CurvOpt-MathFoundations.md

CurvOpt: Mathematical Foundations

Energy-Constrained Precision Allocation via Curvature and Information Theory

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1. Problem Formulation

Let a trained neural network with parameters $\theta \in {\mathbb{R}}^d\backslash theta\; \backslash in\; \backslash mathbb\{R\}^d$ minimize empirical risk:

\[ L(\theta) = \frac{1}{n} \sum_{i=1}^{n} \ell(f_\theta(x_i), y_i) \]

We introduce quantization:

\[ \theta_q = \theta + \varepsilon \]

We seek precision assignments $q_{l}q\_l$ per layer:

\[ \min_{q_l \in \mathcal{Q}} \sum_{l=1}^{L} \mathcal{E}_l(q_l) \quad \text{s.t.} \quad L(\theta_q) - L(\theta) \le \epsilon \]

Reference: Boyd & Vandenberghe (2004), Convex Optimization

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2. Second-Order Loss Perturbation

By Taylor expansion:

\[ L(\theta + \varepsilon)

L(\theta) + \nabla L(\theta)^T \varepsilon + \frac{1}{2} \varepsilon^T H(\theta) \varepsilon + o(|\varepsilon|^2) \]

Near a stationary point:

\[ \nabla L(\theta) \approx 0 \]

Thus:

\[ \Delta L \approx \frac{1}{2} \varepsilon^T H \varepsilon \]

Reference: Nocedal & Wright (2006), Numerical Optimization

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3. Spectral Bound

Since $HH$ is symmetric:

\[ \lambda_{\min}(H) |\varepsilon|^2 \le \varepsilon^T H \varepsilon \le \lambda_{\max}(H) |\varepsilon|^2 \]

Thus:

\[ \Delta L \le \frac{1}{2} \lambda_{\max}(H) |\varepsilon|^2 \]

Reference: Goodfellow et al. (2016), Deep Learning

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4. Hutchinson Trace Estimator

\[ \operatorname{Tr}(H)

\mathbb{E}_{v} \left[ v^T H v \right] \]

where $v_i\sim \{-1,+1\}v\_i\; \backslash sim\; \backslash \{-1,+1\backslash \}$.

Reference: Robert & Casella (2004), Monte Carlo Statistical Methods

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5. Quantization Noise Model

Uniform quantization with step size $\mathrm{\Delta }\backslash Delta$:

\[ \varepsilon \sim \mathcal{U}\left(-\frac{\Delta}{2}, \frac{\Delta}{2}\right) \]

Variance:

\[ \operatorname{Var}(\varepsilon)

\frac{\Delta^2}{12} \]

Expected loss increase:

\[ \mathbb{E}[\Delta L] \approx \frac{1}{2} \operatorname{Tr}(H) \cdot \frac{\Delta^2}{12} \]

Reference: Gallager (1968), Information Theory and Reliable Communication

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6. Mutual Information

\[ I(X_l ; Y_l)

\int p(x,y) \log \frac{p(x,y)}{p(x)p(y)} , dx,dy \]

Data Processing Inequality:

\[ I(X; Y_{l+1}) \le I(X; Y_l) \]

Reference: Cover & Thomas (2006), Elements of Information Theory

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7. Constrained Energy Minimization

\[ \min_{q_l} \sum_l \mathcal{E}_l(q_l) + \lambda \left( \sum_l \frac{1}{24} \operatorname{Tr}(H_l) \Delta_l^2

\epsilon \right) \]

KKT condition:

\[ \nabla_{q_l} \mathcal{L} = 0 \]

Reference: Bertsekas (1999), Nonlinear Programming

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Summary

CurvOpt is grounded in:

• Second-order perturbation theory
• Spectral bounds
• Monte Carlo trace estimation
• Classical quantization noise modeling
• Shannon mutual information
• Constrained nonlinear optimization

All formulas are standard results from established literature.