The sampler that finds more correct reasoning and returns the wrong answer

Ask a language model the same hard question eight times and its reasoning paths may disagree. Self-consistency groups those paths by final answer and returns the answer with the most support. The intuition is jury-like: several independent routes to the same result can outweigh one confident-looking mistake.

Power Sampling changes the jury before it votes. It raises every complete trajectory probability to an exponent greater than one, then renormalizes. Likely paths become likelier. Yang, Sun, and Ma show the surprise: this can put more total probability on correct reasoning while making the final consensus wrong.

A trajectory is one complete generated solution. If the base model assigns it probability p, Power Sampling with exponent α targets a new distribution proportional to pᵅ. This differs from lowering token temperature: token sampling renormalizes at every prefix, while Power renormalizes over whole completed paths.

def power_target(trajectories, alpha):
    raw = [trajectory["probability"] ** alpha for trajectory in trajectories]
    total = sum(raw)
    return [weight / total for weight in raw]

The implementation is only exponentiation and normalization. The difficult part is deciding what the downstream aggregator needs the new distribution to preserve.

Our exact toy distribution has three different correct paths with masses 0.16, 0.15, and 0.14. Together they win with 0.45. One wrong answer has a single 0.20 path; 35 unrelated wrong answers form a long tail at 0.01 each. These are illustrative probabilities chosen to expose the paper's mechanism, not reported benchmark data.

Coverage mismatch is the counterexample above. Power rewards paths one by one, while self-consistency pools paths that share an answer. Exponentiation can erase the diversity that makes aggregation useful. A reachable correct path is not the same thing as broad support for the correct answer.

Dose mismatch is a second problem. The relative log-weight change between two paths equals (α − 1) times their log-likelihood gap. That gap varies by prompt, so one global exponent can barely move a flat distribution and nearly collapse a steep one. The same α is the same formula, not the same intervention.

Relative-Rank SoftSat replaces raw likelihood with a within-prompt percentile from zero to one. It then bends the gain curve: moderate ranks rise, but gains stop increasing above a chosen rank m. The bounded multiplier preserves more support and makes the dose comparable across prompts.

def softsat(rank, saturation_rank):
    scaled = min(rank / saturation_rank, 1.0)
    return 1.0 - (1.0 - scaled) ** 2

def softsat_weights(trajectories, ranks, beta, saturation_rank):
    raw = [trajectory["probability"] * exp(beta * softsat(rank, saturation_rank))
           for trajectory, rank in zip(trajectories, ranks)]
    total = sum(raw)
    return [weight / total for weight in raw]

The paper implements the target as importance weights on one shared pool of base samples. That avoids extra model calls. Our chart uses the exact discrete probability rank; the deployed finite-pool version estimates rank from the sampled candidates and additionally clips weights around mean one.

SoftSat improves five rows, ties one, and trails uniform consensus in three. Its largest loss is 1.004 points, far smaller than Power's 18.474-point failure. The honest conclusion is therefore narrower than “SoftSat wins”: controlling support removes catastrophic regressions while retaining some gains.

Their separate check is strong but narrow: on 1,118 MoreHopQA cases, weighted and repeatedly sampled Power-SMC agree on 99.55% of answers and differ by 0.089 accuracy points, with a 95% confidence interval from −0.268 to 0.447. Mean inference time falls 7.42×. That validates one finite setting, not universal equivalence.

Next, probe ties in empirical rank, larger candidate pools, alternative answer-normalization rules, and prompts whose likelihood spread changes sharply. Most importantly, report answer-level margins alongside pass@k. The model that supplies these path probabilities is a Transformer; follow the computation on the interactive Transformer page.