Informational Control Learning (ICT-ML): A Foundational Algorithmic Prototype Based on the Informational Control Theory (Version 1.0 – Preliminary Release)

Informational Control Learning (ICT-ML): A Foundational Algorithmic Prototype Based on the Informational Control Theory (ICT) Alejandro Cruz-L´opez Universidad Aut´onoma Metropolitana – Unidad Iztapalapa Hypatia Research Institute M´exico [email protected] October 2025 License Notice. ©2025 Alejandro Cruz-L´opez. All rights reserved. Exclusive institutional license for derivative works held by Hypatia Research Institute M´exico. Redistribution or derivative use requires formal affiliation or written consent. Abstract This preprint presents the Informational Control Theory (ICT), a framework that models learning as informational self-regulation instead of parameter optimization. In ICT, system states are probability distributions in the simplex ∆n, and learning emerges from minimizing the Kullback–Leibler divergence through multiplicative corrections that preserve normalization. A prototype algorithm, ICT-ML, demonstrates this principle using the MNIST dataset. Without gradients, weights, or backpropagation, it performs classification as an informational flow toward class archetypes, showing stable convergence and interpretable dynamics. Although preliminary, this work establishes the first computational evidence that learning can arise purely from informational divergence minimization, suggesting a new foundation for gradient-free machine learning. Contents 1 Introduction 3 2 Methodology 4 2.1 Problem Setting and Notation ......................... 4 2.2 Projection to the Probability Simplex ..................... 4 2.3 Class Prototypes (Informational Targets) ................... 4 2.4 Informational Energy: Kullback–Leibler Divergence ............. 5 2.5 Informational Correction Operator (Discrete Flow) ............. 5 2.6 Single-Class Learning Dynamics ........................ 5 2.7 Multiclass Training Procedure ......................... 5 2.8 Prediction by Minimum Informational Divergence .............. 6 2.9 Evaluation Protocol ............................... 6 2.10 Numerical Stability and Implementation Details ............... 6 2.11 Hyperparameters and Recommended Defaults ................ 6 2.12 Algorithmic Summary (Pseudocode) ..................... 7 3 Results 8 3.1 Dataset Summary and Simplex Verification (no figure) ........... 8 3.2 Initial Informational States Grid ........................ 9 3.3 Class Prototypes (Informational Archetypes) ................. 10 3.4 Single-Class Convergence: States Panel .................... 10 3.5 Single-Class Convergence: KL Descent Curve ................. 11 3.6 Multiclass KL Descent (Training Curves) ................... 11 3.7 Prediction Examples (Minimum Informational Divergence) ......... 12 3.8 Confusion Matrix (Global Evaluation) .................... 13 4 Discussion 13 5 Conclusion 13 6 Institutional Statement and Rights 14 2 1 Introduction Over the last decades, Machine Learning (ML) has achieved remarkable empirical success through the systematic optimization of parameterized models. However, this progress has also revealed an intrinsic limitation: most current algorithms rely on externally defined architectures, gradient descent procedures, and extensive hyperparameter tuning. These elements, while operationally efficient, obscure the deeper question of what learning fundamentally is. The conventional paradigm treats learning as an optimization problem, not as an informational process. The Informational Control Theory (ICT) proposes a different foundation. It views learning as a natural consequence of informational self-regulation within a system that continuously minimizes its internal informational disequilibrium. In this framework, the state of a system is not represented by a set of parameters, but by a probability distribution—a normalized informational configuration that evolves according to a law of informational control. Rather than “learning by adjustment,” a ICT system learns by equilibration, progressively aligning its internal informational state with external informational sources through divergence minimization. This conceptual shift transforms the act of learning into a flow of information governed by the geometry of the probability simplex. The present work operationalizes this idea through a computational prototype, called Informational Control Learning (ICTML), which performs classification directly in the simplex ∆n. Each sample is represented as a normalized informational state, and each class as an informational archetype obtained from empirical averages. Learning emerges as a contraction flow toward these archetypes, driven by the minimization of the Kullback–Leibler (KL) divergence. Unlike gradient-based or parameterized models, ICT-ML operates with no weights, backpropagation, or latent feature extraction. Its dynamics are entirely defined by the informational relationships among data points. This structure results in a fully interpretable, geometry-preserving evolution that maintains probabilistic normalization at every step and admits analytical control of stability and convergence. The broader implication of this approach is that intelligence can be modeled as an informational control process, where perception, learning, and decision-making are unified through the regulation of divergence. The current implementation provides the first empirical demonstration of this principle using the MNIST dataset, showing that supervised classification can be reproduced purely through informational flows. This preprint therefore presents the computational kernel of the ICT framework. It does not yet address the full theoretical formalism, but establishes the essential mechanism: a learning process emerging from informational control, rather than optimization. Theoretical extensions and proofs of convergence will be presented in a forthcoming manuscript devoted to the mathematical foundations of ICT. 3 2 Methodology This section specifies the complete algorithmic pipeline of ICT-ML (Informational Control Learning) as an applied, gradient-free learning mechanism operating directly in the probability simplex. We detail data projection, prototype estimation, informational dynamics, prediction, evaluation, and implementation safeguards to ensure numerical stability and reproducibility. 2.1 Problem Setting and Notation Let D={(x(i), y(i))}N i=1 be a labeled dataset with x(i)∈R28×28 (MNIST images) and labels y(i)∈ {1,...,10}, where the digit 0 is mapped to class 10 for convenience. After flattening, each image becomes a nonnegative vector in Rnwith n= 784. The probability simplex of dimension nis ∆n=nX∈Rn Xi≥0, n X i=1 Xi= 1o. Every sample is projected to ∆nand hence interpreted as an informational state. 2.2 Projection to the Probability Simplex Given a flattened, nonnegative image vector ˜ X∈Rn, we define its simplex projection by elementwise normalization: X=˜ X max Pn i=1 ˜ Xi, ε∈∆n, with a small numerical guard ε∈[10−12,10−8] to avoid division by zero. This projection is applied to all training and test samples. No bias term is used anywhere in the ICT-ML pipeline. 2.3 Class Prototypes (Informational Targets) For each class c∈ {1,...,10}, we define its prototype (informational target) as the normalized arithmetic mean in ∆n: P(c)=1 NcX i:y(i)=c X(i)and then P(c)←P(c) Pn j=1 P(c) j , where Ncis the number of training samples in class c. The renormalization enforces exact membership in ∆n. 4 2.4 Informational Energy: Kullback–Leibler Divergence The informational discrepancy between a prototype P∈∆nand a state X∈∆nis measured by the Kullback–Leibler (KL) divergence DKL(P∥X) = n X i=1 PilogPi Xi, computed with clipping Pi←max(Pi, ε) and Xi←max(Xi, ε) to ensure a finite, numerically stable value and preserve positivity. This quantity serves as the informational energy minimized by the algorithm. 2.5 Informational Correction Operator (Discrete Flow) The core update rule is a convex contraction in ∆ntowards a target prototype P: Xt+1 = (1 −α)Xt+α P, (1) with stepsize α∈(0,1]. This operator preserves nonnegativity and the unit-sum constraint, and it decreases DKL(P∥Xt) monotonically. An optional regularized (damped) variant pulls the state slightly towards the uniform distribution U= (1/n)1: Xt+1 = (1 −α−λ)Xt+α P +λ U, (2) with λ∈[0,1−α) controlling the attenuation. Both (1) and (2) leave the simplex invariant. 2.6 Single-Class Learning Dynamics For a fixed class cand some initial state X0∈∆n(e.g., a sample of class c), the iteration X(c) t+1 = (1 −α)X(c) t+α P(c) is executed for t= 0,1,...,T −1. At each step, the algorithm logs DKL(P(c)∥X(c) t). Termination criteria include a maximum number of iterations Tand/or an energy tolerance DKL(P(c)∥X(c) t)≤δ. 2.7 Multiclass Training Procedure The training phase consists of constructing all prototypes {P(c)}10 c=1. Optionally, one may illustrate convergence by running the single-class dynamics for a representative sample per class and plotting the KL trajectories {DKL(P(c)∥X(c) t)}to confirm monotone decay. 5 2.8 Prediction by Minimum Informational Divergence Given an unseen test image ˜ X∈Rn, project it to ∆nas X=˜ X/ max(Pi˜ Xi, ε). The predicted label is the prototype that minimizes the KL divergence: ˆc(X) = arg min c∈{1,...,10}DKLP(c)∥X. Ties (rare under clipping) can be broken by secondary criteria, e.g., ℓ2proximity ∥P(c)− X∥2, a prior over classes, or by selecting the smallest index. 2.9 Evaluation Protocol For a random subset of test samples, predictions ˆcare compared to ground-truth labels to form a 10 ×10 confusion matrix M, where Mij counts test samples of true class i predicted as j. Overall accuracy is Acc = tr(M)/Pi,j Mij. Additional metrics (per-class precision/recall, macro/micro averages) can be computed in the standard way. 2.10 Numerical Stability and Implementation Details •Clipping and epsilons: Use ε∈[10−12,10−8] to clip Pand Xinside the KL; normalize with max(·, ε) to avoid division by zero. •Simplex invariants: Both updates (1) and (2) preserve nonnegativity and unit sum by construction. •Vectorization: Prototype computation and KL evaluations should be vectorized for efficiency (NumPy or similar). •Complexity: Per-sample prediction cost is O(Cn) with C= 10 and n= 784; training cost is dominated by computing means (linear in dataset size). •Reproducibility: Fix random seeds for any stochastic selection (e.g., representative samples); record α,λ,T, and εused. •Visualization: For display only, scale images by a positive factor (e.g., ×200) without altering the stored simplex vectors. 2.11 Hyperparameters and Recommended Defaults •Step size α:α∈[0.1,0.3] typically yields fast, stable convergence. •Damping λ:Optional; λ∈[0,0.1] can smooth dynamics in noisy settings. •Iterations T:T∈[20,40] is sufficient to visually reach the prototype; prediction itself does not require iterations. •Tolerance δ:Use δ∈[10−6,10−4] if an energy-based stopping rule is preferred. 6 2.12 Algorithmic Summary (Pseudocode) Listing 1: ICT-ML: training, prediction, and evaluation 1# TRAINING ( build prototypes ) 2# Input : training set {( tilde_X [i], y[i ])} 3# Output : prototypes P[c] in Delta_n for c=1..10 4 5for each class cin {1..10}: 6S_c = { i : y[i] == c } 7# Project samples to simplex 8Xc = [ project_to_simplex ( tilde_X [i], eps) for iin S_c ] 9P[c] = normalize ( mean (Xc) ) # enforce sum =1 10 11 # ( Optional ) Convergence illustration per class 12 for each class c: 13 X = random choice from Xc 14 for tin range (T): 15 log_KL[c, t] = KL(P[c], X, eps) 16 X = (1 - alpha ) * X + alpha * P[c] # or regularized update 17 18 # PREDICTION ( per test sample ) 19 # Input : unseen tilde_X 20 # Output : predicted class c_hat 21 X = project_to_simplex ( tilde_X , eps ) 22 scores = [ KL(P[c], X, eps) for cin {1..10} ] 23 c_hat = argmin ( scores ) 24 25 # EVALUATION ( subset of test set) 26 confusion = zeros (10 , 10) 27 for each test sample ( tilde_X , y_true ) in subset: 28 c_hat = predict ( tilde_X ) 29 confusion [ y_true -1, c_hat -1] += 1 30 accuracy = trace ( confusion ) / sum( confusion ) Code availability. A prototype implementation of the ICT-ML algorithm was developed in Python for internal validation at the Hypatia Research Institute M´exico. The code is currently under revision and will be released in a future institutional version once standardization and documentation are completed. 7 3 Results This section reports the empirical results of the ICT-ML algorithm. For clarity, each subsection cites the exact image filename used in the figure environment so you can match files unambiguously when compiling. 3.1 Dataset Summary and Simplex Verification (no figure) •Simplex dimension: ∆784. •Training examples: 60,000; Test examples: 10,000. •Each X(i)is normalized to PjX(i) j= 1; typical min/max in an image: [0,0.0093]. •Example baseline divergence with the uniform U:DKL(X0∥U)≈1.7493. •Class counts (train): 1: 6742, 2: 5958, 3: 6131, 4: 5842, 5: 5421, 6: 5918, 7: 6265, 8: 5851, 9: 5949, 10: 5923. 8 3.2 Initial Informational States Grid Figure 1: Initial informational states X(i)in the simplex ∆784 (visual scale). Filename: initial informational states.png. 9