We introduce the Probabilistic Coin Change Problem (PCCP), a novel variant of the classical Combination Coin Change Problem (CCCP), motivated by a real-world scientific inverse task. The goal of CCCP is to enumerate all unordered combinations of coin denominations that sum to a given target. In PCCP, each coin type’s value follows a discrete probability distribution, and the aggregate value of a combination of coins is thus stochastic. Given a set of such coin types and noisy observations of total sums, the task is to infer the most likely latent coin combination. To address the combinatorial and probabilistic complexity of PCCP, we propose DeepProReasoner (Deep Combinatorial Probabilistic Reasoning with Embedded Representations), an unsupervised, end-to-end, deep-learning framework that integrates combinatorial reasoning, latent-space modeling, and differentiable probabilistic reasoning. The model is trained using a reconstruction loss between the observed empirical distribution and a decoded probability mass function (PMF), enabling efficient gradient-based search over a continuous relaxation of the combinatorial space. We evaluate DeepProReasoner on two instances of PCCP: (1) a synthetic Candy Mix problem for ablation studies, and (2) a real-world task of molecular formula inference from ultrahigh resolution mass spectrometry (MS) data. Besides the two given instances, PCCP captures a wide range of inverse settings in biology, chemistry, environmental sciences, and medicine, where latent combinatorial structures give rise to noisy aggregate observations through stochastic processes. Our results show that DeepProReasoner achieves high accuracy and robustness, outperforming state-of-the-art methods.

Addressing global sustainability challenges as outlined by the United Nations (UN) Sustainable Development Goals (SDGs) often requires navigating many potentially conflicting societal objectives simultaneously. For instance, increasing hydropower production enhances renewable energy supply but may adversely impact people and nature. Understanding these trade-offs is crucial, and the Pareto frontier - the set of solutions that cannot be improved with respect to one objective without negatively affecting another - is a valuable framework. Strategic hydropower planning concerns finding energy portfolios that achieve decarbonization targets, while balancing energy production with socioeconomic and environmental impacts. Previous work has considered exact and approximate algorithms for Pareto optimization for tree-structured networks, such as rivers, for hydropower planning. However, such approaches do not account for bounding constraints, such as realistic energy production targets, critical in real-world applications. Herein, we propose a novel approach for constraint-aware Pareto optimization for tree-structured networks, incorporating objective bounds to ensure more realistic and robust solution outcomes. We apply our constraint-aware Pareto approach to the strategic planning of hydropower expansion, considering energy bounds to adhere to the UN's net zero by 2050 decarbonization targets, in the Magdalena River basin, home to more than 80% of Colombia’s population. Our analysis demonstrates how lower and upper bounds can significantly modify the unconstrained Pareto frontier, revealing that feasible Pareto solutions can be dominated by infeasible solutions, and thus may be ignored by constraint-agnostic solvers. Our results highlight the importance of considering real-world constraints in multi-objective problems such as optimizing hydropower expansion to meet both energy and sustainability goals.

Addressing real-world optimization problems becomes particularly challenging when analytic objective functions or constraints are unavailable. While numerous studies have addressed the issue of unknown objectives, limited research has focused on scenarios where feasibility constraints are not given explicitly. Overlooking these constraints can lead to spurious solutions that are unrealistic in practice. To deal with such unknown constraints, we propose to perform optimization within the data manifold using diffusion models. To constrain the optimization process to the data manifold, we reformulate the original optimization problem as a sampling problem from the product of the Boltzmann distribution defined by the objective function and the data distribution learned by the diffusion model. Depending on the differentiability of the objective function, we propose two different sampling methods. For differentiable objectives, we propose a two-stage framework that begins with a guided diffusion process for warm-up, followed by a Langevin dynamics stage for further correction. For non-differentiable objectives, we propose an iterative importance sampling strategy using the diffusion model as the proposal distribution. Comprehensive experiments on a synthetic dataset, six real-world black-box optimization datasets, and a multi-objective molecule optimization dataset show that our method achieves better or comparable performance with previous state-of-the-art baselines.

Combinatorial optimization (CO) problems are pervasive across various domains, but their NP-hard nature often necessitates problem-specific heuristic algorithms. Recent advancements in deep learning have led to the development of learning-based heuristics, yet these approaches often struggle with limited search capabilities. We introduce Explore-and-Exploit GNN ($X^2$GNN, pronounced x-squared GNN), a novel unsupervised neural framework that combines exploration and exploitation for combinatorial search optimization: i) Exploration - $X^2$GNN generates multiple solutions simultaneously, promoting diversity in the search space; (ii) Exploitation - $X^2$GNN employs neural stochastic iterative refinement to exploit partial existing solutions, guiding the search toward promising regions and helping escape local optima. By balancing exploration and exploitation, $X^2$GNN achieves superior performance and generalization on several graph CO problems including Max Cut, Max Independent Set, and Max Clique. Notably, for large Max Clique problems, $X^2$GNN consistently generates solutions within 1.2\% of optimality, while other state-of-the-art learning-based approaches struggle to reach within 22\% of optimal. Moreover, $X^2$GNN consistently generates better solutions than Gurobi on large graphs for all three problems under reasonable time budgets. Furthermore, $X^2$GNN exhibits exceptional generalization capabilities. For the Maximum Independent Set problem, $X^2$GNN outperforms state-of-the-art methods even when trained on smaller or out-of-distribution graphs compared to the test set. Our framework offers a more effective and flexible approach to neural combinatorial optimization, addressing a key challenge in the field and providing a promising direction for future research in learning-based heuristics for combinatorial optimization.

Climate change and biodiversity loss are among humanity’s most pressing challenges. In 2022, under the auspices of the United Nations, over 190 countries reached a historic agreement to address the alarming loss of biodiversity and restore natural ecosystems. Target 3, often referred to as ``30x30'', seeks to effectively protect and manage 30% of the world’s terrestrial, inland water, coastal, and marine areas by 2030. In this work, we address the UN 30x30 target in the context of global freshwater fish conservation. Freshwater ecosystems are disproportionately unprotected, and their biota are declining at an alarming rate. Our goal is to select new protected areas that protect freshwater fish species as much as possible without exceeding total coverage of 30% of land area. To support this goal, we introduce the Expansion of Connected Components from Alternative Terminals Problem, a graph-based optimization problem that captures ecological priorities and connectivity constraints. We analyze its computational complexity, propose novel integer programming formulations, and develop scalable solution methods. We further evaluate its typical-case complexity under diverse settings and demonstrate that our approach scales to a global real-world scope, encompassing approximately 200,000 freshwater basins and 13,000 species, paving the way for implementing the 30x30 target on a worldwide scale.

Deep generative models like GAN and VAE have shown impressive results in generating unconstrained objects like images. However, many design settings arising in industrial design, material science, computer graphics and more require that the generated objects satisfy hard combinatorial constraints or meet objectives in addition to modeling a data distribution. To address this, we propose GenCO, a generative framework that guarantees constraint satisfaction throughout training by leveraging differentiable combinatorial solvers to enforce feasibility. GenCO imposes the generative loss on provably feasible solutions rather than intermediate soft solutions, meaning that the deep generative network can focus on ensuring the generated objects match the data distribution without having to also capture feasibility. This shift enables practitioners to enforce hard constraints on the generated outputs during end-to-end training, enabling assessments of their feasibility and introducing additional combinatorial loss components to deep generative training. We demonstrate the effectiveness of our approach on a variety of generative combinatorial tasks, including game level generation, map creation for path planning, and photonic device design, consistently demonstrating its capability to yield diverse, high-quality solutions that verifiably adhere to user-specified combinatorial properties.

Mixed integer linear programs (MILP) are flexible and powerful tools for modeling and solving many difficult real-world combinatorial optimization problems. In this paper, we propose a novel machine learning (ML)-based framework ConPaS that learns to predict solutions to MILPs with contrastive learning. For training, we collect high-quality solutions as positive samples. We also collect low-quality or infeasible solutions as negative samples using novel optimization-based or sampling approaches. We then learn to make discriminative predictions by contrasting the positive and negative samples. During testing, we predict and fix the assignments for a subset of integer variables and then solve the resulting reduced MILP to find high-quality solutions. Empirically, ConPaS achieves state-of-the-art results compared to other ML-based approaches in terms of the quality of and the speed at which solutions are found.

Modern neural network classifiers achieve remarkable performance across a variety of tasks; however, they frequently exhibit overconfidence in their predictions due to the cross-entropy loss. Inspired by this problem, we propose the Critic Loss for Image Classification (CrtCl, pronounced Critical). CrtCl formulates image classification training in a generator-critic framework, with a base classifier acting as a generator, and a correctness critic imposing a loss on the classifier. The base classifier, acting as the generator, given images, generates the probability distribution over classes and intermediate embeddings. The critic model, given the image, intermediate embeddings, and output predictions of the base model, predicts the probability that the base model has produced the correct classification, which then can be back propagated as a self supervision signal. Notably, the critic does not use the label as input, meaning that the critic can train the base model on both labeled and unlabeled data in semi-supervised learning settings. CrtCl represents a learned loss method for accuracy, alleviating the negative side effects of using cross-entropy loss. Additionally, CrtCl provides a powerful way to select data to be labeled in an active learning setting, by estimating the classification ability of the base model on unlabeled data. We study the effectiveness of CrtCl in low-labeled data regimes, and in the context of active learning. In classification, we find that CrtCl, compared to recent baselines, increases classifier generalization and calibration with various amounts of labeled data. In active learning, we show our method outperforms baselines in accuracy and calibration. We observe consistent results across three image classification datasets.

The ability to form, retrieve, and reason about memories in response to stimuli is central to general intelligence, enabling learning, adaptation, and insight. Large Language Models (LLMs), when given proper memories or context, can reason and respond effectively. However, they still struggle to optimally encode, store, and retrieve memories, a limitation that constrains their full potential as specialized AI agents. Retrieval Augmented Generation (RAG) seeks to address this by enriching LLMs with in-context examples. Inspired by human memory, we introduce Graphical Eigen Memories for Retrieval Augmented Generation (GEM-RAG), which tags information with LLMgenerated “utility” questions, links information together in a graph by similarity, and uses eigendecomposition to form higherlevel summary information. This approach not only enhances RAG tasks but also offers a way to explore text data sets. Using UnifiedQA, GPT-3.5 Turbo, SBERT, and OpenAI text encoders, we show that GEM-RAG outperforms state-of-the-art RAG methods on two standard QA tasks and discuss its implications for robust RAG systems.

Optimization problems with nonlinear cost functions and combinatorial constraints appear in many real-world applications but remain challenging to solve efficiently compared to their linear counterparts. To bridge this gap, we propose $\textbf{\emph{\texttt{SurCo}}}$ that learns linear $\underline{\text{Sur}}$rogate costs which can be used in existing $\underline{\text{Co}}$mbinatorial solvers to output good solutions to the original nonlinear combinatorial optimization problem. The surrogate costs are learned end-to-end with nonlinear loss by differentiating through the linear surrogate solver, combining the flexibility of gradient-based methods with the structure of linear combinatorial optimization. We propose three $\texttt{SurCo}$ variants: $\texttt{SurCo}-\texttt{zero}$ for individual nonlinear problems, $\texttt{SurCo}-\texttt{prior}$ for problem distributions, and $\texttt{SurCo}-\texttt{hybrid}$ to combine both distribution and problem-specific information. We give theoretical intuition motivating $\texttt{SurCo}$, and evaluate it empirically. Experiments show that $\texttt{SurCo}$ finds better solutions faster than state-of-the-art and domain expert approaches in real-world optimization problems such as embedding table sharding, inverse photonic design, and nonlinear route planning.

Integer Linear Programs (ILPs) are powerful tools for modeling and solving a large number of combinatorial optimization problems. Recently, it has been shown that Large Neighborhood Search (LNS), as a heuristic algorithm, can find high-quality solutions to ILPs faster than Branch and Bound. However, how to find the right heuristics to maximize the performance of LNS remains an open problem. In this paper, we propose a novel approach, CL-LNS, that delivers state-of-the-art anytime performance on several ILP benchmarks measured by metrics including the primal gap, the primal integral, survival rates and the best performing rate. Specifically, CL-LNS collects positive and negative solution samples from an expert heuristic that is slow to compute and learns a more efficient one with contrastive learning. We use graph attention networks and a richer set of features to further improve its performance.

Recent works in learning-integrated optimization have shown promise in settings where the optimization problem is only partially observed or where general-purpose optimizers perform poorly without expert tuning. By learning an optimizer $\mathbf{g}$ to tackle these challenging problems with $f$ as the objective, the optimization process can be substantially accelerated by leveraging past experience. The optimizer can be trained with supervision from known optimal solutions or implicitly by optimizing the compound function $f\circ \mathbf{g}$. The implicit approach may not require optimal solutions as labels and is capable of handling problem uncertainty; however, it is slow to train and deploy due to frequent calls to optimizer $\mathbf{g}$ during both training and testing. The training is further challenged by sparse gradients of $\mathbf{g}$, especially for combinatorial solvers. To address these challenges, we propose using a smooth and learnable **Landscape Surrogate** $\mathcal{M}$ as a replacement for $f\circ \mathbf{g}$. This surrogate, learnable by neural networks, can be computed faster than the solver $\mathbf{g}$, provides dense and smooth gradients during training, can generalize to unseen optimization problems, and is efficiently learned via alternating optimization. We test our approach on both synthetic problems, including shortest path and multidimensional knapsack, and real-world problems such as portfolio optimization, achieving comparable or superior objective values compared to state-of-the-art baselines while reducing the number of calls to $\mathbf{g}$. Notably, our approach outperforms existing methods for computationally expensive high-dimensional problems.

Controlling bias in training datasets is vital for ensuring equal treatment, or parity, between different groups in downstream applications. A naive solution is to transform the data so that it is statistically independent of group membership, but this may throw away too much information when a reasonable compromise between fairness and accuracy is desired. Another common approach is to limit the ability of a particular adversary who seeks to maximize parity. Unfortunately, representations produced by adversarial approaches may still retain biases as their efficacy is tied to the complexity of the adversary used during training. To this end, we theoretically establish that by limiting the mutual information between representations and protected attributes, we can assuredly control the parity of any downstream classifier. We demonstrate an effective method for controlling parity through mutual information based on contrastive information estimators and show that they outperform approaches that rely on variational bounds based on complex generative models. We test our approach on UCI Adult and Heritage Health datasets and demonstrate that our approach provides more informative representations across a range of desired parity thresholds while providing strong theoretical guarantees on the parity of any downstream algorithm.

We propose a machine learning approach for quickly solving Mixed Integer Programs (MIP) by learning to prioritize a set of decision variables, which we call pseudo-backdoors, for branching that results in faster solution times. Learning-based approaches have seen success in the area of solving combinatorial optimization problems by being able to flexibly leverage common structures in a given distribution of problems. Our approach takes inspiration from the concept of strong backdoors, which corresponds to a small set of variables such that only branching on these variables yields an optimal integral solution and a proof of optimality. Our notion of pseudo-backdoors corresponds to a small set of variables such that only branching on them leads to faster solve time (which can be solver dependent). A key advantage of pseudo-backdoors over strong backdoors is that they are much amenable to data-driven identification or prediction. Our proposed method learns to estimate the solver performance of a proposed pseudo-backdoor, using a labeled dataset collected on a set of training MIP instances. This model can then be used to identify high-quality pseudo-backdoors on new MIP instances from the same distribution. We evaluate our method on the generalized independent set problems and find that our approach can efficiently identify high-quality pseudo-backdoors. In addition, we compare our learned approach against Gurobi, a state-of-the-art MIP solver, demonstrating that our method can be used to improve solver performance.

Machine learning components commonly appear in larger decision-making pipelines; however, the model training process typically focuses only on a loss that measures average accuracy between predicted values and ground truth values. Decision-focused learning explicitly integrates the downstream decision problem when training the predictive model, in order to optimize the quality of decisions induced by the predictions. It has been successfully applied to several limited combinatorial problem classes, such as those that can be expressed as linear programs (LP), and submodular optimization. However, these previous applications have uniformly focused on problems with simple constraints. Here, we enable decision-focused learning for the broad class of problems that can be encoded as a mixed integer linear program (MIP), hence supporting arbitrary linear constraints over discrete and continuous variables. We show how to differentiate through a MIP by employing a cutting planes solution approach, an algorithm that iteratively tightens the continuous relaxation by adding constraints removing fractional solutions. We evaluate our new end-to-end approach on several real world domains and show that it outperforms the standard two phase approaches that treat prediction and optimization separately, as well as a baseline approach of simply applying decision-focused learning to the LP relaxation of the MIP. Lastly, we demonstrate generalization performance in several transfer learning tasks.

We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint.