min_norm_solvers

Pytorch friendly implementation of minimum norm solvers.

Problem setting

Given a list of vectors ${ v_i }{i=1}^m$, finding weight vector ${ w_i }{i=1}^m$ belonging to a simplex $\Delta_w$, i.e., $\sum_{i=1}^m w_i = 1$ and $w_i \geq 0 ; \forall i$ such that the weighted sum vector $\bar{v} = \sum_{i=1}^m w_i v_i$ obtains the minimum norm. More specifically, solving the optimization as below:

$$w = \underset{w \in \Delta_w}{\text{argmin}};| \sum_{i=1}^m w_i v_i|_2^2$$

Solver

• min_norm_solvers: Mainly copy from the repository https://github.com/isl-org/MultiObjectiveOptimization. My job is just replacing some numpy functions by torch functions. Especially the torch.dot function is not equivalent to the numpy.dot function. Also fix some bugs (i.e., iter_count doesn't change therefore, there can be a forever while loop)
• gradient_descent_solvers: Using soft_max reparameterize and gradient descent to solve the problem. More specifically, we define a trainable variable $\alpha \in \mathbb{R}^m$, then using soft_max function to project $\alpha$ to the simplex $\Delta_w$. Iterative solution

$$\alpha^{t+1} = \alpha^t - \eta ; \nabla_{\alpha} | \sum_{i=1}^m \text{softmax}(\alpha)_i v_i |_2^2$$

How to use

Please refer to the example.py file.