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Optimization · Non-Convex

Kaheon Kim, Rentian Yao, Changbo Zhu, Xiaohui Chen

The optimal transport barycenter (a.k.a. Wasserstein barycenter) is a fundamental notion of averaging that extends from the Euclidean space to the Wasserstein space of probability distributions. Computation of the *unregularized* barycenter for discretized probability distributions on point clouds is a challenging task when the domain dimension $d > 1$. Most practical algorithms for approximating the barycenter problem are based on entropic regularization. In this paper, we introduce a nearly linear time $O(m \log{m})$ and linear space complexity $O(m)$ primal-dual algorithm, the *Wasserstein-Descent $\dot{\mathbb{H}}^1$-Ascent* (WDHA) algorithm, for computing the *exact* barycenter when the input probability density functions are discretized on an $m$-point grid. The key success of the WDHA algorithm hinges on alternating between two different yet closely related Wasserstein and Sobolev optimization geometries for the primal barycenter and dual Kantorovich potential subproblems. Under reasonable assumptions, we establish the convergence rate and iteration complexity of WDHA to its stationary point when the step size is appropriately chosen. Superior computational efficacy, scalability, and accuracy over the existing Sinkhorn-type algorithms are demonstrated on high-resolution (e.g., $1024 \times 1024$ images) 2D synthetic and real data.

Theory · Domain Adaptation and Transfer Learning

Tyler Ingebrand, Adam Thorpe, Ufuk Topcu

A central challenge in transfer learning is designing algorithms that can quickly adapt and generalize to new tasks without retraining. Yet, the conditions of when and how algorithms can effectively transfer to new tasks is poorly characterized. We introduce a geometric characterization of transfer in Hilbert spaces and define three types of inductive transfer: interpolation within the convex hull, extrapolation to the linear span, and extrapolation outside the span. We propose a method grounded in the theory of function encoders to achieve all three types of transfer. Specifically, we introduce a novel training scheme for function encoders using least-squares optimization, prove a universal approximation theorem for function encoders, and provide a comprehensive comparison with existing approaches such as transformers and meta-learning on four diverse benchmarks. Our experiments demonstrate that the function encoder outperforms state-of-the-art methods on four benchmark tasks and on all three types of transfer.

Theory · Optimization

Binghui Peng, Aviad Rubinstein

We revisit the problem of selecting $k$-out-of-$n$ elements with the goal of optimizing an objective function, and ask whether it can be solved approximately with sublinear query complexity. For objective functions that are monotone submodular, [Li, Feldman, Kazemi, Karbasi, NeurIPS'22; Kuhnle, AISTATS'21] gave an $\Omega(n/k)$ query lower bound for approximating to within any constant factor. We strengthen their lower bound to a nearly tight $\tilde{\Omega}(n)$. This lower bound holds even for estimating the value of the optimal subset. When the objective function is additive, we prove that finding an approximately optimal subset still requires near-linear query complexity, but we can estimate the value of the optimal subset in $\tilde{O}(n/k)$ queries, and that this is tight up to polylog factors.

Theory · Reinforcement Learning and Planning

Gaspard Lambrechts, Damien Ernst, Aditya Mahajan

In reinforcement learning for partially observable environments, many successful algorithms have been developed within the asymmetric learning paradigm. This paradigm leverages additional state information available at training time for faster learning. Although the proposed learning objectives are usually theoretically sound, these methods still lack a precise theoretical justification for their potential benefits. We propose such a justification for asymmetric actor-critic algorithms with linear function approximators by adapting a finite-time convergence analysis to this setting. The resulting finite-time bound reveals that the asymmetric critic eliminates error terms arising from aliasing in the agent state.

General Machine Learning · Causality

Quang-Duy Tran, Bao Duong, Phuoc Nguyen, Thin Nguyen

Telling apart the cause and effect between two random variables with purely observational data is a challenging problem that finds applications in various scientific disciplines. A key principle utilized in this task is the algorithmic Markov condition, which postulates that the joint distribution, when factorized according to the causal direction, yields a more succinct codelength compared to the anti-causal direction. Previous approaches approximate these codelengths by relying on simple functions or Gaussian processes (GPs) with easily evaluable complexity, compromising between model fitness and computational complexity. To overcome these limitations, we propose leveraging the variational Bayesian learning of neural networks as an interpretation of the codelengths. Consequently, we can enhance the model fitness while promoting the succinctness of the codelengths, while avoiding the significant computational complexity of the GP-based approaches. Extensive experiments on both synthetic and real-world benchmarks in cause-effect identification demonstrate the effectiveness of our proposed method, surpassing the overall performance of related complexity-based and structural causal model regression-based approaches.

Deep Learning · Other Representation Learning

Zhemin Li, Liyuan Ma, Hongxia Wang, Yaoyun Zeng, 晓龙 韩

Implicit Neural Representations (INRs) rely heavily on architectural choices for good generalization. Developing theoretically grounded approaches for architecture design remains an active area of research. Via theoretical analysis of the infinite-width limit, we establish a methodology that characterizes INR's generalization by means of kernel alignment. We first formulate the optimal kernel that minimizes pointwise expected squared error, then demonstrate that the Neural Tangent Kernel of the composed function (INR with input encoding) can approximate any positive semidefinite dot-product kernels through input feature mapping adjustments. Building upon these insights, we propose a Kernel Alignment Regularizer (KAR) that naturally integrates with existing INR systems to enhance kernel alignment. We further develop Plug-in Encoding for Aligned Kernels (PEAK) to refine INR models with KAR using learnable input encoding. This work contributes to the ongoing research efforts in bridging theory and practice for principled INR architecture design. Code is available at https://github.com/lizhemin15/KAR.

Applications · Everything Else

Aleksandr Kolomeitsev, ANH-HUY PHAN

Topology optimization plays a crucial role in designing efficient and manufacturable structures. Traditional methods often yield free-form voids that, although providing design flexibility, introduce significant manufacturing challenges and require extensive post-processing. Conversely, feature-mapping topology optimization reduces post-processing efforts by constructing topologies using predefined geometric features. Nevertheless, existing approaches are significantly constrained by the limited set of geometric features available, the variety of parameters that each type of geometric feature can possess, and the necessity of employing differentiable signed distance functions. In this paper, we present a novel method that combines Neural Heaviside Signed Distance Functions (Heaviside SDFs) with structured latent shape representations to generate manufacturable voids directly within the optimization framework. Our architecture incorporates encoder and decoder networks to effectively approximate the Heaviside function and facilitate optimization within a unified latent space, thus addressing the feature diversity limitations of current feature-mapping techniques. Experimental results validate the effectiveness of our approach in balancing structural compliance, offering a new pathway to CAD-integrated design with minimal human intervention.

Theory · Reinforcement Learning and Planning

Yilei Chen, Aldo Pacchiano, Ioannis Paschalidis

We study the multiple-policy evaluation problem where we are given a set of $K$ policies and the goal is to evaluate their performance (expected total reward over a fixed horizon) to an accuracy $\epsilon$ with probability at least $1-\delta$. We propose an algorithm named CAESAR for this problem. Our approach is based on computing an approximately optimal sampling distribution and using the data sampled from it to perform the simultaneous estimation of the policy values. CAESAR has two phases. In the first phase, we produce coarse estimates of the visitation distributions of the target policies at a low order sample complexity rate that scales with $\tilde{O}(\frac{1}{\epsilon})$. In the second phase, we approximate the optimal sampling distribution and compute the importance weighting ratios for all target policies by minimizing a step-wise quadratic loss function inspired by the DualDICE objective. Up to low order and logarithmic terms CAESAR achieves a sample complexity $\tilde{O}\left(\frac{H^4}{\epsilon^2}\sum_{h=1}^H\max_{k\in[K]}\sum_{s,a}\frac{(d_h^{\pi^k}(s,a))^2}{\mu^*_h(s,a)}\right)$, where $d^{\pi}$ is the visitation distribution of policy $\pi$, $\mu^*$ is the optimal sampling distribution, and $H$ is the horizon.

Reinforcement Learning

Bang You, Puze Liu, Huaping Liu, Jan Peters, Oleg Arenz

Simplicity is a powerful inductive bias. In reinforcement learning, regularization is used for simpler policies, data augmentation for simpler representations, and sparse reward functions for simpler objectives, all that, with the underlying motivation to increase generalizability and robustness by focusing on the essentials. Supplementary to these techniques, we investigate how to promote simple behavior throughout the episode. To that end, we introduce a modification of the reinforcement learning problem that additionally maximizes the total correlation within the induced trajectories. We propose a practical algorithm that optimizes all models, including policy and state representation, based on a lower-bound approximation. In simulated robot environments, our method naturally generates policies that induce periodic and compressible trajectories, and that exhibit superior robustness to noise and changes in dynamics compared to baseline methods, while also improving performance in the original tasks.

Optimization · Discrete and Combinatorial Optimization

Qixin Zhang, Wei Huang, Can Jin, Puning Zhao, Yao Shu, Li Shen, Dacheng Tao

Identifying the most representative subset for a close-to-submodular objective while satisfying the predefined partition constraint is a fundamental task with numerous applications in machine learning. However, the existing distorted local-search methods are often hindered by their prohibitive query complexities and the rigid requirement for prior knowledge of difficult-to-obtain structural parameters. To overcome these limitations, we introduce a novel algorithm titled **Multinoulli-SCG**, which not only is parameter-free, but also can achieve the same approximation guarantees as the distorted local-search methods with significantly fewer function evaluations. The core of our **Multinoulli-SCG** algorithm is an innovative continuous-relaxation framework named Multinoulli Extension(***ME***), which can effectively convert the discrete subset selection problem subject to partition constraints into a solvable continuous maximization focused on learning the optimal multinoulli priors across the considered partition. In sharp contrast with the well-established multi-linear extension for submodular subset selection, a notable advantage of our proposed ***ME*** is its intrinsic capacity to provide a lossless rounding scheme for any set function. Finally, we validate the practical efficacy of our proposed algorithms by applying them to video summarization, bayesian A-optimal design and coverage maximization.

Deep Learning · Theory

Max Milkert, David Hyde, Forrest Laine

In a neural network with ReLU activations, the number of piecewise linear regions in the output can grow exponentially with depth.However, this is highly unlikely to happen when the initial parameters are sampled randomly, which therefore often leads to the use of networks that are unnecessarily large.To address this problem, we introduce a novel parameterization of the network that restricts its weights so that a depth $d$ network produces exactly $2^d$ linear regions at initialization and maintains those regions throughout training under the parameterization.This approach allows us to learn approximations of convex, one-dimensional functions that are several orders of magnitude more accurate than their randomly initialized counterparts.We further demonstrate a preliminary extension of our construction to multidimensional and non-convex functions, allowing the technique to replace traditional dense layers in various architectures.

General Machine Learning · Clustering

Yicheng Pan, Renjie Chen, Pengyu Long, Bingchen Fan

Overlap and hierarchy are two prevalent phenomena in clustering, and usually coexist in a single system. There are several studies on each of them separately, but it is unclear how to characterize and evaluate the hybrid structures yet. To address this issue, we initiate the study of hierarchical overlapping clustering on graphs by introducing a new cost function for it. We show the rationality of our cost function via several intuitive properties, and develop an approximation algorithm that achieves a constant approximation factor for its dual version. Our algorithm is a recursive process of overlapping bipartition based on local search, which makes a speed-up version of it extremely scalable. Our experiments demonstrate that the speed-up algorithm has significantly better performances than all the baseline methods in both effectiveness and scalability on synthetic and real datasets.

Probabilistic Methods · Bayesian Models and Methods

Emilia Magnani, Marvin Pförtner, Tobias Weber, Philipp Hennig

Neural operators generalize neural networks to learn mappings between function spaces from data. They are commonly used to learn solution operators of parametric partial differential equations (PDEs) or propagators of time-dependent PDEs. However, to make them useful in high-stakes simulation scenarios, their inherent predictive error must be quantified reliably. We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators. Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator's predictions.We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief. Our framework provides a practical yet theoretically sound way to apply existing Bayesian deep learning methods such as the linearized Laplace approximation to neural operators. Just as the underlying neural operator, our approach is resolution-agnostic by design.The method adds minimal prediction overhead, can be applied post-hoc without retraining the network, and scales to large models and datasets.We evaluate these aspects in a case study on Fourier neural operators.

Deep Learning

Mohammad Mozaffari, Amir Yazdanbakhsh, Maryam Mehri Dehnavi

Conventional model compression techniques for LLMs address high memory consumption and slow inference challenges but typically require computationally expensive retraining to preserve accuracy. In contrast, one-shot compression methods eliminate retraining cost, but struggle to achieve accuracy comparable to dense models. This paper presents SLIM, a new one-shot compression framework that holistically integrates hardware-friendly quantization, sparsity, and low-rank approximation into a unified process. First, we formulate the quantization process using a probabilistic approach (SLIM-Quant) that enables us to apply uniform quantization. Then, we use an existing one-shot pruning method to apply semi-structured sparsity on top of the quantized weights. Finally, to compensate for the introduced aggregated quantization and sparsity error, we use a novel saliency function with unique invertible and additive features that enables us tomathematically compute the value of low-rank adapters. SLIM improves model accuracy by up to 5.66% (LLaMA-2-7B) for 2:4 sparsity with 4-bit weight quantization, outperforming prior methods. Models compressed with SLIM achieve up to 4.3× and 3.8× on Nvidia RTX3060 and A100 GPUs, respectively. Additionally, they achieve up to 0.23× end-to-end memory reduction in comparison to their dense counterparts. We also propose an optional PEFT recipe that further improves accuracyby up to 1.66% (LLaMA-2-13B) compared to SLIM without fine-tuning.

General Machine Learning · Representation Learning

Hengrui Luo, Jeremy E. Purvis, Didong Li

Modern datasets often exhibit high dimensionality, yet the data reside in low-dimensional manifolds that can reveal underlying geometric structures critical for data analysis. A prime example of such a dataset is a collection of cell cycle measurements, where the inherently cyclical nature of the process can be represented as a circle or sphere. Motivated by the need to analyze these types of datasets, we propose a nonlinear dimension reduction method, Spherical Rotation Component Analysis (SRCA), that incorporates geometric information to better approximate low-dimensional manifolds. SRCA is a versatile method designed to work in both high-dimensional and small sample size settings. By employing spheres or ellipsoids, SRCA provides a low-rank spherical representation of the data with general theoretic guarantees, effectively retaining the geometric structure of the dataset during dimensionality reduction. A comprehensive simulation study, along with a successful application to human cell cycle data, further highlights the advantages of SRCA compared to state-of-the-art alternatives, demonstrating its superior performance in approximating the manifold while preserving inherent geometric structures.

Deep Learning

Shiba Biswal, Karthik Elamvazhuthi, Rishi Sonthalia

This paper investigates the use of transformers to approximate the mean-field dynamics of interacting particle systems exhibiting collective behavior. Such systems are fundamental in modeling phenomena across physics, biology, and engineering, including opinion formation, biological networks, and swarm robotics. The key characteristic of these systems is that the particles are indistinguishable, leading to permutation-equivariant dynamics. First, we empirically demonstrate that transformers are well-suited for approximating a variety of mean field models, including the Cucker-Smale model for flocking and milling, and the mean-field system for training two-layer neural networks. We validate our numerical experiments via mathematical theory. Specifically, we prove that if a finite-dimensional transformer effectively approximates the finite-dimensional vector field governing the particle system, then the $L_\infty$ distance between the \textit{expected transformer} and the infinite-dimensional mean-field vector field can be bounded by a function of the number of particles observed during training. Leveraging this result, we establish theoretical bounds on the distance between the true mean-field dynamics and those obtained using the transformer.

Theory · Reinforcement Learning and Planning

Kihyuk Hong, Ambuj Tewari

We study reinforcement learning in infinite-horizon average-reward settings with linear MDPs. Previous work addresses this problem by approximating the average-reward setting by discounted setting and employing a value iteration-based algorithm that uses clipping to constrain the span of the value function for improved statistical efficiency. However, the clipping procedure requires computing the minimum of the value function over the entire state space, which is prohibitive since the state space in linear MDP setting can be large or even infinite. In this paper, we introduce a value iteration method with efficient clipping operation that only requires computing the minimum of value functions over the set of states visited by the algorithm. Our algorithm enjoys the same regret bound as the previous work while being computationally efficient, with computational complexity that is independent of the size of the state space.

Applications · Chemistry, Physics, and Earth Sciences

Prateek Gupta, Andrea Ferrari, Nabil Iqbal

The notion of duality -- that a given physical system can have two different mathematical descriptions -- is a key idea in modern theoretical physics. Establishing a duality in lattice statistical mechanics models requires the construction of a dual Hamiltonian and a map from the original to the dual observables. By using neural networks to parameterize these maps and introducing a loss function that penalises the difference between correlation functions in original and dual models, we formulate the process of duality discovery as an optimization problem. We numerically solve this problem and show that our framework can rediscover the celebrated Kramers-Wannier duality for the 2d Ising model, numerically reconstructing the known mapping of temperatures. We further investigate the 2d Ising model deformed by a plaquette coupling and find families of ``approximate duals''. We discuss future directions and prospects for discovering new dualities within this framework.

General Machine Learning

Chao Yang, Shuting Cui, Yang Yang, Shuang Li

Understanding human mental states—such as intentions and desires—is crucial for natural AI-human collaboration. However, this is challenging because human actions occur irregularly over time, and the underlying mental states that drive these actions are unobserved. To tackle this, we propose a novel framework that combines a logic-informed temporal point process (TPP) with amortized variational Expectation-Maximization (EM). Our key innovation is integrating logic rules as priors to guide the TPP’s intensity function, allowing the model to capture the interplay between actions and mental events while reducing dependence on large datasets. To handle the intractability of mental state inference, we introduce a discrete-time renewal process to approximate the posterior. By jointly optimizing model parameters, logic rules, and inference networks, our approach infers entire mental event sequences and adaptively predicts future actions. Experiments on both synthetic and real-world datasets show that our method outperforms existing approaches in accurately inferring mental states and predicting actions, demonstrating its effectiveness in modeling human cognitive processes.

Optimization · Discrete and Combinatorial Optimization

Alex Crane, Thomas Stanley, Blair D. Sullivan, Nate Veldt

We consider a framework for clustering edge-colored hypergraphs, where the goal is to cluster (equivalently, to *color*) objects based on the primary type of multiway interactions they participate in. One well-studied objective is to color nodes to minimize the number of *unsatisfied* hyperedges---those containing one or more nodes whose color does not match the hyperedge color. We motivate and present advances for several directions that extend beyond this minimization problem. We first provide new algorithms for maximizing *satisfied* edges, which is the same at optimality but is much more challenging to approximate, with all prior work restricted to graphs. We develop the first approximation algorithm for hypergraphs, and then refine it to improve the best-known approximation factor for graphs. We then introduce new objective functions that incorporate notions of balance and fairness, and provide new hardness results, approximations, and fixed-parameter tractability results.