This paper introduces a novel decision-making framework that promotes consistency among decisions made by diverse models while utilizing external knowledge. Leveraging the Integer Linear Programming (ILP) framework, we map predictions from various models into globally normalized and comparable values by incorporating information about decisions' prior probability, confidence (uncertainty), and the models' expected accuracy. Our empirical study demonstrates the superiority of our approach over conventional baselines on multiple datasets.

This paper presents Dual Lagrangian Learning (DLL), a principled learning methodology that combines conic duality theory with the representation power of ML models. DLL leverages conic duality to provide dual-feasible solutions, and therefore valid Lagrangian dual bounds, for parametric linear and nonlinear conic optimization problems. The paper introduces differentiable conic projection layers, a systematic dual completion procedure, and a self-supervised learning framework. The effectiveness of DLL is demonstrated on linear and nonlinear parametric optimization problems for which DLL provides valid dual bounds within 0.5% of optimality.

Black-box optimization has emerged as an effective technique in many real-world applications to find the global optimum of expensive, noisy black-box functions. Some notable applications include hyper-parameter optimization in machine learning algorithms Snoek et al. [2012], Bergstra and Bengio [2012], synthesis of short polymer fiber materials, alloy design, 3D bio-printing, and molecule design Greenhill et al. [2020], Shahriari et al. [2015], optimizing design parameters in computational fluid dynamics Morita et al. [2022], and scientific research (e.g., multilayer nanoparticle, photonic crystal topology) Kim et al. [2022]. Bayesian Optimization is a popular example of black-box optimization method. Typically, Bayesian Optimization algorithms use a probabilistic regression model, such as a Gaussian Process (GP), trained on existing function observations. This model is then utilized to create an acquisition function that balances exploration and exploitation to recommend the next evaluation point for the black-box functions. Various options exist for acquisition functions, including improvement-based methods like Probability of Improvement Kushner [1964], Expected Improvement Mockus et al. [1978], the Upper Confidence Bound Srinivas

In this paper, we demonstrate that the PGW problem can be transformed into a variant of the Gromov-Wasserstein problem, akin to the conversion of the partial optimal transport problem into an optimal transport problem. This transformation leads to two new solvers, mathematically and computationally equivalent, based on the Frank-Wolfe algorithm, that provide efficient solutions to the PGW problem. We further establish that the PGW problem constitutes a metric for metric measure spaces. Finally, we validate the effectiveness of our proposed solvers in terms of computation time and performance on shape-matching and positive-unlabeled learning problems, comparing them against existing baselines.

Computing eigenvalue decomposition (EVD) of a given linear operator, or finding its leading eigenvalues and eigenfunctions, is a fundamental task in many machine learning and scientific computing problems. For high-dimensional eigenvalue problems, training neural networks to parameterize the eigenfunctions is considered as a promising alternative to the classical numerical linear algebra techniques. This paper proposes a new optimization framework based on the low-rank approximation characterization of a truncated singular value decomposition, accompanied by new techniques called nesting for learning the top-$L$ singular values and singular functions in the correct order. The proposed method promotes the desired orthogonality in the learned functions implicitly and efficiently via an unconstrained optimization formulation, which is easy to solve with off-the-shelf gradient-based optimization algorithms. We demonstrate the effectiveness of the proposed optimization framework for use cases in computational physics and machine learning.

Recent advancements have introduced machine learning frameworks to enhance the Branch and Bound (B\&B) branching policies for solving Mixed Integer Linear Programming (MILP). These methods, primarily relying on imitation learning of Strong Branching, have shown superior performance. However, collecting expert samples for imitation learning, particularly for Strong Branching, is a time-consuming endeavor. To address this challenge, we propose \textbf{C}ontrastive Learning with \textbf{A}ugmented \textbf{M}ILPs for \textbf{Branch}ing (CAMBranch), a framework that generates Augmented MILPs (AMILPs) by applying variable shifting to limited expert data from their original MILPs. This approach enables the acquisition of a considerable number of labeled expert samples. CAMBranch leverages both MILPs and AMILPs for imitation learning and employs contrastive learning to enhance the model's ability to capture MILP features, thereby improving the quality of branching decisions. Experimental results demonstrate that CAMBranch, trained with only 10\% of the complete dataset, exhibits superior performance. Ablation studies further validate the effectiveness of our method.

Probabilistic circuits (PCs) such as sum-product networks efficiently represent large multi-variate probability distributions. They are preferred in practice over other probabilistic representations such as Bayesian and Markov networks because PCs can solve marginal inference (MAR) tasks in time that scales linearly in the size of the network. Unfortunately, the maximum-a-posteriori (MAP) and marginal MAP (MMAP) tasks remain NP-hard in these models. Inspired by the recent work on using neural networks for generating near-optimal solutions to optimization problems such as integer linear programming, we propose an approach that uses neural networks to approximate (M)MAP inference in PCs. The key idea in our approach is to approximate the cost of an assignment to the query variables using a continuous multilinear function, and then use the latter as a loss function. The two main benefits of our new method are that it is self-supervised and after the neural network is learned, it requires only linear time to output a solution. We evaluate our new approach on several benchmark datasets and show that it outperforms three competing linear time approximations, max-product inference, max-marginal inference and sequential estimation, which are used in practice to solve MMAP tasks in PCs.

Locomotion gaits are fundamental for control of soft terrestrial robots. However, synthesis of these gaits is challenging due to modeling of robot-environment interaction and lack of a mathematical framework. This work presents an environment-centric, data-driven and fault-tolerant probabilistic Model-Free Control (pMFC) framework that allows for soft multi-limb robots to learn from their environment and synthesize diverse sets of locomotion gaits for realizing open-loop control. Here, discretization of factors dominating robot-environment interactions enables an environment-specific graphical representation where the edges encode experimental locomotion data corresponding to the robot motion primitives. In this graph, locomotion gaits are defined as simple cycles that are transformation invariant, i.e., the locomotion is independent of the starting vertex of these periodic cycles. Gait synthesis, the problem of finding optimal locomotion gaits for a given substrate, is formulated as Binary Integer Linear Programming (BILP) problems with a linearized cost function, linear constraints, and iterative simple cycle detection. Experimentally, gaits are synthesized for varying robot-environment interactions. Variables include robot morphology - three-limb and four-limb robots, TerreSoRo-III and TerreSoRo-IV; substrate - rubber mat, whiteboard and carpet; and actuator functionality - simulated loss of robot limb actuation. On an average, gait synthesis improves the translation and rotation speeds by 82% and 97% respectively. The results highlight that data-driven methods are vital to soft robot locomotion control due to the significant influence of unexpected asymmetries in the system and the dependence of optimal gait sequences on the experimental robot-environment interaction.

Correlation clustering is a powerful unsupervised learning paradigm that supports positive and negative similarities. In this paper, we assume the similarities are not known in advance. Instead, we employ active learning to iteratively query similarities in a cost-efficient way. In particular, we develop three effective acquisition functions to be used in this setting. One is based on the notion of inconsistency (i.e., when similarities violate the transitive property). The remaining two are based on information-theoretic quantities, i.e., entropy and information gain.

Diffusion models are a class of generative models that function by progressively introducing noise into data and then methodically demonising it [17, 11]. They have revolutionized high-fidelity creation of complex data, and their applications have rapidly expanded beyond mere image synthesis, finding relevance in areas such as engineering [22, 24], automation [3, 13], chemistry [1, 12], and scientific research [2, 6]. Although diffusion models excel at generating content that is coherent and aligns closely with the original data distribution, their direct application in scenarios requiring stringent adherence to predefined criteria poses significant challenges. Particularly in domains where the generated data needs to not only resemble real-world examples but also rigorously comply with established specifications, physical laws, or engineering principles, conventional diffusion models are unable to ensure this level of precision. Given these limitations, one may consider an alternative approach: training a diffusion model on a data distribution that already aligns with these constraints.