This paper analyzes regret bounds for Gaussian process Thompson sampling.
problem Analyzing the performance of Gaussian process Thompson sampling (GP-TS) in Bayesian optimization.
method The paper derives several regret bounds for GP-TS, including a lower bound, upper bounds on the second moment of cumulative regret, expected lenient regret, and improved cumulative regret.
result The paper provides improved regret upper bounds for GP-TS, showing that it suffers from a polynomial dependence on 1 / δ 1/δ 1/ δ with probability δ δ δ . Optimistic Hedge achieves optimal regret bounds in two-player zero-sum games.
problem Achieving optimal regret bounds for optimistic Hedge in two-player zero-sum games.
method Refined regret analysis and optimization problem formulation.
result Optimistic Hedge achieves O ( log m log n ) O(\sqrt{\log m \log n}) O ( log m log n ) regret bounds, matching upper and lower bounds. Paper improves worst-case regret bounds for RLSVI in reinforcement learning.
problem Minimizing regret in reinforcement learning with randomized value functions.
method Introduces a clipping variant of Thompson Sampling for RLSVI.
result Achieves a i l d e O ( H 2 S A T ) ilde{\mathrm{O}}(H^2S\sqrt{AT}) i l d e O ( H 2 S A T ) worst-case regret bound. This paper considers the stability of online learning algorithms and its implications for learnability (bounded regret). We introduce a novel quantity called {\em forward regret} that intuitively measures how good an online learning algorithm is if it is allowed a one-step look-ahead into the future. We show that given…
We establish linear regret bounds for convex smooth losses using Fenchel-Young losses.
problem Establishing linear regret bounds for convex smooth losses.
method Constructing a convex smooth surrogate loss using Fenchel-Young losses generated by the convolutional negentropy.
result We derive a smooth loss with a linear surrogate regret bound.
New study on regret lower bounds for multi-agent multi-armed bandit problems.
problem Understanding the limits of performance in multi-agent multi-armed bandit problems.
method Comprehensive study on different settings, establishing tight lower bounds.
result First comprehensive study on regret lower bounds across various settings.
The paper proves a regret bound for a sub-Gaussian mixture on unbounded data.
problem Tackles the challenge of achieving regret bounds for sub-Gaussian mixtures on unbounded data.
method Uses path-wise (deterministic) regret bounds and a cumulative variance process to derive the bound.
result Shows that on a specific event, the regret is eventually bounded by ln(ln V_T).
Paper improves regret bounds for Gaussian process upper confidence bound in Bayesian optimization.
problem Minimizing regret in Gaussian process bandit optimization.
method Gaussian process upper confidence bound (GP-UCB) algorithm with refined analysis.
result Achieves O ( T ln 2 T ) O(\sqrt{T \ln^2 T}) O ( T ln 2 T ) cumulative regret under squared exponential kernel. New algorithm reduces online learning regret for bounded recall games.
problem Reducing regret in online learning with limited past information.
method Constructing a stationary bounded-recall algorithm with O ( 1 / M ) O(1/\sqrt{M}) O ( 1/ M ) regret. result Any low regret bounded-recall algorithm must be aware of past losses' order.
New algorithms improve machine learning performance with explicit regret bounds.
problem Improving machine learning performance with explicit regret bounds.
method Projection-based linear regression algorithms with a focus on modern machine-learning models and their algorithmic performance.
result Established a priori regret bounds with explicit λ-dependence.
Improved regret bound for linear ensemble sampling.
problem Closing the gap between theory and practice in linear ensemble sampling.
method General regret analysis framework for linear bandit algorithms, revealing a relationship with LinPHE.
result Achieves a frequentist regret bound of i l d e O ( d 3 / 2 T ) ilde{O}(d^{3/2}\sqrt{T}) i l d e O ( d 3/2 T ) for linear ensemble sampling. Near-optimal per-action regret bounds for sleeping bandits are derived.
problem Optimizing performance in sleeping bandits where arms and losses are chosen by an adversary.
method Directly minimizing per-action regret using generalized versions of EXP3, EXP3-IX, and FTRL with Tsallis entropy.
result Near-optimal bounds of order O ( T A ln K ) O(\sqrt{TA\ln{K}}) O ( T A ln K ) and O ( T A K ) O(\sqrt{T\sqrt{AK}}) O ( T A K ) are obtained. We study the stochastic multi-armed bandit problem when one knows the value μ ( ⋆ ) μ^{(\star)} μ ( ⋆ ) of an optimal arm, as a well as a positive lower bound on the smallest positive gap Δ Δ Δ . We propose a new randomized policy that attains a regret {\em uniformly bounded over time} in this setting. We also prove several lower bound…
Mirror descent linked to information ratio via Bayesian regret bounds.
problem Understanding stability in mirror descent and its relation to information ratio.
method Developed a connection between mirror descent and information ratio using Bayesian regret bounds.
result Mirror descent with suitable estimators and distributions achieves bounds similar to information-directed sampling.
New bounds on IDS for RL show how to balance computation and learning efficiency.
problem Understanding and optimizing information-directed sampling (IDS) for reinforcement learning.
method Developed novel information-theoretic tools to bound information ratio and cumulative information gain.
result Derived prior-free Bayesian regret bounds for IDS in tabular finite-horizon MDPs and improved computational efficiency.
Study on individual regret in cooperative MAB with agents communicating over a graph.
problem Individual regret in cooperative stochastic multi-armed bandits with communication constraints.
method Analyzed COOP-SE algorithm, derived individual regret bounds under various communication constraints.
result First to show an individual regret bound in cooperative stochastic MAB independent of graph diameter.
The paper honors Lai's contributions to multi-armed bandits and establishes new regret bounds.
problem Improving regret bounds in multi-armed bandit problems.
method Establishes non-asymptotic regret bounds for upper confidence bound indices.
result New regret bounds match Lai-Robbins lower bound.
Online learning algorithms are designed to learn even when their input is generated by an adversary. The widely-accepted formal definition of an online algorithm's ability to learn is the game-theoretic notion of regret. We argue that the standard definition of regret becomes inadequate if the adversary is allowed to a…
This guide simplifies high-probability regret bounds in empirical risk minimization.
problem High-probability regret bounds in empirical risk minimization.
method Modular presentation, three-step recipe, localized Rademacher complexity, local maximal inequalities, metric-entropy integrals.
result Recover familiar rates for various function classes and derive regret bounds for nuisance components.
Improved single-pass streaming MAB regret bound to O(K^1/3T^2/3).
problem Minimize regret in single-pass streaming multi-armed bandits with sublinear memory.
method Improved regret lower bound to Ω(K^1/3T^2/3) and upper bound to O(K^1/3T^2/3).
result Achieved tight regret bound of O(K^1/3T^2/3) for algorithms with o(K) memory.
Study finds optimal regret bound for multi-armed bandit problem with expert advice.
problem Optimizing decision-making in a multi-armed bandit problem with expert advice.
method Proved a tight lower bound matching the upper bound of Kale (2014) for minimax expected regret.
result The minimax optimal expected regret is Θ(√(T K log (N/K))) for the problem.
New algorithm for multi-fidelity bandits reduces costs and improves regret.
problem Optimizing decisions with varying costs and accuracy in multi-fidelity bandits.
method Cost complexity bounds, algorithmic framework, elimination-based algorithm.
result New regret definition and matching upper and lower bounds for elimination-based algorithm.
The paper analyzes Q-learning in 2-player Markov games and provides gap-dependent logarithmic regret bounds.
problem Analyzing the cumulative regret of Nash Q-learning in 2-player turn-based stochastic Markov games.
method Proposed gap-dependent logarithmic upper bounds for cumulative regret in episodic tabular setting and discounted game setting.
result The proposed bounds match theoretical lower bounds up to a logarithmic term.
The paper improves bounds on regret in Gaussian process bandits.
problem Sequential optimization of expensive, possibly non-convex functions with noisy feedback.
method Analyzes maximal information gain and decay rates of GP kernel eigenvalues to improve regret bounds.
result General bounds on maximal information gain and improved regret bounds for various settings, including Matérn kernels.
Unified framework for distributional regret in bandits and reinforcement learning.
problem Characterizing the distribution of regret in multi-armed bandits and reinforcement learning.
method Unified framework with a UCBVI-style algorithm and distributional regret bounds.
result Distributional regret bounds with optimal trade-offs between expected and distributional regret.
In online learning, the dynamic regret metric chooses the reference (optimal) solution that may change over time, while the typical (static) regret metric assumes the reference solution to be constant over the whole time horizon. The dynamic regret metric is particularly interesting for applications such as online reco…
New trade-off found in bandit problems with unknown range.
problem Stochastic bandit problems with unknown range.
method Exhibit a strategy achieving new trade-off between distribution-dependent and distribution-free regret bounds.
result Achieves the rates for regret indicated by the new trade-off.
Study on regret minimization in deterministic MDPs.
problem Minimizing regret in deterministic reinforcement learning.
method Logarithmic regret lower bounds, leveraging graph theory and cycles.
result Explicitly quantifies the fundamental limit of performance achievable by any learning algorithm.
Balances and eliminates base algorithms in bandits and RL to bound total regret.
problem Model selection in bandits and reinforcement learning with unknown optimal regret.
method Balances and eliminates base algorithms based on candidate regret bounds.
result Total regret bound is the best valid candidate regret bound times a small multiplicative factor.
New algorithms for linear bandits avoid norm knowledge, reducing regret.
problem Linear bandits require knowledge of norm bound S S S on parameter θ ∗ θ^* θ ∗ , leading to high regret. method Proposes two novel algorithms for changing and fixed arm sets, analyzing their regret bounds.
result Regret bounds show no significant price for not knowing S S S , with no price for fixed arm sets. The paper refines and extends batched kernelized bandits, improving regret bounds and introducing a robust setting.
problem Optimizing black-box functions with noisy batches in Reproducing Kernel Hilbert Space.
method Refined and extended existing regret bounds, including adaptive batch sizes and robust optimization.
result Improved regret bounds for batched kernelized bandits, showing optimal number of batches and adaptive batch sizes.
In this paper, we consider the problem of prediction with expert advice in dynamic environments. We choose tracking regret as the performance metric and develop two adaptive and efficient algorithms with data-dependent tracking regret bounds. The first algorithm achieves a second-order tracking regret bound, which impr…
New algorithm reduces control error in systems with changing dynamics.
problem Online control of systems with time-varying linear dynamics.
method Introduces adaptive regret metric and a novel meta-algorithm.
result First adaptive regret bound for online convex optimization with memory.
New algorithm CROP achieves asymptotic optimality with bounded regret.
problem Optimistic algorithms fail to achieve asymptotic instance-dependent regret optimality.
method CRush Optimism with Pessimism (CROP) algorithm that eliminates optimistic hypotheses.
result CROP achieves constant-factor asymptotic optimality and bounded regret.
New bounds for online portfolio selection without smoothness assumptions.
problem Online portfolio selection with non-Lipschitz, non-smooth losses.
method Data-dependent bounds using novel smoothness characterizations and FTRL with self-concordant regularizers.
result Achieves logarithmic regrets when data is 'easy' and sublinear worst-case regrets.
New method reduces regret and communication costs in federated Q-learning.
problem Worst-case regret and communication cost bounds in federated Q-learning.
method Gap-dependent analysis leveraging MDP structures.
result Achieves log T \log T log T -type regret and communication cost bounds. Kernel-based bandit is an extensively studied black-box optimization problem, in which the objective function is assumed to live in a known reproducing kernel Hilbert space. While nearly optimal regret bounds (up to logarithmic factors) are established in the noisy setting, surprisingly, less is known about the noise-f…
Improved online Q-learning for MDPs with concentration bounds.
problem Online Q-learning in infinite-horizon discounted MDPs with sublinear regret for large gaps.
method Smoothed ε n ε_n ε n -Greedy exploration scheme combining ε n ε_n ε n -greedy and Boltzmann exploration, analyzed using concentration bounds for contractive Markovian stochastic approximation. result Near- i l d e O ( N 9 / 10 ) ilde{O}(N^{9/10}) i l d e O ( N 9/10 ) regret bound for Smoothed ε n ε_n ε n -Greedy exploration scheme. Optimal simple regret bound for Gaussian Process bandits.
problem Sequential optimization of expensive-to-evaluate functions.
method Proved a bound on simple regret for pure exploration algorithms.
result Order optimal bound on simple regret for Gaussian Process bandits.
We revisit the proof by Qin et al. (2014) of bounded regret of the C 2 ^2 2 UCB contextual combinatorial bandit. We demonstrate an error in the proof of volumetric expansion of the moment matrix, used in upper bounding a function of context vector norms. We prove a relaxed inequality that yields the originally-stated regre…
Algorithm optimizes cascaded functions with known structure.
problem Optimizing a function network with known structure.
method GPN-UCB algorithm with upper confidence bounds and theoretical regret bounds.
result Near-optimal cumulative and simple regret bounds.
The paper improves regret bounds for admission control in queueing systems.
problem Improving regret bounds for admission control in queueing systems.
method Proposes an algorithm inspired by UCRL2 and uses problem structure to bound regret.
result Proves an upper bound on the expected total regret of O ( S log T + m T log T ) O(S\log T + \sqrt{mT \log T}) O ( S log T + m T log T ) . New bounds show polyhedral surrogates are optimal for generalization.
problem Proving generalization rates for polyhedral loss functions.
method Developed two general results for polyhedral surrogates.
result Polyhedral surrogates provide linear surrogate regret bounds, translating directly to target rates.
New algorithms reduce regret in online MDPs by adapting to data and variance.
problem Adapting to both adversarial and stochastic environments in online MDPs.
method Develops algorithms based on global optimization and policy optimization, using optimistic follow-the-regularized-leader with log-barrier regularization.
result Achieves refined data-dependent and variance-dependent regret bounds.
Two new algorithms reduce online kernel regression's computational cost while maintaining optimal regret bounds.
problem Trade-off between regret and computational cost in online kernel regression.
method AOGD-ALD and NONS-ALD algorithms dynamically maintain nearly orthogonal basis to approximate kernel mapping and control approximate error.
result Achieves nearly optimal regret bounds at sublinear computational complexity.
New algorithms reduce regret in both stochastic and deterministic environments.
problem Designing algorithms that perform well in both types of MDPs.
method Proposed new environment norms and algorithms with variance-dependent regret bounds.
result First algorithm with simultaneously optimal bounds for both stochastic and deterministic MDPs.
New algorithm minimizes Bayesian regret in offline linear bandits.
problem Minimizing Bayesian regret in offline linear bandits.
method Proposes a new algorithm that directly minimizes upper bounds on Bayesian regret using conic optimization.
result Upper bounds are tight and guarantee superior performance compared to LCB.
We consider undiscounted reinforcement learning in Markov decision processes (MDPs) where both the reward functions and the state-transition probabilities may vary (gradually or abruptly) over time. For this problem setting, we propose an algorithm and provide performance guarantees for the regret evaluated against the…