论文检索

General Machine Learning · Online Learning, Active Learning and Bandits

Yegon Kim, Hyunsu Kim, Gyeonghoon Ko, Juho Lee

Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient alternative, but their development is hindered by the cost of generating sufficient training data from numerical solvers. In this paper, we present a novel framework for active learning (AL) in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This dramatically reduces the cost incurred by each trajectory and thus allows the active learning algorithm to try out a more diverse set of trajectories given the same budget. To accommodate this novel framework, we develop an acquisition function that estimates the utility of a set of time steps by approximating its resulting variance reduction. We demonstrate the effectiveness of our method on several benchmark PDEs, including the Burgers' equation, Korteweg–De Vries equation, Kuramoto–Sivashinsky equation, the incompressible Navier-Stokes equation, and the compressible Navier-Stokes equation.Experiments show that our approach improves performance by large margins over the best existing method. Our method not only reduces average error but also the 99\%, 95\%, and 50\% quantiles of error, which is rare for an AL algorithm. All in all, our approach offers a data-efficient solution to surrogate modeling for PDEs.

Theory · Online Learning and Bandits

Tao Jin, Yue Wu, Quanquan Gu, Farzad Farnoud

We study the problem of efficiently aggregating the preferences of items from multiple information sources (oracles) and infer the ranking under both the weak stochastic transitivity (WST) and the strong stochastic transitivity (SST) conditions. When the underlying preference model satisfies the WST condition, we propose an algorithm named RMO-WST, which has a bi-level design: at the higher level, it actively allocates comparison budgets to all undetermined pairs until the full ranking is recovered; at the lower level, it attempts to compare the pair of items and selects the more accurate oracles simultaneously. We prove that the sample complexity of RMO-WST is $ \tilde O( N\sum_{i=2}^{N}H_{\sigma^{-1}(i),{\sigma^{-1}(i-1)}} )$, where $N$ is the number of items to rank, $H$ is a problem-dependent hardness factor, and $\sigma^{-1}(i)$ represents the $i$-th best item. We also provide a tight lower bound that matches the upper bound of approximate ranking under the WST condition, answering a previously open problem. In addition, when the SST condition is satisfied, we propose an algorithm named RMO-SST, which can achieve an $\tilde{O}(\sum_{i=1}^{N} H_i \log(N))$ sample complexity. This outperforms the best-known sample complexity by a factor of $\log(N)$. The theoretical advantages of our algorithms are verified by empirical experiments in a simulated environment.

General Machine Learning · Online Learning, Active Learning and Bandits

Kiran Thekumparampil, Gaurush Hiranandani, Kousha Kalantari, Shoham Sabach, Branislav Kveton

We study learning human preferences from limited comparison feedback, a core machine learning problem that is at the center of reinforcement learning from human feedback (RLHF). We formulate the problem as learning a Plackett-Luce (PL) model from a limited number of $K$-subset comparisons over a universe of $N$ items, where typically $K \ll N$. Our objective is to select the $K$-subsets such that all items can be ranked with minimal mistakes within the budget. We solve the problem using the D-optimal design, which minimizes the worst-case ranking loss under the estimated PL model. All known algorithms for this problem are computationally infeasible in our setting because we consider exponentially many subsets in $K$. To address this challenge, we propose a randomized Frank-Wolfe algorithm with memoization and sparse updates that has a low $O(N^2 + K^2)$ per-iteration complexity. We analyze it and demonstrate its empirical superiority on synthetic and open-source NLP datasets.

Theory · Online Learning and Bandits

Suho Shin, Chenghao Yang, Haifeng Xu, MohammadTaghi Hajiaghayi

We introduce the tokenized linear bandit (TLB) and multi-armed bandit (TMAB), variants of linear and stochastic multi-armed bandit problems inspired by LLM decoding and alignment. In these problems, at each round $t \in [T]$, a user submits a query (context), and the decision maker (DM) sequentially selects a token irrevocably from a token set. Once the sequence is complete, the DM observes a random utility from the user, whose expectation is presented by a sequence function mapping the chosen token sequence to a nonnegative real value that depends on the query.In both problems, we first show that learning is impossible without any structure on the sequence function.We introduce a natural assumption, diminishing distance with more commons (DDMC), and propose algorithms with regret $\tilde{O}(L\sqrt{T})$ and $\tilde{O}(L\sqrt{T^{2/3}})$ for TLB and TMAB, respectively.As a side product, we obtain an (almost) optimality of the greedy decoding for LLM decoding algorithm under DDMC, which justifies the unresaonable effectiveness of greedy decoding in several tasks.This also has an immediate application to decoding-time LLM alignment, when the misaligned utility can be represented as the frozen LLM's utility and a linearly realizable latent function.We finally validate our algorithm's performance empirically as well as verify our assumptions using synthetic and real-world datasets.

General Machine Learning · Online Learning, Active Learning and Bandits

Haosen Ge, Hamsa Bastani, Osbert Bastani

Conformal prediction has emerged as an effective strategy for uncertainty quantification by modifying a model to output sets of labels instead of a single label. These prediction sets come with the guarantee that they contain the true label with high probability. However, conformal prediction typically requires a large calibration dataset of i.i.d. examples. We consider the online learning setting, where examples arrive over time, and the goal is to construct prediction sets dynamically. Departing from existing work, we assume semi-bandit feedback, where we *only observe the true label if it is contained in the prediction set*. For instance, consider calibrating a document retrieval model to a new domain; in this setting, a user would only be able to provide the true label if the target document is in the prediction set of retrieved documents. We propose a novel conformal prediction algorithm targeted at this setting, and prove that it obtains sublinear regret compared to the optimal conformal predictor. We evaluate our algorithm on a retrieval task, an image classification task, and an auction price-setting task, and demonstrate that it empirically achieves good performance compared to several baselines.

Theory · Online Learning and Bandits

Joseph Lazzaro, Ciara Pike-Burke

Piecewise constant functions describe a variety of real-world phenomena in domains ranging from chemistry to manufacturing. In practice, it is often required to confidently identify the locations of the abrupt changes in these functions as quickly as possible. For this, we introduce a fixed-confidence piecewise constant bandit problem. Here, we sequentially query points in the domain and receive noisy evaluations of the function under bandit feedback. We provide instance-dependent lower bounds for the complexity of change point identification in this problem. These lower bounds illustrate that an optimal method should focus its sampling efforts adjacent to each of the change points, and the number of samples around each change point should be inversely proportional to the magnitude of the change.Building on this, we devise a simple and computationally efficient variant of Track-and-Stop and prove that it is asymptotically optimal in many regimes. We support our theoretical findings with experimental results in synthetic environments demonstrating the efficiency of our method.

Theory · Online Learning and Bandits

Spencer Hutchinson, Mahnoosh Alizadeh

In this work, we study online convex optimization with a fixed constraint function $g : \mathbb{R}^d \rightarrow \mathbb{R}$. Prior work on this problem has shown $O(\sqrt{T})$ regret and cumulative constraint satisfaction $\sum_{t=1}^{T} g(x_t) \leq 0$, while only accessing the constraint value and subgradient at the played actions $g(x_t), \partial g(x_t)$. Using the same constraint information, we show a stronger guarantee of anytime constraint satisfaction $g(x_t) \leq 0 \ \forall t \in [T]$, and matching $O(\sqrt{T})$ regret guarantees. These contributions are thanks to our approach of using Polyak feasibility steps to ensure constraint satisfaction, without sacrificing regret. Specifically, after each step of online gradient descent, our algorithm applies a subgradient descent step on the constraint function where the step-size is chosen according to the celebrated Polyak step-size. We further validate this approach with numerical experiments.

Theory · Online Learning and Bandits

Haoyu Wei, Runzhe Wan, Lei Shi, Rui Song

Many real-world bandit applications are characterized by sparse rewards, which can significantly hinder learning efficiency. Leveraging problem-specific structures for careful distribution modeling is recognized as essential for improving estimation efficiency in statistics. However, this approach remains under-explored in the context of bandits. To address this gap, we initiate the study of zero-inflated bandits, where the reward is modeled using a classic semi-parametric distribution known as the zero-inflated distribution. We develop algorithms based on the Upper Confidence Bound and Thompson Sampling frameworks for this specific structure. The superior empirical performance of these methods is demonstrated through extensive numerical studies.

Theory · Online Learning and Bandits

Qiwei Di, Jiafan He, Quanquan Gu

Learning from human feedback plays an important role in aligning generative models, such as large language models (LLM). However, the effectiveness of this approach can be influenced by adversaries, who may intentionally provide misleading preferences to manipulate the output in an undesirable or harmful direction.To tackle this challenge, we study a specific model within this problem domain--contextual dueling bandits with adversarial feedback, where the true preference label can be flipped by an adversary. We propose an algorithm namely robust contextual dueling bandits ($\texttt{RCDB}$), which is based on uncertainty-weighted maximum likelihood estimation. Our algorithm achieves an $\tilde O(d\sqrt{T}/\kappa+dC/\kappa)$ regret bound, where $T$ is the number of rounds, $d$ is the dimension of the context, $\kappa$ is the lower bound of the derivative of the link function, and $ 0 \le C \le T$ is the total number of adversarial feedback. We also prove a lower bound to show that our regret bound is nearly optimal, both in scenarios with and without ($C=0$) adversarial feedback. Our work is the first to achieve nearly minimax optimal regret for dueling bandits in the presence of adversarial preference feedback. Additionally, for the sigmoid link function, we develop a novel algorithm that takes into account the effect of local derivatives into maximum likelihood estimation (MLE) analysis through a refined method for estimating the link function's derivative. This method helps us to eliminate the $\kappa$ dependence in the leading term with respect to $T$, which reduces the exponential dependence on the parameter radius $B$ to a polynomial dependence. We conduct experiments to evaluate our proposed algorithm $\texttt{RCDB}$ against various types of adversarial feedback. Experimental results demonstrate its superiority over the state-of-the-art dueling bandit algorithms in the presence of adversarial feedback.

General Machine Learning · Online Learning, Active Learning and Bandits

Maximilian Graf, Victor Thuot, Nicolas Verzelen

We study the problem of clustering a set of items based on bandit feedback. Each of the $n$ items is characterized by a feature vector, with a possibly large dimension $d$. The items are partitioned into two unknown groups, such that items within the same group share the same feature vector. We consider a sequential and adaptive setting in which, at each round, the learner selects one item and one feature, then observes a noisy evaluation of the item's feature. The learner's objective is to recover the correct partition of the items, while keeping the number of observations as small as possible. We provide an algorithm which relies on finding a relevant feature for the clustering task, leveraging the Sequential Halving algorithm. With probability at least $1-\delta$, we obtain an accurate recovery of the partition and derive an upper bound on the required budget . Furthermore, we derive an instance-dependent lower bound, which is tight in some relevant cases.

General Machine Learning · Online Learning, Active Learning and Bandits

Yash Kheshwani, Avishek Ghosh, Nikhil Karamchandani

This work investigates the problem of best arm identification for multi-agent multi-armed bandits. We consider $N$ agents grouped into $M$ clusters, where each cluster solves a stochastic bandit problem. The mapping between agents and bandits is \textit{a priori} unknown. Each bandit is associated with $K$ arms, and the goal is to identify the best arm for each agent under a $\delta$-probably correct ($\delta$-PC) framework, while minimizing sample complexity and communication overhead. We propose two novel algorithms: \emph{Clustering then Best Arm Identification} (\texttt{Cl-BAI}) and \emph{Best Arm Identification then Clustering} (\texttt{BAI-Cl}). \texttt{Cl-BAI} employs a two-phase approach that first clusters agents based on the bandit problems they are learning, followed by identifying the best arm for each cluster. \texttt{BAI-Cl} reverses the sequence by identifying the best arms first and then clustering agents accordingly. Both algorithms exploit the successive elimination framework to ensure computational efficiency and high accuracy. Theoretical analysis establishes $\delta$-PC guarantees for both methods, derives bounds on their sample complexity, and provides a lower bound for the problem class. Moreover, when $M$ is small (a constant), we show that the sample complexity of (a variant of) \texttt{BAI-Cl} is (order-wise) minimax optimal. Experiments on synthetic and real-world (Movie Lens, Yelp) data demonstrates the superior performance of the proposed algorithms in terms of sample and communication efficiency, particularly in settings where $M \ll N$.

Theory · Online Learning and Bandits

Artavazd Maranjyan, El Mehdi Saad, Peter Richtarik, Francesco Orabona

Asynchronous methods are fundamental for parallelizing computations in distributed machine learning. They aim to accelerate training by fully utilizing all available resources. However, their greedy approach can lead to inefficiencies using more computation than required, especially when computation times vary across devices. If the computation times were known in advance, training could be fast and resource-efficient by assigning more tasks to faster workers. The challenge lies in achieving this optimal allocation without prior knowledge of the computation time distributions. In this paper, we propose ATA (Adaptive Task Allocation), a method that adapts to heterogeneous and random distributions of worker computation times. Through rigorous theoretical analysis, we show that ATA identifies the optimal task allocation and performs comparably to methods with prior knowledge of computation times. Experimental results further demonstrate that ATA is resource-efficient, significantly reducing costs compared to the greedy approach, which can be arbitrarily expensive depending on the number of workers.

Theory · Online Learning and Bandits

Joe Suk, Jung-hun Kim

We study an infinite-armed bandit problem where actions' mean rewards are initially sampled from a _reservoir distribution_. Most prior works in this setting focused on stationary rewards (Berry et al., 1997; Wang et al., 2008; Bonald and Proutiere, 2013; Carpentier and Valko, 2015) with the more challenging adversarial/non-stationary variant only recently studied in the context of rotting/decreasing rewards (Kim et al., 2022; 2024). Furthermore, optimal regret upper bounds were only achieved using parameter knowledge of non-stationarity and only known for certain regimes of regularity of the reservoir. This work shows the first parameter-free optimal regret bounds while also relaxing these distributional assumptions. We also study a natural notion of _significant shift_ for this problem inspired by recent developments in finite-armed MAB (Suk & Kpotufe, 2022). We show that tighter regret bounds in terms of significant shifts can be adaptively attained. Our enhanced rates only depend on the rotting non-stationarity and thus exhibit an interesting phenomenon for this problem where rising non-stationarity does not factor into the difficulty of non-stationarity.

General Machine Learning · Online Learning, Active Learning and Bandits

Yuxiao Wen

In combinatorial semi-bandits, a learner repeatedly selects from a combinatorial decision set of arms, receives the realized sum of rewards, and observes the rewards of the individual selected arms as feedback. In this paper, we extend this framework to include \emph{graph feedback}, where the learner observes the rewards of all neighboring arms of the selected arms in a feedback graph $G$. We establish that the optimal regret over a time horizon $T$ scales as $\widetilde{\Theta}(S\sqrt{T}+\sqrt{\alpha ST})$, where $S$ is the size of the combinatorial decisions and $\alpha$ is the independence number of $G$. This result interpolates between the known regrets $\widetilde\Theta(S\sqrt{T})$ under full information (i.e., $G$ is complete) and $\widetilde\Theta(\sqrt{KST})$ under the semi-bandit feedback (i.e., $G$ has only self-loops), where $K$ is the total number of arms. A key technical ingredient is to realize a convexified action using a random decision vector with negative correlations. We also show that online stochastic mirror descent (OSMD) that only realizes convexified actions in expectation is suboptimal.

Theory · Online Learning and Bandits

Quan Nguyen, Nishant Mehta, Cristóbal Guzmán

The minimax sample complexity of group distributionally robust optimization (GDRO) has been determined up to a $\log(K)$ factor, where $K$ is the number of groups. In this work, we venture beyond the minimax perspective via a novel notion of sparsity that we call $(\lambda, \beta)$-sparsity. In short, this condition means that at any parameter $\theta$, there is a set of at most $\beta$ groups whose risks at $\theta$ are all at least $\lambda$ larger than the risks of the other groups. To find an $\epsilon$-optimal $\theta$, we show via a novel algorithm and analysis that the $\epsilon$-dependent term in the sample complexity can swap a linear dependence on $K$ for a linear dependence on the potentially much smaller $\beta$. This improvement leverages recent progress in sleeping bandits, showing a fundamental connection between the two-player zero-sum game optimization framework for GDRO and per-action regret bounds in sleeping bandits. We next show an adaptive algorithm which, up to logarithmic factors, obtains a sample complexity bound that adapts to the best $(\lambda, \beta)$-sparsity condition that holds. We also show how to obtain a dimension-free semi-adaptive sample complexity bound with a computationally efficient method.Finally, we demonstrate the practicality of the $(\lambda, \beta)$-sparsity condition and the improved sample efficiency of our algorithms on both synthetic and real-life datasets.

Theory · Online Learning and Bandits

Matei Gabriel Cosa, Marek Elias

Combining algorithms is one of the key techniques in learning-augmented algorithms.We consider the following problem:We are given $\ell$ heuristicsfor Metrical Task Systems (MTS), where each might be tailored to a different typeof input instances.While processing an input instance received online,we are allowed to query the action of only one of the heuristics at each time step.Our goal is to achieve performance comparable to the best of the given heuristics.The main difficulty of our setting comes from the fact thatthe cost paid by a heuristic at time $t$ cannot be estimatedunless the same heuristic was also queried at time $t-1$.This is related to Bandit Learning against memory boundedadversaries (Arora et al., 2012).We show how to achieve regret of $O(\text{OPT}^{2/3})$and prove a tight lower bound based on the constructionof Dekel et al. (2013).

Theory · Online Learning and Bandits

Ilias Diakonikolas, Vasilis Kontonis, Christos Tzamos, Nikos Zarifis

We study the task of online learning in the presence of Massart noise. Specifically, instead of assuming that the online adversary chooses an arbitrary sequence of labels, we assume that the context $\boldsymbol{x}$ is selected adversarially but the label $y$ presented to the learner disagrees with the ground-truth label of $\boldsymbol{x}$ with unknown probability {\em at most} $\eta$. We focus on the fundamental class of $\gamma$-margin linear classifiers and present the first computationally efficient algorithm that achieves mistake bound $\eta T + o(T)$. We point out that the mistake bound achieved by our algorithm is qualitatively tight for computationally efficient algorithms; this follows from the fact that, even in the offline setting, achieving 0-1 error better than $\eta$ requires super-polynomial time under standard complexity assumptions.We extend our online learning model to a $k$-arm contextual bandit setting where the rewards---instead of satisfying commonly used realizability assumptions---are consistent, in expectation, with some linear ranking function with weight vector $\boldsymbol{w}^\ast$. Given a list of contexts $\boldsymbol{x}_1,\ldots \boldsymbol{x}_k$, if $\boldsymbol{w}^*\cdot \boldsymbol{x}_i > \boldsymbol{w}^* \cdot \boldsymbol{x}_j$, the expected reward of action $i$must be larger than that of $j$ by at least $\Delta$. We use our Massart online learner to design an efficient bandit algorithm that obtains expected reward at least $(1-1/k)~ \Delta T - o(T)$ bigger than choosing a random action at every round.

Theory · Online Learning and Bandits

R Majumdar, Mahmoud Salamati, Sadegh Soudjani

Learning to control an unknown dynamical system with respect to high-level temporal specifications is an important problem in control theory. We present the first regret-free online algorithm for learning a controller for linear temporal logic (LTL) specifications for systems with unknown dynamics.We assume that the underlying (unknown) dynamics is modeled by a finite-state and action Markov decision process (MDPs).Our core technical result is a regret-free learning algorithm for infinite-horizon reach-avoid problems on MDPs.For general LTL specifications, we show that the synthesis problem can be reduced to a reach-avoid problem once the graph structure is known.Additionally, we provide an algorithm for learning the graph structure, assuming knowledge of a minimum transition probability, which operates independently of the main regret-free algorithm. Our LTL controller synthesis algorithm provides sharp bounds on how close we are to achieving optimal behavior after a finite number of learning episodes.In contrast, previous algorithms for LTL synthesis only provide asymptotic guarantees, which give no insight into the transient performance during the learning phase.

Theory · Online Learning and Bandits

Junyan Liu, ARNAB MAITI, Artin Tajdini, Kevin Jamieson, Lillian Ratliff

We initiate the study of a repeated principal-agent problem over a finite horizon $T$, where a principal sequentially interacts with $K\geq 2$ types of agents arriving in an *adversarial* order. At each round, the principal strategically chooses one of the $N$ arms to incentivize for an arriving agent of *unknown type*. The agent then chooses an arm based on its own utility and the provided incentive, and the principal receives a corresponding reward. The objective is to minimize regret against the best incentive in hindsight. Without prior knowledge of agent behavior, we show that the problem becomes intractable, leading to linear regret. We analyze two key settings where sublinear regret is achievable. In the first setting, the principal knows the arm each agent type would select greedily for any given incentive. Under this setting, we propose an algorithm that achieves a regret bound of $\mathcal{O}(\min(\sqrt{KT\log N},K\sqrt{T}))$ and provide a matching lower bound up to a $\log K$ factor. In the second setting, an agent's response varies smoothly with the incentive and is governed by a Lipschitz constant $L$. Under this setting, we show that there is an algorithm with a regret bound of $\tilde{\mathcal{O}}((LN)^{1/3}T^{2/3})$ and establish a matching lower bound up to logarithmic factors. Finally, we extend our algorithmic results for both settings by allowing the principal to incentivize multiple arms simultaneously in each round.

Theory · Online Learning and Bandits

François Bachoc, Tommaso Cesari, Roberto Colomboni

We study the role of contextual information in the online learning problem of brokerage between traders.In this sequential problem, at each time step, two traders arrive with secret valuations about an asset they wish to trade.The learner (a broker) suggests a trading (or brokerage) price based on contextual data about the asset and the market conditions.Then, the traders reveal their willingness to buy or sell based on whether their valuations are higher or lower than the brokerage price.A trade occurs if one of the two traders decides to buy and the other to sell, i.e., if the broker's proposed price falls between the smallest and the largest of their two valuations.We design algorithms for this problem and prove optimal theoretical regret guarantees under various standard assumptions.