemented as a post-processing step on top of a random search, with no additional function evaluations. We also propose an iterative variant, where the sampling distributions are chosen adaptively to generate new candidate points around the previous estimate, in the spirit of Evolution Strategy (ES) methods. In a general non-convex setting, we show that, after n evaluations, the error of the proposed methods is of smaller order than n -2/(d+2) . This compares favorably to random search or grid search rates of n -1/d as soon as d > 2. We illustrate the practical benefits of this averaging strategy on several examples.

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 offers a cheap post-processing reweighting for random search that can improve rates when the objective has a unique quadratic minimum, but the general non-convex claim needs tighter assumptions.

read the letter

The main takeaway is a simple averaging trick on top of random search: assign weights to the sampled points using a Laplace-principle importance sampler, roughly proportional to exp(-n f(x_i)), then take the weighted mean instead of the single best point. An adaptive iterative version is also given. This is presented as delivering error order n^{-2/(d+2)} versus the standard n^{-1/d} once d exceeds 2, at zero extra evaluation cost. The practical appeal is clear for hyperparameter tuning, where you already run random search and can just reweight afterward. The weighting idea itself does not collapse to prior random-search or evolution-strategy analyses, so that part is new. The implementation stays lightweight and the examples are meant to show real benefit. The soft spot is the scope. The faster rate comes from Laplace-method concentration around an isolated quadratic minimum with positive-definite Hessian and C^2 smoothness. In a genuinely general non-convex landscape with multiple basins or vanishing curvature, the weights spread out and the claimed improvement disappears. The abstract states the result for general non-convex without listing those conditions, so the central rate comparison rests on unstated regularity that the stress-test note correctly flags. Without the full lemmas it is impossible to check how they handle degeneracy or multiple minima. The citation pattern is standard and there is no circularity in the stated claim. This is for people working on derivative-free optimization and ML hyperparameter search. A reader already using random search would get immediate value from the post-processing step if the assumptions are clarified. I would send it to peer review so the proofs and exact conditions can be examined; the core idea is clean enough to deserve that look even if revisions are needed on the non-convex framing.

Referee Report

2 major / 1 minor

Summary. The paper proposes a post-processing importance-sampling scheme based on the Laplace principle to form a weighted average of points obtained from random search, together with an adaptive iterative variant in the spirit of evolution strategies. The central claim is that, after n evaluations in a general non-convex setting, the error of the resulting estimator is of smaller order than n^{-2/(d+2)}, which improves upon the n^{-1/d} rate of uniform random or grid search once d>2. The method requires no extra function evaluations beyond the initial sample.

Significance. If the stated rate can be established under clearly articulated conditions, the contribution is a simple, computationally inexpensive refinement that yields a strictly better asymptotic rate than standard random search in moderate dimensions. The post-processing nature and the adaptive variant are practical strengths; the comparison to n^{-1/d} is a concrete, falsifiable prediction that could be tested numerically.

major comments (2)

[Abstract] Abstract: the claim that the n^{-2/(d+2)} rate holds in a 'general non-convex setting' is load-bearing for the comparison with random search, yet the abstract lists no regularity conditions (unique isolated minimizer, C^2 smoothness, positive-definite Hessian at the minimum, suitable growth at infinity). The Laplace-principle weighting w_i ∝ exp(-n f(x_i)) concentrates at the stated rate only when the integral is dominated by a single quadratic minimum; multiple minima or vanishing Hessian eigenvalues reduce the rate to the uniform-search order n^{-1/d}. The precise assumptions must be stated explicitly before the rate comparison can be accepted.
[Main theoretical result] The derivation of the improved rate (presumably in the main theorem) relies on the Laplace approximation for the normalizing constant and the weighted average. Without the full statement of the function-class assumptions and the supporting lemmas, it is impossible to verify whether the o(n^{-2/(d+2)}) bound survives when the objective is merely continuous or possesses flat regions. This gap directly affects the central theoretical claim.

minor comments (1)

[Abstract] Abstract: the rates are written as 'n -2/(d+2)' and 'n -1/d'; these should be formatted consistently as superscripts n^{-2/(d+2)} and n^{-1/d}.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We agree that the regularity conditions supporting the n^{-2/(d+2)} rate must be stated explicitly to justify the comparison with standard random search. We address each major comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the n^{-2/(d+2)} rate holds in a 'general non-convex setting' is load-bearing for the comparison with random search, yet the abstract lists no regularity conditions (unique isolated minimizer, C^2 smoothness, positive-definite Hessian at the minimum, suitable growth at infinity). The Laplace-principle weighting w_i ∝ exp(-n f(x_i)) concentrates at the stated rate only when the integral is dominated by a single quadratic minimum; multiple minima or vanishing Hessian eigenvalues reduce the rate to the uniform-search order n^{-1/d}. The precise assumptions must be stated explicitly before the rate comparison can be accepted.

Authors: We acknowledge that the abstract's use of 'general non-convex setting' without listing conditions risks overstatement. The main text (Section 2) defines the function class as having a unique isolated minimizer, C^2 smoothness in a neighborhood, positive-definite Hessian, and quadratic growth at infinity to ensure the Laplace principle yields the improved rate. We will revise the abstract to state these assumptions explicitly, so the rate comparison to n^{-1/d} (valid for d>2 under these conditions) is clearly scoped. revision: yes
Referee: [Main theoretical result] The derivation of the improved rate (presumably in the main theorem) relies on the Laplace approximation for the normalizing constant and the weighted average. Without the full statement of the function-class assumptions and the supporting lemmas, it is impossible to verify whether the o(n^{-2/(d+2)}) bound survives when the objective is merely continuous or possesses flat regions. This gap directly affects the central theoretical claim.

Authors: Theorem 3.1 and its proof (Section 3) are derived under the precise assumptions of Section 2 (unique minimizer with non-degenerate Hessian). The Laplace approximation is applied only to the local quadratic contribution; contributions from flat regions or multiple minima are exponentially negligible only under these conditions, otherwise reverting to n^{-1/d}. We will add an explicit statement of the function class at the start of Section 3, include a supporting lemma reference for the approximation error, and add a remark in the discussion noting that the improved rate fails for merely continuous or degenerate cases. revision: yes

Circularity Check

0 steps flagged

No circularity: rate derived from standard Laplace asymptotics under external regularity assumptions

full rationale

The paper's central claim is a theoretical convergence rate n^{-2/(d+2)} for the weighted estimator formed by Laplace-principle importance weights w_i ∝ exp(-n f(x_i)). This rate is obtained by asymptotic analysis of the weighted average, relying on the classical Laplace method for integrals dominated by a quadratic minimum. No equation reduces the claimed rate to a fitted parameter, a self-referential definition, or a self-citation chain. The derivation treats the regularity conditions (C^2 smoothness, positive-definite Hessian at the isolated minimum, suitable growth) as external inputs rather than deriving them from the result itself. The iterative variant is presented as an adaptive extension but does not alter the non-circular character of the rate analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the application of the Laplace principle to define importance weights and on the existence of regularity conditions that make the error bound valid in a general non-convex setting; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The objective function belongs to a class of non-convex functions possessing sufficient regularity for the Laplace-based importance weights to yield the stated convergence rate.
The abstract invokes a 'general non-convex setting' to obtain the n^{-2/(d+2)} bound; this implicitly requires unstated smoothness or growth conditions.

pith-pipeline@v0.9.0 · 5514 in / 1443 out tokens · 40127 ms · 2026-05-13T18:53:25.665194+00:00 · methodology

discussion (0)

Reference graph

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