Efficient Multivariate Kelly Optimization Reveals Sigmoidal Scaling Laws

Battle Count: 87

Highly relevant to quantitative trading in several ways: (1) The Kelly criterion is fundamental to position sizing and portfolio allocation. (2) The methods enable solving large-scale allocation problems (hundreds of simultaneous bets) that were previously intractable, directly applicable to prediction markets, sports betting, and multi-asset portfolio construction. (3) The scaling laws provide practical guidance on how many assets to include for a desired accuracy level. (4) The bounds framework allows practitioners to quantify the cost of approximation. (5) The connection to Merton's continuous-time portfolio problem bridges discrete and continuous frameworks. However, the binary bet assumption and independence requirement limit direct applicability to general trading scenarios.

The implementation requires: (1) Frullani/Laplace transform formulation with log-space computation and logsumexp tricks for numerical stability. (2) Double-exponential quadrature with careful node selection. (3) Tail subtraction for the near-full-investment regime (w0 ≈ 0). (4) Gradient and Hessian computation via integral representations. (5) Newton-CG optimization in logit (softmax) space with Hessian-vector products. (6) Subproblem decomposition with stepwise/greedy selection and numeraire transformations for bounds. (7) Sigmoid fitting with L-BFGS and linear parameter models. The numerical stability considerations (tail subtraction, log-space arithmetic, gauge fixing) add significant complexity. The paper provides detailed appendix material but no code.

The paper provides detailed mathematical formulations, algorithmic descriptions, and synthetic data generation procedures. However, no code repository is mentioned. The synthetic data generation process is well-described (logit-offset and Beta prior models with specified parameters), and the optimization methods are fully specified. The 1,000 independent instances per configuration and 5 random seeds are described. Reproduction would require implementing the integral transform solver, DE quadrature, tail subtraction, and Newton-CG optimization from scratch.