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emo-v36-paper(ENG).txt

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 Theoretical Foundation of EmoNAVI v3.6: Autonomous Optimization
 
 Improving Regret Bounds via Higher-Order Moment Approximation and Dynamic Distance Estimation
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 1. Introduction
 
 In the optimization of deep learning models, the dynamic adjustment of learning rates is a pivotal challenge determining convergence performance. While conventional optimizers like Adam and AMSGrad utilize the first and second moments of gradients, their ability to directly estimate the local curvature of the loss landscape or the distance D to the optimal solution is limited. This paper proves that the "Emotion Scalar σt​" and "emoDrive" mechanism introduced in EmoNAVI v3.6 function mathematically as an online implementation of higher-order moment approximation and D-adaptation (and COCOB theory). We demonstrate that this approach achieves both extremely low hyperparameter sensitivity and robust convergence.
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 2. Mathematical Redefinition and Higher-Order Moment Approximation
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 2.1 Proxy Indicator Generation via Multi-EMA
 
 EmoNAVI maintains three levels of Exponential Moving Averages (short, medium, and long):
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         5th Moment: Captures the "fluctuation of fluctuations" along the time axis.
 
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 3. Dynamic Distance Estimation (D-adaptation) via emoDrive
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 3.1 Online Approximation of 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 performs this role.
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     Suppression Zone (Low Trust): During abrupt changes where ∣σt​∣>0.75, updates are suppressed at an order of O(1−∣σt​∣). This acts as a safety mechanism against surges in the local Lipschitz constant Lt​, analogous to resetting the betting size in COCOB theory when consecutive losses occur.
 
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 4. Convergence Proof and Regret Analysis
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 4.1 Assumptions and Properties
 
     L-smoothness: The loss function f has a local Lipschitz constant Lt​ and satisfies ∥∇f(w)∥≤G.
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 R(T)≤O​Dt=1∑T​∥gt​∥2⋅(1−∣σt​∣)2​​
 
 As training progresses and σt​→0 (adaptation to the landscape completes), Var(σ) shrinks and the effective learning rate stabilizes. Mathematically, this guarantees "autonomy," significantly reducing dependence on the base learning rate η0​.
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 5. Conclusion
 
 EmoNAVI v3.6 integrates "Landscape Perception via Higher-Order Moments" and "Adaptive Step Control via D-adaptation" into a single loop through the intuitive metaphor of an emotion scalar. This analysis demonstrates that EmoNAVI is a theoretically consistent, next-generation optimizer that merges empirical wisdom with state-of-the-art online learning theories.
 Supplementary Material: Formal Proof of emoDrive Boundedness
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 1. Objective
 
 To prove that emoDrive maintains upper and lower bounds at any step t, ensuring that the update step Δwt​ does not explode and satisfies convergence conditions.
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 2. Lemma: Boundedness of Emotion Scalar σt​
 
 Since σt​=tanh(x), the properties of the tanh function dictate that for any x∈R:
 −1<σt​<1
 
 Thus, ∣σt​∣∈[0,1).
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 3. Theorem: Proof of emoDrive Boundedness
 
 Evaluating the three regions defined in the v3.6.1 implementation:
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 4. Conclusion
 
 For all regions: 0<(1−∣σmax​∣)≤emoDrive≤6.6. This bounded multiplicative factor allows EmoNAVI to maintain the Adam-type convergence rate O(1/T​) while achieving constant-factor acceleration.
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 Summary: The Triple Intelligence of EmoNAVI
 
 EmoNAVI encapsulates three forms of "intelligence" within a single update loop:

emo-v36-paper(JPN).txt

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 ディープラーニングの最適化において、学習率の動的調整は収束性能を決定づける最重要課題である。従来の Adam や AMSGrad は勾配の 1次・2次モーメントを利用するが、局所的な損失地形の急峻さ（曲率）や最適解までの距離 D を直接推定する機能は限定的であった。 本稿では、EmoNAVI v3.6 が導入した「感情スカラー σt​」および「emoDrive」機構が、数学的には高次モーメントの近似と D-adaptation（および COCOB 理論）のオンライン実装として機能し、極めて低いハイパーパラメータ感度と頑健な収束性を両立することを証明する。
 
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 ## 2. 実装の数学的再定義と高次モーメント近似
 
 ### 2.1 Multi-EMA による proxy 指標の生成
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     5次モーメントの履歴化: 感情スカラー σt​=tanh(ΔEMA/scale) は、これらの高次の情報を [−1,1] に非線形圧縮した統計量であり、これを更新式に再帰的に含めることで、長長期的な地形の「滑らかさ」をパラメータ更新に反映させている。
 
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 ## 3. emoDrive による動的距離推定（D-adaptation）
 
 ### 3.1 D-推定のオンライン近似
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 この式は、学習が進み σt​→0（地形への適応が完了）となるにつれ、Var(σ) が縮小し、有効学習率が安定することを示している。結果として、ベース学習率 η0​ への依存性が低減され、ハイパーパラメータ調整を不要とする「自律性」が数学的に保証される。
 
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 ## 5. 結論
 
 EmoNAVI v3.6 は、感情スカラーという直感的なメタファーを通じて、**「高次モーメントによる地形把握」と「D-adaptation による適応的ステップ制御」**を単一のループ内で実現した。 本解析により、EmoNAVI が単なる経験則の集合体ではなく、オンライン学習理論の最先端（COCOB/D-adapt）を高度に融合させた、理論的整合性の高い次世代最適化器であることが示された。