Partial-Match Rewards: Methods & Implications

• Partial-match rewards are reward specifications that assign non-binary credit to outputs satisfying some but not all correctness conditions, thereby capturing valuable intermediate progress.
• They mitigate the sparsity issue inherent in exact-match systems by using fractional execution scores, checklist averaging, and staged credit in applications like Text-to-SQL and code generation.
• These methods facilitate more stable and effective learning by replacing brittle terminal judgments with denser, structured feedback that guides optimization even under ambiguous or delayed conditions.

Searching arXiv for papers on partial-match rewards and closely related reward-shaping formulations. Partial-match rewards are reward specifications that assign non-binary credit to outputs, trajectories, or substructures that satisfy some, but not all, correctness conditions. In recent work, they appear as fractional execution scores in Text-to-SQL, checklist fractions in instruction-following, staged partial credit in code generation, temporally partitioned payoffs in bandits, and relevance-filtered reward decomposition in cooperative multi-agent learning. Their shared purpose is to mitigate reward sparsity relative to exact-match, all-tests-pass, or globally shared rewards, while retaining enough task structure to support efficient optimization under realistic ambiguity, delayed feedback, or incomplete verification (Hao et al., 8 Sep 2025, Dash et al., 27 May 2026, Romano et al., 2022).

1. Conceptual basis and motivation

The central motivation for partial-match rewards is that binary supervision often collapses qualitatively different failures into the same zero signal. In Text-to-SQL, PaVeRL-SQL argues that exact-match and execution-match are often binary, and they miss useful intermediate progress when a query is “mostly right” but not perfectly aligned with the ground truth. The paper emphasizes that this matters when user questions are vague, schemas are noisy or incomplete, and multiple SQL formulations can return materially acceptable answers even if they differ from the gold query. Its core claim is that dense, fractional feedback helps reinforcement learning stabilize and learn better than a pure $0/1$ success signal (Hao et al., 8 Sep 2025).

The same sparsity argument appears in instruction-following and code generation. Soft-RLVR starts from the observation that many prompts are only partially verifiable: a response may satisfy some requirements but not others, and a single holistic pass/fail judgment obscures which constraints were met. SecureCodeRL makes an analogous point for competitive-programming-style generation, where many outputs are “almost right” but fail because of missing print, wrong formatting, runtime crash, syntax issues, or partial algorithmic correctness; under binary reward, all of these failures collapse to the same zero (Dash et al., 27 May 2026, Sijwali et al., 3 Jan 2026).

A common misconception is that every framework adjacent to partial-match rewards provides direct proportional credit for partial success. Some do not. Maximally permissive reward machines, for example, do not reward partial completion directly; they permit every behavior consistent with some valid partial-order realization of the task, but reserve terminal non-negative reward for goal achievement and otherwise apply a shaping penalty of $-1$. This is partial fulfillment handled structurally rather than by a graded reward scale (Varricchione et al., 2024).

2. Canonical design patterns

Recent work instantiates partial-match rewards through several recurring mechanisms.

Mechanism	Representative scoring object	Representative papers
Fractional execution matching	Proportion of matched result columns	(Hao et al., 8 Sep 2025)
Composite SQL subrewards	Execution, judge, syntax, schema, n-gram, format	(Pourreza et al., 29 Mar 2025)
Checklist averaging	Fraction of atomic criteria satisfied	(Dash et al., 27 May 2026)
Staged partial credit	Syntax, execution, output, partial tests	(Sijwali et al., 3 Jan 2026)
Temporally partitioned reward	Reward fragments observed over later rounds	(Romano et al., 2022, Broek et al., 2023)
Relevance-weighted reward decomposition	Rewards from relevant teammates only	(Kapoor et al., 2024)

In PaVeRL-SQL, the dense signal is a fractional execution score, $\text{EX}_f \in [0,1]$, computed from the execution result table rather than from clause-level decomposition. In Soft-RLVR, the reward is the fraction of checklist items judged satisfied: $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$ In SecureCodeRL, partial credit is implemented as a staged ladder in $[0,1]$, with increasing reward for valid syntax, successful execution, producing output, and partial or full test passing. In Reasoning-SQL, several partial signals are linearly combined into a composite reward, with the reported weighting

$$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$

These designs differ in what they treat as the unit of partial correctness—columns, checklist items, execution stages, or symbolic query features—but all replace a brittle terminal judgment with denser supervision (Hao et al., 8 Sep 2025, Dash et al., 27 May 2026, Sijwali et al., 3 Jan 2026, Pourreza et al., 29 Mar 2025).

A second design axis concerns whether partiality is semantic, temporal, or causal. Temporally-partitioned bandits treat one decision’s reward as a sequence of fragments observed across future rounds rather than as a single delayed lump. PRD-MAPPO uses learned attention to identify which teammates are relevant to a given agent’s update, then removes or downweights irrelevant reward contributions. These are not “partial correctness” rewards in the narrow sense, but they exploit partial relevance in the reward structure to improve learning signal quality (Romano et al., 2022, Kapoor et al., 2024).

3. Text-to-SQL as a central case

Text-to-SQL has become a particularly active domain for partial-match reward design because exact SQL matching is both sparse and semantically brittle. PaVeRL-SQL formalizes two execution-based metrics. Binary execution accuracy, $\text{EX}_b \in \{0,1\}$, counts a sample as correct if the execution result table contains all the information from the gold table and the number of extra columns satisfies

$$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$

with $\tau = 5$ by default. Fractional execution accuracy, $\text{EX}_f \in [0,1]$, measures the proportion of gold result columns whose values match in the generated result table, up to row ordering and formatting; extra columns are neutral. The corresponding CoT RL reward is

$-1$0

The paper reports that negative punishments destabilized GRPO training, so the final design uses non-negative rewards only. Mechanistically, the reward is end-to-end and result-table-based: SELECT, WHERE, JOIN, GROUP BY, aggregation, and nested subqueries matter only through their impact on the executed output table (Hao et al., 8 Sep 2025).

PaVeRL-SQL applies this partial-match reward only in its CoT RL track, which trains OmniSQL-7B with a two-stage GRPO schedule. The verbal-RL pipeline, by contrast, is a generate-and-judge workflow: it samples executable SQL candidates, scores them with the same backbone LLM using a scoring prompt, and selects the top-ranked query, but does not present an explicit partial-match reward formula comparable to the CoT reward. Empirically, the paper reports that on Spider2.0-SQLite the verbal-RL pipeline achieves an execution accuracy $-1$1 higher than SOTA, and the CoT pipeline is $-1$2 higher. It also reports PaVeRL-Spider at $-1$3 greedy EX and $-1$4 majority-vote EX on Spider dev, PaVeRL-BIRD at $-1$5 on BIRD dev, and PaVeRL-SynSQL10K at $-1$6 on Spider2.0-SQLite. In a mixed-dialect setting with $-1$7 SynSQL10K SQLite samples and $-1$8 proprietary MariaDB/MySQL samples, the richer SQLite test set does not degrade, while the low-resource MySQL set improves by roughly $-1$9 in execution accuracy (Hao et al., 8 Sep 2025).

Reasoning-SQL pursues the same anti-sparsity objective with a different decomposition. Its partial rewards are schema-linking Jaccard similarity, AI feedback from an LLM judge, n-gram Jaccard similarity with bigrams, syntax check, and format reward. The judge reward is used only when execution accuracy is zero; if execution is correct, the judge reward is set to $\text{EX}_f \in [0,1]$0. The paper states that the weights are chosen so that no incorrect SQL query can obtain a higher overall reward than a correct query. Under this design, the main ablation on BIRD dev reports: base Qwen2.5-Coder-7B at $\text{EX}_f \in [0,1]$1 EX, SFT at $\text{EX}_f \in [0,1]$2, STaR-SFT at $\text{EX}_f \in [0,1]$3, GRPO with execution only at $\text{EX}_f \in [0,1]$4, GRPO with all rewards at $\text{EX}_f \in [0,1]$5, and filtered-schema EX with all rewards at $\text{EX}_f \in [0,1]$6. The same study reports targeted gains of $\text{EX}_f \in [0,1]$7 on Syntax Check from syntax reward, $\text{EX}_f \in [0,1]$8 on Schema Jaccard from schema reward, and $\text{EX}_f \in [0,1]$9 on N-gram Jaccard from n-gram reward (Pourreza et al., 29 Mar 2025).

4. Learned verification and staged partial credit

Soft-RLVR generalizes partial-match rewards to partially verifiable instruction-following. For each prompt $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$0, it constructs a checklist

$$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$1

where each $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$2 is an atomic yes/no criterion. An LLM verifier judges each item separately, potentially with multiple samples, and the reward is the fraction satisfied. The paper formalizes the tradeoff between variance reduction and partial-credit bias. If checklist judgments are conditionally independent, then

$$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$3

so checklist averaging can reduce variance by roughly a factor of $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$4 relative to a single holistic verifier. The cost is a relaxation gap,

$$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$5

which quantifies how much incomplete responses are rewarded beyond strict success. The paper’s main theorem gives conditions under which checklist verification has lower conditional gradient MSE than holistic verification, and its empirical results show that checklist-based Soft-RLVR improves IFEval from $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$6 to $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$7 with a GPT-OSS-20B verifier and to $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$8 with GPT-OSS-120B. With the noisier verifier, checklist training exceeds holistic training by $$R_{\mathrm{soft}}(x,y;\phi) = \frac{1}{K}\sum_{k=1}^K \rho_\phi(x,y,c_k).$$9 points on IFEval; with the stronger verifier, holistic and checklist verification are nearly tied at $[0,1]$0 and $[0,1]$1, respectively (Dash et al., 27 May 2026).

Soft-SVeRL, the self-verifying variant with $[0,1]$2, exposes a failure mode specific to learned partial-match rewards: reward inflation from permissive self-judgments. The paper describes this as “always-yes collapse.” Naive self-verification drives IFEval from $[0,1]$3 at initialization to $[0,1]$4 by step $[0,1]$5. Stabilization requires multiple verifier votes per item, verifier co-training on gold and replay examples, and a partition-style yes-rate penalty in the shared objective

$[0,1]$6

With gold examples, $[0,1]$7, and $[0,1]$8, IFEval reaches $[0,1]$9; lowering $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$0 to $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$1 gives $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$2; adding replay gives $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$3; increasing votes to $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$4 gives $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$5 (Dash et al., 27 May 2026).

SecureCodeRL implements a different partial-credit template for code generation. It optimizes

$$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$6

where the functional term uses a staged reward ladder: $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$7 This separates syntax-valid from syntax-invalid outputs, runnable from crashing programs, output-producing from silent ones, and partially correct from totally wrong solutions. The security term is based on Bandit static analysis. In a pilot evaluation on $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$8 held-out APPS+ prompts per model, PPO-continue improves syntax validity from $$r_i = 3r_{\text{exec}} + 2r_{\text{judge}} + r_{\text{syntax}} + r_{\text{schema}} + r_{\text{ngram}} + r_{\text{format}}.$$9 for the SFT baseline to $\text{EX}_b \in \{0,1\}$0, achieves the only non-zero test success signal at $\text{EX}_b \in \{0,1\}$1 at-least-one-test-pass, and remains $\text{EX}_b \in \{0,1\}$2 Bandit-clean, with mean reward $\text{EX}_b \in \{0,1\}$3 versus $\text{EX}_b \in \{0,1\}$4 for the SFT baseline (Sijwali et al., 3 Jan 2026).

5. Structural, temporal, and combinatorial generalizations

Not all uses of partial-match reward are based on scalar fractional credit. In maximally permissive reward machines, the key move is to synthesize the reward machine from the set of all partial-order plans rather than from a single sequential or partial-order plan. The state space is built from prefixes of all linearisations of all valid partial-order plans, so the agent can progress along any admissible realization. The reward function remains sparse: $\text{EX}_b \in \{0,1\}$5 Theorem 1 states that the optimal value under the all-POPs machine is at least as large as that under a single-POP machine, which is at least as large as that under a sequential-plan machine. This is partial satisfaction by permissiveness rather than by graded intermediate reward (Varricchione et al., 2024).

Temporally-partitioned reward models shift the notion of “partial” from correctness to revelation over time. In TP-MAB, pulling an arm does not reveal its cumulative reward immediately; instead, the reward is partitioned over a finite number of future rounds. Under $\text{EX}_b \in \{0,1\}$6-smoothness, the asymptotic lower bound improves by a factor $\text{EX}_b \in \{0,1\}$7 relative to the unstructured case, and TP-UCB-FR obtains a leading regret term of order $\text{EX}_b \in \{0,1\}$8. The generalized $\text{EX}_b \in \{0,1\}$9-spread model replaces uniform partitioning with an arbitrary discrete distribution over temporal groups, allowing front-loaded or back-loaded reward shapes. TP-UCB-FR-G adapts its confidence radius to both the mean location of reward mass and the index of coincidence $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$0. The reported experiments include regret reductions of about $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$1 in synthetic settings, about $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$2 in some higher-$$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$3 configurations, and about $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$4 on a Spotify playlist recommendation dataset (Romano et al., 2022, Broek et al., 2023).

A more classical stochastic-process formulation appears in renewal theory. There, partial rewards are rewards accrued during the cycle rather than only at renewal epochs. The standard renewal reward theorem extends to this case only under additional regularity assumptions such as boundedness, nonnegativity, monotonicity, or decomposition into a difference of two monotone nonnegative processes. The paper gives a Cauchy-based counterexample showing that $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$5 for cycle rewards is not sufficient to guarantee that $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$6 is well-defined at intermediate times (Vlasiou, 2014).

In cooperative MARL, PRD-MAPPO uses partial reward decoupling to reduce credit-assignment noise. Each agent estimates a relevance set $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$7 using attention weights and computes advantages only from the rewards of relevant teammates; the soft variant reweights all teammates by learned attention rather than hard-thresholding them. The paper reports higher data efficiency, better asymptotic performance, and reduced spikes in gradient estimator variance relative to MAPPO across tasks including StarCraft II (Kapoor et al., 2024).

The language of partial matches also appears outside RL proper. In complex event processing, pSPICE sheds internal partial matches rather than primitive events and assigns each partial match a utility

$$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$8

where completion probability comes from a Markov chain and remaining processing time from a Markov reward process. In dense subhypergraph optimization, SWAMP assigns each hyperedge a monotone reward $$|\mathrm{Cols}(\hat{T}) \setminus \mathrm{Cols}(T)| < \tau,$$9 for including $\tau = 5$0 of its vertices, thereby treating partial hyperedge inclusion itself as a rewarded object. For arbitrary monotone rewards, the paper gives two $\tau = 5$1-approximation algorithms, where $\tau = 5$2 is the maximum hyperedge size (Slo et al., 2020, Bengali et al., 15 Jun 2025).

6. Limits, failure modes, and interpretive issues

The main technical limitation of partial-match rewards is that denser feedback need not be semantically faithful. PaVeRL-SQL’s signal is only column-level. The authors explicitly note that future work should enrich rewards beyond column-level signals, for example with tuple-set comparisons and constraint satisfaction. The current reward may therefore over-credit queries that overlap in columns while differing in rows or constraints. The framework also depends on an executable environment, judge-style scoring in the verbal pipeline can be sensitive to backbone bias, very large schemas remain difficult and require schema retrieval or compression, and the system is not yet extended to multi-turn interactions (Hao et al., 8 Sep 2025).

Soft-RLVR makes the partial-credit bias explicit: rewarding the fraction of satisfied checklist items can favor incomplete responses unless specificity is high and the relaxation gap is controlled. Its theory therefore rejects the simplistic view that “more decomposition is always better.” Larger $\tau = 5$3 alone is not enough; checklist quality and verifier quality both determine whether variance reduction dominates bias. The self-verifying setting adds a further instability, because the model can improve reward either by generating better outputs or by becoming a more permissive judge (Dash et al., 27 May 2026).

SecureCodeRL illustrates another recurrent issue: partial-credit rewards can reveal the first signs of learning even when full success is nearly absent, but they may be difficult to validate conclusively in small pilots. The paper explicitly describes the study as a pilot, notes that only $\tau = 5$4 prompts per model were used, and reports that the security term did not differentiate evaluated outputs because all samples were Bandit-clean. This suggests caution in interpreting mean-reward changes when the dense reward is dominated by early ladder stages (Sijwali et al., 3 Jan 2026).

A further misconception is to conflate partial-match rewards with partial reward observability. These are distinct problems. In monitored MDPs, the environment reward may be hidden unless the agent enters the right monitor state or takes a costly monitor action. “Beyond Optimism” argues that standard optimism can then converge to a suboptimal policy because the agent avoids precisely the actions needed to reveal reward; the proposed remedy is directed exploration via a separate goal-conditioned policy learned from a state-action successor function. This is not a partial-credit design, but a response to the fact that informative rewards may be unobservable rather than merely sparse (Parisi et al., 2024).

Finally, partiality can arise not in the reward signal but in the identification of the reward itself. In reward learning with partial identifiability, multiple rewards may be equally compatible with the feedback. The paper on partial identifiability shows that the best reward for a downstream application may lie outside the feasible set and formalizes this choice as the minimizer of worst-case task-specific mismatch over the feasible set. In the reward-transfer setting, if $\tau = 5$5 and $\tau = 5$6 maximize and minimize the induced preference gap over the feasible set, then the robust choice is

$\tau = 5$7

with worst-case error $\tau = 5$8. This suggests a broader interpretation of partial-match reward: not necessarily the true target reward, but a task-optimal proxy under incomplete identifiability (Lazzati et al., 10 Jan 2025).