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arxiv: 2607.01552 · v1 · pith:UNBF2S6Dnew · submitted 2026-07-02 · 🧮 math.OC

Symbolic Discovery of Iterative Algorithms: A Continuous Latent Space Bayesian Optimization Framework

Tongjia Liu , Ilias Mitrai This is my paper

Pith reviewed 2026-07-03 01:01 UTC · model grok-4.3

classification 🧮 math.OC
keywords algorithm discoveryvariational autoencoderBayesian optimizationiterative algorithmssymbolic update functionslatent space searchoptimization algorithms
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The pith

A variational autoencoder embeds discrete update functions into a continuous latent space where Bayesian optimization finds new symbolic iterative algorithms.

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

The paper sets out to automate discovery of iterative optimization algorithms by searching for their update rules. It trains a variational autoencoder on possible update functions so that the search space becomes continuous rather than discrete. Bayesian optimization then operates in this latent space to propose candidates that decode back into symbolic expressions. The resulting functions are tested on concrete optimization problems, and the whole process runs faster than earlier approaches that relied on mathematical programming over discrete choices. A reader would care because the method removes the need to guess the algebraic form of an algorithm in advance.

Core claim

We formulate the algorithm discovery task as a discrete optimization problem and search for new update functions using latent space Bayesian Optimization. The proposed framework first learns a continuous representation of the discrete space of update functions using variational autoencoders, transforming the algorithm discovery task from a discrete to a continuous search problem. The continuous representation is subsequently used to search for new algorithms using Bayesian optimization. Application to two case studies shows that the proposed approach can discover new update functions in symbolic form without any assumptions on the functional form of the update function. Moreover, the computa

What carries the argument

The variational autoencoder that produces a continuous latent representation of discrete update functions, allowing Bayesian optimization to locate and decode new symbolic algorithms.

If this is right

  • New update functions appear in explicit symbolic form for the optimization problems considered.
  • No assumptions about the algebraic structure of the update functions are required in advance.
  • The discovery process uses less computational time than mathematical programming methods that operate directly on discrete spaces.
  • The same workflow applies across at least two distinct case studies in optimization.

Where Pith is reading between the lines

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

  • The continuous embedding might let researchers interpolate between known algorithms to generate hybrid update rules.
  • If the latent space organizes functions by performance traits, inspecting nearby points could suggest design principles for better algorithms.
  • Applying the discovered functions to optimization problems outside the training cases would test whether the method produces generally useful rules.

Load-bearing premise

The latent space created by the variational autoencoder contains points that decode to update functions which are both novel and effective when applied to the original optimization problems.

What would settle it

Decoding the functions proposed by Bayesian optimization in the latent space and checking whether they actually solve the target optimization problems or reduce computation time relative to known methods.

Figures

Figures reproduced from arXiv: 2607.01552 by Ilias Mitrai, Tongjia Liu.

Figure 1
Figure 1. Figure 1: Proposed Variational Autoencoder-based Bayesian Optimization search for iterative algorithms [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Impact of the latent space dimension on the loss function [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Evolution of gradient norm across iterations for GD and the discovered algorithms in case study 1. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of gradient norm across iterations for GD and the discov [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
read the original abstract

In this paper, we consider the automated discovery of iterative optimization algorithms. We formulate the algorithm discovery task as a discrete optimization problem and search for new update functions using latent space Bayesian Optimization. The proposed framework first learns a continuous representation of the discrete space of update functions using variational autoencoders, transforming the algorithm discovery task from a discrete to a continuous search problem. The continuous representation is subsequently used to search for new algorithms using Bayesian optimization. Application to two case studies shows that the proposed approach can discover new update functions in symbolic form without any assumptions on the functional form of the update function. Moreover, the computational time required to discover the new update functions is lower than existing mathematical programming-based approaches.

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 / 0 minor

Summary. The paper formulates discovery of iterative optimization algorithms as a discrete search over symbolic update functions. It trains a variational autoencoder on a pre-constructed corpus of such functions to obtain a continuous latent space, then runs Bayesian optimization in that space to identify new update rules. The method is applied to two case studies; the abstract claims that new functions are discovered in symbolic form without assumptions on functional form and that the approach is faster than existing mathematical-programming baselines.

Significance. If the latent-space search reliably yields update functions that are both novel relative to the training corpus and demonstrably effective on the target problems, the framework would offer a practical route to automated symbolic algorithm design. The combination of VAE embedding with BO is a reasonable way to convert a combinatorial search into a continuous one, but the significance hinges on whether the decoded functions outperform or meaningfully extend known methods and whether the computational savings are reproducible.

major comments (2)
  1. [Abstract] Abstract: the central claim that the method discovers update functions 'without any assumptions on the functional form' is not supported by the described procedure. The framework first constructs a discrete space of update functions and trains the VAE on it; this construction necessarily selects a finite vocabulary of operators, constants, and expression structures (via grammar or enumeration). Any function outside that pre-specified space cannot be discovered, so the 'no assumptions' phrasing is inaccurate and load-bearing for the paper's novelty argument.
  2. [Abstract] Abstract / method description: the weakest assumption—that the VAE latent space permits BO to locate update functions that are both novel and effective when decoded—receives no supporting evidence in the provided text. Without reported metrics on reconstruction fidelity, latent-space smoothness, or out-of-sample performance of the discovered rules on the original optimization problems, it is impossible to assess whether the continuous relaxation actually solves the discrete discovery task.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We respond to each major comment below and will revise the manuscript accordingly to improve precision and add supporting evidence.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the central claim that the method discovers update functions 'without any assumptions on the functional form' is not supported by the described procedure. The framework first constructs a discrete space of update functions and trains the VAE on it; this construction necessarily selects a finite vocabulary of operators, constants, and expression structures (via grammar or enumeration). Any function outside that pre-specified space cannot be discovered, so the 'no assumptions' phrasing is inaccurate and load-bearing for the paper's novelty argument.

    Authors: We agree that the abstract phrasing is imprecise and overstates the claim. The discrete corpus is generated from a finite grammar and vocabulary, which necessarily bounds the searchable space. We will revise the abstract to state that the method discovers new symbolic update functions within the space defined by the chosen grammar and operators, without presupposing any particular functional form (such as gradient-based or other parametric templates) for the discovered algorithms. This still differentiates the approach from prior work that restricts the search to specific algorithm families. revision: yes

  2. Referee: [Abstract] Abstract / method description: the weakest assumption—that the VAE latent space permits BO to locate update functions that are both novel and effective when decoded—receives no supporting evidence in the provided text. Without reported metrics on reconstruction fidelity, latent-space smoothness, or out-of-sample performance of the discovered rules on the original optimization problems, it is impossible to assess whether the continuous relaxation actually solves the discrete discovery task.

    Authors: The case studies demonstrate that the discovered functions are novel relative to the corpus and effective on the target problems. However, we acknowledge that the manuscript does not report explicit VAE diagnostics such as reconstruction fidelity or latent-space smoothness metrics. In the revision we will add: reconstruction error on held-out functions from the corpus, examples of smooth interpolation between latent points with corresponding decoded expressions, and explicit verification that the final decoded update rules were evaluated on the original optimization tasks. These additions will directly address whether the continuous relaxation is effective. revision: yes

Circularity Check

0 steps flagged

No circularity: method is an independent search over pre-specified symbolic space

full rationale

The paper frames algorithm discovery as a discrete optimization problem solved via VAE latent-space Bayesian optimization. No derivation chain reduces a claimed result to its own fitted inputs by construction, no self-citation is invoked as load-bearing uniqueness, and no prediction is statistically forced from a subset of the same data. The central procedure (train VAE on enumerated expressions, then BO in latent space) is self-contained and externally verifiable; any limitation on the explored function class arises from the initial grammar choice rather than from circular re-use of outputs as inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on the abstract alone, no explicit free parameters, axioms, or invented entities are stated. The approach implicitly assumes standard VAE training and BO acquisition functions function as described in the broader literature.

pith-pipeline@v0.9.1-grok · 5639 in / 1015 out tokens · 33949 ms · 2026-07-03T01:01:02.804749+00:00 · methodology

discussion (0)

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Reference graph

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