[Paper Review] Multistage Stochastic Optimization via Kernels

Dimitris Bertsimas, Kimberly Villalobos Carballo|arXiv (Cornell University)|Mar 11, 2023

Statistical Methods and Inference4 citations

TL;DR

This paper proposes a non-parametric, data-driven approach for multistage stochastic optimization using kernel methods in a reproducing kernel Hilbert space (RKHS), enabling tractable functional optimization via kernel-based functional stochastic gradient descent and sparsification through function subspace projections. The method achieves asymptotic optimality and demonstrates near-optimal performance with scalable computation even on large datasets and high-dimensional problems.

ABSTRACT

We develop a non-parametric, data-driven, tractable approach for solving multistage stochastic optimization problems in which decisions do not affect the uncertainty. The proposed framework represents the decision variables as elements of a reproducing kernel Hilbert space and performs functional stochastic gradient descent to minimize the empirical regularized loss. By incorporating sparsification techniques based on function subspace projections we are able to overcome the computational complexity that standard kernel methods introduce as the data size increases. We prove that the proposed approach is asymptotically optimal for multistage stochastic optimization with side information. Across various computational experiments on stochastic inventory management problems, {our method performs well in multidimensional settings} and remains tractable when the data size is large. Lastly, by computing lower bounds for the optimal loss of the inventory control problem, we show that the proposed method produces decision rules with near-optimal average performance.

Motivation & Objective

• To develop a non-parametric, data-driven framework for multistage stochastic optimization that avoids scenario tree enumeration and the curse of dimensionality.
• To represent decision rules as functions in a reproducing kernel Hilbert space (RKHS) to capture complex, non-linear dependencies from historical data and side information.
• To overcome the computational intractability of standard kernel methods in large-scale settings by introducing sparsification via function subspace projections.
• To ensure asymptotic optimality of the decision rules under mild regularity conditions on the uncertainty distribution.
• To empirically validate the method’s performance in high-dimensional, large-data settings, particularly in inventory and shipment planning problems.

Proposed method

• The decision variables are modeled as elements of a reproducing kernel Hilbert space (RKHS) generated by a universal kernel, enabling non-parametric approximation of complex decision rules.
• Functional stochastic gradient descent is used to minimize an empirical regularized loss function over the RKHS, optimizing over functional parameters rather than discrete decisions.
• Sparsification is achieved through function subspace projections, reducing the number of active parameters per decision variable and mitigating the computational burden of kernel methods.
• The method leverages the Representer Theorem in a functional form but avoids full kernel matrix inversion by projecting onto low-dimensional subspaces of functions.
• The algorithm is trained on historical data and side information, with no assumptions on the correlation structure of uncertainty across stages.
• The approach is extended to multistage problems by sequentially optimizing decision rules conditioned on observed information, using kernel-based function representations at each stage.

(a)

Experimental results

Research questions

• RQ1 Can a non-parametric, kernel-based method achieve asymptotic optimality in multistage stochastic optimization with side information?
• RQ2 How can kernel methods be made computationally tractable for large-scale multistage problems without relying on scenario trees?
• RQ3 To what extent does sparsification via function subspace projections reduce parameter count while preserving solution quality?
• RQ4 How does the proposed method compare to traditional approaches like SAA, SRO, and knn-based methods in terms of out-of-sample performance and computation time?
• RQ5 Can the method maintain high performance in high-dimensional settings and with large data sizes?

Key findings

• The SMOK algorithm achieves the highest average out-of-sample profit (160,737.0) among all methods tested on the shipment planning problem at N=100 data points.
• SMOK maintains high performance with only around 60 parameters per decision, even at N=5000, demonstrating strong scalability.
• Computation time for SMOK grows slowly with data size—only 252 seconds at N=5000—while SRO-knn and SAA-knn become intractable (over 54,000 seconds at N=5000).
• MOK and SMOK outperform SRO, SRO-knn, and SAA-knn in out-of-sample profit across all data sizes, with SMOK achieving the best balance of performance and efficiency.
• The method produces decision rules with near-optimal average performance, as confirmed by lower bounds computed for the inventory control problem.
• Theoretical analysis confirms that the proposed approach is asymptotically optimal under mild conditions, validating its consistency as data size increases.

(c)