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Stochastic Compositional Optimization with Compositional Constraints

Shuoguang Yang, Wei You, Zhe Zhang, Ethan X. Fang

TL;DR

The paper addresses stochastic compositional optimization (SCO) problems with hard, single-level and two-level compositional expected-value constraints. It introduces primal-dual algorithms and a stochastic sequential dual interpretation to tackle unknown, data-driven constraint forms, proving an optimal $O(\frac{1}{\sqrt{N}})$ convergence rate for both the objective and feasibility across single-level and compositional constraints. The framework is extended to two-level compositional EV constraints (CoC-SCO) with a corresponding CC-SCGD algorithm, maintaining the same $O(\frac{1}{\sqrt{N}})$ convergence and handling multiple constraints efficiently. Numerical experiments on CVaR-constrained portfolio optimization validate the theoretical rates and demonstrate practical effectiveness in risk-management contexts, including scenarios with multiple CVaR constraints. Overall, the work establishes new benchmarks for SCO with compositional EV constraints and provides scalable algorithms for large-scale, data-driven risk-management problems.

Abstract

Stochastic compositional optimization (SCO) has attracted considerable attention because of its broad applicability to important real-world problems. However, existing works on SCO assume that the projection within a solution update is simple, which fails to hold for problem instances where the constraints are in the form of expectations, such as empirical conditional value-at-risk constraints. We study a novel model that incorporates single-level expected value and two-level compositional constraints into the current SCO framework. Our model can be applied widely to data-driven optimization and risk management, including risk-averse optimization and high-moment portfolio selection, and can handle multiple constraints. We further propose a class of primal-dual algorithms that generates sequences converging to the optimal solution at the rate of $\cO(\frac{1}{\sqrt{N}})$under both single-level expected value and two-level compositional constraints, where $N$ is the iteration counter, establishing the benchmarks in expected value constrained SCO.

Stochastic Compositional Optimization with Compositional Constraints

TL;DR

The paper addresses stochastic compositional optimization (SCO) problems with hard, single-level and two-level compositional expected-value constraints. It introduces primal-dual algorithms and a stochastic sequential dual interpretation to tackle unknown, data-driven constraint forms, proving an optimal convergence rate for both the objective and feasibility across single-level and compositional constraints. The framework is extended to two-level compositional EV constraints (CoC-SCO) with a corresponding CC-SCGD algorithm, maintaining the same convergence and handling multiple constraints efficiently. Numerical experiments on CVaR-constrained portfolio optimization validate the theoretical rates and demonstrate practical effectiveness in risk-management contexts, including scenarios with multiple CVaR constraints. Overall, the work establishes new benchmarks for SCO with compositional EV constraints and provides scalable algorithms for large-scale, data-driven risk-management problems.

Abstract

Stochastic compositional optimization (SCO) has attracted considerable attention because of its broad applicability to important real-world problems. However, existing works on SCO assume that the projection within a solution update is simple, which fails to hold for problem instances where the constraints are in the form of expectations, such as empirical conditional value-at-risk constraints. We study a novel model that incorporates single-level expected value and two-level compositional constraints into the current SCO framework. Our model can be applied widely to data-driven optimization and risk management, including risk-averse optimization and high-moment portfolio selection, and can handle multiple constraints. We further propose a class of primal-dual algorithms that generates sequences converging to the optimal solution at the rate of under both single-level expected value and two-level compositional constraints, where is the iteration counter, establishing the benchmarks in expected value constrained SCO.
Paper Structure (37 sections, 17 theorems, 185 equations, 2 figures, 2 algorithms)

This paper contains 37 sections, 17 theorems, 185 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivating Applications
  3. Related Work
  4. Stochastic compositional optimization.
  5. Expected value constrained optimization.
  6. Contributions
  7. Notation.
  8. Stochastic Compositional Optimization with Single-level Expected-Value Constraints
  9. Challenges.
  10. A Primal-Dual Algorithm
  11. Stochastic Sequential Dual Interpretation
  12. Convergence Analysis
  13. Convergence of gap function
  14. Boundedness of lambda
  15. Convergence of value and feasibility metrics
  16. ...and 22 more sections

Key Result

Theorem 1

Suppose Assumptions assumption:01, assumption:02, and assumption:03 hold and Algorithm alg:01 generates $\{ (x_t,\lambda_{t}, \pi_{1:2,t},v_t) \}_{t=1}^N$ by setting $\tau_t = t/2$, $\alpha_t = \alpha$, and $\eta_t = \eta$ for $t \leq N$. Let $\lambda \in \mathbb{R}_+^m$ be a nonnegative bounded (r

Figures (2)

  • Figure 1: Empirical convergence of EC-SCGD algorithm for single CVaR constraint. Optimality gap and feasibility residual averaged over 10 runs with shaded 95% confidence intervals.
  • Figure 2: Empirical convergence of EC-SCGD for multiple CVaR constraints. Optimality gap and feasibility residual averaged over 10 runs, with shaded 95% confidence intervals.

Theorems & Definitions (17)

  • Theorem 1
  • Lemma 1
  • Proposition 1
  • Theorem 2
  • Lemma 2
  • Lemma 3
  • Theorem 3
  • Proposition 2
  • Theorem 4
  • Lemma 4: Lemma 3.8 of lan2020first
  • ...and 7 more