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arxiv: 2502.17500 · v3 · submitted 2025-02-21 · 💻 cs.LG · cs.AI

Generalized Euler Logarithm and its Applications in Machine Learning: Natural Gradient, Backpropagation, Generalized EG, Mirror Descent and OLPS

Andrzej Cichocki This is my paper

Pith reviewed 2026-05-23 02:44 UTC · model grok-4.3

classification 💻 cs.LG cs.AI
keywords Euler logarithmgeneralized entropyBregman divergencenatural gradientmirror descentgeneralized cross-entropybackpropagation
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The pith

The two-parameter Euler logarithm unifies many generalized entropies and serves as a link function for generalized exponentiated gradient, mirror descent, and natural gradient methods.

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

The paper establishes the Euler (a,b)-logarithm and its inverse deformed exponential as a common kernel that recovers Tsallis, Kaniadakis, and several other one- and two-parameter logarithms under suitable choices of a and b. It derives the parameter domains that keep the function monotonic, concave, and invertible, then uses these properties to embed the logarithm inside Bregman divergences. This construction yields concrete generalized exponentiated gradient and mirror descent updates, an Euler-based generalized cross-entropy loss whose back-propagation formulas are given explicitly, and a natural-gradient scheme in which the two parameters separately control tail behavior and local curvature via the Fisher information matrix.

Core claim

The Euler (a,b)-logarithm is established as a unifying kernel for a wide family of generalized entropies and divergence measures. On the algorithmic side, generalized Exponentiated Gradient (GEG) and Mirror Descent (MD) schemes are introduced in which the Euler (a,b)-logarithm acts as a flexible link function in the underlying Bregman divergence. An Euler-based Generalized Cross-Entropy (GCE) loss is proposed for deep neural networks, together with its exact backpropagation formulas and seamless integration with Fisher-Rao Natural Gradient descent, where the two deformation parameters decouple tail robustness from local gradient shaping.

What carries the argument

The Euler (a,b)-logarithm, used as the link function inside Bregman divergences to generate the generalized algorithms.

If this is right

  • Generalized EG and MD updates become available for any pair (a,b) inside the valid domain.
  • Exact back-propagation rules exist for the Euler-based GCE loss in deep networks.
  • Fisher-Rao natural gradient descent can be realized with a diagonal approximation that isolates the two deformation parameters.
  • Tail robustness and local gradient shaping are controlled independently by a and b.

Where Pith is reading between the lines

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

  • The same link-function construction could be applied directly to online portfolio selection algorithms listed in the title.
  • Different (a,b) pairs might be selected per layer or per task to match data tail properties without changing the overall optimizer structure.
  • The integral and series representations derived for the logarithm may yield new closed-form expressions for other information-theoretic quantities.

Load-bearing premise

The chosen ranges of a and b must keep the logarithm monotonic, concave, and invertible so that it can serve as a valid link inside Bregman divergences.

What would settle it

Run the proposed GEG or GCE algorithms with parameter pairs lying outside the clarified monotonicity domains and check whether the iterates remain valid probability vectors or whether the loss ceases to decrease.

Figures

Figures reproduced from arXiv: 2502.17500 by Andrzej Cichocki.

Figure 1
Figure 1. Figure 1: Surface plots of the Euler (a, b)-logarithm for various values of hyperparameters a and b. These figures illustrate the (a, b)-logarithm in terms of b and x for fixed a = −0.3 and a = −1.1. Furthermore, by applying nonlinear transformation in (34) x → exp(logT q (x)) (35) 9 [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Surface plots of the Kaniadakis-Scarfone (κ, λ)-logarithm for various values of hyperpa￾rameters λ and κ. These figures illustrate the (λ, κ)-logarithm in terms of α and x for fixed λ = 0.7. The black continuous line represents the reference of the standard natural logarithm, which is ob￾tained for a = b = 0. Similarly, the Tempesta two-parameters (κ, α)-logarithm defined as [54, 14] logTe α,κ(x) = 1 (1 + … view at source ↗
Figure 3
Figure 3. Figure 3: Surface plots of the Tempesta (α, κ)-logarithm for various values of hyperparameters α and kappa. These figures illustrate the (α, κ)-logarithm in terms of α and x for fixed κ = 0.7 and κ = −0.9 The black continuous line represents the reference of the standard logarithm, which is obtained for α = 1 and κ = 0. where λ = (a − b)/a, x˜ = (a − b)x, q = (λ + 1)/λ = (2a − b)/(a − b) and Wq is the Lambert–Tsalli… view at source ↗
read the original abstract

This paper investigates in depth the fundamental properties of the two-parameter generalized Euler logarithm and its inverse, the associated deformed $(a,b)$-exponential function. We systematically clarify the parameter domains that guarantee monotonicity, concavity, and invertibility, derive series and integral representations, and provide explicit links to a broad class of one- and two-parameter deformations, including Tsallis, Kaniadakis, Schw\"ammle--Tsallis, Kaniadakis--Scarfone, and Tempesta-type logarithms and their inverse exponentials. In this way, the Euler $(a,b)$-logarithm is established as a unifying kernel for a wide family of generalized entropies and divergence measures. On the algorithmic side, we extend applications of the Euler logarithm to modern machine learning and optimization. We introduce generalized Exponentiated Gradient (GEG) and Mirror Descent (MD) schemes in which the Euler $(a,b)$-logarithm acts as a flexible link function in the underlying Bregman divergence. In addition, we propose an Euler-based Generalized Cross-Entropy (GCE) loss for deep neural networks, derive its exact backpropagation formulas, and detail its seamless integration with Fisher-Rao Natural Gradient (NG) descent. By isolating the Fisher Information Matrix (FIM) and developing a diagonal NG approximation, we demonstrate how the two deformation parameters successfully decouple tail robustness from local gradient shaping.

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

1 major / 2 minor

Summary. The paper defines the two-parameter Euler (a,b)-logarithm and its inverse (a,b)-exponential, clarifies the (a,b) domains ensuring monotonicity, concavity and invertibility, derives series/integral representations, and shows explicit reductions to Tsallis, Kaniadakis, Schwämmle–Tsallis and related one- and two-parameter deformations. It then uses the Euler logarithm as the link function inside Bregman divergences to define generalized exponentiated gradient and mirror descent updates, introduces an Euler-based generalized cross-entropy loss with exact back-propagation rules, and combines it with a diagonal Fisher-Rao natural-gradient approximation that isolates the two parameters to separately control tail robustness and local gradient shaping.

Significance. If the domain clarification and the associated convexity/invertibility proofs hold, the work supplies a single two-parameter kernel that recovers a broad family of generalized entropies and supplies concrete algorithmic extensions (GEG, MD, GCE loss, exact back-prop, diagonal NG) whose parameter decoupling is potentially useful for robust deep-learning training. The explicit links to existing deformations and the machine-learning instantiations would constitute a genuine unification rather than a reparametrization.

major comments (1)
  1. [Parameter-domain section (and any appendix containing the monotonicity/concavity arguments)] The central load-bearing claim is that the clarified (a,b) domains guarantee strict concavity (hence convexity of the mirror map) and global invertibility so that the Euler logarithm can serve as a valid Bregman link across all listed deformations and algorithms. The manuscript states these domains but supplies no derivative sign charts, second-derivative analysis, or explicit convexity proofs for the interior or boundary points of the claimed region; without such verification the unification and the claimed decoupling of tail robustness from gradient shaping remain unconfirmed.
minor comments (2)
  1. [Introduction and definitions] Notation for the two-parameter exponential and its inverse should be introduced once with a single consistent symbol rather than alternating between “deformed exponential” and “(a,b)-exponential.”
  2. [Natural-gradient subsection] The diagonal FIM approximation is presented without a quantitative error bound or comparison to the full-matrix NG baseline; a short remark on the approximation quality would strengthen the NG section.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the importance of rigorous verification of the domain properties. We address the single major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Parameter-domain section (and any appendix containing the monotonicity/concavity arguments)] The central load-bearing claim is that the clarified (a,b) domains guarantee strict concavity (hence convexity of the mirror map) and global invertibility so that the Euler logarithm can serve as a valid Bregman link across all listed deformations and algorithms. The manuscript states these domains but supplies no derivative sign charts, second-derivative analysis, or explicit convexity proofs for the interior or boundary points of the claimed region; without such verification the unification and the claimed decoupling of tail robustness from gradient shaping remain unconfirmed.

    Authors: We agree that explicit second-derivative analysis and sign charts are necessary to fully substantiate the claimed (a,b) domains. While the manuscript states the domains that ensure monotonicity, concavity, and invertibility, the detailed derivative sign analysis and convexity proofs for the interior and boundary points are only outlined rather than presented with complete charts. In the revision we will expand the parameter-domain section and add a dedicated appendix containing (i) the first- and second-derivative expressions, (ii) sign charts over the interior and boundary of the claimed region, and (iii) explicit verification that the second derivative is strictly negative (hence strict concavity of the Euler logarithm and convexity of the associated mirror map) together with global invertibility. These additions will directly confirm the validity of the Bregman link for all listed deformations and support the parameter-decoupling claims in the algorithmic sections. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations and applications are self-contained

full rationale

The paper defines the Euler (a,b)-logarithm, derives its monotonicity/concavity/invertibility domains, series/integral forms, and explicit mappings to Tsallis/Kaniadakis/etc. deformations directly from its functional form, then uses the resulting object as a link inside newly proposed Bregman-based GEG/MD schemes and GCE loss with exact backprop and diagonal NG. No equations reduce a claimed prediction or uniqueness result to a fitted input or prior self-citation by construction; the ML extensions introduce independent algorithmic content grounded in the derived link function rather than tautological re-use of inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

Review based solely on abstract; free parameters are the two deformation parameters whose domains control key properties. Axioms concern the function's monotonicity and invertibility for valid use in divergences. No invented entities are introduced.

free parameters (1)
  • a, b
    Two deformation parameters whose domains are clarified to ensure monotonicity, concavity, and invertibility for use as link functions.
axioms (1)
  • domain assumption The generalized logarithm is monotonic, concave, and invertible for appropriate parameter domains
    Abstract states that these domains are clarified to establish the function as a valid kernel for entropies and divergences.

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