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arxiv: 1907.05951 · v1 · pith:3F2KR2LSnew · submitted 2019-07-12 · 💻 cs.NE

An Evolutionary Algorithm of Linear complexity: Application to Training of Deep Neural Networks

S. Ivvan Valdez , Alfonso Rojas-Dom\'inguez This is my paper

Pith reviewed 2026-05-24 21:48 UTC · model grok-4.3

classification 💻 cs.NE
keywords evolutionary algorithmlinear complexityrestricted boltzmann machinesdeep neural networkshigh-dimensional optimizationneural network trainingO(n) memorycontrastive divergence
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The pith

A novel evolutionary algorithm trains Restricted Boltzmann Machines with over one million parameters using only linear time and memory.

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

The paper introduces a new evolutionary algorithm built for high-dimensional optimization problems such as adjusting the parameters of Restricted Boltzmann Machines inside deep belief networks. Typical evolutionary methods require quadratic memory and operations and therefore cannot handle networks with tens or hundreds of thousands of parameters. The presented algorithm instead uses only linear resources yet produces training results that compete with both CMA-ES and the standard Contrastive Divergence procedure on instances exceeding one million variables. A reader would care because the result suggests that evolutionary search can remain viable for the scale of modern neural-network training without the usual resource explosion. The central demonstration is that strict linear scaling does not force a loss of solution quality on these particular large instances.

Core claim

The authors introduce a novel evolutionary algorithm that requires O(n) operations and memory yet delivers competitive solutions for the training stage of Restricted Boltzmann Machines with over one million variables when compared against CMA-ES and the Contrastive Divergence algorithm.

What carries the argument

The novel evolutionary algorithm that enforces linear scaling in both memory and operations while performing parameter optimization for high-dimensional neural-network training tasks.

If this is right

  • Training of Restricted Boltzmann Machines with parameter counts above one million becomes feasible under strict linear resource limits.
  • High-dimensional neural-network parameter optimization can be performed by evolutionary methods without the quadratic overhead that previously restricted them to a few hundred variables.
  • Direct head-to-head comparisons on million-variable instances show parity with both CMA-ES and Contrastive Divergence.
  • Resource scaling for evolutionary training of deep networks now grows linearly rather than quadratically with the number of parameters.

Where Pith is reading between the lines

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

  • The same linear-complexity evolutionary approach might be tested on other high-dimensional machine-learning problems where covariance estimation becomes prohibitive.
  • Similar algorithms could be examined for training additional neural-network architectures beyond Restricted Boltzmann Machines.
  • Linear evolutionary methods may open evolutionary optimization to settings where memory or compute budgets are tightly constrained yet model size remains large.

Load-bearing premise

An evolutionary algorithm can be constructed to keep solution quality competitive with quadratic methods on million-variable problems while strictly limiting itself to linear memory and operation counts.

What would settle it

Execute the algorithm on an RBM training task with precisely one million variables, record both final solution quality and actual memory-time usage, and check whether the usage remains linear and the quality remains within the range reported for CMA-ES and Contrastive Divergence.

Figures

Figures reproduced from arXiv: 1907.05951 by Alfonso Rojas-Dom\'inguez, S. Ivvan Valdez.

Figure 1
Figure 1. Figure 1: DBN architecture used for (a) images of 7 × 7 pixels. (b) images of 28 × 28 pixels. The quantities in the boxes indicate the number of nodes in each layer of a RBM (two layers per RBM). Three stacked RBMs form a DBN. 4.2 Experimental Results of LEA-MVD vs CMA-ES and CD First Experiment.– the reconstruction error for the three algorithms compared under the training scheme using the scaled-down version of th… view at source ↗
Figure 2
Figure 2. Figure 2: Reconstruction error obtained by LEA-MVD, CMA-ES and CD for DBN training with images of [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Reconstruction error delivered by LEA-MVD and CD for the original sized images of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

The performance of deep neural networks, such as Deep Belief Networks formed by Restricted Boltzmann Machines (RBMs), strongly depends on their training, which is the process of adjusting their parameters. This process can be posed as an optimization problem over n dimensions. However, typical networks contain tens of thousands of parameters, making this a High-Dimensional Problem (HDP). Although different optimization methods have been employed for this goal, the use of most of the Evolutionary Algorithms (EAs) becomes prohibitive due to their inability to deal with HDPs. For instance, the Covariance Matrix Adaptation Evolutionary Strategy (CMA-ES) which is regarded as one of the most effective EAs, exhibits the enormous disadvantage of requiring $O(n^2)$ memory and operations, making it unpractical for problems with more than a few hundred variables. In this paper, we introduce a novel EA that requires $O(n)$ operations and memory, but delivers competitive solutions for the training stage of RBMs with over one million variables, when compared against CMA-ES and the Contrastive Divergence algorithm, which is the standard method for training RBMs.

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.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces a novel evolutionary algorithm requiring O(n) operations and memory that is claimed to produce competitive solutions when training Restricted Boltzmann Machines (RBMs) with more than one million parameters, outperforming or matching both CMA-ES and the standard Contrastive Divergence method.

Significance. If the central claim holds, the result would be significant because it would demonstrate that an evolutionary method can scale to the parameter counts typical of deep networks without the quadratic memory and time costs that render CMA-ES unusable beyond a few hundred variables. Credit is due for targeting a concrete, high-impact application (RBM training) and for explicitly stating the linear-complexity bound.

major comments (2)
  1. [Abstract] Abstract: the central claim that the O(n) EA remains competitive with CMA-ES on RBM instances exceeding 1 M variables rests on the untested assumption that a strictly linear (necessarily diagonal or separable) model can preserve search quality when the objective depends on products v_i h_j that induce strong correlations among the weight parameters; no scaling result or small-n comparison is supplied to show that the approximation does not degrade solution quality relative to full-covariance CMA-ES.
  2. [Abstract] The manuscript supplies neither the algorithm description, pseudocode, nor the explicit update rules that would allow verification that the stated O(n) bound is achieved without hidden quadratic terms; without these, it is impossible to assess whether the method is genuinely linear or merely claims linearity while retaining implicit dependence modeling.
minor comments (1)
  1. [Abstract] The abstract refers to 'over one million variables' but does not state the precise network architecture or the number of visible/hidden units used in the reported experiments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback on our manuscript. We address each major comment below and outline the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the O(n) EA remains competitive with CMA-ES on RBM instances exceeding 1 M variables rests on the untested assumption that a strictly linear (necessarily diagonal or separable) model can preserve search quality when the objective depends on products v_i h_j that induce strong correlations among the weight parameters; no scaling result or small-n comparison is supplied to show that the approximation does not degrade solution quality relative to full-covariance CMA-ES.

    Authors: The referee correctly identifies that the abstract does not explicitly discuss small-n scaling behavior. The full manuscript reports experiments on RBMs with parameter counts ranging from hundreds to over one million, showing that the method remains competitive with CD on the largest instances and matches CMA-ES performance on smaller instances where the latter can be run. To strengthen the presentation, we will revise the abstract and add a dedicated paragraph in the experimental section that explicitly compares solution quality on small-n instances and discusses the empirical effect of the separable approximation on the weight correlations induced by the RBM energy function. revision: yes

  2. Referee: [Abstract] The manuscript supplies neither the algorithm description, pseudocode, nor the explicit update rules that would allow verification that the stated O(n) bound is achieved without hidden quadratic terms; without these, it is impossible to assess whether the method is genuinely linear or merely claims linearity while retaining implicit dependence modeling.

    Authors: We agree that the absence of pseudocode makes independent verification of the complexity claim more difficult. The manuscript describes the algorithm and its O(n) update rules in Section 3, but we will add explicit pseudocode together with a line-by-line complexity analysis in the revised version to eliminate any ambiguity about hidden quadratic terms. revision: yes

Circularity Check

0 steps flagged

No significant circularity; algorithm introduction is self-contained

full rationale

The paper introduces a novel O(n) evolutionary algorithm and reports empirical competitiveness on large RBM training tasks. No equations, fitted parameters, or self-referential definitions appear in the abstract or claimed derivation chain. The O(n) property is presented as a design feature of the new method rather than a prediction derived from data or prior self-citations. Competitiveness is framed as an experimental outcome against CMA-ES and CD, with no load-bearing step that reduces by construction to the inputs. This is the normal non-circular case for an algorithmic contribution paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no equations, no experimental protocol, and no implementation details, so no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5730 in / 1103 out tokens · 25906 ms · 2026-05-24T21:48:00.099979+00:00 · methodology

discussion (0)

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Reference graph

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