CKKS depth-2 (w·x+b)² circuit: adversarial search finds max-error ~2e-3 (5/5 seeds)

[BAder82t]

CKKS depth-2 (w·x+b)² circuit: adversarial search finds max-error ~2e-3 (5/5 seeds)

Summary

On a depth-2 CKKS circuit, TenSEAL 0.3.16 returns max-errors around
2e-3 under worst-case inputs found by adversarial search
(CMA-ES). Random testing does not surface this; the failing inputs
sit in narrow regions of the input space.

Setup

• Circuit: f(x) = (w · x + b)², depth 2
• Inputs: x ∈ [-3, 3]^8
• CKKS params: ring N = 16384, coeff-mod chain
  [60, 40, 40, 40, 40, 60], scale 2^40
• Search: AutoOracle (CMA-ES + landscape probe), 500-eval budget,
  threshold 1e-2, seeds 41–45
• Smoke test: scalar f([0.5]·8) agrees with the plaintext
  reference to 6 decimal places

Reproduction

pip install fhe-oracle tenseal cma numpy
python benchmarks/library_comparison.py \
    --circuit unified-squared-dot --libs tenseal \
    --seeds 41 42 43 44 45

Source: https://github.com/BAder82t/fhe-oracle-oss

Results (3 independent runs)

Run	Fail rate	Median max-err
Mac native	5/5	2.04e-03
Docker linux/amd64	5/5	2.11e-03
Docker linux/amd64 (Apr 21)	5/5	2.20e-03

Per-seed CSV: library_comparison_unified_ckks.csv in the linked
repo.

Hypothesis

A plausible cause is rescale strategy — deferred rescales on
depth-2 squaring may accumulate more scale-mismatch noise on
adversarial inputs near the domain extremes (search consistently
finds witnesses with |x| near the bound). This is a guess, not
certainty.

What might help

• Confirmation of whether this matches expected behaviour for
  TenSEAL's default rescale configuration.
• Documented best-practice for depth-2+ CKKS circuits (explicit
  rescale calls, alternative presets).
• A regression-style CI test using fhe-oracle (example available on
  request).

Refs

• Repo: https://github.com/BAder82t/fhe-oracle-oss
• Benchmark: benchmarks/library_comparison.py
• Raw CSV: benchmarks/results/library_comparison_unified_ckks.csv