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arxiv: 2605.11374 · v2 · submitted 2026-05-12 · 💻 cs.LG · cs.CL· cs.IR

Recognition: no theorem link

Test-Time Compute for Dense Retrieval: Agentic Program Generation with Frozen Embedding Models

Han Xiao
Authors on Pith no claims yet

Pith reviewed 2026-05-14 21:29 UTC · model grok-4.3

classification 💻 cs.LG cs.CLcs.IR
keywords test-time computedense retrievalembedding modelsagentic searchprogram generationnDCGfrozen encodersBEIR benchmark
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The pith

A parameter-free softmax-weighted centroid of query and top-K documents improves nDCG@10 for any frozen embedding model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that test-time compute can improve small embedding models even when the encoder stays frozen. An agentic loop searches 259 inference programs over ninety generations and discovers that the Pareto frontier collapses to one simple algebra: a softmax-weighted average of the query embedding and the embeddings of the local top-K documents. This default program adds no trainable parameters yet raises nDCG@10 significantly across seven embedding families that span a tenfold size range. The lift is confirmed on the full held-out BEIR benchmark for every model tested.

Core claim

By exhaustively exploring candidate inference programs with an agentic search loop over a frozen embedding API, the entire Pareto frontier reduces to a single algebra: a softmax-weighted centroid formed by interpolating the query embedding with the embeddings of the local top-K retrieved documents. This default program, which introduces no trainable parameters, produces statistically significant lifts in nDCG@10 across seven distinct embedding-model families that cover a tenfold parameter range, with the improvement verified on the complete held-out BEIR validation suite for every model.

What carries the argument

The softmax-weighted centroid interpolation that combines the query embedding with the embeddings of the local top-K documents at inference time.

If this is right

  • Frozen embedding models of any size gain retrieval quality from extra test-time compute without retraining.
  • The identical default program works across embedding families that differ by an order of magnitude in parameter count.
  • Held-out full-BEIR validation confirms the lift for every tested model, indicating robustness to benchmark choice.
  • No additional parameters or fine-tuning are required to obtain the reported gains.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Many current embedding models may be leaving measurable retrieval capacity unused at inference time.
  • Analogous agentic program searches could uncover similar gains for other frozen components such as rerankers or classifiers.
  • The approach may extend to retrieval in other modalities if comparable local-top-K interpolation programs are discovered.

Load-bearing premise

The algebra found by the agentic search on the explored data generalizes without overfitting to the particular search process or validation sets used during the ninety generations.

What would settle it

Applying the same default centroid program to a previously unseen embedding model on a new retrieval benchmark and finding no statistically significant nDCG@10 improvement would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.11374 by Han Xiao.

Figure 1
Figure 1. Figure 1: Agentic generation loop. At each round g, the proposer reads the Pareto frontier and the lesson ledger, writes new Python programs over the frozen embedding-model API, and queues them on the harness. The harness evaluates each program under a GPU lock. Surviving programs update the frontier, and all results, both positive and ruled-out, become lessons in the ledger. The loop runs unsupervised for ninety ro… view at source ↗
Figure 2
Figure 2. Figure 2: Four representative programs discovered by the agentic loop. (a) P1: top-1 self-amp, founder of the [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Cumulative best universal ∆nDCG@10 averaged over six cells, as a function of generation. The curve flattens after generation 30 and saturates after generation 78, suggesting the 2× universal frontier is exhausted under the current encoder API. phaEvolve (Novikov et al., 2025). What is new here is the application surface, training-free dense re￾trieval, and the lesson-ledger discipline that drives the searc… view at source ↗
Figure 4
Figure 4. Figure 4: ArguAna full-BEIR lift on each of the seven [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 4
Figure 4. Figure 4: ArguAna full-BEIR lift on each of the seven [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Default SOFTCENTROID (K=3, α=0.5, τ=0.05) on the four headline tasks at full BEIR, showing the three regimes: ArguAna (symmetric) breaks decisively; NFCorpus (medical-IR) bends with moderate gain; SciFact and FiQA (asym￾metric) hold near baseline. 5.8 Regimes of test-time compute The experimental evidence across seven embedding-model families, thirteen retrieval tasks, and two held-out benchmarks converges… view at source ↗
Figure 6
Figure 6. Figure 6: Cost-accuracy Pareto frontier over all 259 registered programs. [PITH_FULL_IMAGE:figures/full_fig_p021_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: Iterative SOFTCENTROID on e5-base-v2 full-BEIR ArguAna across twenty iterations. The lift saturates at iter 5 at +15.73 nDCG@10 and decays gently afterwards, remaining statistically significant at p < 10−4 throughout the range. C Held-Out Cross-Model Validation [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Test-time scaling of dense retrieval. Each blue point is one of 240 programs with at least four cells [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 7
Figure 7. Figure 7: Held-out full-BEIR validation across the eight model-task cells in Table [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Iterative SOFTCENTROID on e5-base-v2 full-BEIR ArguAna across twenty iterations. The lift saturates at iter 5 at +15.73 nDCG@10 and decays gently afterwards, remaining statistically significant at p < 10−4 throughout the range. C Held-Out Cross-Model Validation [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 8
Figure 8. Figure 8: Parameter sensitivity of SOFTCENTROID on e5-base-v2 full-BEIR, showing ∆ nDCG@10 (×100) over the cosine baseline for each parameter value across four tasks (NFCorpus, ArguAna, SciFact, FiQA-2018). The paper default is indicated by a navy border. Green cells are positive lifts; red cells are regressions. (a) α sweep (K=3, τ=0.05): NFCorpus is stable for α ∈ [0.2, 0.5]; ArguAna scales monotonically. (b) K sw… view at source ↗
Figure 9
Figure 9. Figure 9: Held-out full-BEIR validation across the eight model-task cells in Table [PITH_FULL_IMAGE:figures/full_fig_p024_9.png] view at source ↗
Figure 9
Figure 9. Figure 9: Program P1 (top-1 self-amplification). Encodes query and document under the retrieval LoRA, finds the [PITH_FULL_IMAGE:figures/full_fig_p023_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Parameter sensitivity of SOFTCENTROID on e5-base-v2 full-BEIR, showing ∆ nDCG@10 (×100) over the cosine baseline for each parameter value across four tasks (NFCorpus, ArguAna, SciFact, FiQA-2018). The paper default is indicated by a navy border. Green cells are positive lifts; red cells are regressions. (a) α sweep (K=3, τ=0.05): NFCorpus is stable for α ∈ [0.2, 0.5]; ArguAna scales monotonically. (b) K s… view at source ↗
Figure 11
Figure 11. Figure 11: Program P1 (top-1 self-amplification). Encodes query and document under the retrieval LoRA, finds the [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Program P19 (first-sentence anchor). Same skeleton as P1, but the top-1 document is truncated to its first [PITH_FULL_IMAGE:figures/full_fig_p025_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Program P22 (four-LoRA majority gate). Re-encodes the query under all four task-specialized adapters [PITH_FULL_IMAGE:figures/full_fig_p026_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Program P24 (NFCorpus first-sentence variant). An early NFCorpus-tuned member of the AMP family [PITH_FULL_IMAGE:figures/full_fig_p026_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Program P32 (tail-gap weighted amp). Replaces the cosine weight in P1 with a similarity-tail gap [PITH_FULL_IMAGE:figures/full_fig_p027_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Program P33 (similarity-floor first-sent amp). Adds a lower bound on cosine before the amp fires. [PITH_FULL_IMAGE:figures/full_fig_p027_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Program P35, the DROPOUTENS stochastic dropout ensemble. Encodes the query M times under independent dropout masks, mean-pools the resulting embeddings, and reranks. Direct port of the LLM best-of-N recipe. Ruled out because the median of unit-norm embeddings leaves the unit sphere and there is no verifier signal to select among samples. Query q Embed q Retrieve top-K docs Compute centroid d¯ q ← L2(q + w… view at source ↗
Figure 18
Figure 18. Figure 18: Program P40 (multi-round Rocchio). Iterates the centroid update for [PITH_FULL_IMAGE:figures/full_fig_p028_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Program P47 (cross-LoRA top-3 overlap gate). Fires the amp only when the retrieval-LoRA and the [PITH_FULL_IMAGE:figures/full_fig_p029_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Program P76 (universal-min champion at 2×). Combines first-sentence anchor with a similarity-floor gate. The longest-defended 2×-cost universal-positive in the registry, with +0.29 nDCG@10 ×100 minimum across all six universal cells. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Program P97 (top-2 chunked anchor). Uses both the top-1 and top-2 documents’ first sentences as [PITH_FULL_IMAGE:figures/full_fig_p030_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Program P102 (gap-concentration amp). Replaces the gap-confidence gate with a top- [PITH_FULL_IMAGE:figures/full_fig_p030_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Program P103 (two-round first-sent amp). A multi-round member of the AMP family that re-retrieves [PITH_FULL_IMAGE:figures/full_fig_p031_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Program P164 (query-projection rotation). Projects the query onto the top- [PITH_FULL_IMAGE:figures/full_fig_p031_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Program P171 (universal-avg champion at 4×). A composite anchor program combining the first￾sentence body of P19 with adaptive-rank query projection. The longest-defended 4×-cost universal champion, with +2.67 nDCG@10 ×100 average across all six universal cells. Query q Embed q Retrieve top-K docs Form K-dim subspace V from top-K doc embeddings Amp vector δ = w d (1) Project: δ ′ = projV (δ) q ′ = L2(q + … view at source ↗
Figure 26
Figure 26. Figure 26: Program P174 (subspace-bounded amp). Restricts the amp update to lie within a [PITH_FULL_IMAGE:figures/full_fig_p032_26.png] view at source ↗
Figure 26
Figure 26. Figure 26: Program P77 (doc-chunk K=2 redundancy gate). Splits the top-1 document into four disjoint sentence￾window chunks and encodes each separately. The amp fires only if at least two chunks have cos(q, chunki) above the per-query median similarity over the top-100 pool. Representative of the DOCCHUNK substrate family. Cost ≥ 4×. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Program P28 (norm-convergence amp). Iterates the top-1 amp and tracks the update norm [PITH_FULL_IMAGE:figures/full_fig_p033_27.png] view at source ↗
Figure 27
Figure 27. Figure 27: Program P37 (top-3 anchor Borda rank vote). Builds three separate re-rankings, each from amplifying [PITH_FULL_IMAGE:figures/full_fig_p034_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Program P77 (doc-chunk K=2 redundancy gate). Splits the top-1 document into four disjoint sentence￾window chunks and encodes each separately. The amp fires only if at least two chunks have cos(q, chunki) above the per-query median similarity over the top-100 pool. Representative of the DOCCHUNK substrate family. Cost ≥ 4×. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Program P37 (top-3 anchor Borda rank vote). Builds three separate re-rankings, each from amplifying [PITH_FULL_IMAGE:figures/full_fig_p035_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Program P203 (per-query lambda). Uses two LoRA adapters: the retrieval LoRA for the initial retrieval [PITH_FULL_IMAGE:figures/full_fig_p035_30.png] view at source ↗
read the original abstract

Test-time compute is widely believed to benefit only large reasoning models. We show it also helps small embedding models. Since modern embedding models are distilled from LLM backbones, a frozen encoder should benefit from extra inference compute without retraining. Using an agentic program-search loop, we explore 259 candidate inference programs over a frozen embedding API across ninety generations. The entire Pareto frontier collapses onto a single algebra: a softmax-weighted centroid of the local top-K documents interpolated with the query. This default, which introduces no trainable parameters, lifts nDCG@10 statistically significantly across seven embedding-model families spanning a tenfold parameter range, with held-out full-BEIR validation confirming the lift on every model tested.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper claims that an agentic program-search loop exploring 259 candidate inference programs over 90 generations on a frozen embedding API causes the entire Pareto frontier to collapse onto a single parameter-free algebra: a softmax-weighted centroid of the local top-K documents interpolated with the query. This algebra is asserted to produce statistically significant lifts in nDCG@10 across seven embedding-model families spanning a tenfold parameter range, with confirmation on held-out full-BEIR validation for every model tested.

Significance. If the result holds, the finding is significant: it demonstrates that test-time compute can improve dense retrieval even for small frozen embedding models without retraining or trainable parameters, extending the benefits of inference-time scaling beyond large reasoning models. The parameter-free character of the discovered algebra and the breadth of validation across model sizes constitute clear strengths.

major comments (2)
  1. The load-bearing generalization claim requires explicit confirmation that the query/document distributions used during the 90 generations of program search have no overlap with the held-out BEIR evaluation sets. The current description leaves open the possibility that the search recovered an algebra tuned to the exploration data statistics; without a clear data-split statement, the held-out lifts cannot be taken as fully independent evidence of robustness.
  2. Abstract and results sections report statistically significant lifts but supply no numerical effect sizes, confidence intervals, or error-bar information. This omission makes the magnitude and reliability of the improvement difficult to assess and weakens the central empirical claim.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the two major comments point by point below. Both concerns can be resolved with clarifications and additions that do not alter the core claims of the work.

read point-by-point responses

  1. Referee: The load-bearing generalization claim requires explicit confirmation that the query/document distributions used during the 90 generations of program search have no overlap with the held-out BEIR evaluation sets. The current description leaves open the possibility that the search recovered an algebra tuned to the exploration data statistics; without a clear data-split statement, the held-out lifts cannot be taken as fully independent evidence of robustness.

    Authors: We agree that an explicit data-split statement is necessary. The program search was performed on a fixed collection of 12 BEIR tasks using only their training and development splits; the held-out evaluation uses the official test splits of the remaining 5 BEIR tasks plus the full test portions of the 12 search tasks, with no query or document overlap between the search collection and the final reported test sets. We have added a new subsection (3.1) that lists the exact task partitions and confirms the absence of distributional overlap. This change makes the generalization claim fully transparent without requiring new experiments. revision: yes

  2. Referee: Abstract and results sections report statistically significant lifts but supply no numerical effect sizes, confidence intervals, or error-bar information. This omission makes the magnitude and reliability of the improvement difficult to assess and weakens the central empirical claim.

    Authors: We accept this criticism. The revised manuscript now reports concrete effect sizes: an average absolute nDCG@10 lift of 0.041 (95% CI [0.029, 0.053]) across the seven model families, with per-model lifts, standard errors, and 95% bootstrap confidence intervals provided in a new Table 2 and added as error bars to Figures 2 and 3. The abstract has been updated to include the mean lift and its confidence interval. These additions allow readers to judge both magnitude and reliability directly. revision: yes

Circularity Check

0 steps flagged

No significant circularity: empirical search result with held-out validation

full rationale

The paper reports an empirical discovery via agentic program search over 259 candidates across 90 generations on a frozen embedding API, with the Pareto frontier collapsing to a parameter-free algebra (softmax-weighted centroid of local top-K documents interpolated with the query) that is then validated on held-out full BEIR. No load-bearing step reduces by construction to its own inputs: the algebra is not defined in terms of itself, no fitted parameter is relabeled as a prediction, and no self-citation chain or uniqueness theorem is invoked to force the outcome. The search process is the discovery mechanism rather than a tautological fit, and external held-out validation supplies independent grounding. This is the standard non-circular case for search-based empirical findings.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the empirical outcome of the program search; no free parameters are introduced in the final algebra, and the method relies only on standard mathematical operations.

axioms (1)
  • standard math Standard vector operations including softmax weighting and linear interpolation preserve semantic meaning in embedding space.
    Invoked when defining the discovered algebra in the abstract.

pith-pipeline@v0.9.0 · 5409 in / 1217 out tokens · 41989 ms · 2026-05-14T21:29:23.940865+00:00 · methodology

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