kpshinnik/repro-fw-lower-bounds

Reproduction bundle — Lower Bounds for Frank-Wolfe on Strongly Convex Sets

Independent reproduction of ICML 2026 paper #18157 — Lower Bounds for Frank–Wolfe on Strongly Convex Sets (Halbey, Deza, Zimmer, Roux, Stellato, Pokutta), arXiv:2602.04378, OpenReview 3KX1xU2bCC.

This is a theory/computation paper: a constructive Ω(1/√ε) lower bound for Frank–Wolfe. Reproduction is a self-contained Python re-implementation of the paper's dynamics, backward-reconstruction construction, and lower-bound inequalities, cross-checked against the authors' Julia code. No GPU is applicable; the substantive experiment is high-precision (mpmath, up to 60,000-bit) serial arithmetic, run at the paper's headline scale (10,000-iteration horizon) on a laptop CPU.

Contents

Path	What
scripts/fw_core.py	Direct FW (any dim), forward dynamics (eq 7), backward dynamics (eq 10 / Alg 1), g(r), grid/bisection search, angle reconstruction (eq 22)
scripts/experiments.py	Full 5-claim experiment suite → outputs/results.json + figures/*.png
outputs/results.json	All numeric verdicts
outputs/run_scaled.log	Full-scale run log (horizon 10,000)
figures/	Reproductions of paper Figures 1, 4, 5, 7 + iteration-complexity plot
authors_code/	Authors' reference Julia code (cloned, for cross-checking)
paper.pdf	The paper

Reproduce

pip install numpy mpmath matplotlib
python scripts/experiments.py --horizon 10000 --grid 10000   # ~65 s, laptop CPU
# smoke test: python scripts/experiments.py --horizon 1500 --grid 5000   # ~3 s

Result (all 5 claims reproduce)

Claim	Key check	Result
1 — Ω(1/√ε) lower bound	eq (12) residual bound; Lemma 15 cₜ∈[1,5/2]; T~ε^b	viol 0; 100% in range; b=−0.509 (target −0.5)
2 — hard instance matches Garber–Hazan	μ=L=2, α=β=1; Prop 3 upper bound; worst-case rate	holds; rate −1.998 (target −2)
3 — backward reconstruction, 10k+ horizon	Alg 1 at 60,000-bit; forward re-sim; precision need	20,001 steps, fwd/bwd err 0; 0.31 digits/stable-step (⇒ ~10⁻³⁰⁰ for 1000 iters)
4 — (r,s) dynamics: stable phases + jumps	Lemma 1 (invariant subspace); g(r) partition; Prop 5	off-subspace ~1e-13; partition 100%; jumps 415/415
5 — dimension-independent + ellipsoid	identical gaps d=2..200; FW commutes with Φ=A^½	maxdiff 0; commute 5e-15

See the Trackio logbook for full per-claim detail.