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

Researchers present a comprehensive mathematical framework unifying generalized Euler logarithms with applications to machine learning optimization. The work establishes theoretical foundations for deformed exponential functions and introduces new algorithms—Generalized Exponentiated Gradient and Mirror Descent schemes—alongside an Euler-based loss function for neural networks that integrates with natural gradient descent.

This academic paper addresses fundamental mathematical structures that bridge classical information theory and modern machine learning optimization. The generalized Euler logarithm serves as a unifying kernel connecting previously disparate mathematical frameworks including Tsallis, Kaniadakis, and Tempesta-type logarithms, establishing formal relationships that have been studied separately in literature. This unification has practical implications for algorithmic development.

The research contextualizes within the broader trend of developing mathematically principled optimization methods for neural networks. While gradient descent remains dominant, natural gradient descent and information-geometric approaches have gained traction as alternatives that incorporate Fisher Information structure. By demonstrating how two deformation parameters can independently control tail robustness and local gradient behavior, the authors address a known limitation in existing loss functions: the inability to simultaneously optimize for outlier handling and gradient shaping.

For practitioners in machine learning, the Euler-based Generalized Cross-Entropy loss and its seamless integration with Fisher-Rao Natural Gradient descent offers theoretical justification for specific architectural choices. The explicit backpropagation formulas enable direct implementation. The diagonal Fisher Information Matrix approximation reduces computational overhead, making natural gradient methods more practical for large-scale deep learning.

Further developments depend on empirical validation across diverse neural network architectures and datasets. The paper's mathematical rigor positions it as a reference for optimization researchers, though mainstream adoption will require comparative benchmarking against existing methods like Adam and RMSprop. The unification of previously fragmented mathematical frameworks may inspire new optimization techniques combining properties from multiple deformation families.

• →Generalized Euler logarithm unifies multiple mathematical frameworks (Tsallis, Kaniadakis, Tempesta) into single theoretical structure applicable to machine learning
• →New Generalized Exponentiated Gradient and Mirror Descent algorithms use deformed exponential functions as flexible link functions in Bregman divergences
• →Euler-based loss function for neural networks decouples tail robustness from gradient shaping through two independent deformation parameters
• →Fisher Information Matrix isolation enables computationally efficient diagonal natural gradient approximation for large-scale training
• →Exact backpropagation formulas derived for seamless integration with existing deep learning frameworks