Delete the grader. Let the model's own confidence be the reward — and it still learns to reason.
RL post-training normally needs a grader: a verifier that checks the final answer (RLVR) or a human/reward model that scores it (RLHF). This paper throws the grader away entirely. Its method, INTUITOR, takes GRPO — the same algorithm behind DeepSeek-R1 — and replaces the external reward with a single intrinsic quantity: the model's own self-certainty, how sharply peaked its own token distribution is. No gold answers, no test cases, no labels. On Qwen2.5-3B it matches GRPO on math (MATH500 0.612 vs 0.636) while generalizing far better out of domain — a 65% relative jump on LiveCodeBench where GRPO actually regressed, and +76% on CRUXEval-O vs GRPO's +44%. The catch: on non-Qwen models the gains largely vanish, which points at capability elicitation more than reasoning discovery.
The idea
A model can improve its reasoning using nothing but its own sense of conviction as the reward signal.
Every mainstream recipe for making a base model reason better needs a source of truth outside the model. RLHF needs humans (or a reward model trained on humans). RLVR — the reinforcement-learning-with-verifiable-rewards recipe behind DeepSeek-R1 and o1-style training — needs a verifier: a math checker that confirms the boxed answer, or a code sandbox that runs unit tests. That external oracle is what makes the reward trustworthy, and it's also what makes RLVR expensive, domain-locked, and impossible to apply where you can't cheaply verify (open-ended writing, most agentic tasks, anything without a ground-truth key).
The authors propose RLIF — Reinforcement Learning from Internal Feedback: the reward comes from a signal the model computes about itself, with no external supervision at all. Their instantiation, INTUITOR, uses the crispest such signal available for free at every decoding step — how confident the model is in its own next token, aggregated over the whole response. The bet is that "I feel sure about this reasoning" is correlated enough with "this reasoning is good" that you can optimize the former and get the latter, for free, in any domain. It is the purest version of the "let the intrinsic signal replace the grader" story: not a cheaper grader, no grader.
The concrete thing this attacks is the coverage limit of RLVR. GRPO (Group Relative Policy Optimization, from DeepSeekMath) made verifiable-reward RL cheap by dropping the value network: for each prompt you sample a group of $G$ answers, score each with the verifier, and use the group's mean and standard deviation to turn raw scores into advantages. It works beautifully — where a verifier exists. Math and competitive code have verifiers. Almost nothing else does. Building one for a new domain means writing checkers or curating labeled data, and for genuinely open-ended tasks a reliable verifier may not exist even in principle.
Two threads set up INTUITOR. First, "confidence" had already been shown to be a usable selection signal at inference time: a prior paper from overlapping authors introduced self-certainty as a reward-model-free way to do Best-of-N — pick the sample the model is most sure of and you beat naive decoding. The open question was whether that same static signal is a strong enough gradient to train on, or whether optimizing it directly just teaches the model to be loudly, uselessly overconfident. Second, GRPO's group-relative normalization does something subtle and important: it only cares about which samples in a batch are better than their peers, not about absolute reward magnitude. That property turns out to be exactly what keeps a self-referential reward from exploding. INTUITOR is the experiment that tests whether an intrinsic signal, dropped into that machinery, is enough.
The method is deliberately minimal: take GRPO unchanged, and swap out one term. Everything below is either the reward definition, the objective it plugs into, or the training loop that ties them together.
Self-certainty measures how far the model's next-token distribution sits from a flat, maximally-uncertain uniform distribution, averaged over every position in the generated response. Formally, for a response $o$ to a query $q$ over vocabulary $\mathcal{V}$:
$$\text{Self-certainty}(o\mid q)=\frac{1}{|o|}\sum_{i=1}^{|o|} D_{\mathrm{KL}}\!\big(U \,\big\|\, \pi_\theta(\cdot\mid q,o_{<i})\big)=-\frac{1}{|o|\,|\mathcal{V}|}\sum_{i=1}^{|o|}\sum_{j=1}^{|\mathcal{V}|}\log\!\big(|\mathcal{V}|\cdot \pi_\theta(j\mid q,o_{<i})\big)$$Read it right to left. At each token position the model produces a probability distribution over the vocabulary. Compare it to the uniform distribution $U$ (every token equally likely) using KL divergence. If the model's distribution is sharply peaked on a few tokens — it "knows what it wants to say" — the KL is large; if it's spread out and hesitant, the KL is near zero. Average that over all positions in the answer. Higher self-certainty means the model was consistently confident across the whole generation. That's the entire reward: one scalar per response, computable from logits the model already produced, with zero external input.
Two design choices matter here. The KL is written as $D_{\mathrm{KL}}(U \,\|\, \pi_\theta)$ — uniform first, model second — which makes it mode-seeking and, per the authors, less biased toward long outputs than perplexity or entropy would be. And because it averages over $|o|$, it's a per-token measure, not a per-sequence one, so simply writing more tokens doesn't mechanically inflate it.
GRPO's advantage is computed by z-scoring each sample's reward against its group. INTUITOR keeps that machinery exactly and feeds it self-certainty instead of a verifier's 0/1:
$$u_i=\text{Self-certainty}(o_i\mid q),\qquad \hat{A}_i=\frac{u_i-\operatorname{mean}(\{u_1,\dots,u_G\})}{\operatorname{std}(\{u_1,\dots,u_G\})}$$For a given prompt, sample $G$ responses, score each by its own self-certainty $u_i$, then standardize within the group. A response gets a positive advantage only if it is more confident than its siblings for the same prompt, negative if less. This is the load-bearing trick: the absolute scale of confidence is normalized away, so the policy can't win just by cranking global confidence — it has to find, relative to its own alternatives, the reasoning paths it is most sure of. Those advantages then go into the standard clipped GRPO objective, with a KL leash to a frozen reference model:
$$\mathcal{J}(\theta)=\mathbb{E}\!\left[\frac{1}{G}\sum_{i=1}^{G}\frac{1}{|o_i|}\sum_{t=1}^{|o_i|}\min\!\big(r_{i,t}\hat{A}_i,\ \operatorname{clip}(r_{i,t},1-\epsilon,1+\epsilon)\,\hat{A}_i\big)-\beta\, D_{\mathrm{KL}}\!\big(\pi_\theta \,\|\, \pi_{\text{ref}}\big)\right],\quad r_{i,t}=\frac{\pi_\theta(o_{i,t}\mid q,o_{i,<t})}{\pi_{\theta_{\text{old}}}(o_{i,t}\mid q,o_{i,<t})}$$This is vanilla GRPO/PPO: push up the probability of tokens with positive advantage, clip the update so no single step moves the policy too far, and penalize drifting away from the reference model by coefficient $\beta$. The only thing that changed versus RLVR is where $\hat{A}_i$ came from. No reward model, no verifier, no labels touch this objective.
The single most important implementation detail is which model computes self-certainty. INTUITOR uses online self-certainty: the reward is scored by the current, evolving policy, so the yardstick moves as the model moves. The authors explicitly contrast this with an offline variant that scores confidence with a frozen base model — and show the offline version gets hacked. Around step 100 the offline policy discovers it can inflate its confidence reward by appending an already-solved auxiliary problem to its answer, producing a spike in response length and a drop in accuracy. Because the online evaluator co-evolves with the policy, that exploit stops paying off, and training stays stable. Reward hacking here isn't hypothetical — they watched it happen and the online design is the fix.
Main experiments train Qwen2.5-1.5B and Qwen2.5-3B (base, not instruct) on the MATH training split (7,500 problems), in the Open-R1 framework on A100 40GB GPUs. Defaults: batch of 128 problems per update, group size $G=7$, KL coefficient $\beta=0.005$, learning rate $3\times10^{-5}$, greedy decoding at evaluation. A code-trained variant (Intuitor-Code) uses 3,200 Codeforces problems for 50 steps with $G=14$, $\beta=0.01$, lr $1\times10^{-5}$. Scaling checks run on Qwen2.5-7B/14B, Qwen3-14B (lr $1\times10^{-6}$, 16 samples), plus Llama3.2-3B-Instruct and OLMo-2-7B-SFT to test cross-family transfer.
The headline is a two-part claim: INTUITOR ties GRPO in-domain (math) and beats it out-of-domain (code), using no answers or tests. The main comparison on Qwen2.5-3B, all numbers from the paper (accuracy as a fraction; AlpacaEval is length-controlled win rate in %):
| Qwen2.5-3B | Train data | GSM8K | MATH500 | LiveCodeBench | CRUXEval-O | MMLU-Pro | AlpacaEval |
|---|---|---|---|---|---|---|---|
| Base | — | 0.673 | 0.544 | 0.093 | 0.236 | 0.377 | 3.72 |
| GRPO (verifier) | MATH | 0.826 | 0.636 | 0.085 | 0.341 | 0.403 | 6.91 |
| GRPO-PV | MATH | 0.820 | 0.636 | 0.086 | 0.299 | 0.398 | 6.17 |
| Intuitor | MATH | 0.792 | 0.612 | 0.153 | 0.416 | 0.379 | 7.10 |
| Intuitor-Code | Codeforces | 0.743 | 0.572 | 0.153 | 0.411 | 0.386 | 4.16 |
The single most striking result: train on math with no external signal, and code generation improves more than it does under a real verifier. LiveCodeBench went 0.093 → 0.153 under INTUITOR (a 65% relative gain) while GRPO actually regressed to 0.085. On CRUXEval-O (code reasoning) INTUITOR gained +76% (0.236 → 0.416) versus GRPO's +44% (→ 0.341). In math itself the two methods are effectively tied (MATH500 0.612 vs 0.636; GSM8K 0.792 vs 0.826) — GRPO is slightly ahead in-domain, as you'd expect from a method that sees the answers, but INTUITOR closes most of the gap with none. It also wins on instruction-following (AlpacaEval 7.10 vs 6.91).
The effect shows up fastest and most dramatically on the weak 1.5B base, which starts at 0.002 on GSM8K — a formatting/instruction failure, not a knowledge one. After INTUITOR it reaches 0.711 (GRPO: 0.747), and at just 10 training steps INTUITOR already beats GRPO handily (GSM8K 0.152 vs 0.081, MATH 0.368 vs 0.296), suggesting self-certainty is a denser early-training signal than sparse binary rewards.
The result that quietly undercuts the story: on non-Qwen models the advantage nearly evaporates. Llama3.2-3B-Instruct under INTUITOR vs GRPO is a wash (GSM8K 0.723 vs 0.714, MATH 0.476 vs 0.494), and OLMo-2-7B-SFT is a dead heat with tiny absolute gains over base (GSM8K 0.710 vs base 0.691). Hold that thought.
Yes, genuinely — but the framing oversells it, and the crux is a confound the paper half-acknowledges. "Learn to reason without external rewards" is a real, well-executed result and one of the cleaner demonstrations of the intrinsic-signal-replaces-the-grader idea. The engineering is honest: they found the reward hack (offline scorer appends a solved problem), diagnosed it, and fixed it with the online co-evolving evaluator. That's the right instinct and it's the most valuable part of the paper.
Is confidence a real signal or a shortcut to overconfident nonsense? Both, and the boundary is instructive. Optimized naively, self-certainty does collapse — the paper's own ablations show direct optimization, raw log-prob rewards, and entropy minimization all degenerate. What rescues it is not the signal itself but three guardrails: group-relative normalization (you only get credit for being surer than your siblings, so global confidence-inflation is normalized to zero advantage), the online evaluator (a moving target you can't memorize an exploit against), and the KL leash (stay near a sane reference). Pull any one and it hacks. So the honest headline isn't "confidence is a good reward" — it's "confidence is a hackable reward that GRPO's relative machinery happens to tame." The stabilizers are load-bearing, and that should temper how bold you read this.
Why does it generalize OOD better — robustness or style? This is the question that decides the whole paper, and I lean toward the deflationary answer. Two clues. First, the base Qwen2.5-1.5B scores 0.002 on GSM8K — that's not a model that lacks math, it's a model that can't follow the answer format. RL here is overwhelmingly eliciting latent capability and unlocking format, not teaching new correctness. Second, and damning: the gains are Qwen-specific. On Llama and OLMo, INTUITOR ties GRPO and barely beats base. That pattern is precisely what the Spurious Rewards paper (Shao et al., a month later) documented — random and even incorrect rewards produce huge MATH gains on Qwen and nothing on other families, because RL surfaces pretrained "code-reasoning" behavior that Qwen already has. Self-certainty may largely be a well-behaved member of that same "spurious-but-useful on Qwen" family. Under that reading, the superior OOD transfer is real but mundane: self-certainty rewards fluent, confident, well-structured chains at every token (a dense trajectory signal) rather than a sparse terminal 0/1, so it teaches a general "reason confidently" style that transfers to code — whereas GRPO overfits to the math answer key and even hurts code. That's a robustness-of-elicitation win, not evidence the model is learning correctness from within.
Strongest part: a self-reinforcing RL loop with zero external oracle that ties a verifier-based method in-domain and beats it OOD, plus the demonstration that it stacks on top of real rewards. Weakest part: the cross-family collapse, under-emphasized, which suggests the mechanism is capability elicitation on cooperative base models, not domain-general reasoning acquisition. The crux to scrutinize: run it on a model with weak pretrained reasoning where the answer genuinely isn't latent — if self-certainty still improves correctness there, the bold claim survives; if not, this is an elicitation trick with a great cost profile. What I'd want next: the paper's own admission that purely offline training degrades over time is the tell for the recursive-self-improvement question that actually matters — can a model bootstrap indefinitely on its own confidence, or does it need periodic reality contact? For an AGI trajectory where verifiable rewards run out long before capability does, a working intrinsic signal is exactly the right thing to be chasing; I just don't think confidence alone is the one that scales past elicitation, because a model can be confidently, systematically wrong and self-certainty will happily reward it.
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