In blocking $K$-armed bandits, playing an arm renders it unavailable for a fixed number of future rounds. While this model is relatively well understood in the stochastic regime, much less is known when rewards are generated adversarially. Via a novel reduction, we first show that computing the total reward of the best dynamic policy is NP-hard, even when the blocking time $d > 1$ is identical across arms. We therefore turn to tractable comparators and study the class of $d$-periodic policies, proving that the optimal periodic policy is efficiently computable and always obtains at least a $\frac{1}{K}$ fraction of the dynamic optimum. We also show that this $\frac{1}{K}$ factor is information-theoretically tight: no algorithm can achieve sublinear $\alpha$-regret with respect to the offline optimal dynamic policy for any $\alpha > \frac{1}{K}$. Our main result shows that $T^{2/3}$ is the minimax rate for the regret (against periodic policies) for adversarial blocking bandits with identical blocking times, and that this rate is achievable by an efficient algorithm. Our main technical contribution is the lower bound, which establishes that blocking bandits are at least as hard as bandits with switching costs. The matching upper bound instead follows from a reduction to combinatorial semi-bandits over bipartite matchings. Finally, we show that $\sqrt{T}$ regret rates are efficiently achievable in the full information setting, and more generally via $\alpha$-regret with $\alpha = \frac{1}{2}$.

Follow-the-regularized-leader framework has shown effectiveness and flexibility in online learning problems, where the choice of learning rates are known to be crucial. Recently, adaptive learning rates defined in terms of the arm-selection probabilities, obtained by solving convex optimization, have achieved improved best-of-both-worlds (BOBW) guarantees in various bandit problems. In contrast, BOBW guarantees for its computationally efficient alternative, follow-the-perturbed-leader (FTPL), remain relatively limited since its optimization-free nature ironically makes the design of adaptive, probability-dependent learning rates non-trivial. To address this challenge, we propose an adaptive learning rate for FTPL by introducing surrogate probability functions that can be computed only from the available quantities, without requiring the exact probabilities. Based on these learning rates with surrogate functions, we provide the BOBW guarantee for FTPL with Pareto perturbations for any shape parameter $\alpha >1$, generalizing prior results restricted to specific choices of $\alpha=2$. We further show the BOBW guarantees for FTPL with adaptive learning rates in the bandit problem with expert advices. Our approach preserves the computational simplicity of FTPL while enabling probability-dependent adaptivity, and the surrogate-based methodology may be of independent interest in other algorithmic frameworks beyond FTPL and learning rate designs.

Fine-tuning large deep networks with preference-based human feedback has seen promising results. As user numbers grow and tasks shift to complex datasets like images or videos, distributed approaches become essential for efficiently gathering feedback. To address this, we introduce a multiplayer dueling bandit problem, highlighting that exploring non-informative candidate pairs becomes especially challenging in a collaborative environment. We demonstrate that the use of a Follow Your Leader black-box approach matches the asymptotic regret lower-bound when utilizing known dueling bandit algorithms as a foundation. Additionally, we propose and analyze a message-passing fully distributed approach with a novel Condorcet-Winner recommendation protocol, resulting in expedited exploration in the non-asymptotic regime which reduces regret. Our experimental comparisons reveal that our multiplayer algorithms surpass single-player benchmark algorithms, underscoring their efficacy in addressing the nuanced challenges of this setting.

This paper studies the complexity and optimality of Follow-the-Perturbed-Leader (FTPL) policy in the $K$-armed bandit problems. FTPL is a promising policy that achieves the Best-of-Both-Worlds (BOBW) guarantee without solving an optimization problem unlike Follow-the-Regularized-Leader (FTRL). However, FTPL needs a procedure called geometric resampling to estimate the loss, which needs $O(K^2)$ per-round average complexity, usually worse than that of FTRL. To address this issue, we propose a novel technique, which we call Conditional Geometric Resampling (CGR), for unbiased loss estimation applicable to general perturbation distributions. CGR reduces the average complexity to $O(K\log K)$ without sacrificing the regret bounds. We also propose a biased version of CGR that can control the worst-case complexity while keeping the BOBW guarantee for a certain perturbation distribution. We confirm through experiments that CGR does not only significantly improve the average and worst-case runtime but also achieve better regret thanks to the stable loss estimation.

As sequential learning algorithms are increasingly applied to real life, ensuring data privacy while maintaining their utilities emerges as a timely question. In this context, regret minimisation in stochastic bandits under $\epsilon$-global Differential Privacy (DP) has been widely studied. The present literature poses a significant gap between the best-known regret lower and upper bound in this setting, though they ``match in order''. Thus, we revisit the regret lower and upper bounds of $\epsilon$-global DP bandits and improve both. First, we prove a tighter regret lower bound involving a novel information-theoretic quantity characterising the hardness of $\epsilon$-global DP in stochastic bandits. This quantity smoothly interpolates between Kullback–Leibler divergence and Total Variation distance, depending on the privacy budget $\epsilon$. Then, we choose two asymptotically optimal bandit algorithms, i.e., KL-UCB and IMED, and propose their DP versions using a unified blueprint, i.e., (a) running in arm-dependent phases, and (b) adding Laplace noise to achieve privacy. For Bernoulli bandits, we analyse the regrets of these algorithms and show that their regrets asymptotically match our lower bound up to a constant arbitrary close to 1. At the core of our algorithms lies a new concentration inequality for sums of Bernoulli variables under Laplace mechanism, which is a new DP version of the Chernoff bound. Finally, our numerical experiments validate that DP-KLUCB and DP-IMED achieve lower regret than the existing $\epsilon$-global DP bandit algorithms.

Follow-the-Regularized-Leader (FTRL) policies have achieved Best-of-Both-Worlds (BOBW) results in various settings through hybrid regularizers, whereas analogous results for Follow-the-Perturbed-Leader (FTPL) remain limited due to inherent analytical challenges. To advance the analytical foundations of FTPL, we revisit classical FTRL-FTPL duality for unbounded perturbations and establish BOBW results for FTPL under a broad family of asymmetric unbounded Fréchet-type perturbations, including hybrid perturbations combining Gumbel-type and Fréchet-type tails. These results not only extend the BOBW results of FTPL but also offer new insights into designing alternative FTPL policies competitive with hybrid regularization approaches. Motivated by earlier observations in two-armed bandits, we further investigate the connection between the $1/2$-Tsallis entropy and a Fréchet-type perturbation. Our numerical observations suggest that it corresponds to a symmetric Fréchet-type perturbation, and based on this, we establish the first BOBW guarantee for symmetric unbounded perturbations in the two-armed setting. In contrast, in general multi-armed bandits, we find an instance in which symmetric Fréchet-type perturbations violate the standard condition for BOBW analysis, which is a problem not observed with asymmetric or nonnegative Fréchet-type perturbations. Although this example does not rule out alternative analyses achieving BOBW results, it suggests the limitations of directly applying the relationship observed in two-armed cases to the general case and thus emphasizes the need for further investigation to fully understand the behavior of FTPL in broader settings.

We study the problem of estimating the mean reward of the best arm in a multi-armed bandit (MAB) setting. Specifically, given a target precision $\varepsilon$ and confidence level $1-\delta$, the goal is to return an $\varepsilon$-accurate estimate of the largest mean reward with probability at least $1-\delta$, while minimizing the number of samples. We first establish an instance-dependent lower bound on the sample complexity, which requires handling the infinitely many possible candidates of the estimated best mean. This lower bound is expressed in a non-convex optimization problem, which becomes the main difficulty of this problem, preventing the direct application of standard techniques such as Track-and-Stop to provably achieve optimality. To overcome this difficulty, we introduce several new algorithmic and analytical techniques and propose an algorithm that achieves the asymptotic lower bound with matching constants in the leading term. Our method combines a confidence ellipsoid-based stopping condition with a two-phase sampling strategy tailored to manage non-convexity proposed algorithm is simple, nearly free of hyperparameters, and achieves the instance-dependent, asymptotically optimal sample complexity. Experimental results support our theoretical guarantees and demonstrate the practical effectiveness of our method.

Follow-The-Regularized-Leader (FTRL) is known as an effective and versatile approach in online learning, where appropriate choice of the learning rate is crucial for smaller regret. To this end, we formulate the problem of adjusting FTRL’s learning rate as a sequential decision-making problem and introduce the framework of competitive analysis. We establish a lower bound for the competitive ratio and propose update rules for the learning rate that achieves an upper bound within a constant factor of this lower bound. Specifically, we illustrate that the optimal competitive ratio is characterized by the (approximate) monotonicity of components of the penalty term, showing that a constant competitive ratio is achievable if the components of the penalty term form a monotone non-increasing sequence, and derive a tight competitive ratio when penalty terms are $\xi$-approximately monotone non-increasing. Our proposed update rule, referred to as \textit{stability-penalty matching}, also facilitates the construction of Best-Of-Both-Worlds (BOBW) algorithms for stochastic and adversarial environments. In these environments our results contribute to achieving tighter regret bound and broaden the applicability of algorithms for various settings such as multi-armed bandits, graph bandits, linear bandits, and contextual bandits.

This paper studies the optimality of the Follow-the-Perturbed-Leader (FTPL) policy in both adversarial and stochastic $K$-armed bandits. Despite the widespread use of the Follow-the-Regularized-Leader (FTRL) framework with various choices of regularization, the FTPL framework, which relies on random perturbations, has not received much attention, despite its inherent simplicity. In adversarial bandits, there has been conjecture that FTPL could potentially achieve $\mathcal{O}(\sqrt{KT})$ regrets if perturbations follow a distribution with a Fréchet-type tail. Recent work by Honda et al. (2023) showed that FTPL with Fréchet distribution with shape $\alpha=2$ indeed attains this bound and, notably logarithmic regret in stochastic bandits, meaning the Best-of-Both-Worlds (BOBW) capability of FTPL. However, this result only partly resolves the above conjecture because their analysis heavily relies on the specific form of the Fréchet distribution with this shape. In this paper, we establish a sufficient condition for perturbations to achieve $\mathcal{O}(\sqrt{KT})$ regrets in the adversarial setting, which covers, e.g., Fréchet, Pareto, and Student-$t$ distributions. We also demonstrate the BOBW achievability of FTPL with certain Fréchet-type tail distributions. Our results contribute not only to resolving existing conjectures through the lens of extreme value theory but also potentially offer insights into the effect of the regularization functions in FTRL through the mapping from FTPL to FTRL.

Partial monitoring is a generic framework of online decision-making problems with limited feedback. To make decisions from such limited feedback, it is necessary to find an appropriate distribution for exploration. Recently, a powerful approach for this purpose, exploration by optimization (ExO), was proposed, which achieves optimal bounds in adversarial environments with follow-the-regularized-leader for a wide range of online decision-making problems. However, a naive application of ExO in stochastic environments significantly degrades regret bounds. To resolve this issue in locally observable games, we first establish a new framework and analysis for ExO with a hybrid regularizer. This development allows us to significantly improve existing regret bounds of best-of-both-worlds (BOBW) algorithms, which achieves nearly optimal bounds both in stochastic and adversarial environments. In particular, we derive a stochastic regret bound of $O(\sum_{a \neq a^*} k^2 m^2 \log T / \Delta_a)$, where $k$, $m$, and $T$ are the numbers of actions, observations and rounds, $a^*$ is an optimal action, and $\Delta_a$ is the suboptimality gap for action $a$. This bound is roughly $\Theta(k^2 \log T)$ times smaller than existing BOBW bounds. In addition, for globally observable games, we provide a new BOBW algorithm with the first $O(\log T)$ stochastic bound.

We consider nonstationary multi-armed bandit problems where the model parameters of the arms change over time. We introduce the adaptive resetting bandit (ADR-bandit), a bandit algorithm class that leverages adaptive windowing techniques from literature on data streams. We first provide new guarantees on the quality of estimators resulting from adaptive windowing techniques, which are of independent interest. Furthermore, we conduct a finite-time analysis of ADR-bandit in two typical environments: an abrupt environment where changes occur instantaneously and a gradual environment where changes occur progressively. We demonstrate that ADR-bandit has nearly optimal performance when abrupt or gradual changes occur in a coordinated manner that we call global changes. We demonstrate that forced exploration is unnecessary when we assume such global changes. Unlike the existing nonstationary bandit algorithms, ADR-bandit has optimal performance in stationary environments as well as nonstationary environments with global changes. Our experiments show that the proposed algorithms outperform the existing approaches in synthetic and real-world environments.

The cost of labeling is a significant challenge in practical machine learning. This issue arises not only during the learning phase but also at the model evaluation phase, as there is a need for a substantial amount of labeled test data in addition to the training data. In this study, we address the challenge of active model selection with the goal of minimizing labeling costs for choosing the best-performing model from a set of model candidates. Based on an appropriate test loss estimator, we propose an adaptive labeling strategy that can estimate the difference of test losses with small variance, thereby enabling the estimation of the best model using fewer labeling cost. Experimental results on real-world datasets confirm that our method efficiently selects the best model.

We consider the combinatorial semi-bandit problem and present a new algorithm with a best-of-both-worlds regret guarantee; the regrets are bounded near-optimally in the stochastic and adversarial regimes. In the stochastic regime, we prove a variance-dependent regret bound depending on the tight suboptimality gap introduced by Kveton et al. (2015) with a good leading constant. In the adversarial regime, we show that the same algorithm simultaneously obtains various data-dependent regret bounds. Our algorithm is based on the follow-the-regularized-leader framework with a refined regularizer and adaptive learning rate. Finally, we numerically test the proposed algorithm and confirm its superior or competitive performance over existing algorithms, including Thompson sampling under most settings.

This paper discusses the adversarial and stochastic $K$-armed bandit problems. In the adversarial setting, the best possible regret is known to be $O(\sqrt{KT})$ for time horizon $T$. This bound can be achieved by several policies but they require to explicitly compute the arm-selection probabilities by solving optimization problems at each round, which becomes problematic in some settings. One promising candidate to avoid this issue is the Follow-The-Perturbed-Leader (FTPL) policy, which simply chooses the arm with the minimum cumulative estimated loss with a random perturbation. In particular, it has been conjectured that $O(\sqrt{KT})$ regret might be achieved by FTPL with a Fréchet-type perturbation. This paper affirmatively resolves this conjecture by showing that Fréchet perturbation indeed achieves this bound. We also show that FTPL achieves a logarithmic regret for the stochastic setting, meaning that FTPL achieves best-of-both-worlds regret bounds. The key to these bounds is the novel technique to evaluate the stability of the policy and to express the regret at each round in multiple forms depending on the estimated losses.

This study considers the partial monitoring problem with $k$-actions and $d$-outcomes and provides the first best-of-both-worlds algorithms, whose regrets are favorably bounded both in the stochastic and adversarial regimes. In particular, we show that for non-degenerate locally observable games, the regret is $O(m^2 k^4 \log(T) \log(k_{\Pi} T) / \Delta_{\min})$ in the stochastic regime and $O(m k^{3/2} \sqrt{T \log(T) \log k_{\Pi}})$ in the adversarial regime, where $T$ is the number of rounds, $m$ is the maximum number of distinct observations per action, $\Delta_{\min}$ is the minimum suboptimality gap, and $k_{\Pi}$ is the number of Pareto optimal actions. Moreover, we show that for globally observable games, the regret is $O(c_{\mathcal{G}}^2 \log(T) \log(k_{\Pi} T) / \Delta_{\min}^2)$ in the stochastic regime and $O((c_{\mathcal{G}}^2 \log(T) \log(k_{\Pi} T))^{1/3} T^{2/3})$ in the adversarial regime, where $c_{\mathcal{G}}$ is a game-dependent constant. We also provide regret bounds for a stochastic regime with adversarial corruptions. Our algorithms are based on the follow-the-regularized-leader framework and are inspired by the approach of exploration by optimization and the adaptive learning rate in the field of online learning with feedback graphs.

In the stochastic multi-armed bandit problem, a randomized probability matching policy called Thompson sampling (TS) has shown excellent performance in various reward models. In addition to the empirical performance, TS has been shown to achieve asymptotic problem-dependent lower bounds in several models. However, its optimality has been mainly addressed under light-tailed or one-parameter models that belong to exponential families. In this paper, we consider the optimality of TS for the Pareto model that has a heavy tail and is parameterized by two unknown parameters. Specifically, we discuss the optimality of TS with probability matching priors that include the Jeffreys prior and the reference priors. We first prove that TS with certain probability matching priors can achieve the optimal regret bound. Then, we show the suboptimality of TS with other priors, including the Jeffreys and the reference priors. Nevertheless, we find that TS with the Jeffreys and reference priors can achieve the asymptotic lower bound if one uses a truncation procedure. These results suggest carefully choosing noninformative priors to avoid suboptimality and show the effectiveness of truncation procedures in TS-based policies.

Adaptivity to the difficulties of a problem is a key property in sequential decision-making problems to broaden the applicability of algorithms. Follow-the-regularized-leader (FTRL) has recently emerged as one of the most promising approaches for obtaining various types of adaptivity in bandit problems. Aiming to further generalize this adaptivity, we develop a generic adaptive learning rate, called stability-penalty-adaptive (SPA) learning rate for FTRL. This learning rate yields a regret bound jointly depending on stability and penalty of the algorithm, into which the regret of FTRL is typically decomposed. With this result, we establish several algorithms with three types of adaptivity: sparsity, game-dependency, and best-of-both-worlds (BOBW). Despite the fact that sparsity appears frequently in real problems, existing sparse multi-armed bandit algorithms with $k$-arms assume that the sparsity level $s \leq k$ is known in advance, which is often not the case in real-world scenarios. To address this issue, we first establish $s$-agnostic algorithms with regret bounds of $\tilde{O}(\sqrt{sT})$ in the adversarial regime for $T$ rounds, which matches the existing lower bound up to a logarithmic factor. Meanwhile, BOBW algorithms aim to achieve a near-optimal regret in both the stochastic and adversarial regimes. Leveraging the SPA learning rate and the technique for $s$-agnostic algorithms combined with a new analysis to bound the variation in FTRL output in response to changes in a regularizer, we establish the first BOBW algorithm with a sparsity-dependent bound. Additionally, we explore partial monitoring and demonstrate that the proposed SPA learning rate framework allows us to achieve a game-dependent bound and the BOBW simultaneously.

This paper considers the multi-armed bandit (MAB) problem and provides a new best-of-both-worlds (BOBW) algorithm that works nearly optimally in both stochastic and adversarial settings. In stochastic settings, some existing BOBW algorithms achieve tight gap-dependent regret bounds of $O(\sum_{i: \Delta_i>0} \frac{\log T}{\Delta_i})$ for suboptimality gap $\Delta_i$ of arm $i$ and time horizon $T$. On the other hand, it is shown in Audibert et al. (2007) that the regret bound can be tightened to $O(\sum_{i: \Delta_i>0} (\frac{\sigma_i^2}{\Delta_i} + 1) \log T )$ using the loss variance $\sigma_i^2$ of each arm $i$ in the stochastic environments. In this paper, we propose an algorithm based on the follow-the-regularized-leader method, which employs adaptive learning rates that depend on the empirical prediction error of the loss. This is the first BOBW algorithm with gap-variance-dependent bounds, showing that the variance information can be used even in the possibly adversarial environment. Further, the leading constant factor in our gap-variance dependent bound is only (almost) twice the value for the lower bound. In addition, the proposed algorithm enjoys multiple data-dependent regret bounds in adversarial settings and works well in stochastic settings with adversarial corruptions. Table 1 summarizes the achievable bounds in comparison with UCB-V Audibert et al. (2007), Tsallis-INF (Zimmert and Seldin, 2021) and LB-INF (Ito, 2021).

This study considers online learning with general directed feedback graphs. For this problem, we present best-of-both-worlds algorithms that achieve nearly tight regret bounds for adversarial environments as well as poly-logarithmic regret bounds for stochastic environments. As Alon et al. [2015] have shown, tight regret bounds depend on the structure of the feedback graph: strongly observable graphs yield minimax regret of $\tilde{\Theta}( \alpha^{1/2} T^{1/2} )$, while weakly observable graphs induce minimax regret of $\tilde{\Theta}( \delta^{1/3} T^{2/3} )$, where $\alpha$ and $\delta$, respectively, represent the independence number of the graph and the domination number of a certain portion of the graph. Our proposed algorithm for strongly observable graphs has a regret bound of $\tilde{O}( \alpha^{1/2} T^{1/2} )$ for adversarial environments, as well as of $ {O} ( \frac{\alpha (\ln T)^3 }{\Delta_{\min}} ) $ for stochastic environments, where $\Delta_{\min}$ expresses the minimum suboptimality gap. This result resolves an open question raised by Erez and Koren [2021]. We also provide an algorithm for weakly observable graphs that achieves a regret bound of $\tilde{O}( \delta^{1/3}T^{2/3} )$ for adversarial environments and poly-logarithmic regret for stochastic environments. The proposed algorithms are based on the follow-the-regularized-leader approach combined with newly designed update rules for learning rates.

We consider the fixed-budget best arm identification problem where the goal is to find the arm of the largest mean with a fixed number of samples. It is known that the probability of misidentifying the best arm is exponentially small to the number of rounds. However, limited characterizations have been discussed on the rate (exponent) of this value. In this paper, we characterize the minimax optimal rate as a result of an optimization over all possible parameters. We introduce two rates, $R^{\mathrm{go}}$ and $R^{\mathrm{go}}_{\infty}$, corresponding to lower bounds on the probability of misidentification, each of which is associated with a proposed algorithm. The rate $R^{\mathrm{go}}$ is associated with $R^{\mathrm{go}}$-tracking, which can be efficiently implemented by a neural network and is shown to outperform existing algorithms. However, this rate requires a nontrivial condition to be achievable. To address this issue, we introduce the second rate $R^{\mathrm{go}}_\infty$. We show that this rate is indeed achievable by introducing a conceptual algorithm called delayed optimal tracking (DOT).

Abstract Bayesian optimization (BO) is an approach to optimizing an expensive-to-evaluate black-box function and sequentially determines the values of input variables to evaluate the function. However, it is expensive and in some cases becomes difficult to specify values for all input variables, for example, in outsourcing scenarios where production of input queries with many input variables involves significant cost. In this paper, we propose a novel Gaussian process bandit problem, BO with partially specified queries (BOPSQ). In BOPSQ, unlike the standard BO setting, a learner specifies only the values of some input variables, and the values of the unspecified input variables are randomly determined according to a known or unknown distribution. We propose two algorithms based on posterior sampling for cases of known and unknown input distributions. We further derive their regret bounds that are sublinear for popular kernels. We demonstrate the effectiveness of the proposed algorithms using test functions and real-world datasets.

Ordinary supervised learning is useful when we have paired training data of input $X$ and output $Y$. However, such paired data can be difficult to collect in practice. In this paper, we consider the task of predicting $Y$ from $X$ when we have no paired data of them, but we have two separate, independent datasets of $X$ and $Y$ each observed with some mediating variable $U$, that is, we have two datasets $S_X = \{(X_i, U_i)\}$ and $S_Y = \{(U’_j, Y’_j)\}$. A naive approach is to predict $U$ from $X$ using $S_X$ and then $Y$ from $U$ using $S_Y$, but we show that this is not statistically consistent. Moreover, predicting $U$ can be more difficult than predicting $Y$ in practice, e.g., when $U$ has higher dimensionality. To circumvent the difficulty, we propose a new method that avoids predicting $U$ but directly learns $Y = f(X)$ by training $f(X)$ with $S_{X}$ to predict $h(U)$ which is trained with $S_{Y}$ to approximate $Y$. We prove statistical consistency and error bounds of our method and experimentally confirm its practical usefulness.

We focus on a classic reinforcement learning problem, called a multi-armed bandit, and more specifically in the stochastic setting with reward distributions bounded in $[0,1]$. For this model, an optimal problem-dependent asymptotic regret lower bound has been derived. However, the existing algorithms achieving this regret lower bound all require to solve an optimization problem at each step, inducing a large complexity. In this paper, we propose two new algorithms, which we prove to achieve the problem-dependent asymptotic regret lower bound. The first one, which we call Multinomial TS, is an adaptation of Thompson Sampling for Bernoulli rewards to multinomial reward distributions whose support is included in $\{0, \frac{1}{M}, …, 1\}$. This algorithm achieves the regret lower bound in the case of multinomial distributions with the aforementioned support, and it can be easily generalized to bounded reward distributions in $[0, 1]$ by randomly rounding the observed rewards. The second algorithm we introduce, which we call Non-parametric TS, is a randomized algorithm but not based on the posterior sampling in the strict sense. At each step, it computes an average of the observed rewards with random weight. Not only is it asymptotically optimal, but also it performs very well even for small horizons.

Dense subgraph discovery aims to find a dense component in edge-weighted graphs. This is a fundamental graph-mining task with a variety of applications and thus has received much attention recently. Although most existing methods assume that each individual edge weight is easily obtained, such an assumption is not necessarily valid in practice. In this paper, we introduce a novel learning problem for dense subgraph discovery in which a learner queries edge subsets rather than only single edges and observes a noisy sum of edge weights in a queried subset. For this problem, we first propose a polynomial-time algorithm that obtains a nearly-optimal solution with high probability. Moreover, to deal with large-sized graphs, we design a more scalable algorithm with a theoretical guarantee. Computational experiments using real-world graphs demonstrate the effectiveness of our algorithms.

We investigate finite stochastic partial monitoring, which is a general model for sequential learning with limited feedback. While Thompson sampling is one of the most promising algorithms on a variety of online decision-making problems, its properties for stochastic partial monitoring have not been theoretically investigated, and the existing algorithm relies on a heuristic approximation of the posterior distribution. To mitigate these problems, we present a novel Thompson-sampling-based algorithm, which enables us to exactly sample the target parameter from the posterior distribution. Besides, we prove that the new algorithm achieves the logarithmic problem-dependent expected pseudo-regret $\mathsf{O}(\log T)$ for a linearized variant of the problem with local observability. This result is the first regret bound of Thompson sampling for partial monitoring, which also becomes the first logarithmic regret bound of Thompson sampling for linear bandits.

We study the problem of stochastic multiple-arm identification, where an agent sequentially explores a size-[Formula: see text] subset of arms (also known as a super arm) from given [Formula: see text] arms and tries to identify the best super arm. Most work so far has considered the semi-bandit setting, where the agent can observe the reward of each pulled arm or assumed each arm can be queried at each round. However, in real-world applications, it is costly or sometimes impossible to observe a reward of individual arms. In this study, we tackle the full-bandit setting, where only a noisy observation of the total sum of a super arm is given at each pull. Although our problem can be regarded as an instance of the best arm identification in linear bandits, a naive approach based on linear bandits is computationally infeasible since the number of super arms [Formula: see text] is exponential. To cope with this problem, we first design a polynomial-time approximation algorithm for a 0-1 quadratic programming problem arising in confidence ellipsoid maximization. Based on our approximation algorithm, we propose a bandit algorithm whose computation time is [Formula: see text](log [Formula: see text]), thereby achieving an exponential speedup over linear bandit algorithms. We provide a sample complexity upper bound that is still worst-case optimal. Finally, we conduct experiments on large-scale data sets with more than 10[Formula: see text] super arms, demonstrating the superiority of our algorithms in terms of both the computation time and the sample complexity.

Unsupervised domain adaptation is the problem setting where data generating distributions in the source and target domains are different and labels in the target domain are unavailable. An important question in unsupervised domain adaptation is how to measure the difference between the source and target domains. Existing discrepancy measures for unsupervised domain adaptation either require high computation costs or have no theoretical guarantee. To mitigate these problems, this paper proposes a novel discrepancy measure called source-guided discrepancy (S-disc), which exploits labels in the source domain unlike the existing ones. As a consequence, S-disc can be computed efficiently with a finitesample convergence guarantee. In addition, it is shown that S-disc can provide a tighter generalization error bound than the one based on an existing discrepancy measure. Finally, experimental results demonstrate the advantages of S-disc over the existing discrepancy measures.

We formulate and study a novel multi-armed bandit problem called the qualitative dueling bandit (QDB) problem, where an agent observes not numeric but qualitative feedback by pulling each arm. We employ the same regret as the dueling bandit (DB) problem where the duel is carried out by comparing the qualitative feedback. Although we can naively use classic DB algorithms for solving the QDB problem, this reduction significantly worsens the performance—actually, in the QDB problem, the probability that one arm wins the duel over another arm can be directly estimated without carrying out actual duels. In this paper1, we propose such direct algorithms for the QDB problem. Our theoretical analysis shows that the proposed algorithms significantly outperform DB algorithms by incorporating the qualitative feedback, and experimental results also demonstrate vast improvement over the existing DB algorithms.

We investigate the problem of multiclass classification with rejection, where a classifier can choose not to make a prediction to avoid critical misclassification. First, we consider an approach based on simultaneous training of a classifier and a rejector, which achieves the state-of-the-art performance in the binary case. We analyze this approach for the multiclass case and derive a general condition for calibration to the Bayes-optimal solution, which suggests that calibration is hard to achieve by general loss functions unlike the binary case. Next, we consider another traditional approach based on confidence scores, in which the existing work focuses on a specific class of losses. We propose rejection criteria for more general losses for this approach and guarantee calibration to the Bayes-optimal solution. Finally, we conduct experiments to validate the relevance of our theoretical findings.

Uncoupled regression is the problem to learn a model from unlabeled data and the set of target values while the correspondence between them is unknown. Such a situation arises in predicting anonymized targets that involve sensitive information, e.g., one's annual income. Since existing methods for uncoupled regression often require strong assumptions on the true target function, and thus, their range of applications is limited, we introduce a novel framework that does not require such assumptions in this paper. Our key idea is to utilize \emph{pairwise comparison data, which consists of pairs of unlabeled data that we know which one has a larger target value. Such pairwise comparison data is easy to collect, as typically discussed in the learning-to-rank scenario, and does not break the anonymity of data. We propose two practical methods for uncoupled regression from pairwise comparison data and show that the learned regression model converges to the optimal model with the optimal parametric convergence rate when the target variable distributes uniformly. Moreover, we empirically show that for linear models the proposed methods are comparable to ordinary supervised regression with labeled data.

Abstract We consider a novel stochastic multi-armed bandit problem called good arm identification (GAI), where a good arm is defined as an arm with expected reward greater than or equal to a given threshold. GAI is a pure-exploration problem in which a single agent repeats a process of outputting an arm as soon as it is identified as a good one before confirming the other arms are actually not good. The objective of GAI is to minimize the number of samples for each process. We find that GAI faces a new kind of dilemma, the exploration-exploitation dilemma of confidence, which is different from the best arm identification. As a result, an efficient design of algorithms for GAI is quite different from that for the best arm identification. We derive a lower bound on the sample complexity of GAI that is tight up to the logarithmic factor $$\mathrm {O}(\log \frac{1}{\delta })$$ O ( log 1 δ ) for acceptance error rate $$\delta $$ δ . We also develop an algorithm whose sample complexity almost matches the lower bound. We also confirm experimentally that our proposed algorithm outperforms naive algorithms in synthetic settings based on a conventional bandit problem and clinical trial researches for rheumatoid arthritis.

We propose the first fully-adaptive algorithm for pure exploration in linear bandits—the task to find the arm with the largest expected reward, which depends on an unknown parameter linearly. While existing methods partially or entirely fix sequences of arm selections before observing rewards, our method adaptively changes the arm selection strategy based on past observations at each round. We show our sample complexity matches the achievable lower bound up to a constant factor in an extreme case. Furthermore, we evaluate the performance of the methods by simulations based on both synthetic setting and real-world data, in which our method shows vast improvement over existing ones.

The unfairness of a regressor is evaluated by measuring the correlation between the estimator and the sensitive attribute (e.g., race, gender, age), and the coefficient of determination (CoD) is a natural extension of the correlation coefficient when more than one sensitive attribute exists. As is well known, there is a trade-off between fairness and accuracy of a regressor, which implies a perfectly fair optimizer does not always yield a useful prediction. Taking this into consideration, we optimize the accuracy of the estimation subject to a user-defined level of fairness. However, a fairness level as a constraint induces a nonconvexity of the feasible region, which disables the use of an off-the-shelf convex optimizer. Despite such nonconvexity, we show an exact solution is available by using tools of global optimization theory. Furthermore, we propose a nonlinear extension of the method by kernel representation. Unlike most of existing fairness-aware machine learning methods, our method allows us to deal with numeric and multiple sensitive attributes.

This paper considers the error probability of random coding for memoryless channels in the fixed-rate regime. Random coding is a fundamental tool to construct a capacity-achieving channel code and many studies have been conducted for the asymptotics of the decoding error probability. Gallager derived the exact asymptotics (that is, an evaluation with asymptotically vanishing relative error) of the error probability below the critical rate. On the other hand, the asymptotics for the rate above the critical rate has been unknown except for symmetric channels (in the strong sense) and strongly nonlattice channels. This paper derives the exact asymptotics for general memoryless channels covering all previously unsolved cases. The analysis reveals that strongly symmetric channels and strongly nonlattice channels correspond to two extreme cases and the expression of the asymptotics is much complicated for general channels.

In communication through asymmetric channels the capacity-achieving input distribution is not uniform in general. Homophonic coding is a framework to invertibly convert a (usually uniform) message into a sequence with some target distribution, and is a promising candidate to generate codewords with the nonuniform target distribution for asymmetric channels. In particular, a Variable-to-Fixed length (VF) homophonic code can be used as a suitable component for channel codes to avoid decoding error propagation. However, the existing VF homo-phonic code requires the knowledge of the maximum relative gap of probabilities between two adjacent sequences beforehand, which is an unrealistic assumption for long block codes. In this paper we propose a new VF homophonic code without such a requirement by allowing one-symbol decoding delay. We evaluate this code theoretically and experimentally to verify its asymptotic optimality.

Motivated by online advertising, we study a multiple-play multi-armed bandit problem with position bias that involves several slots and the latter slots yield fewer rewards. We characterize the hardness of the problem by deriving an asymptotic regret bound. We propose the Permutation Minimum Empirical Divergence (PMED) algorithm and derive its asymptotically optimal regret bound. Because of the uncertainty of the position bias, the optimal algorithm for such a problem requires non-convex optimizations that are different from usual partial monitoring and semi-bandit problems. We propose a cutting-plane method and related bi-convex relaxation for these optimizations by using auxiliary variables.

Binary almost instantaneous fixed-to-variable length (AIFV) codes are lossless codes that generalize the class of instantaneous fixed-to-variable length codes. The code uses two code trees and assigns source symbols to incomplete internal nodes as well as to leaves. AIFV codes are empirically shown to attain better compression ratio than Huffman codes. Nevertheless, an upper bound on the redundancy of optimal binary AIFV codes is only known to be 1, which is the same as the bound of Huffman codes. In this paper, the upper bound is improved to 1/2, which is shown to coincide with the worst-case redundancy of the codes. Along with this, the worst-case redundancy is derived for sources with pmax ≥1/2, where pmax is the probability of the most likely source symbol. In addition, we propose an extension of binary AIFV codes, which use m code trees and allow at most m-bit decoding delay. We show that the worst-case redundancy of the extended binary AIFV codes is 1/m for m ≤ 4.

We study the K-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. The hardness of recommending Copeland winners, the arms that beat the greatest number of other arms, is characterized by deriving an asymptotic regret bound. We propose Copeland Winners Deterministic Minimum Empirical Divergence (CW-RMED), an algorithm inspired by the DMED algorithm (Honda and Takemura, 2010), and derive an asymptotically optimal regret bound for it. However, it is not known whether the algorithm can be efficiently computed or not. To address this issue, we devise an efficient version (ECW-RMED) and derive its asymptotic regret bound. Experimental comparisons of dueling bandit algorithms show that ECW-RMED significantly outperforms existing ones.

AIFV codes are lossless codes that generalize the class of instantaneous FV codes. The code uses multiple code trees and assigns source symbols to incomplete internal nodes as well as to leaves. AIFV codes are empirically shown to attain better compression ratio than Huffman codes. Nevertheless, an upper bound on the redundancy of optimal binary AIFV codes is only known to be 1, the same as the bound of Huffman codes. In this paper, the upper bound is improved to 1/2, which is shown to be tight. Along with this, a tight upper bound on the redundancy of optimal binary AIFV codes is derived for the case pmax≥1/2, where pmaxis the probability of the most likely source symbol. This is the first theoretical work on the redundancy of optimal binary AIFV codes, suggesting superiority of the codes over Huffman codes.

We study the K-armed dueling bandit problem, a variation of the standard stochastic bandit problem where the feedback is limited to relative comparisons of a pair of arms. We introduce a tight asymptotic regret lower bound that is based on the information divergence. An algorithm that is inspired by the Deterministic Minimum Empirical Divergence algorithm (Honda and Takemura, 2010) is proposed, and its regret is analyzed. The proposed algorithm is found to be the first one with a regret upper bound that matches the lower bound. Experimental comparisons of dueling bandit algorithms show that the proposed algorithm significantly outperforms existing ones.

We discuss a multiple-play multi-armed bandit (MAB) problem in which several arms are selected at each round. Recently, Thompson sampling (TS), a randomized algorithm with a Bayesian spirit, has attracted much attention for its empirically excellent performance, and it is revealed to have an optimal regret bound in the standard single-play MAB problem. In this paper, we propose the multiple-play Thompson sampling (MP-TS) algorithm, an extension of TS to the multiple-play MAB problem, and discuss its regret analysis. We prove that MP-TS has the optimal regret upper bound that matches the regret lower bound provided by Anantharam et al.\,(1987). Therefore, MP-TS is the first computationally efficient algorithm with optimal regret. A set of computer simulations was also conducted, which compared MP-TS with state-of-the-art algorithms. We also propose a modification of MP-TS, which is shown to have better empirical performance.

Error probabilities of random codes for memoryless channels are considered in this paper. In the area of communication systems, admissible error probability is very small and it is sometimes more important to discuss the relative gap between the achievable error probability and its bound than to discuss the absolute gap. Scarlett et al. derived a good upper bound of a random coding union bound based on the technique of saddlepoint approximation but it is not proved that the relative gap of their bound converges to zero. This paper derives a new bound on the achievable error probability in this viewpoint for a class of memoryless channels. The derived bound is strictly smaller than that by Scarlett et al. and its relative gap with the random coding error probability (not a union bound) vanishes as the block length increases for a fixed coding rate.

Polar codes achieve the rate-distortion bound for nonuniform sources and/or asymmetric distortion measures. However, the performance is not always near optimal for finite code length, especially for short code length. In this paper a new scheme for lossy source coding is proposed. In addition to polar coding, arithmetic coding is applied in the scheme. The source is first encoded by polar coding for lossy compression, then it is further compressed losslessly by arithmetic coding. It is shown that the scheme achieves the rate-distortion bound asymptotically with a good empirical performance. It is also shown that the distortion of the scheme has a better second-order exponent than those of the other polar coding schemes.

Partial monitoring is a general model for sequential learning with limited feedback formalized as a game between two players. In this game, the learner chooses an action and at the same time the opponent chooses an outcome, then the learner suffers a loss and receives a feedback signal. The goal of the learner is to minimize the total loss. In this paper, we study partial monitoring with finite actions and stochastic outcomes. We derive a logarithmic distribution-dependent regret lower bound that defines the hardness of the problem. Inspired by the DMED algorithm (Honda and Takemura, 2010) for the multi-armed bandit problem, we propose PM-DMED, an algorithm that minimizes the distribution-dependent regret. PM-DMED significantly outperforms state-of-the-art algorithms in numerical experiments. To show the optimality of PM-DMED with respect to the regret bound, we slightly modify the algorithm by introducing a hinge function (PM-DMED-Hinge). Then, we derive an asymptotical optimal regret upper bound of PM-DMED-Hinge that matches the lower bound.

We propose almost instantaneous fixed-to-variable length (AIFV) codes such that two (resp. K - 1) code trees are used, if code symbols are binary (resp. K-ary for K ≥ 3), and source symbols are assigned to incomplete internal nodes in addition to leaves. Although the AIFV codes are not instantaneous codes, they are devised such that the decoding delay is at most two bits (resp. one code symbol) in the case of binary (resp. K-ary) code alphabet. The AIFV code can attain better average compression rate than the Huffman code at the expenses of a little decoding delay and a little large memory size to store multiple code trees. We also show for the binary and ternary AIFV codes that the optimal AIFV code can be obtained by solving 0-1 integer programming problems.

In stochastic bandit problems, a Bayesian policy called Thompson sampling (TS) has recently attracted much attention for its excellent empirical performance. However, the theoretical analysis of this policy is difficult and its asymptotic optimality is only proved for one-parameter models. In this paper we discuss the optimality of TS for the model of normal distributions with unknown means and variances as one of the most fundamental examples of multiparameter models. First we prove that the expected regret of TS with the uniform prior achieves the theoretical bound, which is the first result to show that the asymptotic bound is achievable for the normal distribution model. Next we prove that TS with Jeffreys prior and reference prior cannot achieve the theoretical bound. Therefore choice of priors is important for TS and non-informative priors are sometimes risky in cases of multiparameter models.

In this paper, a new variable length coding scheme using a low density parity check (LDPC) code is proposed for the lossy compression of general i.i.d. finite sources. It is proved that the proposed scheme achieves the rate-distortion function asymptotically for an LDPC ensemble. For our setting, Miyake-Muramatsu already proposed an asymptotically optimal LDPC coding scheme. In their scheme, a source sequence is first vector-quantized using an LDPC matrix and then it is compressed losslessly by fixed length coding with another LDPC matrix. However, it is not shown whether their scheme can attain good performance practically. This is mainly because of the difficulty of lossless coding by a fixed length code. In the proposed scheme, the lossless compression is performed by arithmetic coding instead of the fixed length code. Combined with vector-quantization using the reinforced belief propagation, the proposed scheme attains performance near the rate-distortion function practically with time complexity roughly equal to${\bf O}(n\log n)$for length$n$source sequences.

This paper considers polar coding for asymmetric settings, that is, channel coding for asymmetric channels and lossy source coding for nonuniform sources and/or asymmetric distortion measures. The difficulty for asymmetric settings comes from the fact that the optimal symbol distributions of codewords are not always uniform. It is known that such nonuniform distributions can be realized by Gallager's scheme which maps multiple auxiliary symbols distributed uniformly to an actual symbol. However, the complexity of Gallager's scheme increases considerably for the case that the optimal distribution cannot be approximated by simple rational numbers. To overcome this problem for the asymmetric settings, a new polar coding scheme is proposed, which can attain the channel capacity without any alphabet extension by invoking results on polar coding for lossless compression. It is also shown that the proposed scheme achieves a better tradeoff between complexity and decoding error probability in many cases.

We consider channel coding of binary asymmetric memoryless channels with polar codes. The difficulty for asymmetric channels comes from the fact that the optimal input probability distributions are not always uniform. Şaşoğlu et al. realized a nonuniform input distribution by mapping multiple auxiliary symbols distributed uniformly to an actual input symbol. However, the complexity of the scheme increases considerably for the case that the input distribution cannot be approximated by simple rational numbers. To overcome this problem, we propose another polar coding scheme for asymmetric channels, which realizes the optimal nonuniform input distribution by randomizing symbols in the frozen bits with an appropriate probability distribution.

LDPC codes initially studied for channel coding can be applied to source coding, and Miyake-Muramatsu showed theoretically that the rate-distortion function can be achieved asymptotically by using LDPC codes for any stationary memoryless finite source. In their scheme, a source sequence is first vector-quantized by using an LDPC matrix and then it is compressed losslessly by another LDPC matrix. So, their scheme is fixed length coding. Unfortunately, it is not shown that their scheme can attain a good performance practically. In this paper, we propose a new variable length coding scheme, which uses linear programming for vector-quantization and arithmetic coding with probability estimated by belief propagation for lossless coding. The proposed variable length lossy coding can attain the rate-distortion function asymptotically. Furthermore, it can practically attain a performance considerably better than the so-called time sharing bound of the rate-distortion function.