论文检索

Theory · Domain Adaptation and Transfer Learning

Zekun CAI, Guangji Bai, Renhe Jiang, Xuan Song, Liang Zhao

Temporal Domain Generalization (TDG) addresses the challenge of training predictive models under temporally varying data distributions. Traditional TDG approaches typically focus on domain data collected at fixed, discrete time intervals, which limits their capability to capture the inherent dynamics within continuous-evolving and irregularly-observed temporal domains. To overcome this, this work formalizes the concept of Continuous Temporal Domain Generalization (CTDG), where domain data are derived from continuous times and are collected at arbitrary times. CTDG tackles critical challenges including: 1) Characterizing the continuous dynamics of both data and models, 2) Learning complex high-dimensional nonlinear dynamics, and 3) Optimizing and controlling the generalization across continuous temporal domains. To address them, we propose a Koopman operator-driven continuous temporal domain generalization (Koodos) framework. We formulate the problem within a continuous dynamic system and leverage the Koopman theory to learn the underlying dynamics; the framework is further enhanced with a comprehensive optimization strategy equipped with analysis and control driven by prior knowledge of the dynamics patterns. Extensive experiments demonstrate the effectiveness and efficiency of our approach. The code can be found at: https://github.com/Zekun-Cai/Koodos.

Deep Learning · Representation Learning

Li Jiao, Qiuxia LAI, YU LI, Qiang Xu

Continual learning requires to overcome catastrophic forgetting when training a single model on a sequence of tasks. Recent top-performing approaches are prompt-based methods that utilize a set of learnable parameters (i.e., prompts) to encode task knowledge, from which appropriate ones are selected to guide the fixed pre-trained model in generating features tailored to a certain task. However, existing methods rely on predicting prompt identities for prompt selection, where the identity prediction process cannot be optimized with task loss. This limitation leads to sub-optimal prompt selection and inadequate adaptation of pre-trained features for a specific task. Previous efforts have tried to address this by directly generating prompts from input queries instead of selecting from a set of candidates. However, these prompts are continuous, which lack sufficient abstraction for task knowledge representation, making them less effective for continual learning. To address these challenges, we propose VQ-Prompt, a prompt-based continual learning method that incorporates Vector Quantization (VQ) into end-to-end training of a set of discrete prompts. In this way, VQ-Prompt can optimize the prompt selection process with task loss and meanwhile achieve effective abstraction of task knowledge for continual learning. Extensive experiments show that VQ-Prompt outperforms state-of-the-art continual learning methods across a variety of benchmarks under the challenging class-incremental setting.

Optimization

Stephen Boyd, Tetiana Parshakova, Ernest Ryu, Jaewook J. Suh

We present a novel methodology for convex optimization algorithm design using ideas from electric RLC circuits. Given an optimization problem, the first stage of the methodology is to design an appropriate electric circuit whose continuous-time dynamics converge to the solution of the optimization problem at hand. Then, the second stage is an automated, computer-assisted discretization of the continuous-time dynamics, yielding a provably convergent discrete-time algorithm. Our methodology recovers many classical (distributed) optimization algorithms and enables users to quickly design and explore a wide range of new algorithms with convergence guarantees.

Optimization · Discrete and Combinatorial Optimization

Anna Arutyunova, Jan Eube, Heiko Röglin, Melanie Schmidt, Sarah Sturm, Julian Wargalla

As a major unsupervised learning method, clustering has received a lot of attention over multiple decades. The various clustering problems that have been studied intensively include, e.g., the $k$-means problem and the $k$-center problem. However, in applications, it is common that good clusterings should optimize multiple objectives (e.g., visualizing data on a map by clustering districts into areas that are both geographically compact but also homogeneous with respect to the data). We study combinations of different objectives, for example optimizing $k$-center and $k$-means simultaneously or optimizing $k$-center with respect to two different metrics. Usually these objectives are conflicting and cannot be optimized simultaneously, making it necessary to find trade-offs. We develop novel algorithms for computing the set of Pareto-optimal solutions (approximately) for various combinations of two objectives. Our algorithms achieve provable approximation guarantees and we demonstrate in several experiments that the (approximate) Pareto set contains good clusterings that cannot be found by considering one of the objectives separately.

Theory · Statistics

Parnian Kassraie, Aram-Alexandre Pooladian, Michal Klein, James Thornton, Jonathan Niles-Weed, Marco Cuturi

Optimal transport (OT) has profoundly impacted machine learning by providing theoretical and computational tools to realign datasets.In this context, given two large point clouds of sizes $n$ and $m$ in $\mathbb{R}^d$, entropic OT (EOT) solvers have emerged as the most reliable tool to either solve the Kantorovich problem and output a $n\times m$ coupling matrix, or to solve the Monge problem and learn a vector-valued push-forward map. While the robustness of EOT couplings/maps makes them a go-to choice in practical applications, EOT solvers remain difficult to tune because of a small but influential set of hyperparameters, notably the omnipresent entropic regularization strength $\varepsilon$. Setting $\varepsilon$ can be difficult, as it simultaneously impacts various performance metrics, such as compute speed, statistical performance, generalization, and bias. In this work, we propose a new class of EOT solvers (ProgOT), that can estimate both plans and transport maps.We take advantage of several opportunities to optimize the computation of EOT solutions by *dividing* mass displacement using a time discretization, borrowing inspiration from dynamic OT formulations, and *conquering* each of these steps using EOT with properly scheduled parameters. We provide experimental evidence demonstrating that ProgOT is a faster and more robust alternative to *standard solvers* when computing couplings at large scales, even outperforming neural network-based approaches. We also prove statistical consistency of our approach for estimating OT maps.

Learning Theory

Rayna Andreeva, Benjamin Dupuis, Rik Sarkar, Tolga Birdal, Umut Simsekli

We present a novel set of rigorous and computationally efficient topology-based complexity notions that exhibit a strong correlation with the generalization gap in modern deep neural networks (DNNs). DNNs show remarkable generalization properties, yet the source of these capabilities remains elusive, defying the established statistical learning theory. Recent studies have revealed that properties of training trajectories can be indicative of generalization. Building on this insight, state-of-the-art methods have leveraged the topology of these trajectories, particularly their fractal dimension, to quantify generalization. Most existing works compute this quantity by assuming continuous- or infinite-time training dynamics, complicating the development of practical estimators capable of accurately predicting generalization without access to test data. In this paper, we respect the discrete-time nature of training trajectories and investigate the underlying topological quantities that can be amenable to topological data analysis tools. This leads to a new family of reliable topological complexity measures that provably bound the generalization error, eliminating the need for restrictive geometric assumptions. These measures are computationally friendly, enabling us to propose simple yet effective algorithms for computing generalization indices. Moreover, our flexible framework can be extended to different domains, tasks, and architectures. Our experimental results demonstrate that our new complexity measures exhibit a strong correlation with generalization error in industry-standard architectures such as transformers and deep graph networks. Our approach consistently outperforms existing topological bounds across a wide range of datasets, models, and optimizers, highlighting the practical relevance and effectiveness of our complexity measures.

Optimization · Discrete and Combinatorial Optimization

Tam Nguyen, Anh-Dzung Doan, zhipeng cai, Tat-Jun Chin

Neuromorphic computers open up the potential of energy-efficient computation using spiking neural networks (SNN), which consist of neurons that exchange spike-based information asynchronously. In particular, SNNs have shown promise in solving combinatorial optimization. Underpinning the SNN methods is the concept of energy minimization of an Ising model, which is closely related to quadratic unconstrained binary optimization (QUBO). Thus, the starting point for many SNN methods is reformulating the target problem as QUBO, then executing an SNN-based QUBO solver. For many combinatorial problems, the reformulation entails introducing penalty terms, potentially with slack variables, that implement feasibility constraints in the QUBO objective. For more complex problems such as hypergraph minimum vertex cover (HMVC), numerous slack variables are introduced which drastically increase the search domain and reduce the effectiveness of the SNN solver. In this paper, we propose a novel SNN formulation for HMVC. Rather than using penalty terms with slack variables, our SNN architecture introduces additional spiking neurons with a constraint checking and correction mechanism that encourages convergence to feasible solutions. In effect, our method obviates the need for reformulating HMVC as QUBO. Experiments on neuromorphic hardware show that our method consistently yielded high quality solutions for HMVC on real and synthetic instances where the SNN-based QUBO solver often failed, while consuming measurably less energy than global solvers on CPU.

Optimization · Discrete and Combinatorial Optimization

Yannan Chen, Beichen Huang, Licheng Zhao, Kaiming Shen

The Normalized cut (NCut) problem is a fundamental and yet notoriously difficult one in the unsupervised clustering field. Because the NCut problem is fractionally structured, the fractional programming (FP) based approach has worked its way into a new frontier. However, the conventional FP techniques are insufficient: the classic Dinkelbach's transform can only deal with a single ratio and hence is limited to the two-class clustering, while the state-of-the-art quadratic transform accounts for multiple ratios but fails to convert the NCut problem to a tractable form. This work advocates a novel extension of the quadratic transform to the multidimensional ratio case, thereby recasting the fractional 0-1 NCut problem into a bipartite matching problem---which can be readily solved in an iterative manner. Furthermore, we explore the connection between the proposed multidimensional FP method and the minorization-maximization theory to verify the convergence.

Optimization · Non-Convex

Lingkai Kong, Molei Tao

Explicit, momentum-based dynamics that optimize functions defined on Lie groups can be constructed via variational optimization and momentum trivialization. Structure preserving time discretizations can then turn this dynamics into optimization algorithms. This article investigates two types of discretization, Lie Heavy-Ball, which is a known splitting scheme, and Lie NAG-SC, which is newly proposed. Their convergence rates are explicitly quantified under $L$-smoothness and \emph{local} strong convexity assumptions. Lie NAG-SC provides acceleration over the momentumless case, i.e. Riemannian gradient descent, but Lie Heavy-Ball does not. When compared to existing accelerated optimizers for general manifolds, both Lie Heavy-Ball and Lie NAG-SC are computationally cheaper and easier to implement, thanks to their utilization of group structure. Only gradient oracle and exponential map are required, but not logarithm map or parallel transport which are computational costly.

Learning Theory

Lasse Vuursteen

We study distributed goodness-of-fit testing for discrete distribution under bandwidth and differential privacy constraints. Information constraint distributed goodness-of-fit testing is a problem that has received considerable attention recently. The important case of discrete distributions is theoretically well understood in the classical case where all data is available in one "central" location. In a federated setting, however, data is distributed across multiple "locations" (e.g. servers) and cannot readily be shared due to e.g. bandwidth or privacy constraints that each server needs to satisfy. We show how recently derived results for goodness-of-fit testing for the mean of a multivariate Gaussian model extend to the discrete distributions, by leveraging Le Cam's theory of statistical equivalence. In doing so, we derive matching minimax upper- and lower-bounds for the goodness-of-fit testing for discrete distributions under bandwidth or privacy constraints in the regime where number of samples held locally are large.

Optimization · Zero-order and Black-box Optimization

Jaewon Chu, Jinyoung Park, Seunghun Lee, Hyunwoo Kim

Latent Bayesian optimization (LBO) approaches have successfully adopted Bayesian optimization over a continuous latent space by employing an encoder-decoder architecture to address the challenge of optimization in a high dimensional or discrete input space. LBO learns a surrogate model to approximate the black-box objective function in the latent space. However, we observed that most LBO methods suffer from the `misalignment problem', which is induced by the reconstruction error of the encoder-decoder architecture. It hinders learning an accurate surrogate model and generating high-quality solutions. In addition, several trust region-based LBO methods select the anchor, the center of the trust region, based solely on the objective function value without considering the trust region's potential to enhance the optimization process. To address these issues, we propose $\textbf{Inv}$ersion-based Latent $\textbf{B}$ayesian $\textbf{O}$ptimization (InvBO), a plug-and-play module for LBO. InvBO consists of two components: an inversion method and a potential-aware trust region anchor selection. The inversion method searches the latent code that completely reconstructs the given target data. The potential-aware trust region anchor selection considers the potential capability of the trust region for better local optimization. Experimental results demonstrate the effectiveness of InvBO on nine real-world benchmarks, such as molecule design and arithmetic expression fitting tasks. Code is available at https://github.com/mlvlab/InvBO.

Physical Models · Physics

Baiyu Su, Qiang Liu

This paper introduces the Quadratic Quantum Variational Monte Carlo (Q$^2$VMC) algorithm, an innovative algorithm in quantum chemistry that significantly enhances the efficiency and accuracy of solving the Schrödinger equation. Inspired by the discretization of imaginary-time Schrödinger evolution, Q$^2$VMC employs a novel quadratic update mechanism that integrates seamlessly with neural network-based ansatzes. Our extensive experiments showcase Q$^2$VMC's superior performance, achieving faster convergence and lower ground state energies in wavefunction optimization across various molecular systems, without additional computational cost. This study not only advances the field of computational quantum chemistry but also highlights the important role of discretized evolution in variational quantum algorithms, offering a scalable and robust framework for future quantum research.

Optimization · Discrete and Combinatorial Optimization

Mingyang Zhou, Weiji Cao, Hao Liao, Rui Mao

The influence maximization (IM) problem aims to identify a budgeted set of nodes with the highest potential to influence the largest number of users in a cascade model, a key challenge in viral marketing. Traditional \emph{IM} approaches consider each user/node independently as a potential target customer. However, in many scenarios, the target customers comprise motifs, where activating only one or a few users within a motif is insufficient for effective viral marketing, which, nevertheless, receives little attention. For instance, if a motif of three friends planning to dine together, targeting all three simultaneously is crucial for a restaurant advertisement to succeed.In this paper, we address the motif-oriented influence maximization problem under the linear threshold model. We prove that the motif-oriented IM problem is NP-hard and that the influence function is neither supermodular nor submodular, in contrast to the classical \emph{IM} setting.To simplify the problem, we establish the submodular upper and lower bounds for the influence function. By leveraging the submodular property, we propose a natural greedy strategy that simultaneously maximizes both bounds. Our algorithm has an approximation ratio of $\tau\cdot (1-1/e-\varepsilon)$ and a near-linear time complexity of $O((k+l)(m+\eta)\log \eta/\varepsilon^2)$.Experimental results on diverse datasets confirm the effectiveness of our approach in motif maximization.

Deep Learning · Robustness

Thomas Kwa, Drake Thomas, Adrià Garriga-Alonso

When applying reinforcement learning from human feedback (RLHF), the reward is learned from data and, therefore, always has some error. It is common to mitigate this by regularizing the policy with KL divergence from a base model, with the hope that balancing reward with regularization will achieve desirable outcomes despite this reward misspecification. We show that when the reward function has light-tailed error, optimal policies under less restrictive KL penalties achieve arbitrarily high utility. However, if error is heavy-tailed, some policies obtain arbitrarily high reward despite achieving no more utility than the base model--a phenomenon we call catastrophic Goodhart. We adapt a discrete optimization method to measure the tails of reward models, finding that they are consistent with light-tailed error. However, the pervasiveness of heavy-tailed distributions in many real-world applications indicates that future sources of RL reward could have heavy-tailed error, increasing the likelihood of reward hacking even with KL regularization.

Theory · Game Theory

Angelos Assos, Yuval Dagan, Constantinos Daskalakis

Learning algorithms are often used to make decisions in sequential decision-making environments. In multi-agent settings, the decisions of each agent can affect the utilities/losses of the other agents. Therefore, if an agent is good at anticipating the behavior of the other agents, in particular how they will make decisions in each round as a function of their experience that far, it could try to judiciously make its own decisions over the rounds of the interaction so as to influence the other agents to behave in a way that ultimately benefits its own utility. In this paper, we study repeated two-player games involving two types of agents: a learner, which employs an online learning algorithm to choose its strategy in each round; and an optimizer, which knows the learner's utility function and the learner's online learning algorithm. The optimizer wants to plan ahead to maximize its own utility, while taking into account the learner's behavior. We provide two results: a positive result for repeated zero-sum games and a negative result for repeated general-sum games. Our positive result is an algorithm for the optimizer, which exactly maximizes its utility against a learner that plays the Replicator Dynamics --- the continuous-time analogue of Multiplicative Weights Update (MWU). Additionally, we use this result to provide an algorithm for the optimizer against MWU, i.e.~for the discrete-time setting, which guarantees an average utility for the optimizer that is higher than the value of the one-shot game. Our negative result shows that, unless P=NP, there is no Fully Polynomial Time Approximation Scheme (FPTAS) for maximizing the utility of an optimizer against a learner that best-responds to the history in each round. Yet, this still leaves open the question of whether there exists a polynomial-time algorithm that optimizes the utility up to $o(T)$.

Optimization · Discrete and Combinatorial Optimization

Franziska Eberle, Felix Hommelsheim, Alexander Lindermayr, Zhenwei Liu, Nicole Megow, Jens Schlöter

Querying complex models for precise information (e.g. traffic models, database systems, large ML models) often entails intense computations and results in long response times. Thus, weaker models which give imprecise results quickly can be advantageous, provided inaccuracies can be resolved using few queries to a stronger model. In the fundamental problem of computing a maximum-weight basis of a matroid, a well-known generalization of many combinatorial optimization problems, algorithms have access to a clean oracle to query matroid information. We additionally equip algorithms with a fast but dirty oracle. We design and analyze practical algorithms which only use few clean queries w.r.t. the quality of the dirty oracle, while maintaining robustness against arbitrarily poor dirty oracles, approaching the performance of classic algorithms for the given problem. Notably, we prove that our algorithms are, in many respects, best-possible. Further, we outline extensions to other matroid oracle types, non-free dirty oracles and other matroid problems.

Optimization · Discrete and Combinatorial Optimization

Xiang Meng, Kayhan Behdin, Haoyue Wang, Rahul Mazumder

The impressive performance of Large Language Models (LLMs) across various natural language processing tasks comes at the cost of vast computational resources and storage requirements. One-shot pruning techniques offer a way to alleviate these burdens by removing redundant weights without the need for retraining. Yet, the massive scale of LLMs often forces current pruning approaches to rely on heuristics instead of optimization-based techniques, potentially resulting in suboptimal compression. In this paper, we introduce ALPS, an optimization-based framework that tackles the pruning problem using the operator splitting technique and a preconditioned conjugate gradient-based post-processing step. Our approach incorporates novel techniques to accelerate and theoretically guarantee convergence while leveraging vectorization and GPU parallelism for efficiency. ALPS substantially outperforms state-of-the-art methods in terms of the pruning objective and perplexity reduction, particularly for highly sparse models. On the LLaMA3-8B model with 70\% sparsity, ALPS achieves a 29\% reduction in test perplexity on the WikiText dataset and a 8\% improvement in zero-shot benchmark performance compared to existing methods. Our code is available at https://github.com/mazumder-lab/ALPS.

Lenart Treven, Bhavya, Yarden As, Florian Dorfler, Andreas Krause

Reinforcement learning (RL) excels in optimizing policies for discrete-time Markov decision processes (MDP). However, various systems are inherently continuous in time, making discrete-time MDPs an inexact modeling choice. In many applications, such as greenhouse control or medical treatments, each interaction (measurement or switching of action) involves manual intervention and thus is inherently costly. Therefore, we generally prefer a time-adaptive approach with fewer interactions with the system.In this work, we formalize an RL framework, **T**ime-**a**daptive **Co**ntrol \& **S**ensing (**TaCoS**), that tackles this challenge by optimizing over policies that besides control predict the duration of its application. Our formulation results in an extended MDP that any standard RL algorithm can solve.We demonstrate that state-of-the-art RL algorithms trained on TaCoS drastically reduce the interaction amount over their discrete-time counterpart while retaining the same or improved performance, and exhibiting robustness over discretization frequency.Finally, we propose OTaCoS, an efficient model-based algorithm for our setting. We show that OTaCoS enjoys sublinear regret for systems with sufficiently smooth dynamics and empirically results in further sample-efficiency gains.

Optimization · Discrete and Combinatorial Optimization

Will Ma, Pan Xu

Online (bipartite) matching under known stationary arrivals is a fundamental model that has been studied extensively under the objective of maximizing the total number of customers served. We instead study the objective of *maximizing the minimum matching rate across all online types*, which is referred to as long-run (individual) fairness. For Online Matching under long-run Fairness (OM-LF) with a single offline agent, we show that the first-come-first-serve (FCFS) policy is $1$-competitive, i.e., matching any optimal clairvoyant. For the general case of OM-LF: We present a sampling algorithm (SAMP) and show that (1) SAMP is of competitiveness of at least $1-1/e$ and (2) it is asymptotically optimal with competitiveness approaches one in different regimes when either all offline agents have a sufficiently large matching capacity, or all online types have a sufficiently large arrival rate, or highly imbalance between the total offline matching capacity and the number of online arrivals. To complement the competitive results, we show the following hardness results for OM-LF: (1) Any non-rejecting policy (matching every arriving online agent if possible) is no more than $1/2$-competitive; (2) Any (randomized) policy is no more than $(\sqrt{3}-1)$-competitive; (3) SAMP can be no more than $(1-1/e)$-competitive suggesting the tightness of competitive analysis for SAMP. We stress that all hardness results mentioned here are independent of any benchmarks. We also consider a few extensions of OM-LF by proposing a few variants of fairness metrics, including long-run group-level fairness and short-run fairness, and we devise related algorithms with provable competitive performance.

Computer Vision · Classification

Felix Petersen, Christian Borgelt, Stefano Ermon

We consider the training of the first layer of vision models and notice the clear relationship between pixel values and gradient update magnitudes: the gradients arriving at the weights of a first layer are by definition directly proportional to (normalized) input pixel values. Thus, an image with low contrast has a smaller impact on learning than an image with higher contrast, and a very bright or very dark image has a stronger impact on the weights than an image with moderate brightness. In this work, we propose performing gradient descent on the embeddings produced by the first layer of the model. However, switching to discrete inputs with an embedding layer is not a reasonable option for vision models. Thus, we propose the conceptual procedure of (i) a gradient descent step on first layer activations to construct an activation proposal, and (ii) finding the optimal weights of the first layer, i.e., those weights which minimize the squared distance to the activation proposal. We provide a closed form solution of the procedure and adjust it for robust stochastic training while computing everything efficiently. Empirically, we find that TrAct (Training Activations) speeds up training by factors between 1.25x and 4x while requiring only a small computational overhead. We demonstrate the utility of TrAct with different optimizers for a range of different vision models including convolutional and transformer architectures.