Regularization Notes

This page summarizes the entropy-regularized problem solved by the package and explains how the reported quantities should be interpreted.

Role Of Regularization

The exact LP defines the benchmark for the discretized martingale optimal transport problem. The regularized solver introduces an entropy penalty indexed by eps in order to obtain a smoother approximation path.

The regularized layer is useful for:

• tracing numerical behavior as smoothing is reduced
• comparing approximation quality across examples
• studying solver stability and constraint satisfaction

Objects Recorded By The Solver

The regularized output includes the following principal quantities.

Quantity	Meaning	Interpretation
expected_payoff	raw expectation of the payoff under the regularized plan	closest analogue of the exact objective value
regularized_primal	entropy-augmented objective	used for primal-dual consistency checks
dual_value	dual objective corresponding to the regularized problem	should be close to regularized_primal
dual_gap	dual_value - regularized_primal	small magnitude indicates internal consistency
iterations	number of iterations performed	rough measure of computational cost
martingale_error	maximum row-wise martingale deviation	checks enforcement of the defining constraint

The distinction between expected_payoff and regularized_primal is essential. Only the latter includes the entropy contribution.

Dependence On eps

The parameter eps controls the amount of smoothing.

Regime	Typical Effect	Practical Consequence
large eps	smoother plans and easier convergence	more bias relative to the LP benchmark
moderate eps	stable approximation path	useful for exploratory comparison
small eps	closer approach to the LP benchmark	increased stiffness and iteration count

This pattern is visible in the regularization-path figures shipped with the gallery.

Internal Variables

The implementation records three dual-related arrays:

• u1
• u2
• h

Their purpose is summarized below.

Variable	Role In The Solver
u1	first marginal dual potential
u2	second marginal dual potential
h	row-wise martingale multiplier

These variables determine the reconstructed regularized plan and are therefore part of the full diagnostic state.

Why Log-Space Updates Are Used

The numerically delicate update is carried out in log space rather than through a direct exponential form. This improves stability, especially in smaller-eps regimes where exponentials can become poorly scaled.

The current implementation therefore treats the regularized solver as a controlled approximation method rather than as a purely black-box transport routine.

Reading The Regularization Path

A regularization-path figure should typically show:

• expected payoff moving toward the exact LP upper value as eps decreases
• regularized primal remaining distinct from raw expected payoff
• smaller eps values requiring more iterations
• dual gaps remaining small in magnitude

If this pattern breaks down, the diagnostics should be inspected before interpreting the corresponding values.

Practical Use

In the current repository workflow, the regularized solver is most informative when used together with:

• the exact LP benchmark
• the stability diagnostics figure
• the structural diagnostics figure
• the per-example markdown report

This combination makes it possible to compare approximation quality, constraint accuracy, and geometric behavior in a single experiment directory.