Convergence Rates of the Stochastic Alternating Algorithm for Bi-Objective Optimization

Abstract

Stochastic alternating algorithms for bi-objective optimization are considered when optimizing two conflicting functions for which optimization steps have to be applied separately for each function. Such algorithms consist of applying a certain number of steps of gradient or subgradient descent on each single objective at each iteration. In this paper, we show that stochastic alternating algorithms achieve a sublinear convergence rate of ${\mathcal {O}}(1/T)$, under strong convexity, for the determination of a minimizer of a weighted-sum of the two functions, parameterized by the number of steps applied on each of them. An extension to the convex case is presented for which the rate weakens to ${\mathcal {O}}(1/\sqrt{T})$. These rates are valid also in the non-smooth case. Importantly, by varying the proportion of steps applied to each function, one can determine an approximation to the Pareto front.

A Proposition using Intermediate Value Theorem

Based on the intermediate value theorem, we derive the following proposition for the purpose of convergence rate analysis of the SA2GD algorithm.

Proposition A.1

Given a continuous real function $\phi (x): {\mathbb {R}}^n \rightarrow {\mathbb {R}}$ and a set of points $\{x_j\}_{j =1}^m$, there exists $w \in {\mathbb {R}}^n$ such that

$$\begin{aligned}m\phi (w) = \sum _{j = 1}^m \phi (x_j),\end{aligned}$$

where $w = \sum _{j = 1}^m \mu _j x_j, \text{ with } \sum _{j = 1}^m \mu _j = 1, \mu _j \ge 0$, $i = 1, \ldots , m$, is a convex linear combination of $\{x_j\}_{j =1}^m$.

Proof

The proposition is obtained by consecutively applying the intermediate value theorem to $\phi (x)$. First, for the pair of points $x_1$ and $x_2$, there exists a point $w_{12} = \mu _{12} x_1 + (1-\mu _{12}) x_2, \mu _{12} \in [0, 1]$, such that $\phi (w_{12}) = (\phi (x_1) + \phi (x_2))/2$ according to the intermediate value theorem, which implies that $\sum _{j = 1}^m \phi (x_j) = 2\phi (w_{12}) + \sum _{j = 3}^m \phi (x_j)$. Then, there exists $w_{13} = \mu _{13} w_{12} + (1-\mu _{13}) x_3, \mu _{13} \ge 0$, such that $\phi (w_{13}) = (2\phi (w_{12}) + \phi (x_3))/3$ holds given that the average function value $(2\phi (w_{12}) + \phi (x_3))/3$ lies between $\phi (w_{12})$ and $\phi (x_3)$. Notice that $w_{13}$ can also be written as convex linear combination of $\{x_1, x_2, x_3\}$. The proof is concluded by continuing this process until $x_m$ is reached. $\square$