# Training Non-Differentiable Networks via Optimal Transport

arXiv: https://arxiv.org/abs/2605.01928 · HTML: https://mlsanity.com/paper/2605.01928/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 84 references that are in the map.
- First posted on arXiv 3 May 2026.

## Builds on (most cited first, 21 of 21)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Estimating or Propagating Gradients Through Stochastic Neurons for Conditional Computation](https://mlsanity.com/paper/1308.3432/) (2013)
- [The Concrete Distribution: A Continuous Relaxation of Discrete Random Variables](https://mlsanity.com/paper/1611.00712/) (2016)
- [Evolution Strategies as a Scalable Alternative to Reinforcement Learning](https://mlsanity.com/paper/1703.03864/) (2017)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Evolution Strategies at Scale: LLM Fine-Tuning Beyond Reinforcement Learning](https://mlsanity.com/paper/2509.24372/) (2025)
- [Evolution Strategies at the Hyperscale](https://mlsanity.com/paper/2511.16652/) (2025)
- [QuZO: Quantized Zeroth-Order Fine-Tuning for Large Language Models](https://mlsanity.com/paper/2502.12346/) (2025)
- [LOTION: Smoothing the Optimization Landscape for Quantized Training](https://mlsanity.com/paper/2510.08757/) (2025)
- [Quantized Evolution Strategies: High-precision Fine-tuning of Quantized LLMs at Low-precision Cost](https://mlsanity.com/paper/2602.03120/) (2026)
- [Online Pseudo-Zeroth-Order Training of Neuromorphic Spiking Neural Networks](https://mlsanity.com/paper/2407.12516/) (2024)
- [End-to-End On-Device Quantization-Aware Training for LLMs at Inference Cost](https://mlsanity.com/paper/2509.00031/) (2025)
- [Function-free Optimization via Comparison Oracles](https://mlsanity.com/paper/2604.26867/) (2026)
- [Beyond Discreteness: Sample Complexity Analysis of Straight-Through Estimator for 1-bit Quantization](https://mlsanity.com/paper/2505.18113/) (2025)
- [FFGAF-SNN: The Forward-Forward Based Gradient Approximation Free Training Framework for Spiking Neural Networks](https://mlsanity.com/paper/2507.23643/) (2025)
- [Low-rank surrogate modeling and stochastic zero-order optimization for training of neural networks with black-box layers](https://mlsanity.com/paper/2509.15113/) (2025)
- [Mean-Field Model for Two-Layer Neural Networks Trained with Consensus-Based Optimization](https://mlsanity.com/paper/2511.21466/) (2025)
- [Signal-Adaptive Trust Regions for Gradient-Free Optimization of Recurrent Spiking Neural Networks](https://mlsanity.com/paper/2601.21572/) (2026)
- [Nonsmooth Optimization with Zeroth Order Comparison Feedback](https://mlsanity.com/paper/2602.05622/) (2026)
- [Orthogonal Weight Modification Enhances Learning Scalability and Convergence Efficiency without Gradient Backpropagation](https://mlsanity.com/paper/2602.22259/) (2026)

## Built on by (most cited first, 1 of 1)

- [PolyStepOR: Learning to Decide Without Optimal Decisions](https://mlsanity.com/paper/2609.32465/) (2026)

## Abstract (arXiv)

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 ± 0.2%, compared with 79.6 ± 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.