9ce1b36206
Research doc updated with precise 95% CIs per query type. compare.py now computes and reports confidence intervals. Kendall τ = 0.79 (95% CI [0.7873, 0.8006]) confirms raw score merging is not viable; RRF already implemented in merger.rs as mitigation. Follow-up bead created (miroir-zfo) for RRF quality validation. Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
232 lines
8.1 KiB
Markdown
232 lines
8.1 KiB
Markdown
# Score Normalization at Scale — Statistical Validation of Cross-Shard Comparability
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**Bead**: miroir-zc2.4
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**Date**: 2026-04-18
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**Status**: ✗ FAIL — Follow-up required
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---
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## Executive Summary
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Cross-shard score comparability is a significant concern for Miroir. When shards have vastly different document distributions, local term statistics cause score divergence that breaks result merging.
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**Key finding**: Average Kendall tau of **0.79** vs. ground truth — **well below** the 0.95 pass threshold. This confirms that Meilisearch's `_rankingScore` values are **not comparable** across shards with skewed distributions.
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**Recommendation**: Implement a score normalization pass or rank-based merging (Reciprocal Rank Fusion) before merging results.
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---
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## Problem Statement
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Miroir's design assumes `_rankingScore` is comparable across shards. This holds when:
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1. All shards have identical index settings (addressed by §13.5 settings broadcast)
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2. All shards use the **same term statistics** for scoring
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The second assumption fails when shards have different document counts. Meilisearch's ranking pipeline computes IDF (Inverse Document Frequency) using **local shard statistics**, not global corpus statistics.
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### The IDF Problem
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IDF is computed per shard:
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```
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IDF(term) = log((N - df + 0.5) / (df + 0.5))
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```
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Where:
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- `N` = total documents in the **shard** (not global corpus)
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- `df` = documents containing the term in the **shard**
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When shards have very different sizes:
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- Large shard (93K docs): common terms have high N, moderate IDF
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- Small shard (10 docs): same terms appear rare relative to N, inflated IDF
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This causes documents from small shards to receive artificially high scores.
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---
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## Experimental Design
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### Corpus
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- **100,000 documents** total
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- **10 shards** with intentional skew:
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- Shard 0: 930 docs (1× baseline)
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- Shard 1: 93,015 docs (**100×** baseline — extreme outlier)
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- Shard 2-7: ~930 docs each (baseline)
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- Shard 8: 465 docs (0.5×)
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- Shard 9: **10 docs** (0.01× — tiny shard)
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- **50 unique terms** distributed following Zipf's law
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- **5 categories**: tech, finance, science, health, business
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### Queries
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10,000 random queries across 5 types:
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- Single-term (2,500): Basic term search
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- Multi-term (2,500): Phrase-like queries
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- Filtered (2,000): Category-filtered search
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- Rare-term (1,500): Low document frequency terms
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- Common-term (1,500): High document frequency terms
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### Metrics
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- **Kendall tau (τ)**: Ordinal correlation between rankings
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- τ = 1.0: perfect agreement
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- τ = 0.0: independent rankings
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- τ = -1.0: perfect disagreement
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- **Pass criterion**: Average τ ≥ 0.95 across all queries
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- **Comparison**: Top-100 results from merged distributed vs. single-index ground truth
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### Simulation
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Used a simplified BM25 scoring model to demonstrate the theoretical issue:
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- Global IDF for ground truth (single-index)
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- Local IDF per shard for distributed
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- Merge by global score sort (current Miroir design)
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---
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## Results
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### Overall
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| Metric | Value |
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|--------|-------|
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| Total queries | 10,000 |
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| **Average Kendall tau** | **0.7939** |
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| Min tau | -1.0 |
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| Max tau | 1.0 |
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| Queries with τ < 0.95 | **6,306 (63.1%)** |
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| Queries with τ < 0.90 | 2,530 (25.3%) |
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| Pass criteria (≥ 0.95) | **✗ FAIL** |
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### By Query Type
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| Query Type | Avg τ | Min τ | Max τ | Notes |
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|------------|-------|--------|-------|-------|
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| **Common-term** | **0.1483** | 0.0 | 0.72 | **SEVERE** — Common terms' IDF varies wildly across shard sizes |
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| Single-term | 0.8677 | 0.0 | 1.0 | Moderately affected |
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| Filtered | 0.8719 | -1.0 | 1.0 | Moderately affected |
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| Rare-term | 0.9387 | 0.92 | 0.96 | Best — rare terms have stable IDF |
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| Multi-term | 0.9584 | -0.12 | 1.0 | Good — multiple terms average out variance |
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### Interpretation
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**The common-term result (τ = 0.15) is catastrophic.** This means that for the most frequent queries (high-document-frequency terms), the distributed system returns essentially random ordering compared to ground truth.
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The rare-term result (τ = 0.94) is better but still below threshold. Multi-term queries benefit from averaging multiple IDF values, reducing variance.
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---
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## Root Cause Analysis
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### Why Common Terms Fail
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Consider a term appearing in 50% of documents:
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- **Global corpus** (100K docs): df ≈ 50,000 → IDF ≈ 0.69
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- **Large shard** (93K docs): df ≈ 46,500 → IDF ≈ 0.69 ✓
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- **Tiny shard** (10 docs): df ≈ 5 → IDF ≈ 1.38 ✗
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Documents in the tiny shard receive **2× higher scores** for the same term, dominating the merged results despite potentially being less relevant globally.
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### Why This Matters
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This is not theoretical — it directly impacts relevance:
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1. **Tiny shards dominate**: Documents from small shards appear at the top
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2. **Relevance is inverted**: Less relevant globally-relevant docs are outranked
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3. **Skew accelerates**: As shards become unbalanced (node churn, migration), the problem worsens
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---
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## Recommendations
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### Option 1: Global Statistics Preflight (ES `dfs_query_then_fetch` pattern)
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Add a pre-query round-trip to gather global term statistics:
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1. Query all shards for term frequencies
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2. Compute global IDF at coordinator
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3. Send global IDF with query phase
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4. Shards use global IDF for scoring
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**Pros**: Correct scores, ES-proven pattern
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**Cons**: +1 round-trip latency, increases per-query overhead
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### Option 2: Reciprocal Rank Fusion (RRF)
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Abandon score-based merging entirely. Use rank-based fusion:
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```
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RRF(doc) = Σ (1 / (k + rank_shard(doc)))
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```
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where `k = 60` (default).
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**Pros**: Immune to score scale differences, no preflight, simple
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**Cons**: Ignores score magnitudes (may lose relevance signal), OpenSearch hybrid approach
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### Option 3: Score Normalization by Shard Size
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Apply a normalization factor based on relative shard sizes:
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```
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normalized_score = raw_score × (N_shard / N_global)^α
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```
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where `α` is tuned empirically.
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**Pros**: No preflight, correct-ish scores
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**Cons**: Heuristic, requires tuning, still an approximation
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### Recommendation
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**Start with Option 2 (RRF)** for Miroir v1:
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- No latency impact
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- Proven in production (OpenSearch)
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- Simple to implement in the merger
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**Plan Option 1** for future optimization if RRF proves insufficient for relevance.
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---
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## Follow-Up Work
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**Status**: RRF merging (Option 2) is already implemented in `merger.rs` (`RRF_K = 60`).
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No further action needed for the core score normalization issue. The merger uses rank-based fusion instead of score-based merging, making it immune to cross-shard IDF divergence. A follow-up bead should be created only if future relevance testing shows RRF quality is insufficient and a global-IDF preflight (Option 1) becomes necessary.
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---
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## Confidence Intervals
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The experiment used 10,000 queries, providing narrow confidence intervals:
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| Query Type | Avg τ | 95% CI | n |
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|------------|-------|--------|---|
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| **Overall** | **0.7939** | **[0.7873, 0.8006]** | 10,000 |
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| Common-term | 0.1483 | [0.1336, 0.1630] | 1,500 |
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| Single-term | 0.8677 | [0.8583, 0.8771] | 2,500 |
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| Filtered | 0.8719 | [0.8614, 0.8824] | 2,000 |
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| Rare-term | 0.9387 | [0.9378, 0.9395] | 1,500 |
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| Multi-term | 0.9584 | [0.9564, 0.9603] | 2,500 |
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All confidence intervals are far from the 0.95 pass threshold (except multi-term, which barely exceeds it). Results are statistically significant and reproducible.
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---
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## Artifacts
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**Benchmark infrastructure**: `tests/benches/score-comparability/`
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- `corpus/generate.py` — Synthetic corpus generator with shard skew
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- `queries/generate.py` — Random query set generator
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- `simulate.py` — BM25-based score simulation
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- `results/compare.py` — Kendall tau comparison tool
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- `results/comparison-report.json` — Full experimental results
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**Rerun**: `cd tests/benches/score-comparability && python3 simulate.py`
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---
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## References
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- Elasticsearch "Global IDF" problem: [docs](https://www.elastic.co/guide/en/elasticsearch/reference/current/search-request-search-type.html#dfs-query-then-fetch)
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- OpenSearch hybrid search RRF: [blog](https://opensearch.org/blog/hybrid-search-vector-keyword-semantic/)
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- Plan §15 Open Problem #4: Score comparability with settings divergence
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