On The Differential Privacy of Thompson Sampling With Gaussian Prior

We show that Thompson Sampling with Gaussian Prior as detailed by Algorithm 2 in (Agrawal & Goyal, 2013) is already differentially private. Theorem 1 show that it enjoys a very competitive privacy loss of only $\mathcal{O}(\ln^2 T)$ after T rounds. Finally, Theorem 2 show that one can control the privacy loss to any desirable $\epsilon$ level by appropriately increasing the variance of the samples from the Gaussian posterior. And this increases the regret only by a term of $\mathcal{O}(\frac{\ln^2 T}{\epsilon})$. This compares favorably to the previous result for Thompson Sampling in the literature ((Mishra & Thakurta, 2015)) which adds a term of $\mathcal{O}(\frac{K \ln^3 T}{\epsilon^2})$ to the regret in order to achieve the same privacy level. Furthermore, our result use the basic Thompson Sampling with few modifications whereas the result of (Mishra & Thakurta, 2015) required sophisticated constructions.

Nearly optimal exploration-exploitation decision thresholds

While in general trading off exploration and exploitation in reinforcement learning is hard, under some formulations relatively simple solutions exist. In this paper, we first derive upper bounds for the utility of selecting different actions in the multi-armed bandit setting. Unlike the common statistical upper confidence bounds, these explicitly link the planning horizon, uncertainty and the need for exploration explicit. The resulting algorithm can be seen as a generalisation of the classical Thompson sampling algorithm. We experimentally test these algorithms, as well as $\epsilon$-greedy and the value of perfect information heuristics. Finally, we also introduce the idea of bagging for reinforcement learning. By employing a version of online bootstrapping, we can efficiently sample from an approximate posterior distribution.

Learning to Match

Outsourcing tasks to previously unknown parties is becoming more common. One specific such problem involves matching a set of workers to a set of tasks. Even if the latter have precise requirements, the quality of individual workers is usually unknown. The problem is thus a version of matching under uncertainty. We believe that this type of problem is going to be increasingly important. When the problem involves only a single skill or type of job, it is essentially a type of bandit problem, and can be solved with standard algorithms. However, we develop an algorithm that can perform matching for workers with multiple skills hired for multiple jobs with multiple requirements. We perform an experimental evaluation in both single-task and multi-task problems, comparing with the bounded $\epsilon$-first algorithm, as well as an oracle that knows the true skills of workers. One of the algorithms we developed gives results approaching 85\% of oracle's performance. We invite the community to take a closer look at this problem and develop real-world benchmarks.

Calibrated Fairness in Bandits

We study fairness within the stochastic, \emph{multi-armed bandit} (MAB) decision making framework. We adapt the fairness framework of "treating similar individuals similarly" to this setting. Here, an `individual' corresponds to an arm and two arms are `similar' if they have a similar quality distribution. First, we adopt a {\em smoothness constraint} that if two arms have a similar quality distribution then the probability of selecting each arm should be similar. In addition, we define the {\em fairness regret}, which corresponds to the degree to which an algorithm is not calibrated, where perfect calibration requires that the probability of selecting an arm is equal to the probability with which the arm has the best quality realization. We show that a variation on Thompson sampling satisfies smooth fairness for total variation distance, and give an $\tilde{O}((kT)^{2/3})$ bound on fairness regret. This complements prior work, which protects an on-average better arm from being less favored. We also explain how to extend our algorithm to the dueling bandit setting.

Thompson Sampling For Stochastic Bandits with Graph Feedback

We present a novel extension of Thompson Sampling for stochastic sequential decision problems with graph feedback, even when the graph structure itself is unknown and/or changing. We provide theoretical guarantees on the Bayesian regret of the algorithm, linking its performance to the underlying properties of the graph. Thompson Sampling has the advantage of being applicable without the need to construct complicated upper confidence bounds for different problems. We illustrate its performance through extensive experimental results on real and simulated networks with graph feedback. More specifically, we tested our algorithms on power law, planted partitions and Erdo's-Renyi graphs, as well as on graphs derived from Facebook and Flixster data. These all show that our algorithms clearly outperform related methods that employ upper confidence bounds, even if the latter use more information about the graph.

Achieving Privacy in the Adversarial Multi-Armed Bandit

In this paper, we improve the previously best known regret bound to achieve $\epsilon$-differential privacy in oblivious adversarial bandits from $\mathcal{O}{(T^{2/3}/\epsilon)}$ to $\mathcal{O}{(\sqrt{T} \ln T /\epsilon)}$. This is achieved by combining a Laplace Mechanism with EXP3. We show that though EXP3 is already differentially private, it leaks a linear amount of information in $T$. However, we can improve this privacy by relying on its intrinsic exponential mechanism for selecting actions. This allows us to reach $\mathcal{O}{(\sqrt{\ln T})}$-DP, with a regret of $\mathcal{O}{(T^{2/3})}$ that holds against an adaptive adversary, an improvement from the best known of $\mathcal{O}{(T^{3/4})}$. This is done by using an algorithm that run EXP3 in a mini-batch loop. Finally, we run experiments that clearly demonstrate the validity of our theoretical analysis.

Bayesian Differential Privacy through Posterior Sampling

Differential privacy formalises privacy-preserving mechanisms that provide access to a database. We pose the question of whether Bayesian inference itself can be used directly to provide private access to data, with no modification. The answer is affirmative: under certain conditions on the prior, sampling from the posterior distribution can be used to achieve a desired level of privacy and utility. To do so, we generalise differential privacy to arbitrary dataset metrics, outcome spaces and distribution families. This allows us to also deal with non-i.i.d or non-tabular datasets. We prove bounds on the sensitivity of the posterior to the data, which gives a measure of robustness. We also show how to use posterior sampling to provide differentially private responses to queries, within a decision-theoretic framework. Finally, we provide bounds on the utility and on the distinguishability of datasets. The latter are complemented by a novel use of Le Cam's method to obtain lower bounds. All our general results hold for arbitrary database metrics, including those for the common definition of differential privacy. For specific choices of the metric, we give a number of examples satisfying our assumptions.

On the Differential Privacy of Bayesian Inference

We study how to communicate findings of Bayesian inference to third parties, while preserving the strong guarantee of differential privacy. Our main contributions are four different algorithms for private Bayesian inference on proba-bilistic graphical models. These include two mechanisms for adding noise to the Bayesian updates, either directly to the posterior parameters, or to their Fourier transform so as to preserve update consistency. We also utilise a recently introduced posterior sampling mechanism, for which we prove bounds for the specific but general case of discrete Bayesian networks; and we introduce a maximum-a-posteriori private mechanism. Our analysis includes utility and privacy bounds, with a novel focus on the influence of graph structure on privacy. Worked examples and experiments with Bayesian na{\"i}ve Bayes and Bayesian linear regression illustrate the application of our mechanisms.

Algorithms for Differentially Private Multi-Armed Bandits

We present differentially private algorithms for the stochastic Multi-Armed Bandit (MAB) problem. This is a problem for applications such as adaptive clinical trials, experiment design, and user-targeted advertising where private information is connected to individual rewards. Our major contribution is to show that there exist $(\epsilon, \delta)$ differentially private variants of Upper Confidence Bound algorithms which have optimal regret, $O(\epsilon^{-1} + \log T)$. This is a significant improvement over previous results, which only achieve poly-log regret $O(\epsilon^{-2} \log^{2} T)$, because of our use of a novel interval-based mechanism. We also substantially improve the bounds of previous family of algorithms which use a continual release mechanism. Experiments clearly validate our theoretical bounds.

Generalised Entropy MDPs and Minimax Regret

Bayesian methods suffer from the problem of how to specify prior beliefs. One interesting idea is to consider worst-case priors. This requires solving a stochastic zero-sum game. In this paper, we extend well-known results from bandit theory in order to discover minimax-Bayes policies and discuss when they are practical.