SoftOpt

A standalone optimizer that excels against leading market optimizers like Adam, in problems with a known, differentiable computation graph.

Each step is an update followed by an exact Newton correction computed from your own model, through Klein–Maimon soft-number calculus (Foundations of Soft Logic, Klein & Maimon, Springer 2024). Free, open source, and runs locally.

Is SoftOpt the right tool for your problem?

One question decides it, and you can answer it without running anything.

Can you get a residual vector? That is: can you compute model(theta) - data and get back an array, rather than a single number?

• Yes → use scipy.optimize.least_squares. It builds the full Jacobian and exploits the least-squares structure, and it will beat SoftOpt on both accuracy and speed. This holds regardless of how large or how noisy the problem is. We measured it: on solar-cell parameter extraction, least_squares reached 7.9 mA against SoftOpt's 37.3 mA, and did it 21x faster.

• No → SoftOpt is built for this case. Each evaluation returns one number, and it is noisy: a quantum measurement, a laser-tracker sweep, a backtest, a simulator that returns a score. There is no residual vector for least_squares to work with, and finite-difference methods like L-BFGS-B end up differentiating the noise rather than the model.

SoftOpt's advantage comes from taking the derivative from your model instead of from your measurements. That only matters when the measurements are the noisy part. Every validated domain below is of the second kind; that is not a convenience of selection, it is where the mechanism applies.

Where it helps

If your problem has a known computation graph — a quantum circuit, a physical simulator, a projection or measurement model, anything you can write down exactly, even if the measurements of it are noisy — SoftOpt computes an exact directional derivative of that model on every step, plus the curvature along the same direction, and uses them to correct the optimizer's trajectory. Validated, with real hardware-noise-model data, across:

• Quantum chemistry (VQE) — H₂, H₄, BeH₂, HeH⁺, and larger multireference molecules
• Quantum control (GRAPE) — the single cleanest result across every domain tested
• Computer vision — multi-camera bundle adjustment / camera calibration
• Finance — portfolio optimization (Markowitz mean-variance)
• Condensed-matter physics — quasicrystal and spin-chain models (Ising, XY, Heisenberg, SSH, Kitaev)
• Pharmacokinetics, logistic regression, and more

Where it does not help

Full reinforcement learning (or anything else with a continuously moving target — a policy, an adversary, a non-stationary distribution) is outside SoftOpt's validated scope. The mechanism needs a fixed objective to compute a meaningful correction against; a moving target breaks that assumption. Use plain Adam/SGD there.

Installation

pip install softopt

Quick start

import numpy as np
from softopt import SoftOpt, soft_compile

# Write your model once, in plain Python. No special arithmetic.
def my_model(theta):
    x, y = theta
    return (x - 2.0)**2 + np.exp(y) * x

# soft_compile turns it into the exact directional-derivative function
# SoftOpt needs, and verifies the result against your own model.
g_delta = soft_compile(my_model, n_params=2)

opt = SoftOpt(2, g_delta, lr=0.02)

for step in range(num_steps):
    grad_estimate = my_gradient_estimate(theta)   # e.g. from SPSA on real hardware
    theta = opt.step(theta, grad_estimate)

Your model needs to be built from arithmetic and the elementary functions the soft algebra defines: + - * / ** (including negative and fractional exponents, and 2.0 ** x), abs, sqrt, exp, log, log10, sin, cos, tanh, comparisons, and numpy's versions of those. Loops and if/else branching on parameter values work too — the compiled derivative is then valid on the branch your parameters are actually in.

The one thing to avoid is converting a traced value back to a plain number mid-model — float(x), or np.array(params) on the parameter list, both silently discard the derivative. soft_compile's verification catches this and says so.

If you'd rather write the derivative function yourself — because your model lives in a simulator SoftOpt can't trace, or because you want the speed of a hand-tuned implementation — pass any g_delta(theta, delta) that returns the exact directional derivative, and skip soft_compile entirely.

Validated results

All results below use IBM's FakeFez noise model via Qiskit + Aer, or realistic finite-sample/measurement noise for the non-quantum domains, with torch.optim.Adam as the baseline. Improvement is the reduction in gap to the known optimum (or, for Portfolio, the reduction in loss).

Quantum chemistry (VQE)

Quantum control

Condensed-matter & spin models

Computer vision

Finance

See benchmarks/ for the exact, runnable scripts behind every one of these numbers, including the raw per-seed results.

A performance table cannot show why a method wins. The ablation runs the optimizer's own code path while replacing the genuine soft-computed curvature with a frozen constant, or with real curvature measured at an unrelated point — everything else held identical.

On GRAPE, 1000 seeds per arm:

A correction fed a non-genuine value does not merely help less — it lands about 1600× worse than applying no correction at all. Run it yourself:

cd benchmarks/grape && python3 causal_ablation.py --seeds 1000
python3 tests/test_causal.py          # the same test, smaller and faster

Two interfaces to the same algebra

SoftNumber gives you operator syntax (x * y, x.exp()) and is what soft_compile traces your model with. The tuple functions (smul, sadd, ssin, ...) are the same operations applied to whole numpy arrays at once, which is what the benchmark engines use — a quantum circuit propagates 2ⁿ amplitudes per gate, and one Python object per amplitude would not be workable.

They are the same algebra and produce identical numbers; tests/test_causal.py asserts that agreement to 1e-15 across every operation, and also asserts that soft-number operations actually fire during an optimizer step.

The math

The full soft-number algebra is available from the package (from softopt import SoftNumber, sadd, smul, ...), each operation cited to its exact source in Foundations of Soft Logic (Klein & Maimon, Springer 2024):

SoftNumber wraps these with operator syntax (x + y, x * y, x ** 3, x.sqrt()) and is verified, axiom by axiom, against the book's own proofs that bridge numbers form an abelian group under addition and a ring under both operations (tests/test_soft_number.py).

What SoftOpt is not

• Not a claim to beat specialized full-Jacobian second-order solvers (Levenberg–Marquardt, L-BFGS) where those are already practical — SoftOpt's validated niche is genuine improvement over first-order optimizers (Adam, SGD) already in use, particularly where switching to a full second-order method isn't practical (embedded in a larger pipeline, high dimensionality, or measurement noise).
• Not a general-purpose black-box optimizer — it requires a known computation graph, as described above.

License

MIT. See LICENSE.