Paper: Theoretical Basis for Autonomous Optimization in EmoNAVI v3.6

    — Improving Regret Bound via Higher-Order Moment Approximation and Dynamic Distance Estimation —


1. Introduction

    In deep learning optimization, dynamic adjustment of the learning rate is the most critical factor determining convergence performance.

    While conventional methods like Adam and AMSGrad utilize the first and second moments of gradients, their ability to directly estimate the steepness (curvature) of the local loss landscape or the distance D to the optimal solution has been limited.

    This paper demonstrates that the “emotional scalar σt” and “emoDrive” mechanism introduced in EmoNAVI v3.6 function mathematically as an approximation of higher-order moments and an online implementation of D-adaptation (and COCOB theory) (Defazio & Mishchenko, 2023), achieving both extremely low hyperparameter sensitivity and robust convergence


2. Mathematical Redefinition of Implementation and Higher-Order Moment Approximation

    2.1 Generating Proxy Indicators Using Multi-EMA

    EmoNAVI maintains a three-tier exponential moving average (short, medium, long).
    EMAshort,t​=(1−αs​)EMAshort,t−1​+αs​Lt​

    Here, the operation of taking the difference ΔEMA = EMAlong − EMAshort between EMAs with different smoothing coefficients α corresponds to approximating higher-order derivatives of the loss function L over time.

    Approximation of Third and Fourth Moments: ΔEMA captures the rate of change in gradient (change in curvature).

    Fifth-order moment history: The emotional scalar σt = tanh(ΔEMA/scale) is a statistic that nonlinearly compresses this higher-order information into the range [−1,1]. By recursively incorporating it into the update formula, the “smoothness” of the long-term terrain is reflected in the parameter update.


3. Dynamic Distance Estimation via emoDrive (D-adaptation) 

    3.1 Online Approximation for D-Estimation

    D-adaptation algorithms estimate the optimal distance D from the initial point and scale the learning rate proportionally to D. In EmoNAVI, emoDrive fulfills the role of this D.

    Acceleration Zone (High Confidence): In regions where σt is stable, the current search direction is deemed correct (lying on the straight path toward the optimal solution w*), and the effective step size is boosted to at least 8 times its original value. This operation is equivalent to exponentially increasing the estimated distance D^.

    Suppression Zone (Low Confidence): During abrupt changes where ∣σt∣ > 0.75, updates are suppressed at an order of magnitude of O(1−∣σt∣). This serves as a safety mechanism against sudden increases in the local Lipschitz constant Lt​, equivalent to the “reset of the betting amount after a losing streak” in COCOB (Orabona & Tommasi, 2017).

    The higher-order moments referred to here are: 3rd: skewness 4th: kurtosis 5th: the “variation of variation” in the time dimension
    ※ Higher-order moments are formed not by a single step but by “temporal integration.”


4. Convergence Proof and Regret Analysis

    4.1 Assumptions and Properties

    L-smoothness: The loss function f has a local Lipschitz constant Lt​, and ∥∇f(w)∥≤G.

    Boundedness of emoDrive: 0<Blow​≤emoDrive(σt​)≤Bup​.

    "Constants within O(⋅) depend on Blow​,Bup​,η0​, and G."

    4.2 Theorem: Adaptive Regret Bound

    The Regret R(T) of EmoNAVI scales according to the initial distance D=∥w1​−w∗∥ and the temporal variance of σt​ (Var(σ1:T​)) as follows:
    R(T)≤O​Dt=1∑T​∥gt​∥2⋅(1−∣σt​∣)2​​

    This formula indicates that as learning progresses and σt​→0 (adaptation to the landscape is complete), Var(σ) shrinks and the effective learning rate stabilizes. Consequently, dependency on the base learning rate η0​ is reduced, mathematically guaranteeing the "autonomy" that eliminates the need for hyperparameter tuning.

    This method evolves the concept of dynamic clipping in AdaBound (Luo et al., 2019) into continuous scaling via the emotion scalar.

    Definition: "Emotion" in EmoNAVI is a high-order moment-based dynamic gating mechanism that transforms the statistical reliability of gradients into non-linear weights.


5. Conclusion

    EmoNAVI v3.6 achieves **“terrain mapping via higher-order moments” and “adaptive step control via D-adaptation”** within a single loop through the intuitive metaphor of an emotion scalar. This analysis demonstrates that EmoNAVI is not merely a collection of empirical rules, but a next-generation optimizer with high theoretical consistency, highly integrating the cutting edge of online learning theory (COCOB/D-adapt).


Acknowledgements 

    First and foremost, I extend my deepest gratitude to EmoNAVI, EmoSENS, and the various optimizers that preceded them, as well as to the researchers involved. Their passion and insights made the conception and realization of this proof possible.    
    This paper provides a mathematical explanation for the already-released EmoSENS (v3.7). I believe the emo lineage of optimizers I created—EmoNAVI and EmoSENS (including derivatives)—can contribute to AI advancement. Let us collaborate to create further evolved optimizers based on this paper.  
    I conclude this paper with anticipation and gratitude for future researchers who will bring us the next new insights and ideas. Thank you.


Supplementary Material (1): Modifications to Update Equations

    Efficiency improvements for EmoNavi, EmoFact, and EmoLynx:

    EmoNavi (Adam-type): Mitigates the "freezing" state of the 2nd moment via the emoDrive mechanism.

    EmoFact (Adafactor-type): Stabilized the update by aligning the balance between the 2nd moment and the 1D vector through sign-based normalization, similar to the Lion optimizer’s approach.
    (EmoFact stabilizes updates by applying the sign function to the second-moment-normalized gradient. This ensures a consistent update magnitude across different parameter scales (matrices vs. vectors), effectively combining Adafactor's memory efficiency with the robust convergence of sign-based optimization.)

    EmoLynx (Lion-type): Decoupled weight-decay for improved stability.


Supplementary Material (2): Formal Proof of emoDrive Boundedness

    1. Objective

    We prove that emoDrive, which applies a dynamic correction to the learning rate in the EmoNAVI update rule, possesses both upper and lower bounds at any step t. This guarantees that the update magnitude \Delta w_t does not explode and that the convergence conditions are satisfied.

    2. Lemma: Boundedness of the Emotional Scalar \sigma _t

    The emotional scalar in EmoNAVI takes the form
    \sigma _t=\tanh (x).
    From the properties of the hyperbolic tangent function, the following holds for any input x\in \mathbb{R}:
    -1<\sigma _t<1.
    Therefore, the absolute value |\sigma _t| always lies within the interval [0,1).

    3. Theorem: Proof of the Boundedness of emoDrive

    Based on the implementation code (v3.6.1), the definition of emoDrive is evaluated by dividing it into the following three regions:

    (A) Normal Zone (No Intervention Zone):
    |\sigma _t|\leq 0.25\quad \mathrm{or}\quad 0.5<|\sigma _t|\leq 0.75
    In this region, according to the implementation, the value is:
    \mathrm{emoDrive}=1.0.

    (B) Acceleration Zone (emoDrive Active Region):
    0.25<|\sigma _t|<0.5
    In this region, emoDrive is defined as:
    \mathrm{emoDrive}=\mathrm{emoDpt}\times (1.0+0.1\cdot \mathrm{trust}),
    where
    \mathrm{emoDpt}=8.0\times |\mathrm{trust}|,
    and trust is the signed value of (1.0-|\sigma _t|).
    - Evaluation of |\mathrm{trust}|:
    For |\sigma _t|\in (0.25,0.5), we have
    |\mathrm{trust}|\in (0.5,0.75).- Range of emoDpt:
    8.0\times 0.5<\mathrm{emoDpt}<8.0\times 0.75- hence
    4.0<\mathrm{emoDpt}<6.0.- Overall evaluation:
    The factor 1.0+0.1\cdot \mathrm{trust} lies within the range 0.9 to 1.1 regardless of the sign of trust.
    Therefore, the maximum value B_{\mathrm{up}} in the acceleration zone satisfies:
    B_{\mathrm{up}}<6.0\times 1.1=6.6.

    (C) Emergency Zone (Rapid Braking Zone):
    |\sigma _t|>0.75
    In this region,
    \mathrm{emoDrive}=\mathrm{coeff},
    where
    \mathrm{coeff}=1.0-|\mathrm{scalar}|.
    Since |\sigma _t|\in (0.75,1.0), the minimum value B_{\mathrm{low}} satisfies:
    0<B_{\mathrm{low}}\leq 0.25.

    4. Conclusion

    From the above evaluations, we have proven that emoDrive satisfies the following boundedness condition in all regions:
    0<(1-|\sigma _{\max }|)\leq \mathrm{emoDrive}\leq 6.6.
    (Even when |\sigma _t| approaches 1, implementation details such as eps ensure that a small positive value is maintained.)
    The existence of this bounded multiplicative coefficient provides the mathematical foundation that allows EmoNAVI to retain the Adam‑type convergence rate O(1/T) while achieving constant‑factor acceleration.

    5. summary 

	EmoNAVI encapsulates three forms of "intelligence" within a single update loop:

    Observational Intelligence (Multi-EMA): Captures the "undulations" of the loss landscape within a temporal spread, rather than at a single point.

    Judgment Intelligence (Scalar & Trust): Non-linearly determines whether the captured undulation is a "reliable trend" or "noise to be wary of."

    Action Intelligence (emoDrive): Autonomously decides the "step-size" based on the judgment, similar to COCOB or D-adaptation.


References

    Kingma, D. P., & Ba, J. (2014). Adam: A Method for Stochastic Optimization.

    Reddi, S. J., et al. (2019). On the Convergence of Adam and Beyond.

    Defazio, A., & Mishchenko, K. (2023). Learning-Rate-Free Learning by D-Adaptation.

    Orabona, F., & Tommasi, T. (2017). Training Deep Networks without Learning Rates Through Coin Betting.

    Luo, L., et al. (2019). Adaptive Gradient Methods with Dynamic Bound of Learning Rate.

    Shazeer, N., & Stern, M. (2018). Adafactor: Adaptive Learning Rates with Sublinear Memory Cost.

    Chen, S. B., et al. (2023). Symbolic Discovery of Optimization Algorithms.