accbpg 0.1

Accelerated Bregman proximal gradient (ABPG) methods

# Accelerated Bregman Proximal Gradient Methods

Accelerated first-order algorithms for solving relatively-smooth convex optimization problems of the form

minimize { f(x) + P(x) | x in C }

with a reference function h(x), where

• h(x) is convex and essentially smooth on C

• f(x) is convex and differentiable, and L-smooth relative to h(x), that is, f(x)-L*h(x) is convex

• P(x) is convex and closed (lower semi-continuous)

• C is a closed convex set

### Implemented algorithms in [HRX2018](https://arxiv.org/abs/1808.03045)

• BPG_LS (Bregman proximal gradient) method with line search

• ABPG (Accelerated BPG) method

• ABPG-expo (ABPG with exponent adaption)

• ABPG-gain (ABPG with gain adaption)

• ABDA (Accelerated Bregman dual averaging) method

## Installation

Clone or fork from GitHub. Or install from PyPI:

pip install accbpg

## Usage

import accbpg

# generate a random instance of D-optimal design problem f, h, L, x0 = accbpg.D_opt_design(80, 200)

# solve the problem instance using BPG with line search x1, F1, G1 = accbpg.BPG_LS(f, h, L, x0, maxitrs=1000, verskip=100)

# solve it again using ABPG_gain with gamma=2 x2, F2, G2, D2 = accbpg.ABPG_gain(f, h, L, 2, x0, maxitrs=1000, verbskip=100)

compare the two methods by visualization

import matplotlib.pyplot as plt Fmin = min(F1.min(), F2.min()) plt.semilogy(range(len(F1)), F1-Fmin, range(len(F2)), F2-Fmin)

## Examples in [HRX2018](https://arxiv.org/abs/1808.03045)

D-optimal experiment design

import accbpg.ex_D_opt

Nonnegative regression with KL-divergence

import accbpg.ex_KL_regr

Poisson linear inverse problems

import accbpg.ex_PoissonL1 import accbpg.ex_PoissonL2