基于最优传输训练不可微网络
Training Non-Differentiable Networks via Optimal Transport
AI总结：
该研究提出PolyStep方法，通过最优传输框架利用前向传播训练不可微网络，在脉冲网络、MAX-SAT等任务上优于多数无梯度基线，仅前向传播即可实现子空间平稳性。
摘要：
我们对存在跳变的损失函数进行优化：脉冲阈值、量化层和离散路由会在前向传播中引入跳变，而反向传播无法应用于此。有限差分法失效：在导数估计半径处，量化网络上99.5%的探测对的损失值完全相同，而平滑对照组仅为1.6%。在跳变点处，Clarke和保守平稳性未定义，且任意半径h的平滑操作会使梯度陡增至1/h，因此不存在消失半径可定义导数。固定分辨率平稳性仍可成立。PolyStep仅通过前向传播即可达到该平稳性：在旋转多面体上对探测点排序，并沿其方向的softmax加权平均值步进，这是KL惩罚传输程序λ=0的端点。第二个不可能性条件确定了框架：在宽于探测范围的平台上，每条重新加权成本行的规则（包括softmax）都会冻结，仅列约束可逃脱。对于每个有界可测目标，探测方向的成本加权平均值恰好是平滑损失上的梯度步，且无需任何探测点跨越跳变集。经过T步后，在内部最优探测半径的偏差下限下，它以O(T^-(1/2-γ))的速率达到子空间平稳性，在损失为Lipschitz的地方升级为Goldstein平稳性。在匹配优化器步数下，PolyStep在6种架构与6种无梯度基线的36项对比中领先。它在困难的LIF脉冲网络上达到93.0%的准确率，而最佳调优基线为79.6%；在百万变量MAX-SAT问题上满足92.6%的子句，而随机基准为87.5%。匹配总评估次数则反转了argmax注意力和一个MAX-SAT规模，使困难的MoE持平。每子空间维度的一步成本为一次前向传播，使PolyStep在420万参数从头训练时处于随机水平；在存在梯度的地方，Adam更快更准确。
英文摘要：
Hard thresholds, quantization, and discrete routing can produce training losses with flat regions and jumps, where ordinary gradients vanish or are undefined. We introduce PolyStep, a forward-only optimizer that evaluates rotated polytope probes and moves parameter blocks along weighted averages of the probe directions. We derive the weights from one-sided entropic transport and use its uncoupled softmax solution in our primary experiments. Our analysis explains when variation among probe costs produces motion and when that motion decreases the loss. On a regular simplex, nonconstant costs always give a nonzero direction. For monotone ridge losses, the softmax update cannot increase the loss at any positive temperature; a perturbation bound gives sufficient conditions for descent near curved jumps. For bounded measurable losses, we randomize the probe radii and identify an exact smoothing whose gradient equals the expected linear cost-weighted direction up to scale. This identity yields a stationarity bound for an idealized fixed-temperature variant: under regularity and sampling assumptions stronger than those met by our trained configurations, the bound has an $O(T^{-1/2})$ term and a persistent bias floor. We evaluate the practical method on networks with hard operations, discrete optimization, and policy search. On MNIST with hard-threshold spiking neurons, PolyStep reaches $93.0 \pm 0.2\%$, compared with $79.6 \pm 5.2\%$ for the best-tuned gradient-free baseline at matched evaluations. These gains come with a query cost proportional to the search dimension per fresh step, which limits the number of updates available at a fixed budget.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
