Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Latent Objective Induction and Diversity-Constrained Selection: Algorithms for Multi-Locale Retrieval Pipelines

Faruk Alpay, Levent Sarioglu

TL;DR

Latent Objective Induction (LOI), an environment-shaping operator over prompt spaces that steers downstream model behavior without restricting the feasible output set, is formalized and proves its convergence under mild assumptions.

Abstract

We present three algorithms with formal correctness guarantees and complexity bounds for the problem of selecting a diverse, multi-locale set of sources from ranked search results. First, we formulate weighted locale allocation as a constrained integer partition problem and give an $O(n \log n)$ algorithm that simultaneously satisfies minimum-representation, budget-exhaustion, and proportionality-bound constraints; we prove all three hold with a tight deviation bound of $< 1$. Second, we define a cascaded country-code inference function as a deterministic priority chain over heterogeneous signals (TLD structure, model-inferred metadata, language fallback) and prove it satisfies both determinism and graceful degradation. Third, we introduce a $κ$-domain diversity constraint for source selection and give an $O(|K| \cdot R)$ algorithm that maintains the invariant via hash-map lookup, eliminating the aggregator monopolization pathology present in URL-level deduplication. We further formalize Latent Objective Induction (LOI), an environment-shaping operator over prompt spaces that steers downstream model behavior without restricting the feasible output set, and prove its convergence under mild assumptions. Applied to a multi-locale retrieval pipeline, these algorithms yield 62% improvement in first-party source ratio and 89% reduction in same-domain duplication across 120 multilingual queries.

Latent Objective Induction and Diversity-Constrained Selection: Algorithms for Multi-Locale Retrieval Pipelines

TL;DR

Latent Objective Induction (LOI), an environment-shaping operator over prompt spaces that steers downstream model behavior without restricting the feasible output set, is formalized and proves its convergence under mild assumptions.

Abstract

We present three algorithms with formal correctness guarantees and complexity bounds for the problem of selecting a diverse, multi-locale set of sources from ranked search results. First, we formulate weighted locale allocation as a constrained integer partition problem and give an algorithm that simultaneously satisfies minimum-representation, budget-exhaustion, and proportionality-bound constraints; we prove all three hold with a tight deviation bound of . Second, we define a cascaded country-code inference function as a deterministic priority chain over heterogeneous signals (TLD structure, model-inferred metadata, language fallback) and prove it satisfies both determinism and graceful degradation. Third, we introduce a -domain diversity constraint for source selection and give an algorithm that maintains the invariant via hash-map lookup, eliminating the aggregator monopolization pathology present in URL-level deduplication. We further formalize Latent Objective Induction (LOI), an environment-shaping operator over prompt spaces that steers downstream model behavior without restricting the feasible output set, and prove its convergence under mild assumptions. Applied to a multi-locale retrieval pipeline, these algorithms yield 62% improvement in first-party source ratio and 89% reduction in same-domain duplication across 120 multilingual queries.
Paper Structure (38 sections, 11 theorems, 12 equations, 3 tables, 2 algorithms)

This paper contains 38 sections, 11 theorems, 12 equations, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Constrained Selection and Partition Algorithms.
  4. Priority Chains and Cascaded Inference.
  5. Multilingual Retrieval.
  6. LLM-Augmented Retrieval.
  7. Mechanism Design and Nudge Theory.
  8. Notation and Preliminaries
  9. Latent Objective Induction
  10. Motivation: Explicit vs. Latent Steering
  11. Formal Framework
  12. LOI in Pipeline Architecture: Brief Propagation
  13. Weighted Locale Allocation
  14. Problem Statement
  15. Algorithm
  16. ...and 23 more sections

Key Result

Theorem 4.5

Let $\mathcal{M}$ be an LLM with instruction-following capability $\alpha \in (0, 1]$ (the probability that the model follows a correctly understood directive). Let $\phi$ be an objective with semantic clarity $\beta \in (0, 1]$ (the probability that the concept embedding is correctly interpreted). which converges to $1$ as $k \to \infty$.

Theorems & Definitions (27)

  • Definition 4.1: Explicit Constraint
  • Definition 4.2: Latent Objective Induction
  • Example 4.3: First-Party Preference
  • Definition 4.4: LOI Operator Properties
  • Theorem 4.5: Convergence of LOI
  • proof
  • Remark 4.6: Contrast with Explicit Constraints
  • Definition 4.7: Research Brief
  • Proposition 4.8: Decoupling
  • Definition 5.1: Weighted Locale Allocation
  • ...and 17 more