Mitigating Bias in Adaptive Data Gathering via Differential Privacy

Introduction

Many modern data sets consist of data that is gathered adaptively: the choice of whether to collect more data points of a given type depends on the data already collected. For example, it is common in industry to conduct “A/B” tests to make decisions about many things, including ad targeting, user interface design, and algorithmic modifications, and this A/B testing is often conducted using “bandit learning algorithms” Bubeck et al. (2012), which adaptively select treatments to show to users in an effort to find the best treatment as quickly as possible. Similarly, sequential clinical trials may halt or re-allocate certain treatment groups due to preliminary results, and empirical scientists may initially try and test multiple hypotheses and multiple treatments, but then decide to gather more data in support of certain hypotheses and not others, based on the results of preliminary statistical tests.

Unfortunately, as demonstrated by Nie et al. (2017), the data that results from adaptive data gathering procedures will often exhibit substantial bias. As a result, subsequent analyses that are conducted on the data gathered by adaptive procedures will be prone to error, unless the bias is explicitly taken into account. This can be difficult. Nie et al. (2017) give a selective inference approach: in simple stochastic bandit settings, if the data was gathered by a specific stochastic algorithm that they design, they give an MCMC based procedure to perform maximum likelihood estimation to recover de-biased estimates of the underlying distribution means. In this paper, we give a related, but orthogonal approach whose simplicity allows for a substantial generalization beyond the simple stochastic bandits setting. We show that in very general settings, if the data is gathered by a differentially private procedure, then we can place strong bounds on the bias of the data gathered, without needing any additional de-biasing procedure. Via elementary techniques, this connection implies the existence of simple stochastic bandit algorithms with nearly optimal worst-case regret bounds, with very strong bias guarantees. The connection also allows us to derive algorithms for linear contextual bandits with nearly optimal regret guarantees, and strong bias guarantees. Since our connection to differential privacy only requires that the rewards and not the contexts be kept private, we are able to obtain improved accuracy compared to past approaches to private contextual bandit problems. By leveraging existing connections between differential privacy and adaptive data analysis Dwork et al. (2015c); Bassily et al. (2016); Rogers et al. (2016), we can extend the generality of our approach to bound not just bias, but to correct for effects of adaptivity on arbitrary statistics of the gathered data. For example, we can obtain valid $p$-value corrections for hypothesis tests (like $t$-tests) run on the adaptively collected data. Since the data being gathered will generally be useful for some as yet unspecified scientific analysis, rather than just for the narrow problem of mean estimation, our technique allows for substantially broader possibilities compared to past approaches. Experiments explore the bias incurred by conventional bandit algorithms, confirm the reduction in bias obtained by leveraging privacy, and show why correction for adaptivity is crucial to performing valid post-hoc hypothesis tests. In particular we show that for the fundamental primitive of conducting $t$-tests for regression coefficients, naively conducting tests on adaptively gathered data leads to incorrect inference.

Using elementary techniques, we provide explicit bounds on the bias of empirical arm means maintained by bandit algorithms in the simple stochastic setting that make their selection decisions as a differentially private function of their observations. Together with existing differentially private algorithms for stochastic bandit problems, this yields an algorithm that obtains an essentially optimal worst-case regret bound, and guarantees minimal bias (on the order of $O(1/\sqrt{K\cdot T})$) for the empirical mean maintained for every arm.

We then extend our results to the linear contextual bandit problem. We show that algorithms that make their decisions in a way that is differentially private in the observed reward of each arm (but which need not be differentially private in the context) have bounded bias (as measured by the difference between the predicted reward of each arm at each time step, compared to its true reward). We also derive a differentially private algorithm for the contextual bandit problem, and prove new bounds for it. Together with our bound on bias, this algorithm also obtains strong sublinear regret bounds, while having robust guarantees on bias.

We then make a general observation, relating adaptive data gathering to an adaptive analysis of a fixed dataset (in which the choice of which query to pose to the dataset is adaptive). This lets us apply the large existing literature connecting differential privacy to adaptive data analysis Dwork et al. (2015a, c); Bassily et al. (2016). In particular, it lets us apply the max-information bounds of Dwork et al. (2015b); Rogers et al. (2016) to our adaptive data gathering setting. This allows us to give much more general guarantees about the data collected by differentially private collection procedures, that extend well beyond bias. For example, it lets us correct the $p$-values for arbitrary hypothesis tests run on the gathered data.

Finally, we run a set of experiments that measure the bias incurred by the standard UCB algorithm in the stochastic bandit setting, contrast it with the low bias obtained by a private UCB algorithm, and show that there are settings of the privacy parameter that simultaneously can make bias statistically insignificant, while having competitive empirical regret with the non-private UCB algorithm. We also demonstrate in the linear contextual bandit setting how failing to correct for adaptivity can lead to false discovery when applying $t$-tests for non-zero regression coefficients on an adaptively gathered dataset.

2 Related Work

This paper bridges two recent lines of work. Our starting point is two recent papers: Villar et al. (2015) empirically demonstrate in the context of clinical trials that a variety of simple stochastic bandit algorithms produce biased sample mean estimates (Similar results have been empirically observed in the context of contextual bandits Dimakopoulou et al. (2017)). Nie et al. (2017) prove that simple stochastic bandit algorithms that exhibit two natural properties (satisfied by most commonly used algorithms, including UCB and Thompson Sampling) result in empirical means that exhibit negative bias. They then propose a heuristic algorithm which computes a maximum likelihood estimator for the sample means from the empirical means gathered by a modified UCB algorithm which adds Gumbel noise to the decision statistics. Deshpande et al. (2017) propose a debiasing procedure for ordinary least-squares estimates computed from adaptively gathered data that trades off bias for variance, and prove a central limit theorem for their method. In contrast, the methods we propose in this paper are quite different. Rather than giving an ex-post debiasing procedure, we show that if the data were gathered in a differentially private manner, no debiasing is necessary. The strength of our method is both in its simplicity and generality: rather than proving theorems specific to particular estimators, we give methods to correct the $p$-values for arbitrary hypothesis tests that might be run on the adaptively gathered data.

The second line of work is the recent literature on adaptive data analysis Dwork et al. (2015c, b); Hardt and Ullman (2014); Steinke and Ullman (2015); Russo and Zou (2016); Wang et al. (2016); Bassily et al. (2016); Hardt and Blum (2015); Cummings et al. (2016); Feldman and Steinke (2017a, b) which draws a connection between differential privacy Dwork et al. (2006) and generalization guarantees for adaptively chosen statistics. The adaptivity in this setting is dual to the setting we study in the present paper: In the adaptive data analysis literature, the dataset itself is fixed, and the goal is to find techniques that can mitigate bias due to the adaptive selection of analyses. In contrast, here, we study a setting in which the data gathering procedure is itself adaptive, and can lead to bias even for a fixed set of statistics of interest. However, we show that adaptive data gathering can be re-cast as an adaptive data analysis procedure, and so the results from the adaptive data analysis literature can be ported over.

Preliminaries

In a simple stochastic bandit problem, there are $K$ unknown distributions $P_{i}$ over the unit interval , each with (unknown) mean $\mu_{i}$. Over a series of rounds $t\in\{1,\ldots,T\}$, an algorithm $\mathcal{A}$ chooses an arm $i_{t}\in[K]$, and observes a reward $y_{i_{t},t}\sim P_{i_{t}}$. Given a sequence of choices $i_{1},\ldots,i_{T}$, the pseudo-regret of an algorithm is defined to be:

The definition of an algorithm $\mathcal{A}$ induces a sequence of $T$ (possibly randomized) selection functions $f_{t}:\mathcal{H}^{t-1}\rightarrow[K]$, which map histories onto decisions of which arm to pull at each round.

2 Contextual Bandit Problems

In the contextual bandit problem, decisions are endowed with observable features. Our algorithmic results in this paper focus on the linear contextual bandit problem, but our general connection between adaptive data gathering and differential privacy extends beyond the linear case. For simplicity of exposition, we specialize to the linear case here.

3 Data Gathering in the Query Model

Above we’ve characterized a bandit algorithm $\mathcal{A}$ as gathering data adaptively using a sequence of selection functions $f_{t}$, which map the observed history $\Lambda_{t}\in\mathcal{H}^{t-1}$ to the index of the next arm pulled. In this model only after the arm is chosen is a reward drawn from the appropriate distribution. Then the history is updated, and the process repeats.

In this section, we observe that whether the reward is drawn after the arm is “pulled,” or in advance, is a distinction without a difference. We cast this same interaction into the setting where an analyst asks an adaptively chosen sequence of queries to a fixed dataset, representing the arm rewards. The process of running a bandit algorithm $\mathcal{A}$ up to time $T$ can be formalized as the adaptive selection of $T$ queries against a single database of size $T$ - fixed in advance. The formalization consists of two steps:

By the principle of deferred randomness, we view any simple stochastic bandit algorithm as operating in a setting in which $T$ i.i.d. samples from $\prod_{i=1}^{K}P_{i}$ (vectors of length $K$ representing the rewards for each of $K$ arms on each time step $t$) are drawn before the interaction begins. This is the Interact algorithm below.

In the contextual setting, the contexts are also available, and the $T$ draws are not drawn from identical distributions. Instead, the $t^{th}$ draw is from $\prod_{i=1}^{K}P_{i}^{t}$, where each distribution $P_{i}^{t}$ is determined by the context $x_{i}^{t}$.

The choice of arm pulled at time $t$ by the bandit algorithm can be viewed as the answer to an adaptively selected query against this fixed dataset. This is the InteractQuery algorithm below.

Adaptive data analysis is formalized as an interaction in which a data analyst $\mathcal{A}$ performs computations on a dataset $D$, observes the results, and then may choose the identity of the next computation to run as a function of previously computed results Dwork et al. (2015c, a). A sequence of recent results shows that if the queries are differentially private in the dataset $D$, then they will not in general overfit $D$, in the sense that the distribution over results induced by computing $q(D)$ will be “similar” to the distribution over results induced if $q$ were run on a new dataset, freshly sampled from the same underlying distribution Dwork et al. (2015c, a); Bassily et al. (2016); Dwork et al. (2015b); Rogers et al. (2016). We will be more precise about what these results say in Section 5.

Recall that histories $\Lambda$ record the choices of the algorithm, in addition to its observations. It will be helpful to introduce notation that separates out the choices of the algorithm from its observations. In the simple stochastic setting and the contextual setting, given a history $\Lambda_{t}$, an action history $\Lambda_{t}^{\mathcal{A}}=(i_{1},\ldots,i_{t-1})\in[K]^{t-1}$ denotes the portion of the history recording the actions of the algorithm.

Given a bandit tableau and a bandit algorithm $\mathcal{A}$, we have the following interaction:

We now define Algorithms Bandit and InteractQuery. Bandit is a standard contextual bandit algorithm defined by selection functions $f_{t}$, and InteractQuery is the Interact routine that draws the rewards in advance, and at time $t$ selects action $i_{t}$ as the result of query $q_{\Lambda_{t}^{\mathcal{A}}}$. With the above definitions in hand, it is straightforward to show that the two Algorithms are equivalent, in that they induce the same joint distribution on their outputs. In both algorithms for convenience we assume we are in the linear contextual setting, and we write $\eta_{i_{t}}$ to denote the i.i.d. error distributions of the rewards, conditional on the contexts.

Let $P_{1,t}$ be the joint distribution induced by Algorithm Bandit on $\Lambda_{t}$ at time $t$, and let $P_{2,t}$ be the joint distribution induced by Algorithm InteractQuery on $\Lambda_{t}=\Lambda_{t}^{\mathcal{A}}(D)$. Then $\forall t\;P_{1,t}=P_{2,t}$.

The upshot of this equivalence is that we can import existing results that hold in the setting in which the dataset is fixed, and queries are adaptively chosen. There are a large collection of results of this form that apply when the queries are differentially private Dwork et al. (2015c); Bassily et al. (2016); Rogers et al. (2016) which apply directly to our setting. In the next section we formally define differential privacy in the simple stochastic and contextual bandit setting, and leave the description of the more general transfer theorems to Section 5.

4 Differential Privacy

We will be interested in algorithms that are differentially private. In the simple stochastic bandit setting, we will require differential privacy with respect to the rewards. In the contextual bandit setting, we will also require differential privacy with respect to the rewards, but not necessarily with respect to the contexts.

We now define the neighboring relation we need to define bandit differential privacy:

Note that changing a context does not result in a neighboring tableau: this neighboring relation will correspond to privacy for the rewards, but not for the contexts.

Note that we could have equivalently defined reward neighbors to be tableaus that differ in only a single entry, rather than in an entire row. The distinction is unimportant in a bandit setting, because a bandit algorithm will be able to observe only a single entry in any particular row.

If $\delta=0$, we say that $\mathcal{A}$ is $\epsilon$-differentially private.

5 The Binary Mechanism

The Binary mechanism Chan et al. (2011); Dwork et al. (2010) is an online algorithm that continually releases an estimate of a running sum $\sum_{i=1}^{t}y_{i}$ as each $y_{i}$ arrives one at a time, while preserving $\epsilon$-differential privacy of the entire sequence $(y_{i})_{i=1}^{T}$, and guaranteeing worst case error that scales only with $\log(T)$. It does this by using a tree-based aggregation scheme that computes partial sums online using the Laplace mechanism, which are then combined to produce estimates for each sample mean $\hat{Y}_{i}^{t}$. Since the scheme operates via the Laplace mechanism, it extends immediately to the setting when each $y_{i}$ is a vector with bounded $l_{1}$ norm. In our private algorithms we actually use a modified version of the binary mechanism due to Chan et al. (2011) called the hybrid mechanism, which operates without a fixed time horizon $T$. For the rest of the paper we denote the noise added to the $t^{th}$ partial sum by the hybrid mechanism, either in vector or scalar form, as $\eta\sim\text{Hybrid}(t,\epsilon)$.

where $\log^{*}$ denotes the binary iterated logarithm, and Ln is the function defined as $\text{Ln}(n)=\prod_{r=0}^{\log^{*}(n)}\log^{(r)}n$ in Chan et al. (2011).

Privacy Reduces Bias in Stochastic Bandit Problems

We begin by showing that differentially private algorithms that operate in the stochastic bandit setting compute empirical means for their arms that are nearly unbiased. Together with known differentially private algorithms for stochastic bandit problems, the result is an algorithm that obtains a nearly optimal (worst-case) regret guarantee while also guaranteeing that the collected data is nearly unbiased. We could (and do) obtain these results by combining the reduction to answering adaptively selected queries given by Theorem 1 with the standard generalization theorems in adaptive data analysis (e.g. Corollary 3 in its most general form), but we first prove these de-biasing results from first principles to build intuition.

Let $\mathcal{A}$ be an $(\epsilon,\delta)$-differentially private algorithm in the stochastic bandit setting. Then, for all $i\in[K]$, and all $t$, we have:

Note that since $\mu_{i}\in$, and for $\epsilon\ll 1$, $e^{\epsilon}\approx 1+\epsilon$, this theorem bounds the bias by roughly $\epsilon+T\delta$. Often, we will have $\delta=0$ and so the bias will be bounded by roughly $\epsilon$.

where the first equality follows by the linearity of expectation, and the second follows by the law of iterated expectation.

Our goal is to show that the conditioning in the inner expectation does not substantially change the value of the expectation. Specifically, we want to show that all $t$, and any value $y_{it}$, we have

which is what we want (The reverse inequality is symmetric).

as desired. The upper bound on the bias follows symmetrically from Lemma 2. ∎

There are existing differentially private variants of the classic UCB algorithm (Auer et al. (2002); Agrawal (1995); Lai and Robbins (1985)), which give a nearly optimal tradeoff between privacy and regret Mishra and Thakurta (2014); Tossou and Dimitrakakis (2017, 2016). For completeness, we give a simple version of a private UCB algorithm in the Appendix which we use in our experiments. Here, we simply quote the relevant theorem, which is a consequence of a theorem in Tossou and Dimitrakakis (2016):

Tossou and Dimitrakakis (2016) Let $\{\mu_{i}:i\in[K]\}$ be the means of the $k$-arms. Let $\mu^{*}=\max_{k}\mu_{k}$, and for each arm $k$ let $\Delta=\min_{\mu_{k}<\mu^{*}}\mu^{*}-\mu_{k}$. Then there is an $\epsilon$-differentially private algorithm that obtains expected regret bounded by:

where $B=\frac{\sqrt{8}}{2\epsilon}\ln(4T^{4})$. Taking the worst case over instances (values $\Delta_{k}$) and recalling that $\sum_{k}N_{k}^{T}=T$, this implies expected regret bounded by:

Thus, we can take $\epsilon$ to be as small as $\epsilon=O(\frac{\ln^{1.5}T}{\sqrt{kT}})$ while still having a regret bound of $O(\sqrt{kT\log T})$, which is nearly optimal in the worst case (over instances) Audibert and Bubeck (2009).

Combining the above bound with Theorem 2, and letting $\epsilon=O(\frac{\ln^{1.5}T}{\sqrt{kT}})$, we have:

There exists a simple stochastic bandit algorithm that simultaneously guarantees that the bias of the empirical average for each arm $i$ is bounded by $O(\mu_{i}\cdot\frac{\ln^{1.5}T}{\sqrt{kT}})$ and guarantees expected regret bounded by $O(\sqrt{kT\log T})$.

Privacy Reduces Bias in Linear Contextual Bandit Problems

In this section, we extend Theorem 2 to directly show that differential privacy controls a natural measure of “bias” in linear contextual bandit problems as well. We then design and analyze a new differentially private algorithm for the linear contextual bandit problem, based on the Lin-UCB algorithm Li et al. (2010). This will allow us to give an algorithm which simultaneously offers bias and regret bounds.

We write $\Lambda_{i,T}$ to denote the sequence of context/reward pairs for arm $i$ that a contextual bandit algorithm $\mathcal{A}$ has observed through time step $T$. Note that $|\Lambda_{i,T}|=N_{i}^{T}$. It will sometimes be convenient to separate out contexts and rewards: we will write $C_{i,T}$ to refer to just the sequence of contexts observed through time $T$, and $D_{i,T}$ to refer to just the corresponding sequence of rewards observed through time $T$. Note that once we fix $\Lambda_{i,T}$, $C_{i,T}$ and $D_{i,T}$ are determined, but fixing $C_{i,T}$ leaves $D_{i,T}$ a random variable. The randomness in $C_{i,T}$ is over which contexts from arm $i$ $\mathcal{A}$ has selected by round $T$, not over the actual contexts $x_{it}$ - these are fixed. Thus the following results will hold over a worst-case set of contexts, including when the contexts are drawn from an arbitrary distribution. We will denote the sequence of arms pulled by $\mathcal{A}$ up to time $T$ by $\Lambda_{T}^{\mathcal{A}}$. We note that $\Lambda_{T}^{\mathcal{A}}$ fixes $C_{i,T}$ independently of the observed rewards $D_{i,T}$, and so if $\mathcal{A}$ is differentially private in the observed rewards, the post-processing $C_{i,T}$ is as well. First, we define the least squares estimator:

Given a sequence of observations $\Lambda_{i,T}$, a least squares estimator $\hat{\theta}_{i}$ is any vector that satisfies:

Fix a time horizon $T$, a tableau of contexts, an arm $i$, and a contextual bandit algorithm $\mathcal{A}$. Let $\hat{\theta}_{i}$ be the least squares estimator trained on the set of observations $\Lambda_{i,T}$. Then the bias of arm $i$ is defined to be the maximum bias of the predictions made by $\hat{\theta}_{i}$ on the contexts in $C_{i,T}$, over any worst case realization of $C_{i,T}$. The inner expectation is over $D_{i,T}$ since $\hat{\theta}_{i}$ depends on the rewards at arm $i$.

It then follows from an elementary application of differential privacy similar to that in the proof of Theorem 2, that if the algorithm $\mathcal{A}$ makes its arm selection decisions in a way that is differentially private in the observed sequences of rewards, the least squares estimators computed based on the observations of $\mathcal{A}$ have bounded bias as defined above. The proof is deferred to the Appendix.

Let $\mathcal{A}$ be any linear contextual bandit algorithm whose selections are $\epsilon$-differentially private in the rewards. Fix a time horizon $T$, and let $\hat{\theta}_{i}$ be a least squares estimator computed on the set of observations $\Lambda_{i,T}$. Then for every arm $i\in[K]$ and any round $t$:

Below we outline a reward-private variant of the LinUCB algorithm Chu et al. (2011), and state a corresponding regret bound. In combination with Theorem 4 this will give an algorithm that yields a smooth tradeoff between regret and bias. This algorithm is similar to the private linear UCB algorithm presented in Mishra and Thakurta (2014). The main difference compared to the algorithm in Mishra and Thakurta (2014) is that Theorem 4 requires only reward privacy, whereas the algorithm from Mishra and Thakurta (2014) is designed to guarantee privacy of the contexts as well. The result is that we can add less noise, which also makes the regret analysis more tractable — none is given in Mishra and Thakurta (2014) — and the regret bound better. Estimates of the linear function at each arm are based on the ridge regression estimator, which gives a lower bound on the singular values of the design matrix and hence an upper bound on the effect of the noise. As part of the regret analysis we use the self-normalized martingale inequality developed in Abbasi-Yadkori et al. (2011); for details see the proof in the Appendix.

Algorithm 1 is $\epsilon$-reward differentially private and has regret:

The following corollary follows by setting $\lambda=1$ and setting $\epsilon$ to be as small as possible, without it becoming an asymptotically dominant term in the regret bound. We then apply Theorem 4 to convert the privacy guarantee into a bias guarantee.

Setting $\lambda=1$ and $\epsilon=O(\sqrt{\frac{K}{T}})$, Algorithm 1 has regret:

with probability $1-\delta$, and for each arm $i$ satisfies

Readers familiar with the linear contextual bandit literature will remark that the optimal non-private regret bound in the realizable setting scales like $O(\sqrt{Td\log K})$ Chu et al. (2011), as opposed to $O(d\sqrt{TK})$ above. This is an artifact of the fact that for ease of presentation we have analyzed a simpler LinUCB variant using techniques from Abbasi-Yadkori et al. (2011), rather than the more complicated SupLinUCB algorithm of Chu et al. (2011). It is not a consequence of using the binary mechanism to guarantee privacy – it is likely the same technique would give a private variant of SupLinUCB with a tighter regret bound than the one given above.

Max Information & Arbitrary Hypothesis Tests

Up through this point, we have focused our attention on showing how the private collection of data mitigates the effect that adaptivity has on bias, in both the stochastic and contextual bandit problems. In this section, we draw upon more powerful results from the adaptive data analysis literature to go substantially beyond bias: to correct the $p$-values of hypothesis tests applied to adaptively gathered data. These $p$-value corrections follow from the connection between differential privacy and a quantity called max information, which controls the extent to which the dependence of selected test on the dataset can distort the statistical validity of the test (Dwork et al., 2015b; Rogers et al., 2016). We briefly define max information, state the connection to differential privacy, and illustrate how max information bounds can be used to perform adaptive analyses in the private data gathering framework.

Let $X,Z$ be jointly distributed random variables over domain $(\mathcal{X},\mathcal{Z})$. Let $X\otimes Z$ denote the random variable that draws independent copies of $X,Z$ according to their marginal distributions. The max-information between $X,Z$, denoted $I_{\infty}(X,Z)$, is defined:

Similarly, we define the $\beta$-approximate max information

A function $\gamma:\to$ is a valid $p$-value correction function for $\mathcal{A}$ if the procedure:

Select a test statistic $t=\mathcal{A}(D)$

Reject the null hypothesis if $p(t(D))\leq\gamma(\alpha)$

has probability at most $\alpha$ of rejection, when $D\sim P_{0}$.

Then the following theorem gives a valid $p$-value correction function when $(D,A(D))$ have bounded $\beta$-approximate max information.

Let $\mathcal{A}$ be a data-dependent algorithm for selecting a test statistics such that $I_{\beta}(X,\mathcal{A}(X))\leq k$. Then the following function $\gamma$ is a valid $p$-value correction function for $\mathcal{A}$:

Finally, we can connect max information to differential privacy, which allows us to leverage private algorithms to perform arbitrary valid statistical tests.

Let $\mathcal{A}$ be an $\epsilon$-differentially private algorithm, let $P$ be an arbitrary product distribution over datasets of size $n$, and let $D\sim P$. Then for every $\beta>0$:

Rogers et al. (2016) extend this theorem to algorithm satisfying $(\epsilon,\delta)$-differential privacy.

We note that a hypothesis of this theorem is that the data is drawn from a product distribution. In the contextual bandit setting, this corresponds to rows in the bandit tableau being drawn from a product distribution. This will be the case if contexts are drawn from a distribution at each round, and then rewards are generated as some fixed stochastic function of the contexts. Note that contexts (and even rewards) can be correlated with one another within a round, so long as they are selected independently across rounds. In contrast, the regret bound we prove allows the contexts to be selected by an adversary, but adversarially selected contexts would violate the independence assumption needed for Theorem 7.

We now formalize the process of running a hypothesis test against an adaptively collected dataset. A bandit algorithm $\mathcal{A}$ generates a history $\Lambda_{T}\in\mathcal{H}^{T}$. Let the reward portion of the gathered dataset be denoted by $D_{\mathcal{A}}$. We define an adaptive test statistic selector as follows.

Fix the reward portion of a bandit tableau $D$ and bandit algorithm $\mathcal{A}$. An adaptive test statistic selector is a function $s$ from action histories to test statistics such that $s(\Lambda_{T}^{\mathcal{A}})$ is a real-valued function of the adaptively gathered dataset ${D_{\mathcal{A}}}$.

Importantly, the selection of the test statistic $s(\Lambda_{T}^{\mathcal{A}})$ can depend on the sequence of arms pulled by $\mathcal{A}$ (and in the contextual setting, on all contexts observed), but not otherwise on the reward portion of the tableau $D$. For example, $t_{\mathcal{A}}=s(\Lambda_{T}^{\mathcal{A}})$ could be the $t$-statistic corresponding to the null hypothesis that the arm $i^{*}$ which was pulled the greatest number of times has mean $\mu$:

By virtue of Theorems 6 and 7, and our view of adaptive data gathering as adaptively selected queries, we get the following corollary:

Let $\mathcal{A}$ be an $\epsilon$ reward differentially private bandit algorithm, and let $s$ be an adaptive test statistic selector. Fix $\beta>0$, and let $\gamma(\alpha)=\frac{\alpha}{2^{\log(e)(\epsilon^{2}T/2+\epsilon\sqrt{T\log(2/\beta)/2})}},$ for $\alpha\in$. Then for any adaptively selected statistic $t_{\mathcal{A}}=s(\Lambda_{T}^{\mathcal{A}})$, and any product distribution $P$ corresponding to the null hypothesis for $t_{\mathcal{A}}$

If we set $\epsilon=O(1/\sqrt{T})$ in Corollary 3, then $\gamma(\alpha)=O(\alpha)$– i.e. a valid $p$-value correction that only scales $\alpha$ by a constant. For example, in the simple stochastic setting, we can recall corollary 1 to obtain:

Setting $\epsilon=O(\frac{\ln^{1.5}T}{\sqrt{kT}})$ there exists a simple stochastic bandit algorithm that guarantees expected regret bounded by $O(\sqrt{kT\log T})$, such that for any adaptive test statistic $t$ evaluated on the collected data, there exists a valid $p$-value correction function $\gamma(\alpha)=O(\alpha)$.

Of course, our theorems allow us to smoothly trade off the severity of the $p$-value correction with the regret bound.

Experiments

We first validate our theoretical bounds on bias in the simple stochastic bandit setting. As expected the standard UCB algorithm underestimates the mean at each arm, while the private UCB algorithm of Mishra and Thakurta (2015) obtains very low bias. While using the $\epsilon$ suggested by the theory in Corollary 4 effectively reduces bias and achieves near optimal asymptotic regret, the resulting private algorithm only achieves non-trivial regret for large $T$ due to large constants and logarithmic factors in our bounds. This motivates a heuristic choice of $\epsilon$ that provides no theoretical guarantees on bias reduction, but leads to regret that is comparable to the non-private UCB algorithm. We find empirically that even with this large choice of $\epsilon$ we achieve an $8$ fold reduction in bias relative to UCB. This is consistent with the observation that our guarantees hold in the worst-case, and suggests that there is room for improvement in our theoretical bounds — both improving constants in the worst-case bounds on bias and on regret, and for proving instance specific bounds. Finally, we show that in the linear contextual bandit setting collecting data adaptively with a linear UCB algorithm and then conducting $t$-tests for regression coefficients yields incorrect inference (absent a $p$-value correction). These findings confirm the necessity of our methods when drawing conclusions from adaptively gathered data.

In our first stochastic bandit experiment we set $K=20$ and $T=500$. The $K$ arm means are equally spaced between and $1$ with gap $\Delta=.05$, with $\mu_{0}=1$. We run UCB and $\epsilon$-private UCB for $T$ rounds with $\epsilon=.05$, and after each run compute the difference between the sample mean at each arm and the true mean. We repeat this process $10,000$ times, averaging to obtain high confidence estimates of the bias at each arm. The average absolute bias over all arms for private UCB was $.00176$, with the bias for every arm being statistically indistinguishable from (see Figures 2 for confidence intervals) while the average absolute bias (over arms) for UCB was $.0698$, or over $40$ times higher. The most biased arm had a measured bias of roughly $0.14$, and except for the top $4$ arms, the bias of each arm was statistically significant. It is worth noting that private UCB achieves bias significantly lower than the $\epsilon=.05$ guaranteed by the theory, indicating that the theoretical bounds on bias obtained from differential privacy are conservative. Figures 2, 2 show the bias at each arm for private UCB vs. UCB, with $95\%$ confidence intervals around the bias at each arm. Not only is the bias for private UCB an order of magnitude smaller on average, it does not exhibit the systemic negative bias evident in Figure 2.

Noting that the observed reduction in bias for $\epsilon=.05$ exceeded that guaranteed by the theory, we run a second experiment with $K=5,T=100000,\Delta=.05,$ and $\epsilon=400$, averaging results over $1000$ iterations. Figure 3 shows that private UCB achieves sub-linear regret comparable with UCB. While $\epsilon=400$ provides no meaningful theoretical guarantee, the average absolute bias at each arm mean obtained by the private algorithm was $.0015$ (statistically indistinguishable from 0 at 95% confidence for each arm), while the non-private UCB algorithm obtained average bias $.011$, $7.5$ times larger. The bias reduction for the arm with the smallest mean (for which the bias is the worst with the non private algorithm) was by more than a factor of 10. Figures 5,5 show the bias at each arm for the private and non-private UCB algorithms together with 95% confidence intervals; again we observe a negative skew in the bias for UCB, consistent with the theory in Nie et al. (2017).

2 Linear Contextual Bandits

References

Appendix A Differential Privacy Basics

We recall the standard definition of differential privacy, which can be defined over any neighboring relationship on data sets $D,D^{\prime}\in\mathcal{X}^{*}$. The standard relation says that $D,D^{\prime}$ are neighbors (written as $D\sim D^{\prime}$) if they differ in a single element.

Fix $\epsilon\geq 0$. A randomized algorithm $A:\mathcal{X}^{*}\rightarrow\mathcal{O}$ is $(\epsilon,\delta)$-differentially private if for every pair of neighboring data sets $D\sim D^{\prime}\in\mathcal{X}^{*}$, and for every event $S\subseteq\mathcal{O}$:

Differentially private computations enjoy two nice properties:

Let $A:\mathcal{X}^{*}\rightarrow\mathcal{O}$ be any $(\epsilon,\delta)$-differentially private algorithm, and let $f:\mathcal{O}\rightarrow\mathcal{O^{\prime}}$ be any (possibly randomized) algorithm. Then the algorithm $f\circ A:\mathcal{X}^{*}\rightarrow\mathcal{O}^{\prime}$ is also $(\epsilon,\delta)$-differentially private.

Post-processing implies that, for example, every decision process based on the output of a differentially private algorithm is also differentially private.

Let $A_{1}:\mathcal{X}^{*}\rightarrow\mathcal{O}$, $A_{2}:\mathcal{X}^{*}\rightarrow\mathcal{O}^{\prime}$ be algorithms that are $(\epsilon_{1},\delta_{1})$- and $(\epsilon_{2},\delta_{2})$-differentially private, respectively. Then the algorithm $A:\mathcal{X}^{*}\rightarrow\mathcal{O}\times\mathcal{O^{\prime}}$ defined as $A(x)=(A_{1}(x),A_{2}(x))$ is $(\epsilon_{1}+\epsilon_{2}),(\delta_{1}+\delta_{2})$-differentially private.

Two random variables $X,Y$ defined over the same domain $R$ are $(\epsilon,\delta)$-close, written $X\approx_{\epsilon,\delta}Y$ , if for all $S\subseteq R$:

Note that if $A$ is an $(\epsilon,\delta)$-differentially private algorithm, and $D,D^{\prime}$ are neighboring datasets, then $A(D)\approx_{\epsilon,\delta}A(D^{\prime})$. We make use of a simple lemma:

Appendix B Useful Concentration Inequalities

Appendix C A Private UCB algorithm

For completeness, we reproduce a version of the private UCB algorithm of Mishra and Thakurta which we use in our experiments. See algorithm 2.

Appendix D Missing Proofs

Fix any $x_{ik}\in C_{i,T}$. We write ${\dagger}$ to denote the matrix inverse in the case it exists, or else the pseudo-inverse if not. We first expand ${\hat{\theta}_{i}}^{\prime}x_{ik}$:

The reward-privacy claim follows immediately from the privacy of the hybrid mechanism Chan et al. and the post-processing property of differential privacy (Lemma 1). Here we prove the regret bound. We first show that the confidence intervals given by $\hat{y}_{tk}\pm(\frac{1}{\lambda}s_{it}+w_{it})$ are valid $\forall i,t$ with probability $1-\delta$. Then since we always play the action with the highest upper confidence bound, with high probability we can bound our regret at time $T$ by the sum of the widths of the confidence intervals of the chosen actions at each time step.

where the second inequality follows from applying the Cauchy-Schwarz inequality with respsect to the matrix inner product $\langle\cdot,\cdot\rangle_{\hat{V}_{it}^{-1}}$. We also have that $||\eta_{it}||_{\hat{V}_{it}^{-1}}\leq 1/\sqrt{\lambda}||\eta_{it}||_{2}$, and by the utility theorem for the Hybrid mechanism Chan et al. , with probability $1-\delta/2,\;\forall i,t,\;||\eta_{it}||_{2}\leq s_{it}=O(\frac{1}{\epsilon}\log^{1.5}T\log(K/\delta))$. Thus by triangle inequality and a union bound, with probability $1-\delta,\;\forall i,t$:

Let $R(T)$ denote the pseudo-regret at time $T$, and $R_{i}(T)$ denote the sum of the widths of the confidence intervals at arm $i$, over all times in which arm $i$ was pulled. Then with probability $1-\delta$:

The RHS is maximized at $N_{i}^{T}=\frac{T}{K}$ for all $i$, giving:

Reproducing the analysis of Abbasi-Yadkori et al. , made more explicit on page $13$ in the Appendix of Joseph et al. gives:

The crux of their analysis is actually the bound $\sum_{t=1}^{n}||x_{it}||_{\hat{V}_{it}^{-1}}\leq 2d\log(1+n/d\lambda)$, which holds for $\lambda\geq 1$. Letting $n=T/K$ bounds the second summation, giving that with probability $1-\delta$: