In this package are developped different loss functions:

Lagrangian Sub-Problem Loss

loss_LR is the one presented in the paper ^[1] and consist on the bound provided by the Lagrangian Sub-Problem (with a proper sign that allows to write the Lagrangian Dual as minimization problem).

Lagrangian Sub-Problem Loss on GPU

loss_LR_gpu: as LRloss but the sub-problem is solved in GPU.

Warning: for the moment this loss function works only for MCND instances with GPU memorization.

Anyway it is not faster than the one that use CPU.

GAP Loss

loss_GAP this loss function simply consists on the GAP of percentage the value $v$ provided by the Lagrangian sub-Problem and the optimal value of the Lagrangian dual and can be computed as:

\[\frac{v-v^*}{v^*}*100\]

when the Lagrangian Dual is a minimization problem

GAP Closure Loss

loss_GAP_closure this loss function is similar to GAPloss as still consider the value $v$ provided by the Lagrangian sub-Problem and the optimal value of the Lagrangian dual. But it tries to further compare these solutions with the continuous relaxation bound CR. It is computed as:

\[\frac{v}{v^*-CR} * 100\]

Hinge Loss

For an instance $\iota \in I$ with gold solution $(x^*, y^*)$ (more precisely $(x^*(\iota), y^*(\iota))$) of $L(\pi^*)$, the Hinge loss is

\[H(w;\iota) = L(\pi(w), x^{*}, y^{*}) - \min_{x,y} \Big(L(\pi(w), x, y) - \alpha \Delta_{y^*}(y)\Big)\]

where $w$ are the parameters of the model, $\pi(w)$ is the prediction of the model given $w$, $(x^{*},y^{*})$ is the gold solution of the instance, $\Delta_{y^*}(y)$ is the hamming loss between $y$ and $y^*$ and $\alpha$ is a non negative scalar.

The loss_hinge is

\[\frac{1}{|I|}\sum_{\iota \in I} \frac{1}{|A(\iota)|}H(w;\iota)\]

Mean Squared Error

loss_mse is only the MSE between the mean squared error between the predicted and the optimal Lagrangian Multipliers.

Multi Prediction LR loss

loss_multi_LR_factory is a specialized version of loss_LR that is able to handle with Multiple Lagrangian Multipliers Predictions.

More sophisticated variants of this loss function will be provided in the future.

References