H. Ammar
Publications
Risk-Controlled Lean-as-Judge for Natural-Language Mathematical Reasoning
Lean is increasingly used to judge natural-language mathematical answers, but its signal is partial: many answers never formalize, and a failed proof may reflect an ill-typed statement or a missing library fact, not a wrong answer. On MATH-500 we show this signal is (i) sharply coverage-dependent, that is the proof-winning answer is correct 96% of the time at high proved coverage but 20% at low, and (ii) sparse and often unfaithful: a 7B autoformalizer proves a class for only 28% of problems, and a manual audit finds only approximately 43% of those proofs faithful. We propose COVCAL, a selector over Lean-trace diagnostics that certifies a finite-sample selective-risk bound on accepted answers or abstains, under two regimes (a conservative Bonferroni bound and a tighter dev-then-cal rule). Feasibility depends on autoformalization coverage: with the 7B formalizer the signal is too sparse and Bonferroni abstains on all 20 bootstrap partitions, whereas a prover-specialized formalizer reaches 79% coverage and flips it to feasible on 17 of 20, accepting approximately 48% of problems at 0.98 accepted accuracy. Since self-consistency alone is already 91% accurate, our contribution is a precise account of when, and with which formalizer, a partial formal signal can be trusted under risk control.
The Model Knows, the Decoder Finds: Future Value Guided Particle Power Sampling
A recurring pattern in "reasoning without training" is that base LLMs already assign non-trivial probability mass to correct multi-step solutions; the bottleneck is locating these modes efficiently at inference time. Power sampling provides a principled way to bias decoding toward such modes by targeting p_theta(x)^alpha with alpha > 1, but practical approximations must account for future-dependent correction factors that determine which prefixes remain promising. We introduce Auxiliary Particle Power Sampling (APPS), a blockwise particle algorithm for approximating the sequence-level power target with a bounded population of partial solutions. APPS propagates hypotheses in parallel using proposal-corrected power reweighting and refines their survival through future-value-guided selection at resampling boundaries. This redistributes finite compute across competing prefixes rather than committing to a single unfolding path, while providing a direct scaling knob in the particle count and predictable peak memory. We instantiate the future-value signal with short-horizon rollouts and also study an amortized variant that replaces rollouts with a lightweight learned selection head. Across reasoning benchmarks, APPS improves the accuracy-runtime trade-off of training-free decoding and suggests that part of the gap to post-trained systems can be recovered through more faithful inference-time power approximation.