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Sample-and-Bound for Non-Convex Optimization

Yaoguang Zhai, Zhizhen Qin, Sicun Gao

TL;DR

This work targets non-convex global optimization by proposing MCIR, a Sample-and-Bound approach that blends Monte Carlo Tree Search with interval bounds and regional estimation. By storing samples within boxes, performing modified UCT-guided selection, and learning representative regions via Hessian/gradient information, MCIR aggressively focuses search on promising areas while retaining exploration. The method integrates local optimization and backpropagation of bounds to maintain a coherent search, and it is shown to be competitive across synthetic, bound-constrained, and neural-network benchmarks, often outperforming traditional baselines. Overall, MCIR demonstrates a scalable framework for high-dimensional non-convex problems that leverages analytic structure and sampling to improve efficiency and solution quality.

Abstract

Standard approaches for global optimization of non-convex functions, such as branch-and-bound, maintain partition trees to systematically prune the domain. The tree size grows exponentially in the number of dimensions. We propose new sampling-based methods for non-convex optimization that adapts Monte Carlo Tree Search (MCTS) to improve efficiency. Instead of the standard use of visitation count in Upper Confidence Bounds, we utilize numerical overapproximations of the objective as an uncertainty metric, and also take into account of sampled estimates of first-order and second-order information. The Monte Carlo tree in our approach avoids the usual fixed combinatorial patterns in growing the tree, and aggressively zooms into the promising regions, while still balancing exploration and exploitation. We evaluate the proposed algorithms on high-dimensional non-convex optimization benchmarks against competitive baselines and analyze the effects of the hyper parameters.

Sample-and-Bound for Non-Convex Optimization

TL;DR

This work targets non-convex global optimization by proposing MCIR, a Sample-and-Bound approach that blends Monte Carlo Tree Search with interval bounds and regional estimation. By storing samples within boxes, performing modified UCT-guided selection, and learning representative regions via Hessian/gradient information, MCIR aggressively focuses search on promising areas while retaining exploration. The method integrates local optimization and backpropagation of bounds to maintain a coherent search, and it is shown to be competitive across synthetic, bound-constrained, and neural-network benchmarks, often outperforming traditional baselines. Overall, MCIR demonstrates a scalable framework for high-dimensional non-convex problems that leverages analytic structure and sampling to improve efficiency and solution quality.

Abstract

Standard approaches for global optimization of non-convex functions, such as branch-and-bound, maintain partition trees to systematically prune the domain. The tree size grows exponentially in the number of dimensions. We propose new sampling-based methods for non-convex optimization that adapts Monte Carlo Tree Search (MCTS) to improve efficiency. Instead of the standard use of visitation count in Upper Confidence Bounds, we utilize numerical overapproximations of the objective as an uncertainty metric, and also take into account of sampled estimates of first-order and second-order information. The Monte Carlo tree in our approach avoids the usual fixed combinatorial patterns in growing the tree, and aggressively zooms into the promising regions, while still balancing exploration and exploitation. We evaluate the proposed algorithms on high-dimensional non-convex optimization benchmarks against competitive baselines and analyze the effects of the hyper parameters.
Paper Structure (26 sections, 3 equations, 5 figures, 4 tables)

This paper contains 26 sections, 3 equations, 5 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminary
  4. Problem Formulation.
  5. Interval Arithmetic.
  6. Monte Carlo Tree Search.
  7. Monte Carlo Tree Search with Interval Bounds and Regional Estimation
  8. Overview.
  9. Sub-domain Marking.
  10. Path Selection.
  11. Tree Expansion.
  12. Local Optimization.
  13. Backward Propagation.
  14. Experiments
  15. Benchmarks.
  16. ...and 11 more sections

Figures (5)

  • Figure 1: Steps in each iteration of the MCIR algorithm
  • Figure 2: Overall performance of the baselines and MCIR on tested synthetic functions.
  • Figure 3: Overall performance of the baselines and MCIR on Biggsbi1, Harkerp, Watson, and Neural Networks.
  • Figure 4: Ablation studies on function Michalewicz-50d for the number of children at expansion (a), the effectiveness of local optimization (b), on function Watson-31d for the effectiveness of local optimizaiton (c), and on function Ackley-50d for $C_{lb}$ (d), $C_{v}$ (e), and $C_{x}$ (f).
  • Figure 5: Landscape of test functions: Ackley, Levy, and Michalewicz.