Yiheng Su, Emmanouil-Vasileios Vlatakis-Gkaragkounis
Recent work has shown that, in zero-sum games, an optimizer can extract $Ω(\sqrt{T})$ strategic surplus---cumulative payoff above the minimax benchmark---against noiseless no-regret learners with a constant step size. However, in practice, learners often act on perturbed payoff scores and use decaying learning rates $η_t$. Since existing extraction mechanisms rely on exact scores, a natural question is whether noise shields the learner from exploitation and eliminates surplus altogether. We address this question for the widely used class of Follow-the-Regularized-Leader (FTRL) algorithms. Under standard regularity conditions, we establish three results, none of which assumes independence of the noise. First, under summability conditions, passive play of a strategy fixed independently of the horizon earns at most constant surplus. Second, we design a new mechanism that exploits the learner's noise-dependent response: a clairvoyant optimizer who knows the update rule and the limiting mean response secures $Ω(1/η_T)$ surplus whenever that mean is profitable. Third, even when that mean is not profitable, active alternating play extracts $Ω(\sum_{t<T}η_t)$ surplus by exploiting the learner's movement rather than its location, under suitable responsiveness and stability conditions. Finally, at the canonical learning rate $η_t\propto t^{-1/2}$, we recover the noiseless $Ω(\sqrt{T})$ rate for mean-zero, finite-variance perturbations---whether fresh with arbitrary temporal dependence or accumulated from i.i.d. increments---provided the limiting mean response is profitable.
Open-weight large language models (LLMs) can generate function-level programs from natural-language prompts, but plausible candidates still fail on hidden semantics and repeat mistakes across repair attempts. We present ReMCTS, an execution-grounded, memory-augmented, LLM-guided MCTS-style search framework. It organizes program candidates as tree states, retains branch-local debugging context, retrieves failure experience across branches, and distinguishes failed checks from unavailable evidence. On HumanEval and MBPP-Sanitized, visible-test ReMCTS improves over direct generation in 8 of 10 model-dataset pairs under held-out evaluation, whereas proxy-only search is less stable. Controlled tree-search, sampling, repair, and memory ablations characterize the source and limits of these gains. A 30-task HumanEval-X C++ pilot further demonstrates compatibility with compiler-backed execution, but does not constitute a broad multilingual evaluation.
Yiheng Su, Emmanouil-Vasileios Vlatakis-Gkaragkounis
We study the computational complexity of clairvoyant strategy optimization against a known online learner in finite-horizon repeated games. In general-sum games with bounded payoffs, unless $\mathsf{P}=\mathsf{NP}$, no polynomial-time optimizer can achieve a reward within $cT$ of the reward of the optimal pure-action sequence for an absolute constant $c>0$, even when allowed to output explicitly represented mixed strategies at every round. Our hardness result strengthens prior work along two complementary axes: it extends to a broader class of learning dynamics, including widely used mean-based algorithms, answering an open question of Assos, Dagan, and Rajaraman (2025) for learners beyond multiplicative weights; and it permits mixed outputs while retaining the weaker pure-sequence benchmark. Thus, neither mixed strategies nor algorithmic specificity removes the computational obstruction in general-sum games. In sharp contrast, zero-sum games exhibit the reverse phenomenon: deterministic pure actions suffice for efficiently recovering the leading surplus guarantees previously obtained through mixed play against Follow-the-Regularized-Leader (FTRL) opponents. Thus, the two regimes form a striking mirror image: mixed play cannot overcome general-sum hardness even against a pure benchmark, whereas pure play retains the full surplus guarantees of mixed strategies in zero-sum games.