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Optimization/Discrete and Combinatorial Optimization

Jacobus van der Linden, Mathijs de Weerdt, Emir Demirović

Global optimization of decision trees has shown to be promising in terms of accuracy, size, and consequently human comprehensibility. However, many of the methods used rely on general-purpose solvers for which scalability remains an issue.Dynamic programming methods have been shown to scale much better because they exploit the tree structure by solving subtrees as independent subproblems. However, this only works when an objective can be optimized separately for subtrees.We explore this relationship in detail and show the necessary and sufficient conditions for such separability and generalize previous dynamic programming approaches into a framework that can optimize any combination of separable objectives and constraints.Experiments on five application domains show the general applicability of this framework, while outperforming the scalability of general-purpose solvers by a large margin.

Optimization/Discrete and Combinatorial Optimization

Leonard Papenmeier, Luigi Nardi, Matthias Poloczek

Impactful applications such as materials discovery, hardware design, neural architecture search, or portfolio optimization require optimizing high-dimensional black-box functions with mixed and combinatorial input spaces.While Bayesian optimization has recently made significant progress in solving such problems, an in-depth analysis reveals that the current state-of-the-art methods are not reliable. Their performances degrade substantially when the unknown optima of the function do not have a certain structure. To fill the need for a reliable algorithm for combinatorial and mixed spaces, this paper proposes Bounce that relies on a novel map of various variable types into nested embeddings of increasing dimensionality.Comprehensive experiments show that Bounce reliably achieves and often even improves upon state-of-the-art performance on a variety of high-dimensional problems.

Optimization/Discrete and Combinatorial Optimization

Yuxin Pan, Yize Chen, Fangzhen Lin

Designing effective policies for the online 3D bin packing problem (3D-BPP) has been a long-standing challenge, primarily due to the unpredictable nature of incoming box sequences and stringent physical constraints. While current deep reinforcement learning (DRL) methods for online 3D-BPP have shown promising results in optimizing average performance over an underlying box sequence distribution, they often fail in real-world settings where some worst-case scenarios can materialize. Standard robust DRL algorithms tend to overly prioritize optimizing the worst-case performance at the expense of performance under normal problem instance distribution. To address these issues, we first introduce a permutation-based attacker to investigate the practical robustness of both DRL-based and heuristic methods proposed for solving online 3D-BPP. Then, we propose an adjustable robust reinforcement learning (AR2L) framework that allows efficient adjustment of robustness weights to achieve the desired balance of the policy's performance in average and worst-case environments. Specifically, we formulate the objective function as a weighted sum of expected and worst-case returns, and derive the lower performance bound by relating to the return under a mixture dynamics. To realize this lower bound, we adopt an iterative procedure that searches for the associated mixture dynamics and improves the corresponding policy. We integrate this procedure into two popular robust adversarial algorithms to develop the exact and approximate AR2L algorithms. Experiments demonstrate that AR2L is versatile in the sense that it improves policy robustness while maintaining an acceptable level of performance for the nominal case.

Optimization/Discrete and Combinatorial Optimization

Haoran Ye, Jiarui Wang, Zhiguang Cao, Helan Liang, Yong Li

Ant Colony Optimization (ACO) is a meta-heuristic algorithm that has been successfully applied to various Combinatorial Optimization Problems (COPs). Traditionally, customizing ACO for a specific problem requires the expert design of knowledge-driven heuristics. In this paper, we propose DeepACO, a generic framework that leverages deep reinforcement learning to automate heuristic designs. DeepACO serves to strengthen the heuristic measures of existing ACO algorithms and dispense with laborious manual design in future ACO applications. As a neural-enhanced meta-heuristic, DeepACO consistently outperforms its ACO counterparts on eight COPs using a single neural model and a single set of hyperparameters. As a Neural Combinatorial Optimization method, DeepACO performs better than or on par with problem-specific methods on canonical routing problems. Our code is publicly available at https://github.com/henry-yeh/DeepACO.

Optimization/Discrete and Combinatorial Optimization

Seunghun Lee, Jaewon Chu, Sihyeon Kim, Juyeon Ko, Hyunwoo Kim

Bayesian optimization is a powerful method for optimizing black-box functions with limited function evaluations. Recent works have shown that optimization in a latent space through deep generative models such as variational autoencoders leads to effective and efficient Bayesian optimization for structured or discrete data. However, as the optimization does not take place in the input space, it leads to an inherent gap that results in potentially suboptimal solutions. To alleviate the discrepancy, we propose Correlated latent space Bayesian Optimization (CoBO), which focuses on learning correlated latent spaces characterized by a strong correlation between the distances in the latent space and the distances within the objective function. Specifically, our method introduces Lipschitz regularization, loss weighting, and trust region recoordination to minimize the inherent gap around the promising areas. We demonstrate the effectiveness of our approach on several optimization tasks in discrete data, such as molecule design and arithmetic expression fitting, and achieve high performance within a small budget.

Optimization/Discrete and Combinatorial Optimization

Sebastian Sanokowski, Wilhelm Berghammer, Sepp Hochreiter, Sebastian Lehner

Several recent unsupervised learning methods use probabilistic approaches to solve combinatorial optimization (CO) problems based on the assumption of statistically independent solution variables. We demonstrate that this assumption imposes performance limitations in particular on difficult problem instances. Our results corroborate that an autoregressive approach which captures statistical dependencies among solution variables yields superior performance on many popular CO problems. We introduce Subgraph Tokenization in which the configuration of a set of solution variables is represented by a single token. This tokenization technique alleviates the drawback of the long sequential sampling procedure which is inherent to autoregressive methods without sacrificing expressivity. Importantly, we theoretically motivate an annealed entropy regularization and show empirically that it is essential for efficient and stable learning.

Optimization/Discrete and Combinatorial Optimization

Xingchen Wan, Pierre Osselin, Henry Kenlay, Binxin Ru, Michael A Osborne, Xiaowen Dong

The increasing availability of graph-structured data motivates the task of optimising over functions defined on the node set of graphs. Traditional graph search algorithms can be applied in this case, but they may be sample-inefficient and do not make use of information about the function values; on the other hand, Bayesian optimisation is a class of promising black-box solvers with superior sample efficiency, but it has scarcely been applied to such novel setups. To fill this gap, we propose a novel Bayesian optimisation framework that optimises over functions defined on generic, large-scale and potentially unknown graphs. Through the learning of suitable kernels on graphs, our framework has the advantage of adapting to the behaviour of the target function. The local modelling approach further guarantees the efficiency of our method. Extensive experiments on both synthetic and real-world graphs demonstrate the effectiveness of the proposed optimisation framework.

Optimization/Discrete and Combinatorial Optimization

Darko Drakulic, Sofia Michel, Florian Mai, Arnaud Sors, Jean-Marc Andreoli

Despite the success of neural-based combinatorial optimization methods for end-to-end heuristic learning, out-of-distribution generalization remains a challenge. In this paper, we present a novel formulation of Combinatorial Optimization Problems (COPs) as Markov Decision Processes (MDPs) that effectively leverages common symmetries of COPs to improve out-of-distribution robustness. Starting from a direct MDP formulation of a constructive method, we introduce a generic way to reduce the state space, based on Bisimulation Quotienting (BQ) in MDPs. Then, for COPs with a recursive nature, we specialize the bisimulation and show how the reduced state exploits the symmetries of these problems and facilitates MDP solving. Our approach is principled and we prove that an optimal policy for the proposed BQ-MDP actually solves the associated COPs. We illustrate our approach on five classical problems: the Euclidean and Asymmetric Traveling Salesman, Capacitated Vehicle Routing, Orienteering and Knapsack Problems. Furthermore, for each problem, we introduce a simple attention-based policy network for the BQ-MDPs, which we train by imitation of (near) optimal solutions of small instances from a single distribution. We obtain new state-of-the-art results for the five COPs on both synthetic and realistic benchmarks. Notably, in contrast to most existing neural approaches, our learned policies show excellent generalization performance to much larger instances than seen during training, without any additional search procedure. Our code is available at: [link](https://github.com/naver/bq-nco).

Optimization/Learning for Optimization

Eeshaan Jain, Tushar Nandy, Gaurav Aggarwal, Ashish Tendulkar, Rishabh Iyer, Abir De

Existing subset selection methods for efficient learning predominantly employ discrete combinatorial and model-specific approaches, which lack generalizability--- for each new model, the algorithm has to be executed from the beginning. Therefore, for an unseen architecture, one cannot use the subset chosen for a different model. In this work, we propose $\texttt{SubSelNet}$, a non-adaptive subset selection framework, which tackles these problems. Here, we first introduce an attention-based neural gadget that leverages the graph structure of architectures and acts as a surrogate to trained deep neural networks for quick model prediction. Then, we use these predictions to build subset samplers. This naturally provides us two variants of $\texttt{SubSelNet}$. The first variant is transductive (called Transductive-$\texttt{SubSelNet}$), which computes the subset separately for each model by solving a small optimization problem. Such an optimization is still super fast, thanks to the replacement of explicit model training by the model approximator. The second variant is inductive (called Inductive-$\texttt{SubSelNet}$), which computes the subset using a trained subset selector, without any optimization. Our experiments show that our model outperforms several methods across several real datasets.

Optimization/Discrete and Combinatorial Optimization

Akshaykumar Gattani, Sharath Raghvendra, Pouyan Shirzadian

Algorithms for the minimum-cost bipartite matching can be used to estimate Wasserstein distance between two distributions.Given two sets $A$ and $B$ of $n$ points in a $2$-dimensional Euclidean space, one can use a fast implementation of the Hungarian method to compute a minimum-cost bipartite matching of $A$ and $B$ in $\tilde{O}(n^2)$ time. Let $\Delta$ be the spread, i.e., the ratio of the distance of the farthest to the closest pair of points in $A\cup B$. In this paper, we present a new algorithm to compute a minimum-cost bipartite matching of $A$ and $B$ with a similar worst-case execution time of $\tilde{O}(n^2 \log \Delta)$. However, when $A$ and $B$ are drawn independently and identically from a fixed distribution that is not known to the algorithm, the execution time of our algorithm is, in expectation, $\tilde{O}(n^{7/4}\log \Delta)$.To the best of our knowledge, our algorithm is the first one to achieve a sub-quadratic execution time even for stochastic point sets with real-valued coordinates.Our algorithm extends to any dimension $d$, where it runs in $\tilde{O}(n^{2-\frac{1}{2d}}\Phi(n))$ time for stochastic point sets $A$ and $B$; here $\Phi(n)$ is the query/update time of a dynamic weighted nearest neighbor data structure. Our algorithm can be seen as a careful adaptation of the Hungarian method in the geometric divide-and-conquer framework.

Optimization/Discrete and Combinatorial Optimization

Wenjing Chen, Victoria Crawford

In this paper, we consider the optimization problem Submodular Cover (SCP), which is to find a minimum cardinality subset of a finite universe $U$ such that the value of a submodular function $f$ is above an input threshold $\tau$. In particular, we consider several variants of SCP including the general case, the case where $f$ is additionally assumed to be monotone, and finally the case where $f$ is a regularized monotone submodular function. Our most significant contributions are that: (i) We propose a scalable algorithm for monotone SCP that achieves nearly the same approximation guarantees as the standard greedy algorithm in significantly faster time; (ii) We are the first to develop an algorithm for general SCP that achieves a solution arbitrarily close to being feasible; and finally (iii) we are the first to develop algorithms for regularized SCP. Our algorithms are then demonstrated to be effective in an extensive experimental section on data summarization and graph cut, two applications of SCP.