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Langevin Multiplicative Weights Update with Applications in Polynomial Portfolio Management
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Proposes Langevin Multiplicative Weights Update (LMWU), a noisy MWU variant derived from Shahshahani-geometry Brownian motion for nonconvex optimization over product of simplices, with a non-asymptotic convergence analysis and a polynomial portfolio application.
Problem
Global optimization of nonconvex objectives over a simplex or product of simplices, where MWU and projected gradient methods tend to converge only to local minima or second-order stationary points.
Approach
Adds a Langevin noise scaled by Shahshahani (non-Euclidean) geometry of the simplex to the linear multiplicative weights update, so the iterate can escape local minima while staying in the simplex with a cheap normalization-based projection.
Key results
- On the Degenerate risk preference, LMWU scores 70.8930 versus 74.7203 for MWU (lower is better).
- On the MV preference, LMWU scores 0.8464 versus 0.8561 for MWU; on MVS, 0.8314 versus 0.8391.
- On the MVSK preference, LMWU scores 14.9585 versus 16.1104 for MWU and 15.6559 for AMWU, the best of the table.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Replace MWU with LMWU | Polynomial portfolio optimization · Degenerate preference | loss: 74.7203 -> 70.8930 | Helped |
| Replace MWU with LMWU | Polynomial portfolio optimization · Increasing preference | loss: 17.4527 -> 16.7684 | Helped |
| Replace MWU with LMWU | Polynomial portfolio optimization · MV preference | loss: 0.8561 -> 0.8464 | Helped |
| Replace MWU with LMWU | Polynomial portfolio optimization · MVS preference | loss: 0.8391 -> 0.8314 | Helped |
| Replace MWU with LMWU | Polynomial portfolio optimization · MVSK preference | loss: 16.1104 -> 14.9585 | Helped |
| Replace MWU with LMWU | Polynomial portfolio optimization · Equal preference | loss: 44.5159 -> 42.8307 | Helped |
| Use a larger beta | LMWU · Non-convex test functions | convergence-point quality | Helped |
| Choose an inappropriate beta | LMWU · Non-convex test functions | function value | Hurt |
Surprising findings
- Projected Langevin outperforms LMWU on the MVS (0.8846 vs 0.8314) and Increasing-preference-related comparisons in the portfolio table, and LMWU underperforms MWU when beta is poorly chosen.
Limitations
- Convergence guarantee relies on several strong assumptions (log-Sobolev inequality, entropy bound, smoothness) that are hard to verify in practice.
- Performance is sensitive to the inverse temperature beta; an inappropriate beta makes LMWU underperform MWU.
- Time-varying settings and optimal beta selection are left to future work.
Experimental setup
Datasets: daily returns of 10 NASDAQ stocks, 2011-2021 (T=2517)
Benchmarks: polynomial portfolio optimization out-of-sample loss on NASDAQ data synthetic nonconvex test functions f1-f6
Metrics: average out-of-sample loss score function value convergence
Baselines: MWU Accelerated MWU Projected Langevin Perturbed RGD Langevin GD
Builds on: Multiplicative Weights Update Langevin dynamics Accelerated MWU Projected Langevin Perturbed Riemannian gradient descent
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