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Primal-dual algorithm for contextual stochastic combinatorial optimization

Louis Bouvier, Thibault Prunet, Vincent Leclère, Axel Parmentier

TL;DR

The paper addresses contextual stochastic combinatorial optimization by learning fast, policy-based decisions using neural networks with combinatorial optimization layers. It introduces a surrogate, Fenchel-Young loss-based learning objective and an alternating primal-dual scheme that decomposes across scenarios, enabling tractable updates via a distribution-space regularization (including sparse perturbations). The authors prove conditions under which the surrogate converges at a rate of ${\mathcal{O}}(1/t)$ and bound the discrepancy between surrogate and empirical risks under strong convexity, demonstrating competitive performance on a contextual two-stage minimum weight spanning tree problem compared to imitation-based baselines. Practically, the approach offers scalable training and online applicability while achieving accuracy close to expensive Lagrangian-based heuristics, with a clear pathway for generalization and theoretical extension in future work.

Abstract

This paper introduces a novel approach to contextual stochastic optimization, integrating operations research and machine learning to address decision-making under uncertainty. Traditional methods often fail to leverage contextual information, which underscores the necessity for new algorithms. In this study, we utilize neural networks with combinatorial optimization layers to encode policies. Our goal is to minimize the empirical risk, which is estimated from past data on uncertain parameters and contexts. To that end, we present a surrogate learning problem and a generic primal-dual algorithm that is applicable to various combinatorial settings in stochastic optimization. Our approach extends classic Fenchel-Young loss results and introduces a new regularization method using sparse perturbations on the distribution simplex. This allows for tractable updates in the original space and can accommodate diverse objective functions. We demonstrate the linear convergence of our algorithm under certain conditions and provide a bound on the non-optimality of the resulting policy in terms of the empirical risk. Experiments on a contextual stochastic minimum weight spanning tree problem show that our algorithm is efficient and scalable, achieving performance comparable to imitation learning of solutions computed using an expensive Lagrangian-based heuristic.

Primal-dual algorithm for contextual stochastic combinatorial optimization

TL;DR

The paper addresses contextual stochastic combinatorial optimization by learning fast, policy-based decisions using neural networks with combinatorial optimization layers. It introduces a surrogate, Fenchel-Young loss-based learning objective and an alternating primal-dual scheme that decomposes across scenarios, enabling tractable updates via a distribution-space regularization (including sparse perturbations). The authors prove conditions under which the surrogate converges at a rate of and bound the discrepancy between surrogate and empirical risks under strong convexity, demonstrating competitive performance on a contextual two-stage minimum weight spanning tree problem compared to imitation-based baselines. Practically, the approach offers scalable training and online applicability while achieving accuracy close to expensive Lagrangian-based heuristics, with a clear pathway for generalization and theoretical extension in future work.

Abstract

This paper introduces a novel approach to contextual stochastic optimization, integrating operations research and machine learning to address decision-making under uncertainty. Traditional methods often fail to leverage contextual information, which underscores the necessity for new algorithms. In this study, we utilize neural networks with combinatorial optimization layers to encode policies. Our goal is to minimize the empirical risk, which is estimated from past data on uncertain parameters and contexts. To that end, we present a surrogate learning problem and a generic primal-dual algorithm that is applicable to various combinatorial settings in stochastic optimization. Our approach extends classic Fenchel-Young loss results and introduces a new regularization method using sparse perturbations on the distribution simplex. This allows for tractable updates in the original space and can accommodate diverse objective functions. We demonstrate the linear convergence of our algorithm under certain conditions and provide a bound on the non-optimality of the resulting policy in terms of the empirical risk. Experiments on a contextual stochastic minimum weight spanning tree problem show that our algorithm is efficient and scalable, achieving performance comparable to imitation learning of solutions computed using an expensive Lagrangian-based heuristic.
Paper Structure (58 sections, 19 theorems, 140 equations, 5 figures, 4 tables, 1 algorithm)

This paper contains 58 sections, 19 theorems, 140 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Stochastic optimization policies.
  3. Learning approach.
  4. Policies based on a combinatorial optimization layer.
  5. Fenchel-Young losses.
  6. Alternating minimization algorithm.
  7. Related works.
  8. Contributions and outline.
  9. General notations.
  10. Moment polytope.
  11. Distribution polytope.
  12. Main results
  13. Policies over combinatorial spaces and empirical risk minimization
  14. Legendre-type functions (Rockafellar+1970)
  15. Mapping scores to distributions.
  16. ...and 43 more sections

Key Result

Proposition 1

Using the notations of Section subsec:damped_primal_dual_algo and the shortened expression $\gamma_i$ for $\gamma(x_i, \xi_i)$, Equations eq:decomposition_product_w-eq:coordination_product_w can be recast as where $\mathcal{L}_{\Omega_{\mathcal{C}(x)}}$ is the Fenchel-Young loss on the moment polytope $\mathcal{C}(x) = \mathop{\mathrm{conv}}\nolimits(\mathcal{Y}(x))$ generated by

Figures (5)

  • Figure 1: Policies over combinatorial spaces leveraging Fenchel duality. Left: Neural network architecture. Right: Cost function and predicted linear objective in the score space, and their mapping to the distribution space.
  • Figure 2: Primal-dual maps for non-full-dimensional $\mathop{\mathrm{dom}}\nolimits(\Omega)$.
  • Figure 3: Proportion of $\theta$ values giving the optimal solution of problem \ref{['eq:CSO_toy_pb']} when $\varepsilon$ varies.
  • Figure 4: Two-stage minimum spanning neural network.
  • Figure 5: Validation and test average gaps of policies over the iterations of our primal-dual algorithm, for the two-stage minimum weight spanning tree.

Theorems & Definitions (42)

  • Remark 1
  • Example 1: Entropy
  • Example 2: Sparse perturbation
  • Proposition 1
  • Proposition 2
  • Lemma 0
  • Theorem 1
  • Proposition 3: Convexity of the Jensen gap of a separable regularization
  • Theorem 2
  • Proposition 4
  • ...and 32 more