Permutation symmetries of deep networks make basic operations like model merging and similarity estimation challenging. In many cases, aligning the weights of the networks, i.e., finding optimal permutations between their weights, is necessary. Unfortunately, weight alignment is an NP-hard problem. Prior research has mainly focused on solving relaxed versions of the alignment problem, leading to either time-consuming methods or sub-optimal solutions. To accelerate the alignment process and improve its quality, we propose a novel framework aimed at learning to solve the weight alignment problem, which we name Deep-Align. To that end, we first prove that weight alignment adheres to two fundamental symmetries and then, propose a deep architecture that respects these symmetries. Notably, our framework does not require any labeled data. We provide a theoretical analysis of our approach and evaluate Deep-Align on several types of network architectures and learning setups. Our experimental results indicate that a feed-forward pass with Deep-Align produces better or equivalent alignments compared to those produced by current optimization algorithms. Additionally, our alignments can be used as an effective initialization for other methods, leading to improved solutions with a significant speedup in convergence.

Concretely, we prove that polynomial-sized message-passing algorithms can represent the most powerful polynomial time algorithms for Max Constraint Satisfaction Problems assuming the Unique Games Conjecture. We leverage this result to construct efficient graph neural network architectures, OptGNN, that obtain highquality approximate solutions on landmark combinatorial optimization problems such as Max-Cut, Min-Vertex-Cover, and Max-3-SAT. Our approach achieves strong empirical results across a wide range of real-world and synthetic datasets against solvers and neural baselines. Finally, we take advantage of OptGNN's ability to capture convex relaxations to design an algorithm for producing bounds on the optimal solution from the learned embeddings of OptGNN. Combinatorial Optimization (CO) is the class of problems that optimize functions subject to constraints over discrete search spaces. They are often NP-hard to solve and to approximate, owing to their typically exponential search ...

This paper considers the best policy identification (BPI) problem in online Constrained Markov Decision Processes (CMDPs). We are interested in algorithms that are model-free, have low regret, and identify an approximately optimal policy with a high probability. Existing model-free algorithms for online CMDPs with sublinear regret and constraint violation do not provide any convergence guarantee to an optimal policy and provide only average performance guarantees when a policy is uniformly sampled at random from all previously used policies. In this paper, we develop a new algorithm, named Pruning-Refinement-Identification (PRI), based on a fundamental structural property of CMDPs proved before, which we call limited stochasticity. The property says for a CMDP with $N$ constraints, there exists an optimal policy with at most $N$ stochastic decisions. The proposed algorithm first identifies at which step and in which state a stochastic decision has to be taken and then fine-tunes the distributions of these stochastic decisions. PRI achieves trio objectives: (i) PRI is a model-free algorithm; and (ii) it outputs an approximately optimal policy with a high probability at the end of learning; and (iii) PRI guarantees $\tilde{\mathcal{O}}(H\sqrt{K})$ regret and constraint violation, which significantly improves the best existing regret bound $\tilde{\mathcal{O}}(H^4 \sqrt{SA}K^{\frac{4}{5}})$ under a model-free algorithm, where $H$ is the length of each episode, $S$ is the number of states, $A$ is the number of actions, and the total number of episodes during learning is $2K+\tilde{\cal O}(K^{0.25}).$

Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower's best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer linear bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for the bilevel knapsack interdiction problem, the "critical node game" from network security, a donor-recipient healthcare problem, and discrete network design from transportation planning. These problems are diverse in that they have linear or non-linear objectives/constraints and integer or mixed-integer variables, making Neur2BiLO unique in its versatility.

Accurate modeling of system dynamics holds intriguing potential in broad scientific fields including cytodynamics and fluid mechanics. This task often presents significant challenges when (i) observations are limited to cross-sectional samples (where individual trajectories are inaccessible for learning), and moreover, (ii) the behaviors of individual particles are heterogeneous (especially in biological systems due to biodiversity). To address them, we introduce a novel framework dubbed correlational Lagrangian Schr\"odinger bridge (CLSB), aiming to seek for the evolution "bridging" among cross-sectional observations, while regularized for the minimal population "cost". In contrast to prior methods relying on \textit{individual}-level regularizers for all particles \textit{homogeneously} (e.g. restraining individual motions), CLSB operates at the population level admitting the heterogeneity nature, resulting in a more generalizable modeling in practice. To this end, our contributions include (1) a new class of population regularizers capturing the temporal variations in multivariate relations, with the tractable formulation derived, (2) three domain-informed instantiations based on genetic co-expression stability, and (3) an integration of population regularizers into data-driven generative models as constrained optimization, and a numerical solution, with further extension to conditional generative models. Empirically, we demonstrate the superiority of CLSB in single-cell sequencing data analyses such as simulating cell development over time and predicting cellular responses to drugs of varied doses.

The proposed method shows a stateof-the-art score of 16.213, significantly outperforming the reported best score in the benchmark genetic algorithms (e.g., Jensen, 2019). of 15.185, in practical molecular optimization The recent work of Gao et al. (2022a) proposes a practical (PMO), which is an official benchmark for molecular optimization (PMO) benchmark, emphasizing sample-efficient molecular optimization. Remarkably, the importance of sample efficiency in de novo molecular ours exceeds all baselines, including reinforcement optimization for practical applicability. The benchmark is learning, Bayesian optimization, generative reasonable because real-world applications of molecule optimization models, GFlowNets, and genetic algorithms, (e.g., drug discovery) require expensive scoring in 14 out of 23 tasks. Our code is available at processes such as wet lab experiments.

Controllable Pareto front learning (CPFL) approximates the Pareto solution set and then locates a Pareto optimal solution with respect to a given reference vector. However, decision-maker objectives were limited to a constraint region in practice, so instead of training on the entire decision space, we only trained on the constraint region. Controllable Pareto front learning with Split Feasibility Constraints (SFC) is a way to find the best Pareto solutions to a split multi-objective optimization problem that meets certain constraints. In the previous study, CPFL used a Hypernetwork model comprising multi-layer perceptron (Hyper-MLP) blocks. With the substantial advancement of transformer architecture in deep learning, transformers can outperform other architectures in various tasks. Therefore, we have developed a hyper-transformer (Hyper-Trans) model for CPFL with SFC. We use the theory of universal approximation for the sequence-to-sequence function to show that the Hyper-Trans model makes MED errors smaller in computational experiments than the Hyper-MLP model.

Stochastic dominance models risk-averse preferences for decision making with uncertain outcomes, which naturally captures the intrinsic structure of the underlying uncertainty, in contrast to simply resorting to the expectations. Despite theoretically appealing, the application of stochastic dominance in machine learning has been scarce, due to the following challenges: $\textbf{i)}$, the original concept of stochastic dominance only provides a $\textit{partial order}$, therefore, is not amenable to serve as an optimality criterion; and $\textbf{ii)}$, an efficient computational recipe remains lacking due to the continuum nature of evaluating stochastic dominance.%, which barriers its application for machine learning. In this work, we make the first attempt towards establishing a general framework of learning with stochastic dominance. We first generalize the stochastic dominance concept to enable feasible comparisons between any arbitrary pair of random variables. We next develop a simple and computationally efficient approach for finding the optimal solution in terms of stochastic dominance, which can be seamlessly plugged into many learning tasks. Numerical experiments demonstrate that the proposed method achieves comparable performance as standard risk-neutral strategies and obtains better trade-offs against risk across a variety of applications including supervised learning, reinforcement learning, and portfolio optimization.

It is widely known that state-of-the-art machine learning models, including vision and language models, can be seriously compromised by adversarial perturbations. It is therefore increasingly relevant to develop capabilities to certify their performance in the presence of the most effective adversarial attacks. Our paper offers a new approach to certify the performance of machine learning models in the presence of adversarial attacks with population level risk guarantees. In particular, we introduce the notion of $(\alpha,\zeta)$ machine learning model safety. We propose a hypothesis testing procedure, based on the availability of a calibration set, to derive statistical guarantees providing that the probability of declaring that the adversarial (population) risk of a machine learning model is less than $\alpha$ (i.e. the model is safe), while the model is in fact unsafe (i.e. the model adversarial population risk is higher than $\alpha$), is less than $\zeta$. We also propose Bayesian optimization algorithms to determine efficiently whether a machine learning model is $(\alpha,\zeta)$-safe in the presence of an adversarial attack, along with statistical guarantees. We apply our framework to a range of machine learning models including various sizes of vision Transformer (ViT) and ResNet models impaired by a variety of adversarial attacks, such as AutoAttack, SquareAttack and natural evolution strategy attack, to illustrate the operation of our approach. Importantly, we show that ViT's are generally more robust to adversarial attacks than ResNets, and ViT-large is more robust than smaller models. Our approach goes beyond existing empirical adversarial risk-based certification guarantees. It formulates rigorous (and provable) performance guarantees that can be used to satisfy regulatory requirements mandating the use of state-of-the-art technical tools.

This paper introduces Dual Interior Point Learning (DIPL) and Dual Supergradient Learning (DSL) to learn dual feasible solutions to parametric linear programs with bounded variables, which are pervasive across many industries. DIPL mimics a novel dual interior point algorithm while DSL mimics classical dual supergradient ascent. DIPL and DSL ensure dual feasibility by predicting dual variables associated with the constraints then exploiting the flexibility of the duals of the bound constraints. DIPL and DSL complement existing primal learning methods by providing a certificate of quality. They are shown to produce high-fidelity dual-feasible solutions to large-scale optimal power flow problems providing valid dual bounds under 0.5% optimality gap.