# A Stochastic Primal-Dual Method for Optimization with Conditional Value at Risk Constraints

## TL;DR

This paper introduces a stochastic primal-dual method for optimizing problems with CVaR constraints, providing convergence guarantees and analyzing the impact of risk parameters on computational complexity.

## Contribution

It proposes a simple modification to existing primal-dual algorithms that avoids complex bounds, enabling efficient risk-constrained optimization with theoretical guarantees.

## Key findings

- Achieves $rac{ ext{constant}}{\sqrt{K}}$-approximate feasibility and optimality within $K$ iterations.
- Provides bounds on step sizes and characterizes the computational cost related to risk parameters.
- Draws parallels between sample complexity of CVaR optimization and chance-constrained programming.

## Abstract

We study a first-order primal-dual subgradient method to optimize risk-constrained risk-penalized optimization problems, where risk is modeled via the popular conditional value at risk (CVaR) measure. The algorithm processes independent and identically distributed samples from the underlying uncertainty in an online fashion, and produces an $\eta/\sqrt{K}$-approximately feasible and $\eta/\sqrt{K}$-approximately optimal point within $K$ iterations with constant step-size, where $\eta$ increases with tunable risk-parameters of CVaR. We find optimized step sizes using our bounds and precisely characterize the computational cost of risk aversion as revealed by the growth in $\eta$. Our proposed algorithm makes a simple modification to a typical primal-dual stochastic subgradient algorithm. With this mild change, our analysis surprisingly obviates the need for a priori bounds or complex adaptive bounding schemes for dual variables assumed in many prior works. We also draw interesting parallels in sample complexity with that for chance-constrained programs derived in the literature with a very different solution architecture.