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Regularization

Regularization schemes penalize undesirable portfolio weight structures during optimization. They help control concentration, instability, and pathological solutions while trading off optimality for robustness.

Formally, a regularizer adds a penalty term to the cost objective \(\mathcal{L}(\mathbf{w})\) as

\[ \min_{\mathbf{w}} \ \mathcal{L}(\mathbf{w}) + \lambda \cdot R(\mathbf{w}) \]

where \(R(\mathbf{w})\) encodes structural preferences over the weights \(\mathbf{w}\) and \(\lambda\) is the strength of the preference.

Regularization Schemes

Regularization Scheme Identifier Formulation
Taxicab Norm l1 \(\sum_i \lvert \mathbf{w}_i\rvert\)
Euclidean Norm l2 \(\sum_i \mathbf{w}_i^2\)
Chebyshev Norm l-inf \(\max_i \lvert \mathbf{w}_i\rvert\)
Negative Entropy of Weights entropy \(\sum_i \mathbf{w}_i \log \mathbf{w}_i\)
Jensen-Shannon Divergence from Uniform Weights jsd \(\text{D}_{\text{JSD}}(\mathbf{w} \| \mathbf{u})\)
Variance of Weights variance \(\text{Var}(\mathbf{w})\)
Mean Pairwise Absolute Deviation mpad \(\frac{1}{n^2} \sum_{i}^n \sum_{j}^n \lvert \mathbf{w}_i - \mathbf{w}_j\rvert\)
Maximum Pairwise Deviation mpd \(\max_{i,j} \lvert \mathbf{w}_i - \mathbf{w}_j \rvert\)
Wasserstein-1 Distance from Uniform Weights wass-1 \(\text{W}_{1}(\mathbf{w}, \mathbf{u})\)

Notes

  • l1 regularization is mainly used for long-short portfolios to encourage less extreme allocations to meet the net exposure of 1. Using it on long-only portfolios is redundant.
  • For long-short portfolios, mathematically grounded regularizers such as entropy, jsd and wass-1 first normalize the weights and constrain them to the simplex before applying the regularization, ensuring mathematical coherence is not violated.

Portfolios without Regularization Support

Not all portfolio objectives in OPES support regularization. Some are inherently robust, while others rely on non-optimization-based objectives, making regularization inapplicable. The following objectives do not support regularization:

  • Uniform
  • InverseVolatility
  • SoftmaxMean
  • HierarchicalRiskParity
  • UniversalPortfolios
  • ExponentialGradient
  • KLRobustMaxMean
  • KLRobustKelly
  • WassRobustMaxMean
  • WassRobustMinVariance
  • WassRobustMeanVariance

Example

# Importing an optimizer which supports regularization
from opes.objectives import MaxMean

# Initializing different portfolios with various regularization schemes
maxmean_taxi = MaxMean(reg='l1', strength=0.01)             # Taxicab norm
maxmean_eucl = MaxMean(reg='l2', strength=0.01)             # Euclidean norm
maxmean_cheby = MaxMean(reg='l-inf', strength=0.01)         # Chebyshev norm
maxmean_entropy = MaxMean(reg='entropy', strength=0.01)     # Negative entropy of weights
maxmean_jsd = MaxMean(reg='jsd', strength=0.01)             # Jensen-Shannon divergence
maxmean_var = MaxMean(reg='variance', strength=0.01)        # Variance of weights
maxmean_mpad = MaxMean(reg='mpad', strength=0.01)           # Mean pairwise absolute deviation
maxmean_mpd = MaxMean(reg='mpd', strength=0.01)             # Maximum pairwise deviation
maxmean_wass = MaxMean(reg='wass-1', strength=0.01)         # Wasserstein-1