arXiv：cs.LG· An T. Le·· 13 小时前
PolyStep：用最优传输训练不可微网络
Training Non-Differentiable Networks via Optimal Transport
AI 导读
PolyStep 是一种仅前向的优化器，通过对旋转多面体探针进行评估，并沿探针方向的加权平均移动参数块，权重由单侧熵最优传输导出。在 MNIST 硬阈值脉冲神经元网络上，PolyStep 达到 93.0 ± 0.2%，优于最佳调优的无梯度基线 79.6 ± 5.2%。该方法的查询成本与每步搜索维度成正比，限制了固定预算下可用的更新次数。
正文
Abstract: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.
| Comments: | 95 pages, 29 tables, 8 figures. Published in Transactions on Machine Learning Research (TMLR). Code: this https URL |
| Subjects: | Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE); Robotics (cs.RO); Optimization and Control (math.OC) |
| Cite as: | arXiv:2605.01928 [cs.LG] |
| (or arXiv:2605.01928v4 [cs.LG] for this version) | |
| https://doi.org/10.48550/arXiv.2605.01928 arXiv-issued DOI via DataCite |
Submission history
From: An Thai Le [view email]
[v1]
Sun, 3 May 2026 15:20:56 UTC (546 KB)
[v2]
Thu, 13 Aug 2026 05:19:45 UTC (467 KB)
[v3]
Mon, 21 Sep 2026 15:54:02 UTC (621 KB)
[v4]
Thu, 8 Oct 2026 19:57:13 UTC (621 KB)
来源：arXiv：cs.LG · arxiv.org