学习到的扩散提议何时有助于约束求解？针对连续代数系统的对照研究
When Do Learned Diffusion Proposals Help Constraint Solving? A Controlled Study on Continuous Algebraic Systems
AI总结：
本研究针对连续代数系统开展对照实验，发现学习到的扩散提议仅在高维、变量未耦合的场景下，才比随机多起点更有助于约束求解，且其优势并非适用于所有真实系统。
AI中文摘要：
求解连续代数约束系统需要两个决策：哪些值满足约束，以及哪种结构增强能使不可解系统变得可解。经典求解器能很好地回答第一个问题，但第二个问题仅通过枚举实现。在该离散决策上，选择K种增强方式的候选条件修复排序器在仅需少量调用时就达到了穷举搜索的上限，表现优于随机方法（平衡非线性菜单准确率为0.997 vs 0.236；p < 10^-70；各随机种子下为0.982 ± 0.006），且在准确率和成本上优于预算匹配的每候选探测方法。MARC将此类系统转化为因子图，基于该图，图神经网络扩散去噪器会提出赋值，对精确计算机代数能量的下降操作会优化这些赋值，而精确符号检查器会验证解。基于扩散的提议评估很少包含一项对照：相同优化预算下的随机多起点方法。将其应用于我们的系统后，它大幅削弱了学习到的提议在值决策上的贡献。它在选择满足赋值方面是否优于随机多起点？仅在可预测的范围内勉强胜出。在陷入低维族时，它与随机重启表现相当，但在高维（随机搜索在此失效）时占优。一旦变量耦合，优势就消失了。由于所有方法共享一次优化和一次检查，K次随机多起点的最优解成功概率恰好为1 - (1 - q(n))^K，其中q(n)是单起点可达性；一个实测常数（无自由参数）可复现整个曲线（平均绝对误差0.012）。有利范围并非我们合成族所特有：在机器人学、定位、优化和代数领域的8个真实系统中，经典多起点求解了全部8个，而在学习有利范围内的系统中，没有一个被求解。我们明确了学习到的提议改进求解器的适用范围。
英文摘要：
Solving a continuous algebraic constraint system requires two decisions: which values satisfy the constraints, and which structural augmentation renders an unsolvable system solvable. Classical solvers answer the first well and the second only by enumeration. On that discrete decision, a candidate-conditioned repair ranker choosing among K augmentations reaches the exhaustive-search ceiling at a fraction of the calls, outperforming random (0.997 vs 0.236 balanced nonlinear menu accuracy; p < 10^-70; 0.982 +/- 0.006 across seeds) and beating a budget-matched per-candidate probe on accuracy and cost. MARC turns such a system into a factor graph, over which a graph-neural diffusion denoiser proposes assignments, descent on an exact computer-algebra energy polishes them, and an exact symbolic checker certifies solutions. Evaluations of diffusion-based proposals rarely include one control: random multi-start under the same refinement budget. Applied to our system, it sharply curtails what the learned proposal contributes on the value decision. Does it beat random multi-start at choosing satisfying assignments? Only narrowly, in a predictable regime. Across trapped low-dimensional families it ties with random restart, but dominates in high dimension, where random search fails. Once variables couple, the advantage is gone. Since all methods share one polish and one checker, best-of-K random multi-start succeeds with probability exactly 1 - (1 - q(n))^K, where q(n) is single-start reachability; one measured constant, with no free parameters, reproduces the entire curve (mean absolute error 0.012). The favorable regime is not specific to our synthetic families: across eight real-world systems in robotics, positioning, optimization, and algebra, classical multi-start solved all eight, none in the learning-favorable regime. We map the regimes in which learned proposals improve solvers.
