[Paper] Conserved active information

Source: arXiv - 2512.21834v1

Overview

The paper introduces conserved active information (I^\oplus), a new way to measure how much “knowledge” a search or optimization algorithm actually brings to a problem. Unlike the classic active‑information metric, (I^\oplus) respects the No‑Free‑Lunch (NFL) theorem by accounting for information gain and loss across the entire search space. The authors show that this symmetric measure uncovers hidden regimes—situations where strong prior knowledge can reduce global disorder—something that standard KL‑divergence‑based analyses miss.

Key Contributions

• Definition of conserved active information (I^\oplus) – a symmetric extension of active information that quantifies net information change (gain − loss) over the whole search space.
• Formal proof of NFL conservation for (I^\oplus), demonstrating that the metric obeys the “no free lunch” principle by construction.
• Analytical examples (Bernoulli and uniform‑baseline cases) that illustrate regimes where strong knowledge decreases overall disorder, a phenomenon invisible to KL divergence.
• Theoretical results distinguishing “disorder” (adding mild knowledge to an already ordered system) from “order‑imposing” strong knowledge, under a uniform baseline.
• Applied case studies:
  1. Markov‑chain search dynamics
  2. A toy model of cosmological fine‑tuning, showing how (I^\oplus) can be used beyond classic optimization problems.
• Resolution of a long‑standing critique of the original active‑information framework, paving the way for broader practical adoption.

Methodology

1. Baseline Construction – The authors start with a reference (baseline) distribution that represents a completely uninformed search (e.g., uniform over all possible solutions).

2. Active Information Extension – Classic active information measures the KL‑divergence between the actual search distribution and the baseline, but only in one direction (gain). The new metric adds a loss term, yielding a symmetric quantity:

   [ I^\oplus = D_{\text{KL}}(P_{\text{actual}};|;P_{\text{baseline}}) - D_{\text{KL}}(P_{\text{baseline}};|;P_{\text{actual}}) ]

   This captures net information flow while guaranteeing that the sum over the whole space stays constant (the “conserved” part).

3. Analytical Toy Models – The paper works through Bernoulli‑type problems and uniform‑baseline scenarios to derive closed‑form expressions for (I^\oplus) and to highlight when the metric becomes negative (indicating net loss of disorder).

4. Proof Sketch – Using combinatorial arguments, the authors prove that the total (I^\oplus) summed over all possible target states is zero, satisfying the NFL conservation law.

5. Case Studies – They embed the metric into a Markov‑chain search process and a simplified cosmological fine‑tuning model, computing (I^\oplus) at each step to illustrate how the metric behaves in dynamic settings.

Results & Findings

• Symmetry & Conservation – (I^\oplus) is mathematically symmetric and its global sum is zero, confirming that any information gain in one region of the space must be offset by a loss elsewhere.
• Hidden Regimes Detected – In the Bernoulli example, strong prior knowledge (high probability mass on a narrow region) leads to negative (I^\oplus) for the rest of the space, indicating a reduction of global disorder that KL divergence alone would label as “pure gain.”
• Disorder vs. Order‑Imposing Knowledge – Under a uniform baseline, the authors formally separate two regimes:
  • Disorder: Adding mild knowledge to an already ordered system increases entropy (positive (I^\oplus)).
  • Order‑imposing: Adding strong knowledge to a disordered system reduces entropy (negative (I^\oplus)).
• Markov‑Chain Illustration – When a search process is biased toward high‑fitness states, the chain’s transient distribution shows a clear dip in (I^\oplus) for low‑fitness regions, confirming the metric’s sensitivity to where information is being concentrated.
• Cosmological Fine‑Tuning – A toy universe model demonstrates that a highly constrained set of physical constants yields a large negative (I^\oplus) for the complementary parameter space, offering a quantitative lens on “fine‑tuning” arguments.

Practical Implications

• Algorithm Diagnostics – Developers can compute (I^\oplus) for evolutionary algorithms, reinforcement‑learning policies, or hyper‑parameter search to see not just how much they improve performance, but where they are sacrificing exploration.
• Fairness & Bias Auditing – In recommendation systems, a negative (I^\oplus) in under‑served user segments signals that the model is “stealing” information from those groups—useful for bias detection.
• Resource Allocation – For distributed search (e.g., cloud‑based hyper‑parameter sweeps), (I^\oplus) can guide where to allocate compute: regions with high positive values may benefit from more exploration, while negative regions indicate over‑exploitation.
• Explainable AI – By visualizing the net information flow across feature spaces, engineers can better explain why a model prefers certain decision boundaries.
• Cross‑Domain Transfer – Since the metric is baseline‑agnostic, it can be applied to non‑traditional search problems such as network routing, automated theorem proving, or scientific model selection (e.g., cosmology, biology).

Limitations & Future Work

• Baseline Dependency – The interpretation of (I^\oplus) hinges on the chosen baseline distribution; selecting an inappropriate baseline can obscure meaningful signals.
• Scalability – Computing the full KL terms over massive, high‑dimensional search spaces may be computationally expensive; approximations or sampling strategies are needed for real‑world systems.
• Empirical Validation – The paper’s case studies are largely analytical or toy‑model based; extensive benchmarking on large‑scale optimization suites (e.g., OpenAI Gym, NAS benchmarks) remains to be done.
• Extension to Non‑Probabilistic Settings – Adapting the metric to deterministic heuristics or discrete combinatorial solvers is an open question.
• Dynamic Baselines – Future work could explore baselines that evolve with the search process, potentially yielding a more nuanced view of information flow over time.

Authors

• Yanchen Chen
• Daniel Andrés Díaz-Pachón

Paper Information

• arXiv ID: 2512.21834v1
• Categories: cs.NE, cs.CC, cs.HC, cs.IT
• Published: December 26, 2025
• PDF: Download PDF