Thompson Sampling for Complex Bandit Problems

Introduction

The stochastic Multi-Armed Bandit (MAB) is a classical framework in machine learning and optimization. In the basic MAB setting, there is a finite set of actions, each of which has a reward derived from some stochastic process, and a learner selects actions to optimize long-term performance. The MAB model gives a crystallized abstraction of a fundamental decision problem – whether to explore or exploit in the face of uncertainty. Bandit problems have been extensively studied and several well-performing methods are known for optimizing the reward . However, the requirement that the actions’ rewards be independent is often a severe limitation, as seen in these examples:

Web Advertising: Consider a website publisher selecting at each time a subset of ads to be displayed to the user. As the publisher is paid per click, it would like to maximize its revenue, but dependencies between different ads could mean that the problem does not “decompose nicely”. For instance, showing two car ads might not significantly increase the click probability over a single car ad.

Job Scheduling: Assume we have a small number of resources or machines, and in each time step we receive a set of jobs (the “basic arms”), where the duration of each job follows some fixed but unknown distribution. The latency of a machine is the sum of the latencies of the jobs (basic arms) assigned to it, and the makespan of the system is the maximum latency across machines. Here, the decision maker’s complex action is to partition the jobs (basic arms) between the machines, to achieve the least makespan on average.

Routing: Consider a multi-commodity flow problem, where for each source-destination pair, we need to select a route (a complex action). In this setting the capacities of the edges (the basic arms) are random variables, and the reward is the total flow in the system at each time. In this example, the rewards of different paths are inter-dependent, since the flow on one path depends on which other paths where selected.

These examples motivate settings where a model more complex than the simple MAB is required. Our high-level goal is to describe a methodology that can tackle bandit problems with complex action/reward structure, and also guarantee high performance. A crucial complication in the problems above is that it is unlikely that we will get to observe the reward of each basic action chosen. Rather, we can hope to receive only an aggregate reward for the complex action taken. Our approach to complex bandit problems stems from the idea that when faced with uncertainty, pretending to be Bayesian can be advantageous. A purely Bayesian view of the MAB assumes that the model parameters (i.e., the arms’ distributions) are drawn from a prior distribution. We argue that even in a frequentist setup, in which the stochastic model is unknown but fixed, working with a fictitious prior over the model (i.e., being pseudo-Bayesian) helps solve very general bandit problems with complex actions and observations.

Our algorithmic prescription for complex bandits is Thompson sampling : Start with a fictitious prior distribution over the parameters of the basic arms of the model, whose posterior gets updated as actions are played. A parameter is randomly drawn according to the posterior and the (complex) action optimal for the parameter is played. The rationale behind this is twofold: (1) Updating the posterior adds useful information about the true unknown parameter. (2) Correlations among complex bandit actions (due to their dependence on the basic parameters) are implicitly captured by posterior updates on the space of basic parameters.

The main advantage of a pseudo-Bayesian approach like Thompson sampling, compared to other MAB methodologies such as UCB, is that it can handle a wide range of information models that go beyond observing the individual rewards alone. For example, suppose we observe only the final makespan in the multi-processor job scheduling problem above. In Thompson sampling, we merely need to compute a posterior given this observation and its likelihood. In contrast, it seems difficult to adapt an algorithm such as UCB for this case without a naive exponential dependence on the number of basic armsThe work of Dani et al. first extended the UCB framework to the case of linear cost functions. However, for more complex, nonlinear rewards (e.g., multi-commodity flows or makespans), it is unclear how UCB-like algorithms can be applied other than to treat all complex actions independently.. Besides, the deterministic approach of optimizing over regions of the parameter space that UCB-like algorithms follow is arguably harder to apply in practice, as opposed to optimizing over the action space given a sampled parameter in Thompson sampling – often an efficient polynomial-time routine like a sort. The Bayesian view that motivates Thompson sampling also allows us to use efficient numerical algorithms such as particle filtering to approximate complicated posterior distributions in practice.

Our main analytical result is a general regret bound for Thompson sampling in complex bandit settings. No specific structure is imposed on the initial (fictitious) prior, except that it be discretely supported and put nonzero mass on the true model. The bound for this general setting scales logarithmically with timeMore precisely, we obtain a bound of the form $\mathsf{B}+\mathsf{C}\log T$, in which $\mathsf{C}$ is a non-trivial preconstant that captures precisely the structure of correlations among actions and thus is often better than the decoupled sum-of-inverse-KL-divergences bounds seen in literature . The additive constant (wrt time) $\mathsf{B}$, though potentially large and depending on the total number of complex actions, appears to be merely an artifact of our proof technique tailored towards extracting the time scaling $\mathsf{C}$. This is borne out, for instance, from numerical experiments. We remark that such additive constants, in fact, often appear in regret analyses of basic Thompson sampling ., as is well-known. But more interestingly, the preconstant for this logarithmic scaling can be explicitly characterized in terms of the bandit’s KL divergence geometry and represents the information complexity of the bandit problem. The standard MAB imposes no structure among the actions, thus its information complexity simply becomes a sum of terms, one for each separate action. However, in a complex bandit setting, rewards are often more informative about other parameters of the model, in which case the bound reflects the resulting coupling across complex actions.

Recent work has shown the regret-optimality of Thompson sampling for the basic MAB , and has even provided regret bounds for a specific complex bandit setting – the linear bandit case where the reward is a linear function of the actions . However, the analysis of complex bandits in general poses challenges that cannot be overcome using the specialized techniques in these works. Indeed, these existing analyses rely crucially on the conjugacy of the prior and posterior distributions – either independent Beta or exponential family distributions for basic MAB or standard normal distributions for linear bandits. These methods break down when analyzing the evolution of complicated posterior distributions which often lack even a closed form expression.

In contrast to existing regret analyses, we develop a novel proof technique based on looking at the form of the Bayes posterior. This allows us to track the posterior distributions that result from general action and feedback sets, and to express the concentration of the posterior as a constrained optimization problem in path space. It is rather surprising that, with almost no specific structural assumptions on the prior, our technique yields a regret bound that reduces to Lai and Robbins’ classic lower bound for standard MAB, and also gives non-trivial and improved regret scalings for complex bandits. In this vein, our results represent a generalization of existing performance results for Thompson sampling.

We have complemented our theoretical findings with numerical studies of Thompson sampling. The algorithm is implemented using a simple particle filter to maintain and sample from posterior distributions. We evaluated the performance of the algorithm on two complex bandit scenarios – subset selection from a bandit and job scheduling.

Related Work: Bayesian ideas for the multi-armed bandit date back nearly 80 years ago to the work of W. R. Thompson , who introduced an elegant algorithm based on posterior sampling. However, there has been relatively meager work on using Thompson sampling in the control setup. A notable exception is that develops general Bayesian control rules and demonstrates them for classic bandits and Markov decision processes (i.e., reinforcement learning). On the empirical side, a few recent works have demonstrated the success of Thompson sampling . Recent work has shown frequentist-style regret bounds for Thompson sampling in the standard bandit model , and Bayes risk bounds in the purely Bayesian setting . Our work differs from this literature in that we go beyond simple, decoupled actions/observations – we focus on the performance of Thompson setting in a general action/feedback model, and show novel frequentist regret bounds that account for the structure of complex actions.

Regarding bandit problems with actions/rewards more complex than the basic MAB, a line of work that deserves particular mention is that of linear bandit optimization . In this setting, actions are identified with decision vectors in a Euclidean space, and the obtained rewards are random linear functions of actions, drawn from an unknown distribution. Here, we typically see regret bounds for generalizations of the UCB algorithm that show polylogarithmic regret for this setting. However, the methods and bounds are highly tailored to the specific linear feedback structure and do not carry over to other kinds of feedback.

Setup and Notation

The goal is to play an action at each time $t$ to minimize the (expected) regret over $T$ rounds: $R_{T}:=\sum_{t=1}^{T}h(X_{t},a^{*}(\theta^{*}))-h(X_{t},A_{t})$, or alternatively, the number of plays of suboptimal actionsWe refer to the latter objective as regret since, under bounded rewards, both the objectives scale similarly with the problem size.: $\sum_{t=1}^{T}\mathbf{1}\{A_{t}\neq a^{*}\}$.

For each action $a\in\mathcal{A}$, define $S_{a}:=\{\theta\in{\Theta}:a^{*}(\theta)=a\}$ to be the decision region of $a$, i.e., the set of models in $\Theta$ whose optimal action is $a$. Within $S_{a}$, let $S_{a}^{\prime}$ be the models that exactly match $\theta^{*}$ in the sense of the marginal distribution of action $a^{*}$, i.e., $S_{a}^{\prime}:=\{\theta\in S_{a}:D(\theta^{*}_{a^{*}}||\theta_{a^{*}})=0\}$. Let $S_{a}^{\prime\prime}$ be the remaining models in $S_{a}$.

Regret Performance: Overview

We propose using Thompson sampling (Algorithm 1) to play actions in the general bandit model. Before formally stating the regret bound, we present an intuitive explanation of how Thompson sampling learn to play good actions in a general setup where observations, parameters and actions are related via a general likelihood. To this end, let us assume that there are finitely many actions $\mathcal{A}$. Let us also index the actions in $\mathcal{A}$ as $\{1,2,\ldots,|\mathcal{A}|\}$, with the index $|\mathcal{A}|$ denoting the optimal action $a^{*}$ (we will require this indexing later when we associate each coordinate of $|\mathcal{A}|$-dimensional space with its respective action). Denote by $D(\theta^{*}_{a}||\theta_{a})$ the marginal Kullback-Leibler divergence between the output distributions of parameters $\theta^{*}$ and $\theta$ upon playing action $a$, i.e., between the distributions $\{l(y;a,\theta^{*}):y\in\mathcal{Y}\}$ and $\{l(y;a,\theta):y\in\mathcal{Y}\}$.

When action $A_{t}$ is played at time $t$, the prior density gets updated to the posterior as $\pi_{t}(d\theta)\propto\exp\left(-\log\frac{l(Y_{t};A_{t},\theta^{*})}{l(Y_{t};A_{t},\theta)}\right)\pi_{t-1}(d\theta)$. Observe that the conditional expectation of the “instantaneous” log-likelihood ratio $\log\frac{l(Y_{t};A_{t},\theta^{*})}{l(Y_{t};A_{t},\theta)}$, is simply the appropriate marginal KL divergence, i.e.,

with $N_{t}(a):=\sum_{i=1}^{t}\mathbf{1}\{A_{i}=a\}$ denoting the play count of $a$. The quantity in the exponent can be interpreted as a “loss” suffered by the model $\theta$ up to time $t$, and each time an action $a$ is played, $\theta$ incurs an additional loss of essentially the marginal KL divergence $D(\theta^{*}_{a}||\theta_{a})$.

Upon closer inspection, the posterior approximation (1) yields detailed insights into the dynamics of posterior-based sampling. First, since $\exp\left(-\sum_{a\in\mathcal{A}}N_{t}(a)D(\theta^{*}_{a}||\theta_{a})\right)\leq 1$, the true model $\theta^{*}$ always retains a significant share of posterior mass: $\pi_{t}(d\theta^{*})\gtrsim\frac{\exp(0)\;\pi_{0}(d\theta^{*})}{\int_{\Theta}1\;\pi_{0}(d\theta)}=\pi_{0}(d\theta^{*})$. This means that Thompson sampling samples $\theta^{*}$, and hence plays $a^{*}$, with at least a constant probability each time, so that $N_{t}(a^{*})=\Omega(t)$.

Suppose we can show that each model in any $S_{a}^{\prime\prime}$, $a\neq a^{*}$, is such that $D(\theta^{*}_{a^{*}}||\theta_{a^{*}})$ is bounded strictly away from with a gap of $\xi>0$. Then, our preceding calculation immediately tells us that any such model is sampled at time $t$ with a probability exponentially decaying in $t$: $\pi_{t}(d\theta)\lesssim\frac{e^{-\xi\Omega(t)}\pi_{0}(d\theta)}{\pi_{0}(d\theta^{*})}$; the regret from such $S_{a}^{\prime\prime}$-sampling is negligible. On the other hand, how much does the algorithm have to work to make models in $S_{a}^{\prime}$, $a\neq a^{*}$ suffer large ($\approx\log T$) losses and thus rid them of significant posterior probability?Note: Plays of $a^{*}$ do not help increase the losses of these models.

A model $\theta\in S_{a}^{\prime}$ suffers loss whenever the algorithm plays an action $a$ for which $D(\theta^{*}_{a}||\theta_{a})>0$. Hence, several actions can help in making a bad model (or set of models) suffer large enough loss. Imagine that we track the play count vector $N_{t}:=(N_{t}(a))_{a\in\mathcal{A}}$ in the integer lattice from $t=0$ through $t=T$, from its initial value $N_{0}=(0,\ldots,0)$. There comes a first time $\tau_{1}$ when some action $a_{1}\neq a^{*}$ is eliminated (i.e., when all its models’ losses exceed $\log T$). The argument of the preceding paragraph indicates that the play count of $a_{1}$ will stay fixed at $N_{\tau_{1}}(a_{1})$ for the remainder of the horizon up to $T$. Moving on, there arrives a time $\tau_{2}\geq\tau_{1}$ when another action $a_{2}\notin\{a*,a_{1}\}$ is eliminated, at which point its play count ceases to increase beyond $N_{\tau_{2}}(a_{2})$, and so on.

To sum up: Continuing until all actions $a\neq a^{*}$ (i.e., the regions $S_{a}^{\prime}$) are eliminated, we have a path-based bound for the total number of times suboptimal actions can be played. If we let $z_{k}=N_{\tau_{k}}$, i.e., the play counts of all actions at time $\tau_{k}$, then for all $i\geq k$ we must have the constraint $z_{i}(a_{k})=z_{k}(a_{k})$ as plays of $a_{k}$ do not occur after time $\tau_{k}$. Moreover, $\min_{\theta\in S_{a_{k}}^{\prime}}\langle z_{k},D_{\theta}\rangle\approx\log T$: action $a_{k}$ is eliminated precisely at time $\tau_{k}$. A bound on the total number of bad plays thus becomes

The final constraint above ensures that an action $a_{k}$ is eliminated at time $\tau_{k}$, and the penultimate constraint encodes the fact that the eliminated action $a_{k}$ is not played after time $\tau_{k}$. The bound not only depends on $\log T$ but also on the KL-divergence geometry of the bandit, i.e., the marginal divergences $D(\theta^{*}_{a}||\theta_{a})$. Notice that no specific form for the prior or posterior was assumed to derive the bound, save the fact that $\pi_{0}(d\theta^{*})\gtrsim 0$, i.e., that the prior puts “enough” mass on the truth.

In fact, all our approximate calculations leading up to the bound (2) hold rigorously – Theorem 1, to follow, states that under reasonable conditions on the prior, the number of suboptimal plays/regret scales as (2) with high probability. We will also see that the general bound (2) is non-trivial in that (a) for the standard multi-armed bandit, it gives essentially the optimum known regret scaling, and (b) for a family of complex bandit problems, it can be significantly less than the one obtained by treating all actions separately.

Regret Performance: Formal Results

Our main result is a high-probability large-horizon regret boundMore precisely, we bound the number of plays of suboptimal actions. A bound on the standard regret can also be obtained easily from this, via a self-normalizing concentration inequality we use in this paper (Appendix A). However, we avoid stating this in the interest of minimizing clutter in the presentation, since there will be additional $O(\sqrt{\log T})$ terms in the bound on standard regret. for Thompson sampling. The bound holds under the following mild assumptions about the parameter space $\Theta$, action space $|\mathcal{A}|$, observation space $|\mathcal{Y}|$, and the fictitious prior $\pi$.

Remark: We emphasize that the finiteness assumption on the prior is made primarily for technical tractability, without compromising the key learning dynamics of Thompson sampling perform well. In a sense, a continuous prior can be approximated by a fine enough discrete prior without affecting the geometric structure of the parameter space. The core ideas driving our analysis explain why Thompson sampling provably performs well in very general action-observation settings, and, we believe, can be made general enough to handle even continuous priors/posteriors. However, the issues here are primarily measure-theoretic: much finer control will be required to bound and track posterior probabilities in the latter case, perhaps requiring the design of adaptive neighbourhoods of $\theta^{*}$ with sufficiently large posterior probabiity that depend on the evolving history of the algorithm. It is not clear to us how such regions may be constructed for obtaining regret guarantees in the case of continuous priors. We thus defer this highly nontrivial task to future work.

For each action $a\in\mathcal{A}$, let $S_{a}:=\{\theta\in{\Theta}:a^{*}(\theta)=a\}$ be the set of parameters for which playing $a$ is optimal. For any suboptimal action $a\neq a^{*}$, let $S_{a}^{\prime}:=\{\theta\in S_{a}:D(\theta^{*}_{a^{*}}||\theta_{a^{*}})=0\}$, $S_{a}^{\prime\prime}:=S_{a}\setminus S_{a}^{\prime}$, and $\xi:=\inf_{\theta\in S_{a}^{\prime\prime}}D(\theta^{*}_{a^{*}}||\theta_{a^{*}})$.

We now state the regret bound for Thompson sampling for general stochastic bandits. The bound is a rigorous version of the path-based bound presented earlier, in Section 3.

Under Assumptions 1-3, the following holds for the Thompson Sampling algorithm. For $\delta,\epsilon\in(0,1)$, there exists $T^{\star}\geq 0$ such that for all $T\geq T^{\star}$, with probability at least $1-\delta$, $\sum_{a\neq a^{*}}N_{T}(a)\leq\mathsf{B}+\mathsf{C}(\log T)$, where $\mathsf{B}\equiv\mathsf{B}(\delta,\epsilon,\mathcal{A},\mathcal{Y},\Theta,\pi)$ is a problem-dependent constant that does not depend on $T$, and $\mathsf{C}(\log T)\equiv\mathsf{C}(T,\delta,\epsilon,\mathcal{A},\mathcal{Y},\Theta,\pi)$ in general, but we suppress the dependence on the problem parameters $\delta,\epsilon,\mathcal{A},\mathcal{Y},\Theta,\pi$ as we are chiefly concerned with the time scaling.:

The proof is in Appendix A of the supplementary material, and uses a recently developed self-normalized concentration inequality to help track the sample path evolution of the posterior distribution in its general form. The power of Theorem 1 lies in the fact that it accounts for coupling of information across complex actions and give improved structural constants for the regret scaling than the standard decoupled case, as we showWe remark that though the non-scaling (with $T$) additive constant $\mathsf{B}$ might appear large, we believe it is an artifact of our proof technique tailored to extract the time scaling of the regret. Indeed, numerical results show practically no additive factor behaviour. in Corollaries 1 and 2. We also prove Proposition 2, which explicitly quantifies the improvement over the naive regret scaling for general complex bandit problems as a function of marginal KL-divergence separation in the parameter space $\Theta$.

A natural finite prior for this problem can be obtained by discretizing each of the $N$ basic dimensions and putting uniform mass over all points: $\Theta=\left\{\beta,2\beta,\ldots\left(\lfloor\frac{1}{\beta}\rfloor-1\right)\beta\right\}^{N}$, $\beta\in(0,1)$, and $\pi(\theta)=\frac{1}{|\Theta|}$ $\forall\theta\in\Theta$. We can then show, using Theorem 1, that

Suppose $\mu\equiv(\mu_{1},\mu_{2},\ldots,\mu_{N})\in\Theta$ and $\mu_{N-M}<\mu_{N-M+1}$. Then, the following holds for the Thompson sampling algorithm for $\mathcal{Y}$, $\mathcal{A}$, $f$, $g$, $\Theta$ and $\pi$ as above. For $\delta,\epsilon\in(0,1)$, there exists $T^{\star}\geq 0$ such that for all $T\geq T^{\star}$, with probability at least $1-\delta$, $\sum_{a\neq a^{*}}N_{T}(a)\leq\mathsf{B}_{2}+\left(\frac{1+\epsilon}{1-\epsilon}\right)\sum_{i=1}^{N-M}\frac{1}{D(\mu_{i}||\mu_{N-M+1})}\log T$, where $\mathsf{B}_{2}\equiv\mathsf{B}_{2}(\delta,\epsilon,\mathcal{A},\mathcal{Y},\Theta,\pi)$ is a problem-dependent constant that does not depend on $T$.

This result, proved in Appendix B of the supplementary material, illustrates the power of additional information from observing several arms of a bandit at once. Even though the total number of actions ${N\choose M}$ is at worst exponential in $M$, the regret bound scales only as $O((N-M)\log T)$. Note also that for $M=1$ (the standard MAB setting), the regret scaling is essentially $\sum_{i=1}^{N-M}\frac{1}{D(\mu_{i}||\mu_{N-M+1})}\log T$, which is interestingly the optimal regret scaling for standard Bernoulli bandits obtained by specialized algorithms for decoupled bandit arms such as KL-UCB and, more recently, Thompson Sampling with the independent Beta prior .

2 A General Regret Improvement Result & Application to MAX Subset Regret

Using the same setting and size-$M$ subset actions as before but not being able to observe all the individual arms’ rewards results in much more challenging bandit settings. Here, we consider the case where we get to observe as the reward only the maximum value of $M$ chosen arms of a standard $N$-armed Bernoulli bandit. The feedback is still aggregated across basic arms, but at the same time very different from the full information case, e.g., observing a reward of is very uninformative whereas a value of $1$ is highly informative about the constituent arms.

We can again apply the general machinery provided by Theorem 1 to obtain a non-trivial regret bound for observing the highly nonlinear MAX reward. Along the way, we derive the following consequence of Theorem 1, useful in its own right, that explicitly guarantees an improvement in regret directly based on the Kullback-Leibler resolvability of parameters in the parameter space – a measure of coupling across complex actions.

Let $T$ be large enough such that $\max_{\theta\in\Theta,a\in\mathcal{A}}D(\theta^{*}_{a}||\theta_{a})\leq\frac{1+\epsilon}{1-\epsilon}\log T$. Suppose $\Delta\leq\min_{a\neq a^{*},\theta\in S_{a}^{\prime}}D(\theta^{*}_{a}||\theta_{a})$, and that the integer $L$ is such that for every $a\neq a^{*}$ and $\theta\in S_{a}^{\prime}$, $|\{\hat{a}\in\mathcal{A}:\hat{a}\neq a^{*},D(\theta^{*}_{\hat{a}}||\theta_{\hat{a}})\geq\Delta\}|\geq L$, i.e., at least $L$ coordinates of $D_{\theta}$ (excluding the $|\mathcal{A}|$-th coordinate $a^{*}$) are at least $\Delta$. Then, $\mathsf{C}(\log T)\leq\left(\frac{|\mathcal{A}|-L}{\Delta}\right)\frac{2(1+\epsilon)}{1-\epsilon}\log T$.

Note that the result assures a non-trivial additive reduction of $\Omega\left(\frac{L}{\Delta}\log T\right)$ from the naive decoupled regret, whenever any suboptimal model in $\Theta$ can be resolved apart from $\theta^{*}$ by at least $L$ actions in the sense of marginal KL-divergences of their observations. Its proof is contained in Appendix C in the supplementary material.

The following holds for the Thompson sampling algorithm for $\mathcal{Y}$, $\mathcal{A}$, $f$, $g$, $\Theta$ and $\pi$ as above. For $0\leq M\leq N$, $M\neq\frac{N}{2}$, $\delta,\epsilon\in(0,1)$, there exists $T^{\star}\geq 0$ such that for all $T\geq T^{\star}$, with probability at least $1-\delta$, $\sum_{a\neq a^{*}}N_{T}(a)\leq\mathsf{B}_{3}+(\log 2)\left(\frac{1+\epsilon}{1-\epsilon}\right)\left[1+{N-1\choose M}\right]\frac{\log T}{\mu^{2}_{\min}(1-\beta)}$.

Observe that this regret bound is of the order of ${N-1\choose M}\frac{\log T}{\mu^{2}_{\min}}$, which is significantly less than the usual, decoupled bound of $|\mathcal{A}|\frac{\log T}{\mu^{2}_{\min}}={N\choose M}\frac{\log T}{\mu^{2}_{\min}}$ by a multiplicative factor of $\frac{{N-1\choose M}}{{N\choose M}}=\frac{N-M}{N}$, or by an additive factor of ${N-1\choose M-1}\frac{\log T}{\mu^{2}_{\min}}$. In fact, though this is a provable reduction in the regret scaling, the actual reduction is likely to be much better in practice – experimental results attest to this. The proof of this result uses sharp combinatorial estimates relating to vertices on the $N$-dimensional hypercube , and can be found in Appendix C, in the supplementary material.

Discussion & Future Work

We applied Thompson sampling to balance exploration and exploitation in bandit problems where the action/observation space is complex. Using a novel technique of viewing posterior evolution as a path-based optimization problem, we developed a generic regret bound for Thompson sampling with improved constants that capture the structure of the problem. In practice, the algorithm is easy to implement using sequential Monte-Carlo methods such as particle filters.

Moving forward, the technique of converting posterior concentration to an optimization involving exponentiated KL divergences could be useful in showing adversarial regret bounds for Bayesian-inspired algorithms. It is reasonable to posit that Thompson sampling would work well in a range of complex learning settings where a suitable point estimate is available. As an example, optimal bidding for online repeated auctions depending on continuous bid reward functions can be potentiallly learnt by constructing an estimate of the bid curve.

Another unexplored direction is handling large scale reinforcement learning problems with complex, state-dependent Markovian dynamics. It would be promising if computationally demanding large-state space MDPs could be solved using a form of Thompson sampling by policy iteration after sampling from a parameterized set of MDPs; this has previously been shown to work well in practice . We can also attempt to develop a theoretical understanding of pseudo-Bayesian learning for complex spaces like the $X$-armed bandit problem with a continuous state space. At a fundamental level, this could result in a rigorous characterization of Thompsn sampling/pseudo-Bayesian procedures in terms of the value of information per learning step.

References

Appendix A Proof of Theorem 1

Sampling from the posterior as proportional to exponential weights: Let $N_{t}(a)$ be the number of times action $a$ has been played up to (and including) time $t$. At any time $t$, the posterior distribution $\pi_{t}$ over $\Theta$ is given by Bayes’ rule:

with the weight $W_{t}(\theta)$ of each $\theta$ being the likelihood of observing the history under $\theta$:

Here, for any $\theta\in\Theta$ and $a\in\mathcal{A}$, $\theta_{a}$ is used to denote the “marginal” probability distribution $\{l(y;a,\theta)\}_{y\in\mathcal{Y}}$ of the output of action $a$ when the bandit has parameter $\theta$. For probability measures $\nu,\mu$ over $\mathcal{Y}$, $D(\nu||\mu)$ measures the Kullback-Leibler (KL) divergence of $\nu$ wrt $\mu$.

Note that by definition, $W_{t}(\theta^{*})=1$ at all times $t$ – a fact that we use often in the analysis.

For any action-observation sequence $(a_{t},y_{t})$, $t=1,\ldots,T$ of a bandit algorithm,

Let $N_{t}^{\prime}(a)$ (resp. $N_{t}^{\prime\prime}(a)$) be the number of times that a parameter has been drawn from $S_{a}^{\prime}$ (resp. $S_{a}^{\prime\prime}$), so that $N_{t}(a)=N_{t}^{\prime}(a)+N_{t}^{\prime\prime}(a)$.

The following self-normalized, uniform deviation bound controls the empirical distribution of each row $Q_{a}(\cdot)$ of the random reward matrix $Q$. It is a version of a bound proved in .

Let $a\in\mathcal{A}$, $y\in\mathcal{Y}$ and $\delta\in(0,1)$. Then, with probability at least $1-{\delta}{\sqrt{2}}$,

Put $c:=\log\frac{|\mathcal{Y}||\mathcal{A}|}{\delta}$, and $\rho(x)\equiv\rho_{c}(x):=4\sqrt{c+\frac{\log x}{2}}$ for $x>0$. It follows that the following “good data” event occurs with probability at least $(1-\delta\sqrt{2})$:

Fix $\epsilon\in(0,1)$. There exist $\lambda,n^{\star}\geq 0$, not depending on $T$, so that the following is true. For any $\theta\in\Theta$, $a\in\mathcal{A}$ and $y\in\mathcal{Y}$, under the event $G$,

To show the second part, notice again that for fixed $\theta\in\Theta$ and $a\in\mathcal{A}$, there exists $n^{\star}_{\theta,a}\geq 0$ such that

since $\rho(x)=o(x)$. Setting $n^{\star}:=\max_{\theta\in\Theta,a\in\mathcal{A}}n^{\star}_{\theta,a}$ then completes the proof of the second part. ∎

The result of Lemma 3 implies that under the event $G$, and at all times $t\geq 1$:

Also, under the event $G$, the posterior probability of $\theta\in S_{a}^{\prime\prime}$ at all times $t$ can be bounded above using Lemma 3 and the basic bound in (6):

In the above, the penultimate inequality is by Lemma 3 applied to all actions $a\neq a^{*}$, and the final inequality follows in a manner similar to (6), for action $a^{*}$. Letting $d:=\frac{e^{\lambda|\mathcal{A}|}}{\pi(\theta^{*})}$, we have that under the event $G$, for $a\neq a^{*}$ and $\theta\in S_{a}^{\prime\prime}$,

Recall that by definition, any $\theta\in S_{a}^{\prime\prime}$ with $a\neq a^{*}$ can be resolved apart from $\theta^{*}$ in the action $a^{*}$, i.e., $D(\theta^{*}_{a^{*}}||\theta_{a^{*}})\geq\xi$. Moreover, the discrete prior assumption (Assumption 2) implies that $\xi>0$. Using this, we can bound the right-hand side of (8) further under the event $G$:

Integrating (9) over $\theta\in S_{a}^{\prime\prime}$ and noticing that $\pi(S_{a}^{\prime\prime})\leq 1$ gives, under $G$,

We can now estimate, using the conditional version of Markov’s inequality, the number of times that parameters from $S_{a}^{\prime\prime}$ are sampled under “good data” $G$:

where the final inequality is by (10) and the fact that $\pi_{t}(S_{a}^{\prime\prime})\leq 1$.$a\wedge b$ denotes the minimum of $a$ and $b$.

At each time $t$, if we let $\mathcal{F}_{t}$ denote the $\sigma$-algebra generated by the random variables $\{(\theta_{i},A_{i},Y_{i}):i\leq t\}$, then

where, in the penultimate step, we use $\pi_{t}(\theta^{*})\geq p^{*}\cdot\mathbf{1}_{G}$ from (7). Iterating this estimate and using it in (11) together with the trivial bound $\sqrt{N_{t}(a^{*})}\leq\sqrt{t}$ gives

Since $p^{*}e^{-\xi}+1-p^{*}<1$ and $\rho(t)\sqrt{t}=o(t)$, the sum above is dominated by a geometric series after finitely many $t$, and is thus a finite quantity $\alpha<\infty$, say. (Note that $\alpha$ does not depend on $T$.) Replacing $\delta$ by $\frac{\delta}{|\mathcal{A}|}$ and taking a union bound over all $a\neq a^{*}$, this proves

where $\lambda$ and $n^{\star}$ satisfy the assertion of Lemma 3. Thus, by Lemma 3, under $G$, and for all $\theta\in\Theta$,

where the last inequality is because $N_{t}(a)=N_{t}^{\prime}(a)+N_{t}^{\prime\prime}(a)$, and because $b_{\theta,a}(x)$ is monotone non-decreasing in $x$.

Note: In what follows, we assume that $T>0$ is large enough such that $\log T\geq\frac{\lambda|\mathcal{A}|}{\epsilon}$ holds.

We proceed to define the following sequence of non-decreasing stopping times, and associated sets of actions, for the time horizon $1,2,\ldots,T$.

Let $\tau_{0}:=1$ and $\mathcal{A}_{0}:=\emptyset$. For each $k=1,\ldots,|\mathcal{A}|-1$, let

In other words, for each $k$, $\mathcal{A}_{k}$ represents a set of “eliminated” suboptimal actions. $\tau_{k}$ is the first time after $\tau_{k-1}$, when some suboptimal action (which is not already eliminated) gets eliminated in the sense of satisfying the inequality in (12). Essentially, the inequality checks whether the condition

is met for all particles $\theta\in S_{\mathsf{a}_{k}}^{\prime}$ at time $t$, with a slight modification in that the play count $N_{t}^{\prime}(a)$ is “frozen” to $N_{\tau_{m}}(a_{m})$ if action $a$ has been eliminated at an earlier time $\tau_{m}\leq t$, and the introduction of the factor $\frac{1+\epsilon}{1-\epsilon}$ to the right hand side.

In case more than one suboptimal action is eliminated for the first time at $\tau_{k}$, we use a fixed tie-breaking rule in $\mathcal{A}$ to resolve the tie. We then put

Thus, $\tau_{0}\leq\tau_{1}\leq\ldots\leq\tau_{|\mathcal{A}|-1}\leq T$, and $\mathcal{A}_{0}\subseteq\mathcal{A}_{1}\subseteq\ldots\subseteq\mathcal{A}_{|\mathcal{A}|-1}=\mathcal{A}$.

For each action $a\neq a^{*}$, by definition, there exists a unique $\tau_{k}$ for which $a$ is first eliminated at $\tau_{k}$, i.e., $\mathcal{A}_{k}\setminus\mathcal{A}_{k-1}=a$. Let $\tau(a):=\tau_{k}$.

The following lemma states that after an action $a$ is eliminated, the algorithm does not sample from $S_{a}^{\prime}$ more than a constant number of times.

If $\log T\geq\lambda|\mathcal{A}|$, then

Observe that under $G$, whenever $T\geq t>\tau_{k}$, every $\theta\in S_{\mathsf{a}_{k}}^{\prime}$ satisfies

The second inequality above is because the definition of $b_{\theta,a}(x)$ implies that $\forall x\geq 0\;(1-\epsilon)xD(\theta^{*}_{a}||\theta_{a})-b_{\theta,a}(x)\leq\lambda$. The penultimate inequality above is due to the fact that for any $m\leq k$, we have $\tau_{m}\leq\tau_{k}\leq t$, implying that $N_{t}^{\prime}(a_{m})\geq N_{\tau_{m}}^{\prime}(a_{m})$. We now estimate

Replacing $\delta$ by $\frac{\delta}{|\mathcal{A}|}$ and taking a union bound over $k=1,2,\ldots,|\mathcal{A}|-1$ proves the lemma. ∎

Now we bound the number of plays of suboptimal actions under the event

which, according to the results of Theorem 3, Lemma 4 and Lemma 5, occurs with probability at least $1-(\delta\sqrt{2}+2\delta)$. Under the event $H$, we have

Under $H$, $\sum_{k=1}^{|\mathcal{A}|-1}N_{\tau_{k}}^{\prime}(\mathsf{a}_{k})\leq\mathsf{C}_{T}$, where $\mathsf{C}_{T}$ solves

With regard to the definition of the $\tau_{k}$ and $\mathsf{a}_{k}$ in (12), if we take

then it follows, from (12), that the $z_{k}$ and $a_{k}$ satisfy all the constraints of the optimization problem (13). We also have $\sum_{k=1}^{|\mathcal{A}|-1}z_{k}(k)=\sum_{k=1}^{|\mathcal{A}|-1}N_{\tau_{k}}^{\prime}(\mathsf{a}_{k})$. This proves the lemma. ∎

Appendix B Proof of Corollary 1

The optimal action (in this case a subset) is $a^{*}=\{N-M+1,\ldots,N\}$. It can be checked that the assumptions 1-3 are verified, thus the bound (3) applies and we will be done if we estimate $\mathsf{C}(\log T)$.

The essence of the proof is to first partition the space of suboptimal actions (subsets) according to the least-index basic arm that they contain, i.e., for $i=1,2,\ldots,N-M$, let

be all the actions whose least-index (or “weakest”) arm is $i$ This covers all of $\mathcal{A}\setminus\{a^{*}\}$ since every suboptimal set must contain a basic arm of index $N-M$ or lesser..

Take any sequence $\{z_{k}\}_{k=1}^{|\mathcal{A}|-1}$, $\{a_{k}\}_{k=1}^{|\mathcal{A}|-1}$ feasible for (3). Fix $1\leq i\leq N-M$ and consider the sum $\sum_{k:a_{k}\in\mathcal{A}_{i}}z_{k}(a_{k})$. We claim that this does not exceed $1+\left(\frac{1+\epsilon}{1-\epsilon}\right)\frac{1}{D(\mu_{i}||\mu_{N-M+1})}\log T$. If, on the contrary, it does, then put $\hat{k}:=\max\{k:a_{k}\in\mathcal{A}_{i}\}$. Take any model $\theta\in S_{a_{\hat{k}}}^{\prime}$. We must have $D(\mu_{a^{*}}||\theta_{a^{*}})=0$. Since the KL divergence due to observing a tuple of $M$ independent rewards is simply the sum of the $M$ individual (binary) KL divergences, we get that $\theta_{j}=\mu_{j}$ for all $j\geq N-M+1$. However, the optimal action for $\theta$ is $a_{\hat{k}}$ containing the basic arm $i$. Hence, we get that $\theta_{i}\geq\mu_{N-M+1}\geq\mu_{i}$, which implies that $D(\mu_{i}||\theta_{i})\geq D(\mu_{i}||\mu_{N-M+1})$.

by hypothesis. This violates the final inequality of (3) and yields the desired contradiction. Since the above argument is valid for any $1\leq i\leq N-M$, summing over all such $i$ completes the proof.

Appendix C Proof of Proposition 2 & Corollary 2

Let $T$ be large enough such that $\max_{\theta\in\Theta,a\in\mathcal{A}}D(\theta^{*}_{a}||\theta_{a})\leq\frac{1+\epsilon}{1-\epsilon}\log T$. Then, the optimization problem (3) admits the following upper bound:

Take a feasible solution $\{z_{k},a_{k}\}_{k=1}^{|\mathcal{A}|-1}$ for the optimization problem (3). We will show that $z=z_{|\mathcal{A}|-1}$ and $a=a_{|\mathcal{A}|-1}$ satisfy the constraints (14) above and yield the same objective function value in both optimization problems.

as $z_{|\mathcal{A}|-1}(a_{k})\geq z_{k}(a_{k})$, for all $k\leq|\mathcal{A}|-1$, by (3). This shows that the objective functions of both (3) and (14) are equal at $\{z_{k},a_{k}\}_{k=1}^{|\mathcal{A}|-1}$ and $(z,a)$ respectively.

Next, for any $1\leq j\leq|\mathcal{A}|-1$ and the unit vector $e^{(j)}$, we have

This shows that the penultimate constraint in (14) is satisfied. For the final constraint in (14), fix $1\leq j\leq|\mathcal{A}|-1$, so that we have

exactly as in the preceding derivation. This implies that $z(\hat{a})\leq\frac{2}{\delta_{\hat{a}}}\left(\frac{1+\epsilon}{1-\epsilon}\right)\log T$ for all $\hat{a}\neq a^{*}$. ∎

Let $T$ be large enough such that $\max_{\theta\in\Theta,a\in\mathcal{A}}D(\theta^{*}_{a}||\theta_{a})\leq\frac{1+\epsilon}{1-\epsilon}\log T$. Suppose

i.e., at least $L$ coordinates of $D_{\theta}$ (excluding the $|\mathcal{A}|$-th coordinate $a^{*}$) are at least $\Delta$. Then,

Consider a solution $(z,a)$ to a relaxation of the optimization problem (14) obtained by replacing $\delta_{\hat{a}}$ with $\Delta$ and $D_{\theta}$ with $D_{\theta}^{\prime}:=\min(D_{\theta},\Delta\cdot\mathbf{1})\preceq D_{\theta}$ Here $\mathbf{1}$ represents an all-ones vector of dimension $\mathcal{A}$, and the minimum is taken coordinatewise. Also, a solution exists since the objective is continuous and the feasible region is compact.. We claim that $||z||_{1}\equiv\langle\mathbf{1},z\rangle\leq\left(\frac{|\mathcal{A}|-L}{\Delta}\right)\chi$ where $\chi:=\frac{2(1+\epsilon)}{1-\epsilon}\log T$. If not, let $y=\chi\left(\frac{1}{\Delta},\ldots,\frac{1}{\Delta},0\right)$, and observe that

since $D_{\theta}^{\prime}\preceq\Delta\cdot\mathbf{1}$ by definition and $z\preceq y$ by hypothesis. This is a contradiction. ∎

Playing Subsets with Max reward: Let $\beta\in(0,1)$, and suppose that $\Theta=\{1-\beta^{R},1-\beta^{R-1},\ldots,1-\beta^{2},1-\beta\}^{N}$, for positive integers $R$ and $N$. Consider an $N$ armed Bernoulli bandit with arm parameters $\mu\in\Theta$. The complex actions are all size $M$ subsets of the $N$ basic arms, $M\leq\frac{N-1}{2}$. Let $\mu_{min}:=\min_{a\in\mathcal{A}}\prod_{i\in a}(1-\mu_{i})$.

Since the reward from playing a subset $a$ is the maximum (equivalently, the Boolean OR) value, the marginal KL divergence along action $a$ is simply the Bernoulli KL divergence for the product of the parameters: $D(\theta^{*}_{a}||\theta_{a})=D(\mu_{a}||\theta_{a})=D\left(\prod_{i\in a}(1-\mu_{i})||\prod_{i\in a}(1-\theta_{i})\right)$.

If $\mu_{i}=1-\beta^{r_{i}}$ and $\theta_{i}=1-\theta^{s_{i}}$ for integers $r_{i},s_{i}$, $i=1,2,\ldots,N$, then Pinsker’s inequality yields

$D(\mu_{a}||\theta_{a})>0$ if and only if $|\sum_{i\in a}s_{i}-\sum_{i\in a}r_{i}|\geq 1$. This implies, together with the above, that

Next, we claim that for any $\mu\neq\theta\in\Theta$, $D(\mu_{a}||\theta_{a})>0$ for at least $L={N-1\choose M-1}-1$ size $M$ subsets/actions $a$. This is because if otherwise, then $\sum_{i\in a}r_{i}=\sum_{i\in a}s_{i}$ for at least ${N\choose M}-L={N\choose M}-{N-1\choose M-1}+1={N-1\choose M}+1$ subsets $a$. However, a combinatorial result states that the maximum number of weight $M$ vertices of the $N$ dimensional hypercube (in our case, a size $M$ subset corresponds to a weight $M$ vertex) that do not span $N$ dimensions is ${N-1\choose M}$. This forces $r_{i}=r_{i}$ for all $i\in[N]$ and hence $\mu=\theta$, a contradiction.

Now, we can apply Proposition 2 with $\Delta$ and $L$ as above. This gives us that for $T$ large enough, the total number of arm plays is bounded above, with probability at least $1-\delta$, by