Mastane Achab

Mastane Achab

Nurturing the future

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Bregman divergence

Beyond Convexity #2: Barygradient flow

1 minute read

Published:

In my latest paper, I introduced a generalized proximal point algorithm (PPA): given a point $(x,q) \in \mathbb{R}^m \times \Delta_S$ ($m\ge 1$, $S \ge 2$ and $\Delta_S$ the probability simplex), the next iterate $(x’,q’)$ is given by: \(( \nabla f + \lambda A )(x',q') = \nabla f(x,q)\) for step-size $\lambda>0$, $f(x,q)=\frac12 ||x||^2 + h(q)$ with $h(q)=\sum_{s=1}^S q_s \log(q_s)$ the negentropy, and \(A(x,q) = \begin{pmatrix} J_\ell(x)^\intercal q \\ -\ell(x) \end{pmatrix}\) where $J_\ell$ denotes the Jacobian matrix of $\ell=(\ell_1,\dots,\ell_S):\mathbb{R}^m \rightarrow \mathbb{R}^S$ with each $\ell_s$ convex ($\forall 1\le s \le S$).

baryconvex optimization

Beyond Convexity #2: Barygradient flow

1 minute read

Published:

In my latest paper, I introduced a generalized proximal point algorithm (PPA): given a point $(x,q) \in \mathbb{R}^m \times \Delta_S$ ($m\ge 1$, $S \ge 2$ and $\Delta_S$ the probability simplex), the next iterate $(x’,q’)$ is given by: \(( \nabla f + \lambda A )(x',q') = \nabla f(x,q)\) for step-size $\lambda>0$, $f(x,q)=\frac12 ||x||^2 + h(q)$ with $h(q)=\sum_{s=1}^S q_s \log(q_s)$ the negentropy, and \(A(x,q) = \begin{pmatrix} J_\ell(x)^\intercal q \\ -\ell(x) \end{pmatrix}\) where $J_\ell$ denotes the Jacobian matrix of $\ell=(\ell_1,\dots,\ell_S):\mathbb{R}^m \rightarrow \mathbb{R}^S$ with each $\ell_s$ convex ($\forall 1\le s \le S$).

generalized convexity inequality

gradient flow

Beyond Convexity #2: Barygradient flow

1 minute read

Published:

In my latest paper, I introduced a generalized proximal point algorithm (PPA): given a point $(x,q) \in \mathbb{R}^m \times \Delta_S$ ($m\ge 1$, $S \ge 2$ and $\Delta_S$ the probability simplex), the next iterate $(x’,q’)$ is given by: \(( \nabla f + \lambda A )(x',q') = \nabla f(x,q)\) for step-size $\lambda>0$, $f(x,q)=\frac12 ||x||^2 + h(q)$ with $h(q)=\sum_{s=1}^S q_s \log(q_s)$ the negentropy, and \(A(x,q) = \begin{pmatrix} J_\ell(x)^\intercal q \\ -\ell(x) \end{pmatrix}\) where $J_\ell$ denotes the Jacobian matrix of $\ell=(\ell_1,\dots,\ell_S):\mathbb{R}^m \rightarrow \mathbb{R}^S$ with each $\ell_s$ convex ($\forall 1\le s \le S$).

mixture of log-concave distributions