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, a softmax solver, and convergence analysis for piecewise-smooth losses.

How it works

1. Sample polytope vertices around the 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.
4. Update the parameters by barycentric projection from the weighted vertices.

Want to play around with parameters? Viet T. Nguyen built a gorgeous interactive walkthrough that animates every step of the method -> explore the PolyStep visualization.

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_iterations=50)
state = solver.run(torch.randn(100, 10))
print(f"Best cost: {min(state.costs):.4f}")

Neural network training

import torch
import torch.nn as nn
from torch.utils.data import DataLoader, TensorDataset

from polystep import PolyStepOptimizer, train, TrainConfig
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))

# Replace this with your real DataLoader.
train_loader = DataLoader(
    TensorDataset(torch.randn(1024, 784), torch.randint(0, 10, (1024,))),
    batch_size=64,
)

# 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))

Drop-in gradient-free optimizer (ask/tell)

PolyStep also exposes the ask/tell interface used by evolution-strategy libraries, so it drops into ES benchmark harnesses (evosax, NeuroEvoBench) and any black-box loop:

import torch
from polystep import PolyStepES

es = PolyStepES(dim=20, epsilon=0.1, step_radius=0.3, x0=torch.full((20,), 2.0))
for _ in range(300):
    candidates = es.ask()           # (popsize, dim) points to evaluate
    es.tell(objective(candidates))  # lower fitness is better
print(es.best_fitness, es.mean)

experiments/bench_ask_tell.py compares it head-to-head with a Gaussian ES on the standard synthetic suite under a matched evaluation budget.

See examples/ for runnable demos covering SNN, RL, MAX-SAT, MNIST, a Loihi 2 on-chip adaptation skeleton, STE-free binary-net training vs OpenAI-ES, and direct F1 minimization where PolyStep beats both a biased gradient (Adam+STE) and OpenAI-ES.

Spiking net (hard LIF thresholds) trained with forward passes only

CartPole policy search: no value function, no gradients

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

Best accuracy (%), 5 seeds. Bold marks the best method on each row. Adam is gradient-based (backprop on a smoothed surrogate), so it is an unfair reference upper bound, not a gradient-free peer of PolyStep, CMA-ES, OpenAI-ES, and SPSA; it is shown only to bound how far the gradient-free methods sit from a gradient method. A dash means the run is not in this release.

Task	PolyStep	Adam (surrogate)	CMA-ES	OpenAI-ES	SPSA	Non-diff op
SNN/LIF (MNIST)	93.4 ± 0.2	80.5	16.2	33.1	29.4	threshold()
Int8 quantized	97.1 ± 0.1	98.1	80.7	78.1	91.2	round()
Argmax attention	86.8 ± 0.4	89.1	72.6	75.7	77.7	argmax()
Staircase activation	93.2 ± 0.3	97.6	72.8	85.5	49.3	floor()
Hard MoE routing	90.7 ± 0.2	-	62.8	63.5	69.3	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

Adam here is gradient-based (backprop), an unfair reference rather than a peer; PolyStep stays gradient-free and forward-only. These rows only check that PolyStep lands close to a gradient method when gradients do exist.

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	4.2x
T=400	51.6 MB	1,538 MB	29.8x

Among gradient-free optimizers PolyStep is the strongest on every task here, at 100 to 10,000x more evaluations than the ES and SPSA baselines. Against the gradient surrogate, PolyStep wins where the non-differentiability is hard (SNN hard LIF, hard MoE routing) and loses where an accurate smooth surrogate exists (int8, argmax, staircase), so its niche is hard non-differentiability rather than non-differentiability in general. On MAX-SAT the domain-specialized probSAT and WalkSAT-style SLS solvers beat every general optimizer (probSAT reaches about 99.6% at 100K variables and 98.9% at 1M); PolyStep is the strongest general-purpose gradient-free method here, not a replacement for a domain solver.

Features

• Softmax OT solver with an entropic Sinkhorn alternative.
• Subspace compression: HybridSubspace (recommended), AdaptiveSubspace, and sparse projection for very large models.
• Block-wise OT for per-layer decomposition.
• torch.compile opt-in on hot paths.
• Vmap-safe layers: drop-in attention and LSTM that play nicely with torch.vmap.
• Sub-linear memory: forward-only evaluation, no BPTT activation tape (~30x savings at long SNN horizons).
• CMA-ES inspired adaptation of subspace covariance (experimental, monolithic step only; use_adaptive_radius is the stable default).
• MLP fast path using batched torch.bmm instead of vmap for pure-MLP nn.Sequential models.
• Ask/tell API: PolyStepES drops into evosax / NeuroEvoBench-style ES harnesses and black-box loops.

Limitations

• Compute cost. Roughly tens of millions of forward passes (on the SNN benchmark, around 30M) vs. tens of thousands of Adam gradient steps for the same MNIST accuracy. This is inherent to zeroth-order methods.
• High-dimensional NLP. Near-random accuracy on SST-2 (4.2M parameters trained from scratch). Gradient-free methods do not scale to this regime in our experiments.
• Adam baseline. On a smoothed surrogate, Adam beats PolyStep on int8 (98.1 vs 97.1), argmax (89.1 vs 86.8), and staircase (97.6 vs 93.2), and only loses where the non-differentiability is hard (SNN hard LIF 93.4 vs 80.5; hard MoE routing). The stronger surrogate-gradient / BPTT baseline for SNNs (paper §5.3) is not bundled with this release; see the arXiv preprint.

See LIMITATIONS.md for the full discussion.

Project structure

src/polystep/          Core library (optimizer, solvers, subspaces, geometry)
tests/                 Unit, integration, and regression tests
examples/              8 runnable demos (quickstart, SNN, RL, MAX-SAT, MNIST, Loihi 2, STE-free binary net, direct loss minimization)
experiments/           Paper reproduction: runners, results, baselines
docs/                  API overview, reproducibility guide

Documentation

Resource	Description
examples/	8 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 find this work useful, please consider citing:

@article{le2026training,
  title={Training Non-Differentiable Networks via Optimal Transport},
  author={Le, An T},
  journal={arXiv preprint arXiv:2605.01928},
  year={2026}
}

Acknowledgments

A huge thank you to Viet for building a beautiful interactive PolyStep visualization, it brings the method to life and makes every step click!

License

Apache License 2.0. See LICENSE.