Source-linked AI summary
Networked Information Aggregation for Binary Classification
MohammadHossein Bateni, Zahra Hadizadeh, MohammadTaghi Hajiaghayi, Mahdi JafariRaviz, Shayan Taherijam
TL;DR
The paper asks whether sequential agents with partial feature views can aggregate information into a classifier matching the best full-feature logistic predictor. It analyzes logit passing under BCE using tools that replace squared-loss variance arguments, proving depth-dependent upper and lower bounds. These bounds identify network depth as a fundamental bottleneck for information aggregation.
Problem
The paper studies whether networked agents with partial feature access can achieve information aggregation under logistic regression, extending a question previously analyzed for squared loss.
Method
The paper analyzes sequential DAG agents that optimize logistic regression with local features and incoming logits, using KL/Bregman progress, Pinsker-style error control, BCE residual orthogonality, and stability arguments.
Results
The protocol has an O(M/√D) excess-loss upper bound under M-coverage and hard instances with Ω(k/D) excess loss.
Takeaways & Limitations
The results identify network depth as a fundamental bottleneck for information aggregation in networked logistic regression.
Takeaways & Limitations
The upper-bound guarantee assumes that every M contiguous agents collectively observe all features.
Abstract
from arXiv · showhide
We study networked binary classification on a directed acyclic graph (DAG) where each agent observes only a subset of the feature columns of a shared dataset. Agents act sequentially along the DAG: each receives prediction columns from its parents (if any), augments its local features with these columns, fits a logistic predictor by minimizing binary cross-entropy (BCE), and forwards its prediction column to its outgoing neighbors. We ask whether this sequential distributed training procedure achieves information aggregation, meaning that some agent attains small excess loss compared to the best logistic predictor trained with access to all feature columns. This question was studied for linear regression under squared loss by Kearns, Roth, and Ryu (SODA 2026). Extending their guarantees to classification is nontrivial because their analysis relies on quadratic structure that does not directly transfer to BCE with a logistic link. We analyze the resulting sequential logit-passing protocol and prove: (i) an excess loss upper bound of $O(M/\sqrt{D})$ on depth-$D$ paths under the condition that every $M$ contiguous subsequence of $M$ agents collectively observe all features, and (ii) a close lower bound showing instances with excess loss of at least $\Omega(k/D)$ where $k$ is the dimension of the feature space. Together, these results identify network depth as a fundamental bottleneck for information aggregation in networked logistic regression.
1. Introduction
This work studies whether agents with partial, heterogeneous feature information can aggregate observations into accurate global classifiers on DAGs. It extends networked learning guarantees to logistic regression, establishing depth-dependent upper and lower bounds under BCE.
- Introduction: The problem connects classical social learning and networked machine learning, where agents collaboratively reconstruct high-dimensional predictive relationships from dispersed information.The framework builds on the DAG formulation of Kearns et al. (2026).
- Upper Bounds: O(M/√D) excess loss is achieved on depth-D paths when every M contiguous agents collectively observe all features.The result concerns the final agent under the stated coverage condition.
- Our Contributions: The proof replaces squared-loss variance decomposition with KL/Bregman loss progress, a Pinsker-style error link, BCE residual orthogonality, and a stability argument.These tools identify path segments where improvement saturates and bound deviation from the global optimum.
- Lower Bounds and Hard Instances: Ω(k/D) excess loss occurs on hard instances, showing that network depth is a fundamental bottleneck for sequential information aggregation.Here k is the feature-space dimension and D is network depth.
- From Regression to Classification: The protocol passes logits rather than probabilities because sigmoid nonlinearity makes probability-space aggregation unsuitable for directly extending squared-loss arguments.The classification setting lacks the orthogonality and Pythagorean variance decomposition available under squared loss.
- Introduction: Distributed classification matters for applications including medical diagnosis and decentralized fraud detection, motivating guarantees under BCE.The paper situates its theoretical goal within distributed applications.
2. Preliminaries
The paper formulates binary classification with logistic predictors evaluated by BCE, then defines a DAG in which agents sequentially combine local features with incoming logits. The sink agent provides the system’s final prediction.
- 2.1. Binary Classification and Logistic Regression: The logistic predictor models P(y = 1|x) through the sigmoid function σ(z) = 1/(1+e^-z).Parameters θ define the predictor through a linear score z.
- 2.1. Binary Classification and Logistic Regression: The global MLE p* minimizes expected BCE over the full feature space.The loss is L(θ) = −E[y log p(θ)(x) + (1−y) log(1−p(θ)(x))].
- 2.2. Distributed Learning Setup: Agents form a DAG whose edges specify which parent information each agent receives, with learning ordered by a topological sort.The sink or last topologically ordered agent supplies the system output.
- 2.2. Distributed Learning Setup: Each agent observes a subset of feature columns, receives parent logits, and trains a logistic model minimizing local BCE.The local model uses both x_Si and incoming parent outputs.
- 2.3. Sequential Learning Protocol: Agents communicate logits rather than probabilities to preserve the information geometry of the exponential family.A parent logit z_j corresponds to prediction σ(z_j).
3. Upper Bounds
Sequential BCE training can approach the global logistic predictor when a path repeatedly covers all features, using KL-based loss decomposition and residual orthogonality rather than squared-loss variance identities.
- Proof ingredients: BCE optimizer residuals are orthogonal in expectation to the input features, enabling error decompositions around the optimal logistic predictor.The optimality condition is E[x(p*(x) − y)] = 0.
- Proof ingredients: Loss differences are characterized through expected Bernoulli KL divergence, while a Pinsker-style inequality links KL progress to prediction error.This replaces the exact variance decomposition available for squared loss.
- Protocol: Sequentially minimizing BCE produces non-increasing loss because each agent can pass through its predecessor’s logits while optimizing over an expanded feature subspace.The protocol therefore compares successive agents through nested logistic predictor spaces.
- Global convergence: M-coverage requires every contiguous subsequence of M agents to collectively observe all features.This condition supplies repeated opportunities for feature information to enter the sequential protocol.
- Global convergence: The final agent’s excess risk is bounded under an M-coverage path of length D, assuming bounded feature and optimal-logit coefficients.The path is partitioned into length-M blocks, and a stable block is identified by a pigeonhole argument.
4. Lower Bound Analysis
A cyclic feature assignment creates a hard instance where sequential logit passing reveals only one additional relevant feature per pass, yielding an excess-loss lower bound of Ω(k/D).
- Lower-bound result: Ω(k/D) excess loss shows that network depth is a fundamental bottleneck for sequential information aggregation.The construction uses a path of depth D and feature dimension k.
- Problem construction: The hard instance makes early features uninformative about the label in isolation, so they become useful only after repeated passes through the feature cycle.The final global logit is Zk, which requires all k features to cancel intermediate noise terms.
- Information capacity: At the end of pass p, the optimal predictor depends only on Ip = {xk, xk−1, ..., xk−p+1}, and each subsequent pass expands this information set by exactly one feature.Features independent of the current information set are forwarded without modification; the next relevant feature can partially cancel latent noise.
- Information capacity: A linear predictor at pass p has the form cZk + ξ, with Gaussian noise independent of Zk, and minimizing the variance determines the residual-information limit.The analysis represents the predictor using coefficients over the relevant features and derives the variance through the residual noise term.
- Connection to excess loss: For fixed scaling, greater logit noise strictly increases BCE loss, while the optimal scaling factor remains in (0, 1).These properties connect the information restriction to the excess-loss lower bound.
A. Omitted Proofs
The omitted proof establishes a pointwise convexity inequality that lower-bounds expected Bernoulli KL divergence by a quadratic probability error.
- Omitted proof: The proof verifies the KL inequality pointwise by defining a difference function whose second derivative is nonnegative on [0, 1].Since the function is convex and minimized at p = q, the inequality holds for every probability pair.