polystep

Gradient-free neural network training via optimal transport.

PolyStep optimizes neural networks without backpropagation. At each step, it samples polytope vertices around current parameters, evaluates losses via forward passes only, and computes softmax-weighted projections to find descent directions. This enables training models with non-differentiable components - spiking networks, quantized layers, blackbox modules - where gradients are unavailable or undefined.

Based on the Sinkhorn Step algorithm (Le et al., NeurIPS 2023), extended with subspace compression, softmax solver, and convergence guarantees for piecewise-smooth losses.

How it works

1. Sample polytope vertices around current parameters in a compressed subspace
2. Evaluate the loss at each vertex (forward pass only - no gradients)
3. Compute softmax weights over the cost matrix (which directions are good?)
4. Update parameters via barycentric projection from the weighted vertices

_{Training a spiking neural network with hard-threshold LIF spikes. Left: loss landscape in a 2D parameter slice - blocky contours from the spike threshold make the gradient zero almost everywhere. Polytope probes contract toward a basin via softmax-OT. Right: training loss over PolyStep steps.}

Installation

pip install -e .                      # core library (torch + numpy)
pip install -e ".[examples]"          # + torchvision, matplotlib
pip install -e ".[dev]"               # + pytest, ruff (development)
pip install -e ".[experiments]"       # + scipy, pandas, python-sat (paper reproduction)

GPU: pip install torch --index-url https://download.pytorch.org/whl/cu130 (or pick the CUDA build that matches your driver from the PyTorch install page).

Quickstart

Synthetic optimization

import torch
from polystep import PolyStep, Ackley

solver = PolyStep.create(Ackley(dim=10), epsilon=0.5, max_iteration=50)
state = solver.run(torch.randn(100, 10))
print(f"Best cost: {min(state.costs):.4f}")

Neural network training

import torch.nn as nn
from polystep import PolyStepOptimizer, train, TrainConfig, LoggingCallback
from polystep.epsilon import CosineEpsilon
from polystep.hybrid_subspace import HybridSubspace
from polystep.transform import ParamLayout

model = nn.Sequential(nn.Linear(784, 128), nn.ReLU(), nn.Linear(128, 10))

# HybridSubspace compresses the parameter space per-layer
layout = ParamLayout.from_module(model)
subspace = HybridSubspace.from_layout(layout, rank=8)

# Cosine schedules: broad exploration → fine exploitation
optimizer = PolyStepOptimizer(
    model, subspace=subspace, solver="softmax",
    epsilon=CosineEpsilon(init=10.0, target=0.1, decay=0.02),
    step_radius=CosineEpsilon(init=5.0, target=1.0, decay=0.008),
    probe_radius=CosineEpsilon(init=10.0, target=2.0, decay=0.016),
)

train(model, train_loader, nn.CrossEntropyLoss(), optimizer, TrainConfig(epochs=5))

See examples/ for 6 runnable demos (SNN, RL, MAX-SAT, MNIST, Loihi 2 on-chip adaptation skeleton).

When to use PolyStep

PolyStep is designed for models where gradients are unavailable or unreliable:

• Spiking neural networks - hard LIF thresholds, discrete spike events
• Quantized layers - int8 weights, binary/ternary networks
• Blackbox modules - external simulators, API-based models, hardware-in-the-loop
• Hard routing - argmax gating, hard mixture-of-experts
• Combinatorial optimization - MAX-SAT, discrete assignment problems

If your model is fully differentiable, Adam/SGD will be faster and more accurate.

Benchmarks

5-seed mean ± std (seeds: 42, 123, 456, 789, 1337). Hardware: NVIDIA RTX 5090.

Non-differentiable tasks (primary contribution)

Task	PolyStep	CMA-ES	OpenAI-ES	SPSA	Non-diff op
SNN/LIF (MNIST)	93.4% ± 0.2	16.2%	33.1%	29.4%	threshold()
Int8 quantized	97.1% ± 0.1	80.7%	78.1%	-	round()
Argmax attention	86.8% ± 0.4	-	-	-	argmax()
Staircase activation	93.2% ± 0.3	-	-	-	floor()
Hard MoE routing	90.7% ± 0.2	62.8%	63.5%	-	argmax()
MAX-SAT 100K vars	98.0% ± 0.01	90.1%	88.9%	-	round()
MAX-SAT 1M vars	92.6% ± 0.02	-	87.8%	-	round()

Differentiable sanity checks

Task	PolyStep	Adam	Architecture
MNIST	96.0% ± 0.1	97.9%	2-layer MLP (101K)
ETTh1 timeseries	MSE 0.121 ± 0.004	MSE 0.187	LSTM (23K)

SNN memory scaling (forward-only vs. BPTT)

Timesteps	PolyStep	BPTT (surrogate)	Savings
T=25	31.8 MB	132 MB	4.2×
T=400	51.6 MB	1,538 MB	29.8×

PolyStep dominates gradient-free competitors by 10–60+ points on non-differentiable tasks. On MAX-SAT, domain-specialized probSAT achieves 99.6% - PolyStep is the best general-purpose gradient-free optimizer.

Features

• Softmax OT solver - 2.8× faster and 5.3× better convergence than Sinkhorn (Sinkhorn also available)
• Subspace compression - HybridSubspace (recommended), AdaptiveSubspace, sparse projection for 100M+ params
• Block-wise OT - per-layer decomposition for memory efficiency
• torch.compile support - GPU acceleration on hot paths
• VmapSafe layers - drop-in attention and LSTM compatible with torch.vmap
• Sub-linear memory - forward-only: no BPTT activation storage (29.8× savings at T=400)
• CMA-ES adaptation - covariance-adapted subspace rotations
• MLP fast path - batched torch.bmm kernels bypass vmap for pure-MLP models (~3× speedup)

Limitations

• Compute cost: ~20M forward passes vs. ~14K gradient steps for Adam on MNIST. Inherent to all zeroth-order methods.
• High-dimensional NLP: Near-random on SST-2 (4.2M params). Gradient-free methods fail at this parameter scale from scratch.
• Adam-surrogate ceiling: On SNN tasks, Adam with surrogate gradients achieves 97.8% vs. PolyStep's 93.4%. The value proposition is zero-engineering, not raw accuracy.

See LIMITATIONS.md for detailed discussion.

Project structure

src/polystep/          Core library (optimizer, solvers, subspaces, geometry)
tests/                 Unit, integration, and regression tests
examples/              6 runnable demos (quickstart, SNN, RL, MAX-SAT, MNIST, Loihi 2)
experiments/           Paper reproduction: runners, results, baselines
docs/                  API overview, reproducibility guide

Documentation

Resource	Description
examples/	6 runnable demos with output figures
experiments/	Full paper reproduction harness
docs/api_overview.md	API reference
LIMITATIONS.md	Known limitations
CONTRIBUTING.md	Contribution guidelines
CHANGELOG.md	Release history

Citation

If you use PolyStep in your research, please cite:

@article{le2026polystep,
  title   = {Training Non-Differentiable Networks via Optimal Transport},
  author  = {Le, An T.},
  journal = {arXiv preprint arXiv:2605.01928},
  year    = {2026},
  url     = {https://arxiv.org/abs/2605.01928}
}

License

Apache License 2.0. See LICENSE.