Info-Theoretic & Sensitivity-Driven Allocation

• The topic is defined by methodologies that use quantitative metrics like mutual information, KL divergence, and Fisher information to guide optimal resource allocation.
• It integrates sensitivity analysis into allocation strategies across domains such as neural networks, control systems, and experimental design to reduce uncertainty.
• It underpins practical approaches in fields including communications, molecular dynamics, and cloud security by maximizing marginal information gain under resource constraints.

Information-theoretic and sensitivity-driven allocation encompasses a family of methodologies for optimally distributing finite resources—such as computational budget, data acquisition, sensing precision, or parameter calibration—according to metrics grounded in information theory and parameter sensitivity analysis. These approaches have been rigorously developed in statistics, machine learning, control theory, communications, molecular dynamics, cloud security, experimental design, and reliability theory. Unifying these diverse domains is the principle that resources should be directed where the marginal gain in information (or reduction in uncertainty or increased decision utility) is maximized, often under stringent constraints.

1. Core Information-Theoretic Principles

At the foundation of information-theoretic allocation are quantitative measures such as mutual information, Kullback-Leibler (KL) divergence, differential entropy, minimum mean-square error (MMSE), and Fisher information. These function as proxies for model complexity, statistical identifiability, or attainable accuracy.

• Mutual Information ($I(U;V)$): Quantifies the reduction in uncertainty in $U$ gained from knowledge of $V$, ubiquitous in sample complexity analysis and experimental design.
• Kullback-Leibler Divergence ($d(P \Vert Q)$): Used to measure the "distance" between observed/estimated and true probability laws, central to model selection, parameter estimation, and risk quantification.
• Fisher Information ($\mathcal{I}_F$): Encodes the local curvature of the likelihood, directly modulating estimation error, and tightly linked with sensitivity indices via the Cramér–Rao bound.
• MMSE and I-MMSE Identity: The derivative of mutual information with respect to SNR in estimation problems directly gives the MMSE, establishing a sensitivity channel for resource optimization (Ramos et al., 2012).

These quantities underpin allocation rules that match the marginal information-gain or marginal reduction in error/distance across resources.

2. Neural Networks and Compute/Data Allocation

In neural scaling regimes, the optimal division of compute between model capacity ($p$ parameters) and dataset size ($N$ samples) can be formalized via information-theoretic decompositions. Jeon & Van Roy (Jeon et al., 28 Jun 2024) derive an explicit scaling law:

$$E(p,N) \leq \frac{I(F; \tilde F)}{N} + \mathbb{E}_X\left[ d( P(Y|F,X) \| P(Y|\tilde{F},X) ) \right]$$

where

• $I(F;\tilde{F})$: information content of the model class (grows with $p$),
• Misspecification error: model-data mismatch (decays with $p$, independent of $N$).

With a fixed compute budget ($C_{\text{compute}} = pN$), minimization yields the near-linear regime

$$N^* \sim p^* \sim \mathcal{O}(\sqrt{C_{\text{compute}}})$$

indicating optimal proportionality between data and model size up to log factors.

When parameter sensitivity (quantified by the trace of the inverse Fisher information) dominates,

$$E_{\mathrm{est}} \sim \frac{\mathrm{Tr}[\mathcal{I}_F^{-1}]}{N}$$

and the optimal allocation (partitioning $C=pN$) shifts towards higher $N$ for parameter directions with low Fisher information, i.e., high estimation variance. If $\mathrm{Tr}[\mathcal{I}_F^{-1}] \sim p^\gamma$, the optimal regime becomes $p^* \sim C^{1/(2+\gamma)}$, $N^* \sim C^{(1+\gamma)/(2+\gamma)}$ (Jeon et al., 28 Jun 2024).

3. Sensitivity and Information Functions in Dynamical Systems

For parameter inference in dynamical systems, Information Sensitivity Functions (ISFs) quantify the temporal and parametric distribution of information contributed by observations. The ISF framework (Pant, 2017) operates via:

• Sensitivity matrix $S(t) = \partial y(t)/\partial\theta$ computed from the ODE system,
• Instantaneous and accumulated Fisher-like information $F_n = S_n^\top R_n^{-1} S_n$, $D(t) = \int F(\tau)d\tau$,
• Posterior covariance update: $$\Sigma_{\text{post}}(t) = [\Sigma_0^{-1} + D(t)]^{-1}$$,
• Differential-entropy information gain per parameter: $$\Delta h_i = \frac{1}{2}\log(\sigma_{0,i}^2/\sigma_{\text{post},i}^2)$$,
• Inter-parameter mutual information via conditional covariance or block determinant ratios.

Optimal allocation entails concentrating measurement effort at times/variables where $d\,\Delta h_i/dt$ is largest or where posterior variability and parameter correlations indicate remaining uncertainty. Experiment design thus maximizes information gain or minimizes posterior entropy under resource constraints.

4. Allocation in Communications and Control via Information and Sensitivity

In constrained wireless communications, information-assisted allocation reformulates hard combinatorial problems into tractable dynamic programs by softening constraints via KL-divergence "information-to-go" penalties. Optimal solutions balance reward against constraint-alignment by optimizing

$G^\pi(X;\beta) = I_g^\pi(X) - \beta f^\pi(X)$

over policies $\pi$, where $I_g^\pi$ measures deviation of future-paths from constraint-aligned priors (Ahmed et al., 2021). This restores dynamic programming solvability, dramatically reducing computational cost for NP-hard problems such as bit allocation in 5G massive MIMO.

In continuous-time control systems, Bode-like integral constraints on sensitivity functions reveal that resource (e.g., SNR, sensor precision) should be allocated to spectral regions corresponding to system instabilities and non-minimum-phase behavior. These allocations (solved via water-filling variational problems) achieve the mutual information rate lower bounds required by plant pole/zero structure (Wan et al., 2018).

5. Resource Allocation across Parameters in Stochastic and Physical Systems

Sensitivity-driven allocation in stochastic molecular dynamics employs the relative entropy rate (RER) and pathwise Fisher information matrix:

• RER: $$\mathcal{H}(Q^\theta\|Q^{\theta+\epsilon}) = \frac{1}{2} \mathbb{E}\left[ (F^{\theta+\epsilon}(q)-F^\theta(q))^\top (\sigma\sigma^\top)^{-1} (F^{\theta+\epsilon}(q)-F^\theta(q)) \right]$$
• Pathwise FIM: quantifies per-parameter or blockwise sensitivity with respect to perturbations in $\theta$.

Parameters are ranked by FIM diagonals. Allocation is driven by selecting parameters with the largest information-gain-per-cost ratio, and further refined to account for parameter correlations (principal subspaces of FIM) (Tsourtis et al., 2014).

6. Decision-Theoretic and Risk-Adaptive Allocation

The value-of-information paradigm extends sensitivity-driven allocation into decision theory. Expected Value of Partial Perfect Information (EVPPI) directly quantifies the expected reduction in decision loss (e.g., failure or cost) achieved by resolving uncertainty in each input. For reliability and design problems,

• Allocate inspection or experimental budget ($n$ samples, measurement cost) to the variable with largest EVPPI, not necessarily the steepest local sensitivity.
• EVPPI subsumes the Sobol’ first-order index under quadratic loss; for binary loss or non-Gaussian scenarios it provides direct operational utility (Straub et al., 2021).

This extends to budgeted screening-allocation in social and economic settings, where marginal value of additional information for each candidate (relative to thresholded decision rules) is equated to its marginal cost via linear-program frameworks, again enforcing resource deployment where information has greatest utility impact (Cai et al., 2019).

7. Risk-Aware Sensitivity-Driven Allocation in Security and Cloud Systems

Information-theoretic risk allocation in cloud and datacenter security is driven by quantifying leakage using KL-divergence and mutual information between partial and global data distributions. The assignment problem is formulated as a combinatorial program minimizing total expected risk across roles and resources. Heuristic algorithms (top-down or neighbor-based clustering) operationalize allocation by minimizing per-role property leakage as measured by $D_{KL}$ or $|I_A - I_G|$, allocating the least-sensitive or highest-risk roles to the most isolated or secure resources (Felemban et al., 4 Feb 2025).

Summary Table: Key Allocation Principles across Domains

Domain	Information/Sensitivity Metric	Optimality Rule
Neural scaling	$I(F;\tilde F)/N$, $\mathcal{I}_F^{-1}$	$N^* \sim p^*$ up to logs (match estimation and approximation terms)
Dynamical system ID	ISFs, $\Delta h_i$, mutual info	Allocate measurements to maximize $\sum_i \Delta h_i$
Stochastic molecular	Pathwise FIM, RER	Rank/allocate by FIM diagonal, blockwise if correlated
Communications/control	MMSE, mutual information, KL divergence	Water-filling/spectral allocation, equalize marginal info gain
Reliability/design	EVPPI/Sobol’, variance reduction	Allocate budget to maximize reduction in expected loss
Cloud/datacenter risk	KL-divergence, mutual info	Assign roles/VMs to minimize total expected (info-theoretic) risk

Allocation strategies, irrespective of domain, consistently exploit the marginal utility of information or reduction in uncertainty, operationalized by local or global sensitivity indices. These indices are grounded in information theory and parameter sensitivity, ensuring that finite resources are deployed where they most effectively enhance model performance, inference accuracy, control, or security. Empirical and theoretical validations across domains confirm the universality and rigor of these allocation principles (Jeon et al., 28 Jun 2024, Pant, 2017, Tsourtis et al., 2014, Ahmed et al., 2021, Wan et al., 2018, Straub et al., 2021, Felemban et al., 4 Feb 2025, Cai et al., 2019, Ramos et al., 2012).