Can solution traces substitute for process-level reward signals in math reasoning?
This explores whether full worked-out solutions (demonstration traces) can stand in for fine-grained, step-by-step reward signals — the kind a process reward model (PRM) gives — when training math reasoning models.
This question pits two ways of teaching a model to reason against each other: showing it complete solution traces versus scoring each intermediate step with a process-level reward. The corpus suggests the honest answer is *partly, but the two aren't interchangeable — because they carry different information.* The most direct challenge to traces-as-substitute comes from work showing that traces don't actually carry the reasoning we assume they do. Models trained on deliberately corrupted, irrelevant reasoning traces perform comparably to those trained on correct ones, and sometimes generalize better Do reasoning traces need to be semantically correct? — which implies traces often function as computational *scaffolding* rather than verified logic. A related finding goes further: intermediate tokens are generated identically to any other output and carry no special execution semantics, so invalid traces routinely produce correct answers Do reasoning traces actually cause correct answers?. If a trace's individual steps aren't causally doing the reasoning, then leaning on them to supply step-level supervision is shakier than it looks.
Process rewards, by contrast, are valued precisely for the per-step *diagnostic* signal traces lack. One line of work shows that plain numerical rewards hit performance plateaus because they never tell the model *why* a step failed — and feeding back chain-of-thought critiques breaks through those plateaus Can natural language feedback overcome numerical reward plateaus?. This is sharpened by the observation that feedback decomposes into two orthogonal channels: *evaluative* (how good was this) and *directive* (how should it change), and a scalar reward captures only the first Can scalar rewards capture all the information in agent feedback?. A solution trace is rich in directive content but, as a static demonstration, it doesn't grade the model's own faltering steps — so it can't fully replace the corrective signal a good process reward provides.
The twist is that the frontier of process supervision is itself moving *toward* traces — but traces of judgment, not of the solution. Generative step-wise judges that reason about each reasoning step outperform classifier-style reward models, with far less training data Can judges that reason about reasoning outperform classifier rewards?, and reward models that produce their own reasoning before scoring beat outcome-only evaluators Can reward models benefit from reasoning before scoring?. So the field's best 'process signal' is increasingly a generated reasoning trace *about* the solution — collapsing the question's neat dichotomy. There's also a hard caveat for anyone hoping process supervision over raw thinking traces is straightforward: standard PRMs degrade on real thinking traces because those traces branch, backtrack, and revisit, and a step that looks wrong may be useful exploration rather than an error Why do standard process reward models fail on thinking traces?.
Stepping back, a cluster of RLVR findings reframes what either signal is even *doing*. A single training example can lift math accuracy from 36% to 73.6% Can a single training example unlock mathematical reasoning?, and spurious rewards work nearly as well as correct ones — because reward learning mostly *activates* reasoning strategies already latent in pretraining rather than teaching new ones What does reward learning actually do to model reasoning?. If the reward's main job is activation, then a handful of solution traces and a sparse outcome reward may land in roughly the same place — which partly explains why traces can substitute at all. But that same literature warns against reading the wins too generously: much apparent RLVR gain on popular math benchmarks turns out to be memorization of contaminated data, and on clean benchmarks only genuinely correct rewards help Does RLVR success on math benchmarks reflect genuine reasoning improvement?. Trace length is similarly misleading — it tracks how close a problem is to the training distribution, not how hard it is Does longer reasoning actually mean harder problems?.
The sharpest practical resolution in the corpus is to stop treating these as rivals and assign each its strength. Rather than converting rich rubric judgments into dense per-token rewards (which invites reward hacking), one approach uses rubrics as *gates* that accept or reject whole rollouts while token-level rewards optimize within the survivors Can rubrics and dense rewards work together without hacking?. That's the shape of the answer to your question: solution traces are good at supplying valid solution structure to imitate; process-level signals are good at the step-wise correction and directive feedback that demonstrations can't give on their own — and the strongest systems use each for what it does best rather than substituting one for the other.
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Models trained on systematically irrelevant traces maintain solution accuracy and sometimes improve out-of-distribution generalization, suggesting traces function as computational scaffolding rather than meaningful reasoning steps.
R1's intermediate tokens carry no special execution semantics and are generated identically to other LLM output. Invalid traces frequently produce correct answers, proving traces are not causally necessary—they correlate with answers via learned formatting, not functional reasoning.
Critique-GRPO shows that models stuck on performance plateaus can generate correct solutions when given chain-of-thought critiques, revealing that numerical rewards lack critical information about why failures occur and how to improve.
Natural feedback carries two orthogonal types of information: evaluative (how well an action performed) and directive (how it should change). Scalar rewards capture evaluation but discard directional specifics that token-level distillation can recover, making the two complementary rather than redundant.
StepWiser demonstrates that training judges to produce reasoning chains about policy reasoning—rather than classify steps—yields better judgment accuracy and data efficiency. Independent confirmation from GenPRM and ThinkPRM shows generative PRMs outperform discriminative ones with orders of magnitude less training data.
Three independent teams (RRM, RM-R1, DeepSeek-GRM) discovered that adding chain-of-thought reasoning before reward scoring enables adaptive test-time compute scaling for evaluation. Reasoning-based approaches raise the capability ceiling of reward models beyond what outcome-based evaluation achieves.
Standard PRMs degrade on trajectory format because thinking traces include branching, backtracking, and weaker coherence than polished responses. ReasonFlux-PRM addresses this by supervising both trajectories and responses, treating failed steps as informative exploration rather than errors.
A single example in RLVR boosts math performance from 36% to 73.6% and enables test accuracy to improve for 1,400 steps after training accuracy reaches 100%, revealing that minimal activation signals unlock latent reasoning capability.
Research shows RLVR improves sampling efficiency within existing capability boundaries without expanding them. A single training example suffices for activation, and spurious rewards work nearly as well as correct ones for models with appropriate pretraining.
Qwen2.5-Math-7B reconstructs 54.6% of MATH-500 from partial prompts but scores 0.0% on post-release LiveMathBench, revealing dataset contamination. On clean benchmarks, only correct rewards improve performance; random and inverse rewards fail or degrade reasoning ability.
Controlled A* maze experiments show trace length correlates with difficulty only in-distribution but decouples entirely out-of-distribution. Trace length primarily reflects recall of training schemas, not adaptive computation.
DRO shows that using rubrics to accept or reject rollout groups—rather than converting rubric scores into dense rewards—prevents reward hacking. This separation preserves the categorical strength of rubrics while letting token-level rewards optimize within valid answers.