Haitham Bou-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.
The $\mathbf{Y}$-Combinator for LLMs: Solving Long-Context Rot with $λ$-Calculus
LLMs are increasingly used as general-purpose reasoners, but long inputs remain bottlenecked by a fixed context window. Recursive Language Models (RLMs) address this by externalising the prompt and recursively solving subproblems. Yet existing RLMs depend on an open-ended read-eval-print loop (REPL) in which the model generates arbitrary control code, making execution difficult to verify, predict, and analyse. We introduce $λ$-RLM, a framework for long-context reasoning that replaces free-form recursive code generation with a typed functional runtime grounded in $λ$-calculus. It executes a compact library of pre-verified combinators and uses neural inference only on bounded leaf subproblems, turning recursive reasoning into a structured functional program with explicit control flow. We show that $λ$-RLM admits formal guarantees absent from standard RLMs, including termination, closed-form cost bounds, controlled accuracy scaling with recursion depth, and an optimal partition rule under a simple cost model. Empirically, across four long-context reasoning tasks and nine base models, $λ$-RLM outperforms standard RLM in 29 of 36 model-task comparisons, improves average accuracy by up to +21.9 points across model tiers, and reduces latency by up to 4.1x. These results show that typed symbolic control yields a more reliable and efficient foundation for long-context reasoning than open-ended recursive code generation. The complete implementation of $λ$-RLM, is open-sourced for the community at: https://github.com/lambda-calculus-LLM/lambda-RLM.
Why the Brain Consolidates: Predictive Forgetting for Optimal Generalisation
Standard accounts of memory consolidation emphasise the stabilisation of stored representations, but struggle to explain representational drift, semanticisation, or the necessity of offline replay. Here we propose that high-capacity neocortical networks optimise stored representations for generalisation by reducing complexity via predictive forgetting, i.e. the selective retention of experienced information that predicts future outcomes or experience. We show that predictive forgetting formally improves information-theoretic generalisation bounds on stored representations. Under high-fidelity encoding constraints, such compression is generally unattainable in a single pass; high-capacity networks therefore benefit from temporally separated, iterative refinement of stored traces without re-accessing sensory input. We demonstrate this capacity dependence with simulations in autoencoder-based neocortical models, biologically plausible predictive coding circuits, and Transformer-based language models, and derive quantitative predictions for consolidation-dependent changes in neural representational geometry. These results identify a computational role for off-line consolidation beyond stabilisation, showing that outcome-conditioned compression optimises the retention-generalisation trade-off.
Decoding as Optimisation on the Probability Simplex: From Top-K to Top-P (Nucleus) to Best-of-K Samplers
Decoding sits between a language model and everything we do with it, yet it is still treated as a heuristic knob-tuning exercise. We argue decoding should be understood as a principled optimisation layer: at each token, we solve a regularised problem over the probability simplex that trades off model score against structural preferences and constraints. This single template recovers greedy decoding, Softmax sampling, Top-K, Top-P, and Sparsemax-style sparsity as special cases, and explains their common structure through optimality conditions. More importantly, the framework makes it easy to invent new decoders without folklore. We demonstrate this by designing Best-of-K (BoK), a KL-anchored coverage objective aimed at multi-sample pipelines (self-consistency, reranking, verifier selection). BoK targets the probability of covering good alternatives within a fixed K-sample budget and improves empirical performance. We show that such samples can improve accuracy by, for example, +18.6% for Qwen2.5-Math-7B on MATH500 at high sampling temperatures.
Multi-Task GRPO: Reliable LLM Reasoning Across Tasks
RL-based post-training with GRPO is widely used to improve large language models on individual reasoning tasks. However, real-world deployment requires reliable performance across diverse tasks. A straightforward multi-task adaptation of GRPO often leads to imbalanced outcomes, with some tasks dominating optimization while others stagnate. Moreover, tasks can vary widely in how frequently prompts yield zero advantages (and thus zero gradients), which further distorts their effective contribution to the optimization signal. To address these issues, we propose a novel Multi-Task GRPO (MT-GRPO) algorithm that (i) dynamically adapts task weights to explicitly optimize worst-task performance and promote balanced progress across tasks, and (ii) introduces a ratio-preserving sampler to ensure task-wise policy gradients reflect the adapted weights. Experiments on both 3-task and 9-task settings show that MT-GRPO consistently outperforms baselines in worst-task accuracy. In particular, MT-GRPO achieves 16-28% and 6% absolute improvement on worst-task performance over standard GRPO and DAPO, respectively, while maintaining competitive average accuracy. Moreover, MT-GRPO requires 50% fewer training steps to reach 50% worst-task accuracy in the 3-task setting, demonstrating substantially improved efficiency in achieving reliable performance across tasks.
Scalable Power Sampling: Unlocking Efficient, Training-Free Reasoning for LLMs via Distribution Sharpening
Reinforcement learning (RL) post-training is a dominant approach for improving the reasoning performance of large language models (LLMs), yet growing evidence suggests that its gains arise primarily from distribution sharpening rather than the acquisition of new capabilities. Recent work has shown that sampling from the power distribution of LLMs using Markov chain Monte Carlo (MCMC) can recover performance comparable to RL post-training without relying on external rewards; however, the high computational cost of MCMC makes such approaches impractical for widespread adoption. In this work, we propose a theoretically grounded alternative that eliminates the need for iterative MCMC. We derive a novel formulation showing that the global power distribution can be approximated by a token-level scaled low-temperature one, where the scaling factor captures future trajectory quality. Leveraging this insight, we introduce a training-free and verifier-free algorithm that sharpens the base model's generative distribution autoregressively. Empirically, we evaluate our method on math, QA, and code tasks across four LLMs, and show that our method matches or surpasses one-shot GRPO without relying on any external rewards, while reducing inference latency by over 10x compared to MCMC-based sampling.
A Brain-like Synergistic Core in LLMs Drives Behaviour and Learning
The independent evolution of intelligence in biological and artificial systems offers a unique opportunity to identify its fundamental computational principles. Here we show that large language models spontaneously develop synergistic cores -- components where information integration exceeds individual parts -- remarkably similar to those in the human brain. Using principles of information decomposition across multiple LLM model families and architectures, we find that areas in middle layers exhibit synergistic processing while early and late layers rely on redundancy, mirroring the informational organisation in biological brains. This organisation emerges through learning and is absent in randomly initialised networks. Crucially, ablating synergistic components causes disproportionate behavioural changes and performance loss, aligning with theoretical predictions about the fragility of synergy. Moreover, fine-tuning synergistic regions through reinforcement learning yields significantly greater performance gains than training redundant components, yet supervised fine-tuning shows no such advantage. This convergence suggests that synergistic information processing is a fundamental property of intelligence, providing targets for principled model design and testable predictions for biological intelligence.