PAC-Bayesian analysis is a frequentist framework for incorporating prior knowledge into learning. It was inspired by Bayesian learning, which allows sequential data processing and naturally turns posteriors from one processing step into priors for the next. However, despite two and a half decades of research, the ability to update priors sequentially without losing confidence information along the way remained elusive for PAC-Bayes. While PAC-Bayes allows construction of data-informed priors, the final confidence intervals depend only on the number of points that were not used for the construction of the prior, whereas confidence information in the prior, which is related to the number of points used to construct the prior, is lost. This limits the possibility and benefit of sequential prior updates, because the final bounds depend only on the size of the final batch. We present a novel and, in retrospect, surprisingly simple and powerful PAC-Bayesian procedure that allows sequential prior updates with no information loss. The procedure is based on a novel decomposition of the expected loss of randomized classifiers. The decomposition rewrites the loss of the posterior as an excess loss relative to a downscaled loss of the prior plus the downscaled loss of the prior, which is bounded recursively. As a side result, we also present a generalization of the split-kl and PAC-Bayes-split-kl inequalities to discrete random variables, which we use for bounding the excess losses, and which can be of independent interest. In empirical evaluation the new procedure significantly outperforms state-of-the-art.

We propose a new best-of-both-worlds algorithm for bandits with variably delayed feedback. In contrast to prior work, which required prior knowledge of the maximal delay $d_{\max}$ and had a linear dependence of the regret on it, our algorithm can tolerate arbitrary excessive delays up to order $T$ (where $T$ is the time horizon). The algorithm is based on three technical innovations, which may all be of independent interest: (1) We introduce the first implicit exploration scheme that works in best-of-both-worlds setting. (2) We introduce the first control of distribution drift that does not rely on boundedness of delays. The control is based on the implicit exploration scheme and adaptive skipping of observations with excessive delays. (3) We introduce a procedure relating standard regret with drifted regret that does not rely on boundedness of delays. At the conceptual level, we demonstrate that complexity of best-of-both-worlds bandits with delayed feedback is characterized by the amount of information missing at the time of decision making (measured by the number of outstanding observations) rather than the time that the information is missing (measured by the delays).

We study a $K$-armed bandit with delayed feedback and intermediate observations. We consider a model, where intermediate observations have a form of a finite state, which is observed immediately after taking an action, whereas the loss is observed after an adversarially chosen delay. We show that the regime of the mapping of states to losses determines the complexity of the problem, irrespective of whether the mapping of actions to states is stochastic or adversarial. If the mapping of states to losses is adversarial, then the regret rate is of order $\sqrt{(K+d)T}$ (within log factors), where $T$ is the time horizon and $d$ is a fixed delay. This matches the regret rate of a $K$-armed bandit with delayed feedback and without intermediate observations, implying that intermediate observations are not helpful. However, if the mapping of states to losses is stochastic, we show that the regret grows at a rate of $\sqrt{\big(K+\min\\{|\mathcal{S}|,d\\}\big)T}$ (within log factors), implying that if the number $|\mathcal{S}|$ of states is smaller than the delay, then intermediate observations help. We also provide refined high-probability regret upper bounds for non-uniform delays, together with experimental validation of our algorithms.

We present a new concentration of measure inequality for sums of independent bounded random variables, which we name a split-kl inequality. The inequality combines the combinatorial power of the kl inequality with ability to exploit low variance. While for Bernoulli random variables the kl inequality is tighter than the Empirical Bernstein, for random variables taking values inside a bounded interval and having low variance the Empirical Bernstein inequality is tighter than the kl. The proposed split-kl inequality yields the best of both worlds. We discuss an application of the split-kl inequality to bounding excess losses. We also derive a PAC-Bayes-split-kl inequality and use a synthetic example and several UCI datasets to compare it with the PAC-Bayes-kl, PAC-Bayes Empirical Bernstein, PAC-Bayes Unexpected Bernstein, and PAC-Bayes Empirical Bennett inequalities.

We present a modified tuning of the algorithm of Zimmert and Seldin [2020] for adversarial multiarmed bandits with delayed feedback, which in addition to the minimax optimal adversarial regret guarantee shown by Zimmert and Seldin [2020] simultaneously achieves a near-optimal regret guarantee in the stochastic setting with fixed delays. Specifically, the adversarial regret guarantee is $\mathcal{O}(\sqrt{TK} + \sqrt{dT\log K})$, where $T$ is the time horizon, $K$ is the number of arms, and $d$ is the fixed delay, whereas the stochastic regret guarantee is $\mathcal{O}\left(\sum_{i \neq i^*}(\frac{1}{\Delta_i} \log(T) + \frac{d}{\Delta_{i}}) + d K^{1/3}\log K\right)$, where $\Delta_i$ are the suboptimality gaps. We also present an extension of the algorithm to the case of arbitrary delays, which is based on an oracle knowledge of the maximal delay $d_{max}$ and achieves $\mathcal{O}(\sqrt{TK} + \sqrt{D\log K} + d_{max}K^{1/3} \log K)$ regret in the adversarial regime, where $D$ is the total delay, and $\mathcal{O}\left(\sum_{i \neq i^*}(\frac{1}{\Delta_i} \log(T) + \frac{\sigma_{max}}{\Delta_{i}}) + d_{max}K^{1/3}\log K\right)$ regret in the stochastic regime, where $\sigma_{max}$ is the maximal number of outstanding observations. Finally, we present a lower bound that matches regret upper bound achieved by the skipping technique of Zimmert and Seldin [2020] in the adversarial setting.

We consider online learning with feedback graphs, a sequential decision-making framework where the learner's feedback is determined by a directed graph over the action set. We present a computationally-efficient algorithm for learning in this framework that simultaneously achieves near-optimal regret bounds in both stochastic and adversarial environments. The bound against oblivious adversaries is $\tilde{O} (\sqrt{\alpha T})$, where $T$ is the time horizon and $\alpha$ is the independence number of the feedback graph. The bound against stochastic environments is $O\big((\ln T)^2 \max_{S\in \mathcal I(G)} \sum_{i \in S} \Delta_i^{-1}\big)$ where $\mathcal I(G)$ is the family of all independent sets in a suitably defined undirected version of the graph and $\Delta_i$ are the suboptimality gaps.The algorithm combines ideas from the EXP3++ algorithm for stochastic and adversarial bandits and the EXP3.G algorithm for feedback graphs with a novel exploration scheme. The scheme, which exploits the structure of the graph to reduce exploration, is key to obtain best-of-both-worlds guarantees with feedback graphs. We also extend our algorithm and results to a setting where the feedback graphs are allowed to change over time.

We derive improved regret bounds for the Tsallis-INF algorithm of Zimmert and Seldin (2021). We show that in adversarial regimes with a $(\Delta,C,T)$ self-bounding constraint the algorithm achieves $\mathcal{O}\left(\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{\left(\sum_{i\neq i^*} \frac{1}{\Delta_i}\right)^2}\right)+\sqrt{C\left(\sum_{i\neq i^*}\frac{1}{\Delta_i}\right)\log_+\left(\frac{(K-1)T}{C\sum_{i\neq i^*}\frac{1}{\Delta_i}}\right)}\right)$ regret bound, where $T$ is the time horizon, $K$ is the number of arms, $\Delta_i$ are the suboptimality gaps, $i^*$ is the best arm, $C$ is the corruption magnitude, and $\log_+(x) = \max\left(1,\log x\right)$. The regime includes stochastic bandits, stochastically constrained adversarial bandits, and stochastic bandits with adversarial corruptions as special cases. Additionally, we provide a general analysis, which allows to achieve the same kind of improvement for generalizations of Tsallis-INF to other settings beyond multiarmed bandits.

We propose an algorithm for stochastic and adversarial multiarmed bandits with switching costs, where the algorithm pays a price $\lambda$ every time it switches the arm being played. Our algorithm is based on adaptation of the Tsallis-INF algorithm of Zimmert and Seldin (2021) and requires no prior knowledge of the regime or time horizon. In the oblivious adversarial setting it achieves the minimax optimal regret bound of $ O( (\lambda K)^{1/3}T^{2/3} + \sqrt{KT})$, where $T$ is the time horizon and $K$ is the number of arms. In the stochastically constrained adversarial regime, which includes the stochastic regime as a special case, it achieves a regret bound of $O((\lambda K)^{2/3} T^{1/3} + \ln T)\sum_{i \neq i^*} \Delta_i^{-1})$, where $\Delta_i$ are suboptimality gaps and $i^*$ is the unique optimal arm. In the special case of $\lambda = 0$ (no switching costs), both bounds are minimax optimal within constants. We also explore variants of the problem, where switching cost is allowed to change over time. We provide experimental evaluation showing competitiveness of our algorithm with the relevant baselines in the stochastic, stochastically constrained adversarial, and adversarial regimes with fixed switching cost.

We present a new second-order oracle bound for the expected risk of a weighted majority vote. The bound is based on a novel parametric form of the Chebyshev-Cantelli inequality (a.k.a. one-sided Chebyshev’s), which is amenable to efficient minimization. The new form resolves the optimization challenge faced by prior oracle bounds based on the Chebyshev-Cantelli inequality, the C-bounds [Germain et al., 2015], and, at the same time, it improves on the oracle bound based on second order Markov’s inequality introduced by Masegosa et al. [2020]. We also derive a new concentration of measure inequality, which we name PAC-Bayes-Bennett, since it combines PAC-Bayesian bounding with Bennett’s inequality. We use it for empirical estimation of the oracle bound. The PAC-Bayes-Bennett inequality improves on the PAC-Bayes-Bernstein inequality of Seldin et al. [2012]. We provide an empirical evaluation demonstrating that the new bounds can improve on the work of Masegosa et al. [2020]. Both the parametric form of the Chebyshev-Cantelli inequality and the PAC-Bayes-Bennett inequality may be of independent interest for the study of concentration of measure in other domains.

We derive an algorithm that achieves the optimal (within constants) pseudo-regret in both adversarial and stochastic multi-armed bandits without prior knowledge of the regime and time horizon. The algorithm is based on online mirror descent (OMD) with Tsallis entropy regularization with power $\alpha=1/2$ and reduced-variance loss estimators. More generally, we define an adversarial regime with a self-bounding constraint, which includes stochastic regime, stochastically constrained adversarial regime, and stochastic regime with adversarial corruptions as special cases, and show that the algorithm achieves logarithmic regret guarantee in this regime and all of its special cases simultaneously with the optimal regret guarantee in the adversarial regime. The algorithm also achieves adversarial and stochastic optimality in the utility-based dueling bandit setting. We provide empirical evaluation of the algorithm demonstrating that it significantly outperforms UCB1 and EXP3 in stochastic environments. We also provide examples of adversarial environments, where UCB1 and Thompson Sampling exhibit almost linear regret, whereas our algorithm suffers only logarithmic regret. To the best of our knowledge, this is the first example demonstrating vulnerability of Thompson Sampling in adversarial environments. Last but not least, we present a general stochastic analysis and a general adversarial analysis of OMD algorithms with Tsallis entropy regularization for $\alpha\in[0,1]$ and explain the reason why $\alpha=1/2$ works best.

We propose a new algorithm for adversarial multi-armed bandits with unrestricted delays. The algorithm is based on a novel hybrid regularizer applied in the Follow the Regularized Leader (FTRL) framework. It achieves $\mathcal{O}(\sqrt{kn}+\sqrt{D\log(k)})$ regret guarantee, where $k$ is the number of arms, $n$ is the number of rounds, and $D$ is the total delay. The result matches the lower bound within constants and requires no prior knowledge of $n$ or $D$. Additionally, we propose a refined tuning of the algorithm, which achieves $\mathcal{O}(\sqrt{kn}+\min_{S}(|S|+\sqrt{D_{\bar S}\log(k)}))$ regret guarantee, where $S$ is a set of rounds excluded from delay counting, $\bar S = [n]\setminus S$ are the counted rounds, and $D_{\bar S}$ is the total delay in the counted rounds. If the delays are highly unbalanced, the latter regret guarantee can be significantly tighter than the former. The result requires no advance knowledge of the delays and resolves an open problem of Thune et al. (2019). The new FTRL algorithm and its refined tuning are anytime and require no doubling, which resolves another open problem of Thune et al. (2019).

We consider a variation of the multi-armed bandit problem, introduced by Avner et al. (2012), in which the forecaster is allowed to choose one arm to explore and one arm to exploit at every round. The loss of the exploited arm is blindly suffered by the forecaster, while the loss of the explored arm is observed without being suffered. The goal of the learner is to minimize the regret. We derive a new algorithm using regularization by Tsallis entropy to achieve best of both worlds guarantees. In the adversarial setting we show that the algorithm achieves the minimax optimal $O(\sqrt{KT})$ regret bound, slightly improving on the result of Avner et al.. In the stochastic regime the algorithm achieves a time-independent regret bound, significantly improving on the result of Avner et al.. The algorithm also achieves the same time-independent regret bound in the more general stochastically constrained adversarial regime introduced by Wei and Luo (2018).

We present a novel analysis of the expected risk of weighted majority vote in multiclass classification. The analysis takes correlation of predictions by ensemble members into account and provides a bound that is amenable to efficient minimization, which yields improved weighting for the majority vote. We also provide a specialized version of our bound for binary classification, which allows to exploit additional unlabeled data for tighter risk estimation. In experiments, we apply the bound to improve weighting of trees in random forests and show that, in contrast to the commonly used first order bound, minimization of the new bound typically does not lead to degradation of the test error of the ensemble.

We derive an algorithm that achieves the optimal (up to constants) pseudo-regret in both adversarial and stochastic multi-armed bandits without prior knowledge of the regime and time horizon. The algorithm is based on online mirror descent with Tsallis entropy regularizer. We provide a complete characterization of such algorithms and show that Tsallis entropy with power $\alpha = 1/2$ achieves the goal. In addition, the proposed algorithm enjoys improved regret guarantees in two intermediate regimes: the moderately contaminated stochastic regime defined by Seldin and Slivkins [22] and the stochastically constrained adversary studied by Wei and Luo [26]. The algorithm also obtains adversarial and stochastic optimality in the utility-based dueling bandit setting. We provide empirical evaluation of the algorithm demonstrating that it outperforms Ucb1 and Exp3 in stochastic environments. In certain adversarial regimes the algorithm significantly outperforms Ucb1 and Thompson Sampling, which exhibit close to linear regret.

We investigate multiarmed bandits with delayed feedback, where the delays need neither be identical nor bounded. We first prove that "delayed" Exp3 achieves the $O(\sqrt{(KT + D)\ln K})$ regret bound conjectured by Cesa-Bianchi et al. [2016] in the case of variable, but bounded delays. Here, $K$ is the number of actions and $D$ is the total delay over $T$ rounds. We then introduce a new algorithm that lifts the requirement of bounded delays by using a wrapper that skips rounds with excessively large delays. The new algorithm maintains the same regret bound, but similar to its predecessor requires prior knowledge of $D$ and $T$. For this algorithm we then construct a novel doubling scheme that forgoes the prior knowledge requirement under the assumption that the delays are available at action time (rather than at loss observation time). This assumption is satisfied in a broad range of applications, including interaction with servers and service providers. The resulting oracle regret bound is of order $\min_\beta (|S_\beta|+\beta \ln K + (KT + D_\beta)/\beta)$, where $|S_\beta|$ is the number of observations with delay exceeding $\beta$, and $D_\beta$ is the total delay of observations with delay below $\beta$. The bound relaxes to $O(\sqrt{(KT + D)\ln K})$, but we also provide examples where $D_\beta \ll D$ and the oracle bound has a polynomially better dependence on the problem parameters.

We derive an online learning algorithm with improved regret guarantees for ``easy'' loss sequences. We consider two types of ``easiness'': (a) stochastic loss sequences and (b) adversarial loss sequences with small effective range of the losses. While a number of algorithms have been proposed for exploiting small effective range in the full information setting, Gerchinovitz and Lattimore [2016] have shown the impossibility of regret scaling with the effective range of the losses in the bandit setting. We show that just one additional observation per round is sufficient to circumvent the impossibility result. The proposed Second Order Difference Adjustments (SODA) algorithm requires no prior knowledge of the effective range of the losses, $\varepsilon$, and achieves an $O(\varepsilon \sqrt{KT \ln K}) + \tilde{O}(\varepsilon K \sqrt[4]{T})$ expected regret guarantee, where $T$ is the time horizon and $K$ is the number of actions. The scaling with the effective loss range is achieved under significantly weaker assumptions than those made by Cesa-Bianchi and Shamir [2018] in an earlier attempt to circumvent the impossibility result. We also provide a regret lower bound of $\Omega(\varepsilon\sqrt{T K})$, which almost matches the upper bound. In addition, we show that in the stochastic setting SODA achieves an $O\left(\sum_{a:\Delta_a>0} \frac{K\varepsilon^2}{\Delta_a}\right)$ pseudo-regret bound that holds simultaneously with the adversarial regret guarantee. In other words, SODA is safe against an unrestricted oblivious adversary and provides improved regret guarantees for at least two different types of ``easiness'' simultaneously.

We introduce the factored bandits model, which is a framework for learning with limited (bandit) feedback, where actions can be decomposed into a Cartesian product of atomic actions. Factored bandits incorporate rank-1 bandits as a special case, but significantly relax the assumptions on the form of the reward function. We provide an anytime algorithm for stochastic factored bandits and up to constants matching upper and lower regret bounds for the problem. Furthermore, we show that with a slight modification the proposed algorithm can be applied to utility based dueling bandits. We obtain an improvement in the additive terms of the regret bound compared to state of the art algorithms (the additive terms are dominating up to time horizons which are exponential in the number of arms).

We propose a new PAC-Bayesian bound and a way of constructing a hypothesis space, so that the bound is convex in the posterior distribution and also convex in a trade-off parameter between empirical performance of the posterior distribution and its complexity. The complexity is measured by the Kullback-Leibler divergence to a prior. We derive an alternating procedure for minimizing the bound. We show that the bound can be rewritten as a one-dimensional function of the trade-off parameter and provide sufficient conditions under which the function has a single global minimum. When the conditions are satisfied the alternating minimization is guaranteed to converge to the global minimum of the bound. We provide experimental results demonstrating that rigorous minimization of the bound is competitive with cross-validation in tuning the trade-off between complexity and empirical performance. In all our experiments the trade-off turned to be quasiconvex even when the sufficient conditions were violated.

We present a new strategy for gap estimation in randomized algorithms for multiarmed bandits and combine it with the EXP3++ algorithm of Seldin and Slivkins (2014). In the stochastic regime the strategy reduces dependence of regret on a time horizon from $(\ln t)^3$ to $(\ln t)^2$ and eliminates an additive factor of order $∆e^1/∆^2$, where $∆$ is the minimal gap of a problem instance. In the adversarial regime regret guarantee remains unchanged.

Online ranker evaluation focuses on the challenge of efficiently determining, from implicit user feedback, which ranker out of a finite set of rankers is the best. It can be modeled by dueling bandits, a mathematical model for online learning under limited feedback from pairwise comparisons. Comparisons of pairs of rankers is performed by interleaving their result sets and examining which documents users click on. The dueling bandits model addresses the key issue of which pair of rankers to compare at each iteration.

Online ranker evaluation is a key challenge in information retrieval. An important task in the online evaluation of rankers is using implicit user feedback for inferring preferences between rankers. Interleaving methods have been found to be efficient and sensitive, i.e. they can quickly detect even small differences in quality. It has recently been shown that multileaving methods exhibit similar sensitivity but can be more efficient than interleaving methods. This paper presents empirical results demonstrating that existing multileaving methods either do not scale well with the number of rankers, or, more problematically, can produce results which substantially differ from evaluation measures like NDCG. The latter problem is caused by the fact that they do not correctly account for the similarities that can occur between rankers being multileaved. We propose a new multileaving method for handling this problem and demonstrate that it substantially outperforms existing methods, in some cases reducing errors by as much as 50%.

We study two problems of online learning under restricted information access. In the first problem, \emphprediction with limited advice, we consider a game of prediction with expert advice, where on each round of the game we query the advice of a subset of M out of N experts. We present an algorithm that achieves O(\sqrt(N/M)T\ln N) regret on T rounds of this game. The second problem, the \emphmultiarmed bandit with paid observations, is a variant of the adversarial N-armed bandit game, where on round t of the game we can observe the reward of any number of arms, but each observation has a cost c. We present an algorithm that achieves O((cN\ln N)^1/3 T^2/3 + \sqrtT \ln N) regret on T rounds of this game in the worst case. Furthermore, we present a number of refinements that treat arm- and time-dependent observation costs and achieve lower regret under benign conditions. We present lower bounds that show that, apart from the logarithmic factors, the worst-case regret bounds cannot be improved.

We present an algorithm for multiarmed bandits that achieves almost optimal performance in both stochastic and adversarial regimes without prior knowledge about the nature of the environment. Our algorithm is based on augmentation of the EXP3 algorithm with a new control lever in the form of exploration parameters that are tailored individually for each arm. The algorithm simultaneously applies the “old” control lever, the learning rate, to control the regret in the adversarial regime and the new control lever to detect and exploit gaps between the arm losses. This secures problem-dependent “logarithmic” regret when gaps are present without compromising on the worst-case performance guarantee in the adversarial regime. We show that the algorithm can exploit both the usual expected gaps between the arm losses in the stochastic regime and deterministic gaps between the arm losses in the adversarial regime. The algorithm retains “logarithmic” regret guarantee in the stochastic regime even when some observations are contaminated by an adversary, as long as on average the contamination does not reduce the gap by more than a half. Our results for the stochastic regime are supported by experimental validation.

Adversarial multiarmed bandits with expert advice is one of the fundamental problems in studying the exploration-exploitation trade-off. It is known that if we observe the advice of all experts on every round we can achieve O\left(\sqrtKT \ln N\right) regret, where K is the number of arms, T is the number of game rounds, and N is the number of experts. It is also known that if we observe the advice of just one expert on every round, we can achieve regret of order O\left(\sqrtNT\right). Our open problem is what can be achieved by asking M experts on every round, where 1 Cite this Paper BibTeX @InProceedings{pmlr-v30-Seldin13, title = {Open Problem: Adversarial Multiarmed Bandits with Limited Advice }, author = {Seldin, Yevgeny and Crammer, Koby and Bartlett, Peter}, booktitle = {Proceedings of the 26th Annual Conference on Learning Theory}, pages = {1067--1072}, year = {2013}, editor = {Shalev-Shwartz, Shai and Steinwart, Ingo}, volume = {30}, series = {Proceedings of Machine Learning Research}, address = {Princeton, NJ, USA}, month = {12--14 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v30/Seldin13.pdf}, url = {https://proceedings.mlr.press/v30/Seldin13.html}, abstract = {Adversarial multiarmed bandits with expert advice is one of the fundamental problems in studying the exploration-exploitation trade-off. It is known that if we observe the advice of all experts on every round we can achieve O\left(\sqrtKT \ln N\right) regret, where K is the number of arms, T is the number of game rounds, and N is the number of experts. It is also known that if we observe the advice of just one expert on every round, we can achieve regret of order O\left(\sqrtNT\right). Our open problem is what can be achieved by asking M experts on every round, where 1 Copy to Clipboard Download Endnote %0 Conference Paper %T Open Problem: Adversarial Multiarmed Bandits with Limited Advice %A Yevgeny Seldin %A Koby Crammer %A Peter Bartlett %B Proceedings of the 26th Annual Conference on Learning Theory %C Proceedings of Machine Learning Research %D 2013 %E Shai Shalev-Shwartz %E Ingo Steinwart %F pmlr-v30-Seldin13 %I PMLR %P 1067--1072 %U https://proceedings.mlr.press/v30/Seldin13.html %V 30 %X Adversarial multiarmed bandits with expert advice is one of the fundamental problems in studying the exploration-exploitation trade-off. It is known that if we observe the advice of all experts on every round we can achieve O\left(\sqrtKT \ln N\right) regret, where K is the number of arms, T is the number of game rounds, and N is the number of experts. It is also known that if we observe the advice of just one expert on every round, we can achieve regret of order O\left(\sqrtNT\right). Our open problem is what can be achieved by asking M experts on every round, where 1 Copy to Clipboard Download RIS TY - CPAPER TI - Open Problem: Adversarial Multiarmed Bandits with Limited Advice AU - Yevgeny Seldin AU - Koby Crammer AU - Peter Bartlett BT - Proceedings of the 26th Annual Conference on Learning Theory DA - 2013/06/13 ED - Shai Shalev-Shwartz ED - Ingo Steinwart ID - pmlr-v30-Seldin13 PB - PMLR DP - Proceedings of Machine Learning Research VL - 30 SP - 1067 EP - 1072 L1 - http://proceedings.mlr.press/v30/Seldin13.pdf UR - https://proceedings.mlr.press/v30/Seldin13.html AB - Adversarial multiarmed bandits with expert advice is one of the fundamental problems in studying the exploration-exploitation trade-off. It is known that if we observe the advice of all experts on every round we can achieve O\left(\sqrtKT \ln N\right) regret, where K is the number of arms, T is the number of game rounds, and N is the number of experts. It is also known that if we observe the advice of just one expert on every round, we can achieve regret of order O\left(\sqrtNT\right). Our open problem is what can be achieved by asking M experts on every round, where 1 Copy to Clipboard Download APA Seldin, Y., Crammer, K. & Bartlett, P.. (2013). Open Problem: Adversarial Multiarmed Bandits with Limited Advice . Proceedings of the 26th Annual Conference on Learning Theory, in Proceedings of Machine Learning Research 30:1067-1072 Available from https://proceedings.mlr.press/v30/Seldin13.html. Copy to Clipboard Download Related Material Download PDF This site last compiled Sun, 05 Jul 2026 15:04:15 +0000 Github Account Copyright © The authors and PMLR 2026. MLResearchPress

We study the problem of online learning Markov Decision Processes (MDPs) when both the transition distributions and loss functions are chosen by an adversary. We present an algorithm that, under a mixing assumption, achieves $O(\sqrt{T\log|\Pi|}+\log|\Pi|)$ regret with respect to a comparison set of policies $\Pi$. The regret is independent of the size of the state and action spaces. When expectations over sample paths can be computed efficiently and the comparison set $\Pi$ has polynomial size, this algorithm is efficient. We also consider the episodic adversarial online shortest path problem. Here, in each episode an adversary may choose a weighted directed acyclic graph with an identified start and finish node. The goal of the learning algorithm is to choose a path that minimizes the loss while traversing from the start to finish node. At the end of each episode the loss function (given by weights on the edges) is revealed to the learning algorithm. The goal is to minimize regret with respect to a fixed policy for selecting paths. This problem is a special case of the online MDP problem. For randomly chosen graphs and adversarial losses, this problem can be efficiently solved. We show that it also can be efficiently solved for adversarial graphs and randomly chosen losses. When both graphs and losses are adversarially chosen, we present an efficient algorithm whose regret scales linearly with the number of distinct graphs. Finally, we show that designing efficient algorithms for the adversarial online shortest path problem (and hence for the adversarial MDP problem) is as hard as learning parity with noise, a notoriously difficult problem that has been used to design efficient cryptographic schemes.

We present PAC-Bayes-Empirical-Bernstein inequality. The inequality is based on combination of PAC-Bayesian bounding technique with Empirical Bernstein bound. It allows to take advantage of small empirical variance and is especially useful in regression. We show that when the empirical variance is significantly smaller than the empirical loss PAC-Bayes-Empirical-Bernstein inequality is significantly tighter than PAC-Bayes-kl inequality of Seeger (2002) and otherwise it is comparable. PAC-Bayes-Empirical-Bernstein inequality is an interesting example of application of PAC-Bayesian bounding technique to self-bounding functions. We provide empirical comparison of PAC-Bayes-Empirical-Bernstein inequality with PAC-Bayes-kl inequality on a synthetic example and several UCI datasets.

We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. Our results extend the PAC-Bayesian (probably approximately correct) analysis in learning theory from the i.i.d. setting to martingales opening the way for its application to importance weighted sampling, reinforcement learning, and other interactive learning domains, as well as many other domains in probability theory and statistics, where martingales are encountered. We also present a comparison inequality that bounds the expectation of a convex function of a martingale difference sequence shifted to the$[0, 1]$interval by the expectation of the same function of independent Bernoulli random variables. This inequality is applied to derive a tighter analog of Hoeffding–Azuma's inequality.

We derive an instantaneous (per-round) data-dependent regret bound for stochastic multiarmed bandits with side information (also known as contextual bandits). The scaling of our regret bound with the number of states (contexts) $N$ goes as $\sqrt{N I_{\rho_t}(S;A)}$, where $I_{\rho_t}(S;A)$ is the mutual information between states and actions (the side information) used by the algorithm at round $t$. If the algorithm uses all the side information, the regret bound scales as $\sqrt{N \ln K}$, where $K$ is the number of actions (arms). However, if the side information $I_{\rho_t}(S;A)$ is not fully used, the regret bound is significantly tighter. In the extreme case, when $I_{\rho_t}(S;A) = 0$, the dependence on the number of states reduces from linear to logarithmic. Our analysis allows to provide the algorithm large amount of side information, let the algorithm to decide which side information is relevant for the task, and penalize the algorithm only for the side information that it is using de facto. We also present an algorithm for multiarmed bandits with side information with computational complexity that is a linear in the number of actions.

We derive a generalization bound for multi-classification schemes based on grid clustering in categorical parameter product spaces. Grid clustering partitions the parameter space in the form of a Cartesian product of partitions for each of the parameters. The derived bound provides a means to evaluate clustering solutions in terms of the generalization power of a built-on classifier. For classification based on a single feature the bound serves to find a globally optimal classification rule. Comparison of the generalization power of individual features can then be used for feature ranking. Our experiments show that in this role the bound is much more precise than mutual information or normalized correlation indices.

We present a general model-independent approach to the analysis of data in cases when these data do not appear in the form of co-occurrence of two variables X, Y , but rather as a sample of values of an unknown (stochastic) function Z (X, Y ). For example, in gene expression data, the expression level Z is a function of gene X and condition Y ; or in movie ratings data the rating Z is a function of viewer X and movie Y . The approach represents a consistent extension of the Information Bottleneck method that has previously relied on the availability of co-occurrence statistics. By altering the relevance variable we eliminate the need in the sample of joint distribution of all input variables. This new formulation also enables simple MDL-like model complexity control and prediction of missing values of Z . The approach is analyzed and shown to be on a par with the best known clustering algorithms for a wide range of domains. For the prediction of missing values (collaborative filtering) it improves the currently best known results.

MOTIVATION: Characterization of a protein family by its distinct sequence domains is crucial for functional annotation and correct classification of newly discovered proteins. Conventional Multiple Sequence Alignment (MSA) based methods find difficulties when faced with heterogeneous groups of proteins. However, even many families of proteins that do share a common domain contain instances of several other domains, without any common underlying linear ordering. Ignoring this modularity may lead to poor or even false classification results. An automated method that can analyze a group of proteins into the sequence domains it contains is therefore highly desirable. RESULTS: We apply a novel method to the problem of protein domain detection. The method takes as input an unaligned group of protein sequences. It segments them and clusters the segments into groups sharing the same underlying statistics. A Variable Memory Markov (VMM) model is built using a Prediction Suffix Tree (PST) data structure for each group of segments. Refinement is achieved by letting the PSTs compete over the segments, and a deterministic annealing framework infers the number of underlying PST models while avoiding many inferior solutions. We show that regions of similar statistics correlate well with protein sequence domains, by matching a unique signature to each domain. This is done in a fully automated manner, and does not require or attempt an MSA. Several representative cases are analyzed. We identify a protein fusion event, refine an HMM superfamily classification into the underlying families the HMM cannot separate, and detect all 12 instances of a short domain in a group of 396 sequences. CONTACT: [email protected]; [email protected].