arXiv subjects
Publications and source records attributed to Aditya Gopalan.
We present two algorithms for Bayesian optimization in the batch feedback setting, based on Gaussian process upper confidence bound and Thompson sampling approaches, along with frequentist regret guarantees and numerical results.
We develop algorithms with low regret for learning episodic Markov decision processes based on kernel approximation techniques. The algorithms are based on both the Upper Confidence Bound (UCB) as well as Posterior or Thompson Sampling (PSRL) philosophies, and work in the general setting of continuous state and action spaces when the true unknown transition dynamics are assumed to have smoothness induced by an appropriate Reproducing Kernel Hilbert Space (RKHS).
We consider black box optimization of an unknown function in the nonparametric Gaussian process setting when the noise in the observed function values can be heavy tailed. This is in contrast to existing literature that typically assumes sub-Gaussian noise distributions for queries. Under the assumption that the unknown function belongs to the Reproducing Kernel Hilbert Space (RKHS) induced by a kernel, we first show that an adaptation of the well-known GP-UCB algorithm with reward truncation enjoys sublinear $\tilde{O}(T^{\frac{2 + α}{2(1+α)}})$ regret even with only the $(1+α)$-th moments, $α\in (0,1]$, of the reward distribution being bounded ($\tilde{O}$ hides logarithmic factors). However, for the common squared exponential (SE) and Matérn kernels, this is seen to be significantly larger than a fundamental $Ω(T^{\frac{1}{1+α}})$ lower bound on regret. We resolve this gap by developing novel Bayesian optimization algorithms, based on kernel approximation techniques, with regret bounds matching the lower bound in order for the SE kernel. We numerically benchmark the algorithms on environments based on both synthetic models and real-world data sets.
We introduce the probably approximately correct (PAC) \emph{Battling-Bandit} problem with the Plackett-Luce (PL) subset choice model--an online learning framework where at each trial the learner chooses a subset of $k$ arms from a fixed set of $n$ arms, and subsequently observes a stochastic feedback indicating preference information of the items in the chosen subset, e.g., the most preferred item or ranking of the top $m$ most preferred items etc. The objective is to identify a near-best item in the underlying PL model with high confidence. This generalizes the well-studied PAC \emph{Dueling-Bandit} problem over $n$ arms, which aims to recover the \emph{best-arm} from pairwise preference information, and is known to require $O(\frac{n}{ε^2} \ln \frac{1}δ)$ sample complexity \citep{Busa_pl,Busa_top}. We study the sample complexity of this problem under various feedback models: (1) Winner of the subset (WI), and (2) Ranking of top-$m$ items (TR) for $2\le m \le k$. We show, surprisingly, that with winner information (WI) feedback over subsets of size $2 \leq k \leq n$, the best achievable sample complexity is still $O\left( \frac{n}{ε^2} \ln \frac{1}δ\right)$, independent of $k$, and the same as that in the Dueling Bandit setting ($k=2$). For the more general top-$m$ ranking (TR) feedback model, we show a significantly smaller lower bound on sample complexity of $Ω\bigg( \frac{n}{mε^2} \ln \frac{1}δ\bigg)$, which suggests a multiplicative reduction by a factor ${m}$ owing to the additional information revealed from preferences among $m$ items instead of just $1$. We also propose two algorithms for the PAC problem with the TR feedback model with optimal (upto logarithmic factors) sample complexity guarantees, establishing the increase in statistical efficiency from exploiting rank-ordered feedback.
We consider the problem of probably approximately correct (PAC) ranking $n$ items by adaptively eliciting subset-wise preference feedback. At each round, the learner chooses a subset of $k$ items and observes stochastic feedback indicating preference information of the winner (most preferred) item of the chosen subset drawn according to a Plackett-Luce (PL) subset choice model unknown a priori. The objective is to identify an $ε$-optimal ranking of the $n$ items with probability at least $1 - δ$. When the feedback in each subset round is a single Plackett-Luce-sampled item, we show $(ε, δ)$-PAC algorithms with a sample complexity of $O\left(\frac{n}{ε^2} \ln \frac{n}δ \right)$ rounds, which we establish as being order-optimal by exhibiting a matching sample complexity lower bound of $Ω\left(\frac{n}{ε^2} \ln \frac{n}δ \right)$---this shows that there is essentially no improvement possible from the pairwise comparisons setting ($k = 2$). When, however, it is possible to elicit top-$m$ ($\leq k$) ranking feedback according to the PL model from each adaptively chosen subset of size $k$, we show that an $(ε, δ)$-PAC ranking sample complexity of $O\left(\frac{n}{m ε^2} \ln \frac{n}δ \right)$ is achievable with explicit algorithms, which represents an $m$-wise reduction in sample complexity compared to the pairwise case. This again turns out to be order-wise unimprovable across the class of symmetric ranking algorithms. Our algorithms rely on a novel {pivot trick} to maintain only $n$ itemwise score estimates, unlike $O(n^2)$ pairwise score estimates that has been used in prior work. We report results of numerical experiments that corroborate our findings.
We consider online learning for minimizing regret in unknown, episodic Markov decision processes (MDPs) with continuous states and actions. We develop variants of the UCRL and posterior sampling algorithms that employ nonparametric Gaussian process priors to generalize across the state and action spaces. When the transition and reward functions of the true MDP are members of the associated Reproducing Kernel Hilbert Spaces of functions induced by symmetric psd kernels (frequentist setting), we show that the algorithms enjoy sublinear regret bounds. The bounds are in terms of explicit structural parameters of the kernels, namely a novel generalization of the information gain metric from kernelized bandit, and highlight the influence of transition and reward function structure on the learning performance. Our results are applicable to multidimensional state and action spaces with composite kernel structures, and generalize results from the literature on kernelized bandits, and the adaptive control of parametric linear dynamical systems with quadratic costs.
We consider a restless multi-armed bandit in which each arm can be in one of two states. When an arm is sampled, the state of the arm is not available to the sampler. Instead, a binary signal with a known randomness that depends on the state of the arm is available. No signal is available if the arm is not sampled. An arm-dependent reward is accrued from each sampling. In each time step, each arm changes state according to known transition probabilities which in turn depend on whether the arm is sampled or not sampled. Since the state of the arm is never visible and has to be inferred from the current belief and a possible binary signal, we call this the hidden Markov bandit. Our interest is in a policy to select the arm(s) in each time step that maximizes the infinite horizon discounted reward. Specifically, we seek the use of Whittle's index in selecting the arms. We first analyze the single-armed bandit and show that in general, it admits an approximate threshold-type optimal policy when there is a positive reward for the `no-sample' action. We also identify several special cases for which the threshold policy is indeed the optimal policy. Next, we show that such a single-armed bandit also satisfies an approximate-indexability property. For the case when the single-armed bandit admits a threshold-type optimal policy, we perform the calculation of the Whittle index for each arm. Numerical examples illustrate the analytical results.
We consider prediction with expert advice when the loss vectors are assumed to lie in a set described by the sum of atomic norm balls. We derive a regret bound for a general version of the online mirror descent (OMD) algorithm that uses a combination of regularizers, each adapted to the constituent atomic norms. The general result recovers standard OMD regret bounds, and yields regret bounds for new structured settings where the loss vectors are (i) noisy versions of points from a low-rank subspace, (ii) sparse vectors corrupted with noise, and (iii) sparse perturbations of low-rank vectors. For the problem of online learning with structured losses, we also show lower bounds on regret in terms of rank and sparsity of the source set of the loss vectors, which implies lower bounds for the above additive loss settings as well.
We consider the stochastic bandit problem with a continuous set of arms, with the expected reward function over the arms assumed to be fixed but unknown. We provide two new Gaussian process-based algorithms for continuous bandit optimization-Improved GP-UCB (IGP-UCB) and GP-Thomson sampling (GP-TS), and derive corresponding regret bounds. Specifically, the bounds hold when the expected reward function belongs to the reproducing kernel Hilbert space (RKHS) that naturally corresponds to a Gaussian process kernel used as input by the algorithms. Along the way, we derive a new self-normalized concentration inequality for vector- valued martingales of arbitrary, possibly infinite, dimension. Finally, experimental evaluation and comparisons to existing algorithms on synthetic and real-world environments are carried out that highlight the favorable gains of the proposed strategies in many cases.
We consider the problem of online learning in misspecified linear stochastic multi-armed bandit problems. Regret guarantees for state-of-the-art linear bandit algorithms such as Optimism in the Face of Uncertainty Linear bandit (OFUL) hold under the assumption that the arms expected rewards are perfectly linear in their features. It is, however, of interest to investigate the impact of potential misspecification in linear bandit models, where the expected rewards are perturbed away from the linear subspace determined by the arms features. Although OFUL has recently been shown to be robust to relatively small deviations from linearity, we show that any linear bandit algorithm that enjoys optimal regret performance in the perfectly linear setting (e.g., OFUL) must suffer linear regret under a sparse additive perturbation of the linear model. In an attempt to overcome this negative result, we define a natural class of bandit models characterized by a non-sparse deviation from linearity. We argue that the OFUL algorithm can fail to achieve sublinear regret even under models that have non-sparse deviation.We finally develop a novel bandit algorithm, comprising a hypothesis test for linearity followed by a decision to use either the OFUL or Upper Confidence Bound (UCB) algorithm. For perfectly linear bandit models, the algorithm provably exhibits OFULs favorable regret performance, while for misspecified models satisfying the non-sparse deviation property, the algorithm avoids the linear regret phenomenon and falls back on UCBs sublinear regret scaling. Numerical experiments on synthetic data, and on recommendation data from the public Yahoo! Learning to Rank Challenge dataset, empirically support our findings.
We consider a restless multi-armed bandit (RMAB) in which there are two types of arms, say A and B. Each arm can be in one of two states, say $0$ or $1.$ Playing a type A arm brings it to state $0$ with probability one and not playing it induces state transitions with arm-dependent probabilities. Whereas playing a type B arm leads it to state $1$ with probability $1$ and not playing it gets state that dependent on transition probabilities of arm. Further, play of an arm generates a unit reward with a probability that depends on the state of the arm. The belief about the state of the arm can be calculated using a Bayesian update after every play. This RMAB has been designed for use in recommendation systems where the user's preferences depend on the history of recommendations. This RMAB can also be used in applications like creating of playlists or placement of advertisements. In this paper we formulate the long term reward maximization problem as infinite horizon discounted reward and average reward problem. We analyse the RMAB by first studying discounted reward scenario. We show that it is Whittle-indexable and then obtain a closed form expression for the Whittle index for each arm calculated from the belief about its state and the parameters that describe the arm. We next analyse the average reward problem using vanishing discounted approach and derive the closed form expression for Whittle index. For a RMAB to be useful in practice, we need to be able to learn the parameters of the arms. We present an algorithm derived from Thompson sampling scheme, that learns the parameters of the arms and also illustrate its performance numerically.
We study the task of maximizing rewards from recommending items (actions) to users sequentially interacting with a recommender system. Users are modeled as latent mixtures of C many representative user classes, where each class specifies a mean reward profile across actions. Both the user features (mixture distribution over classes) and the item features (mean reward vector per class) are unknown a priori. The user identity is the only contextual information available to the learner while interacting. This induces a low-rank structure on the matrix of expected rewards r a,b from recommending item a to user b. The problem reduces to the well-known linear bandit when either user or item-side features are perfectly known. In the setting where each user, with its stochastically sampled taste profile, interacts only for a small number of sessions, we develop a bandit algorithm for the two-sided uncertainty. It combines the Robust Tensor Power Method of Anandkumar et al. (2014b) with the OFUL linear bandit algorithm of Abbasi-Yadkori et al. (2011). We provide the first rigorous regret analysis of this combination, showing that its regret after T user interactions is $\tilde O(C\sqrt{BT})$, with B the number of users. An ingredient towards this result is a novel robustness property of OFUL, of independent interest.
We consider a collaborative online learning paradigm, wherein a group of agents connected through a social network are engaged in playing a stochastic multi-armed bandit game. Each time an agent takes an action, the corresponding reward is instantaneously observed by the agent, as well as its neighbours in the social network. We perform a regret analysis of various policies in this collaborative learning setting. A key finding of this paper is that natural extensions of widely-studied single agent learning policies to the network setting need not perform well in terms of regret. In particular, we identify a class of non-altruistic and individually consistent policies, and argue by deriving regret lower bounds that they are liable to suffer a large regret in the networked setting. We also show that the learning performance can be substantially improved if the agents exploit the structure of the network, and develop a simple learning algorithm based on dominating sets of the network. Specifically, we first consider a star network, which is a common motif in hierarchical social networks, and show analytically that the hub agent can be used as an information sink to expedite learning and improve the overall regret. We also derive networkwide regret bounds for the algorithm applied to general networks. We conduct numerical experiments on a variety of networks to corroborate our analytical results.
We describe and study a model for an Automated Online Recommendation System (AORS) in which a user's preferences can be time-dependent and can also depend on the history of past recommendations and play-outs. The three key features of the model that makes it more realistic compared to existing models for recommendation systems are (1) user preference is inherently latent, (2) current recommendations can affect future preferences, and (3) it allows for the development of learning algorithms with provable performance guarantees. The problem is cast as an average-cost restless multi-armed bandit for a given user, with an independent partially observable Markov decision process (POMDP) for each item of content. We analyze the POMDP for a single arm, describe its structural properties, and characterize its optimal policy. We then develop a Thompson sampling-based online reinforcement learning algorithm to learn the parameters of the model and optimize utility from the binary responses of the users to continuous recommendations. We then analyze the performance of the learning algorithm and characterize the regret. Illustrative numerical results and directions for extension to the restless hidden Markov multi-armed bandit problem are also presented.
We consider a server serving a time-slotted queued system of multiple packet-based flows, with exogenous packet arrivals and time-varying service rates. At each time, the server can observe instantaneous service rates for only a subset of flows (from within a fixed collection of observable subsets) before scheduling a flow in the subset for service. We are interested in queue-length aware scheduling to keep the queues short, and develop scheduling algorithms that use only partial service rate information from subsets of channels to minimize the likelihood of queue overflow in the system. Specifically, we present a new joint subset-sampling and scheduling algorithm called Max-Exp that uses only the current queue lengths to pick a subset of flows, and subsequently schedules a flow using the Exponential rule. When the collection of observable subsets is disjoint, we show that Max-Exp achieves the best exponential decay rate, among all scheduling algorithms using partial information, of the tail of the longest queue in the system. Towards this, we employ novel analytical techniques for studying the performance of scheduling algorithms using partial state, which may be of independent interest. These include new sample-path large deviations results for processes obtained by non-random, predictable sampling of sequences of independent and identically distributed random variables. A consequence of these results is that scheduling with partial state information yields a rate function significantly different from scheduling with full channel information. In the special case when the observable subsets are singleton flows, i.e., when there is effectively no a priori channel-state information, Max-Exp reduces to simply serving the flow with the longest queue; thus, our results show that to always serve the longest queue in the absence of any channel-state information is large-deviations optimal.
We consider reinforcement learning in parameterized Markov Decision Processes (MDPs), where the parameterization may induce correlation across transition probabilities or rewards. Consequently, observing a particular state transition might yield useful information about other, unobserved, parts of the MDP. We present a version of Thompson sampling for parameterized reinforcement learning problems, and derive a frequentist regret bound for priors over general parameter spaces. The result shows that the number of instants where suboptimal actions are chosen scales logarithmically with time, with high probability. It holds for prior distributions that put significant probability near the true model, without any additional, specific closed-form structure such as conjugate or product-form priors. The constant factor in the logarithmic scaling encodes the information complexity of learning the MDP in terms of the Kullback-Leibler geometry of the parameter space.
The problem of finding an optimal sensing schedule for a mobile device that encounters an intermittent WiFi access opportunity is considered. At any given time, the WiFi is in any of the two modes, ON or OFF, and the mobile's incentive is to connect to the WiFi in the ON mode as soon as possible, while spending as little sensing energy. We introduce a dynamic programming framework which enables the characterization of an explicit solution for several models, particularly when the OFF periods are exponentially distributed. While the problem for non-exponential OFF periods is ill-posed in general, a usual workaround in literature is to make the mobile device aware if one ON period is completely missed. In this restricted setting, using the DP framework, the deterministic nature of the optimal sensing policy is established, and value iterations are shown to converge to the optimal solution. Finally, we address the blind situation where the distributions of ON and OFF periods are unknown. A continuous bandit based learning algorithm that has vanishing regret (loss compared to the optimal strategy with the knowledge of distributions) is presented, and comparisons with the optimal schemes are provided for exponential ON and OFF times.
We study epidemic spreading processes in large networks, when the spread is assisted by a small number of external agents: infection sources with bounded spreading power, but whose movement is unrestricted vis-à-vis the underlying network topology. For networks which are `spatially constrained', we show that the spread of infection can be significantly speeded up even by a few such external agents infecting randomly. Moreover, for general networks, we derive upper-bounds on the order of the spreading time achieved by certain simple (random/greedy) external-spreading policies. Conversely, for certain common classes of networks such as line graphs, grids and random geometric graphs, we also derive lower bounds on the order of the spreading time over all (potentially network-state aware and adversarial) external-spreading policies; these adversarial lower bounds match (up to logarithmic factors) the spreading time achieved by an external agent with a random spreading policy. This demonstrates that random, state-oblivious infection-spreading by an external agent is in fact order-wise optimal for spreading in such spatially constrained networks.