Wasserstein Distributionally Robust Optimization

参考资料

Wasserstein distance https://zhuanlan.zhihu.com/p/...

2019 TutORial: Wasserstein Distributionally Robust Optimization https://www.informs.org/Resou...

The optimize of curse: EVen if input estimates are unbiased, the output are biased.

The nominal distribution $\hat{P_N}$.

• dependes on $\hat{\xi_1}, \hat{\xi_2},...,\hat{\xi_N}$ and is thus random;
• differs from the data-generating distribution $P$

How to measure estimation errors?

Use the Monge/ Kantorovich/ Wasserstein distance

$$ W_p(Q, Q') = (inf_{\pi \in \Pi(Q,Q')} \int_{R^m*R^m}\mid \mid \xi-\xi'\mid\mid^p\pi(d\xi, d\xi'))^{1/p} $$ $\Pi(Q,Q')=$ set of couplings with marginals $Q$ and $Q'$ $\pi(A*B)=$ mass moved from source region A to target region B $\mid \mid \xi-\xi'\mid \mid= $ Price paid for moving mass from $\xi$ to $\xi'$. **Theorem** $W_p^p(Q,Q')=sup\int_{R^m} \phi(\xi')Q'(d\xi')-\int_{R^m} \psi(\xi)Q(d\xi)$ s.t. $\psi$ and $\phi$ are bounded and continuous $\psi(\xi)-\phi(\xi')\leq \mid \mid \xi-\xi'\mid \mid ^p$ $\forall \xi,\xi'\in R^m$ Pricing problem of $3^{rd}$ party: $$ \phi(\xi)= \text{price paid for mass } Q(d\xi)\text{ bought at } \xi $$ $$ \psi(\xi')= \text{price paid for mass } Q'(d\xi')\text{ bought at } \xi'$$ **Kanotorovich & Rubinstein Theorem** $$ W_1(Q,Q')=sup\int_{R^m} \psi(\xi')Q'(d\xi')-\int_{R^m} \psi(\xi)Q(d\xi) $$ p=1 optimaility: price fof buying and selling equal at each location?? Lipschitz modulus: $$Lip(\phi)= sup_{\xi \not= \xi'}\frac{\phi(\xi)-\phi(\xi')}{\mid\mid\xi-\xi'\mid\mid}$$ **dd**