Local Differential Privacy for Sequential Decision Making in a Changing Environment

Several practically relevant applications including recommender systems, Internet advertising have been formulated as sequential decision making problems using the framework of multi-armed bandits. The importance of privacy in such sequential decision making problems has been extensively discussed in the literature (see for example, DBLP:conf/nips/ThakurtaS13; DBLP:conf/uai/MishraT15; tossou:aaai2016).

Differential privacy, introduced by Dwork06calibratingnoise, is one of the popular approaches to address such privacy concerns. In sequential decision making problems, algorithms providing differential privacy preserve data privacy by adding appropriate statistical noise to the data. Duchi:2014:PAL:2700084.2666468 extend this notion to local differential privacy in which data remains private even from the algorithm. The main difference between global and local differential privacy is whether privacy is to be maintained from the algorithm or the (possibly unintended) recipient of the output of the algorithm. In global differential privacy, noise is added by the algorithm so the output does not reveal private information about the input. In local differential privacy, noise is added to the input of the algorithm so that privacy is maintained even from the algorithm.

To understand the motivation for local differential privacy, let us consider the practical application of Internet advertising ¹¹1We consider a simplistic scenario for illustrative purposes.. An advertising system receives, as input, feedback from the users which may reveal private information about them. The advertising system employs a suitable learning algorithm and selects ads for the users tailored to the feedback given by them. These selected ads are then given to the advertisers as output. While using global differential privacy, privacy is maintained from the advertisers by ensuring that the output of the learning algorithms does not reveal information about the input (i.e., user information). Typically, advertising systems are established by leading social media networks, web browsers and other popular websites. DBLP:conf/icdm/Korolova10; kosinski2013private show that it is possible to accurately predict a range of highly sensitive personal attributes including age, sexual orientation, relationship status, political and religious affiliation using the feedback available to the advertising systems. Such possible breach of privacy necessitates us to protect personal user information not only from the advertisers but also from the advertising systems. Local differential privacy is able to achieve this objective unlike global differential privacy.

In this article, we propose to use low privacy regime using local differential privacy. In low privacy regime, the noise added to the data is small and the aim of the privacy mechanism is to send as much information about data as allowed, but no more (NIPS2014_5392). This is in alignment with our dual goal of using privacy in recommendation systems or Internet advertising, and other similar applications: provide useful recommendations/ads to the users while respecting their privacy as much as possible.

We measure the utility of our proposed algorithm using regret which is a measure of the total mistake cost (precise definitions will follow in the next Section). When rewards are bounded (as assumed in most works in the literature), the regret of any algorithm is trivially bounded linearly in the number of time steps $T$. An algorithm is said to be learning if its regret is bounded sub-linearly in $T$.

A non-stationary stochastic corrupt bandits problem is formally characterized by a set of arms $A=\{1,\dots,K\}$ on which are indexed a list of unknown sub-Gaussian reward distributions $\{\boldsymbol{\nu}_{a}(1)\}_{a\in A},\dots,\{\boldsymbol{\nu}_{a}(L_{T})\}_{a\in A}$, a list of unknown sub-Gaussian feedback distributions $\{\boldsymbol{\varsigma}_{a}(1)\}_{a\in A},\dots,\{\boldsymbol{\varsigma}_{a}(L_{T})\}_{a\in A}$, and a list of known mean-corruption functions $\{g_{a}\}_{a\in A}$. Here, the total number of time steps (i.e., the horizon) is indicated as $T$. The environment undergoes $L_{T}$ abrupt changes at unknown time steps called as breakpoints and it remains constant in the intervals between two successive breakpoints.

For notational convenience, we assume that the first breakpoint occurs at $t=1$. From $i^{\text{th}}$ breakpoint till the subsequent breakpoint (or the horizon, in case of the last breakpoint), if the learner pulls an arm $a\in A$ at time $t$, they receive a (hidden) reward $R_{t}$ drawn from the distribution $\boldsymbol{\nu}_{a}(i)$ with mean $\mu_{a}(i)$ and observe a feedback $F_{t}$ drawn from the distribution $\boldsymbol{\varsigma}_{a}(i)$ with mean $\lambda_{a}(i)$. We assume that, for each arm, there exists a loose link between the reward and the feedback through a known corruption function $g_{a}$ which maps the mean of the reward distribution to the mean of the feedback distribution : $g_{a}(\mu_{a}(i))=\lambda_{a}(i),\forall a\in A$ and $1\leq i\leq L_{T}$. Our proposed algorithm and the proven regret bound also work if the corruption function for an arm changes across time as long as the current corruption function is known.

Note that these $g_{a}$ functions may be completely different from one arm to another. For Bernoulli distributions, the reward distributions and the feedback distributions are in $[0,1]$ for all $a\in A$ and we assume all the corruption functions $\{g_{a}\}_{a\in A}$ to be continuous in this interval. We also assume the corruption functions $\{g_{a}\}_{a\in A}$ to be strictly monotonic and denote the corresponding inverse functions by $g_{a}^{-1}$. The assumption of monotonicity is required for efficient learning as proved in pmlr-v83-gajane18a.

Another way to define the link between the reward and the feedback is to provide a corruption scheme operator $\tilde{g}_{a}$ which maps the rewards into feedback distributions.

To solve the problem at hand, we propose SW-KLUCB-CF, an adaptation of the $\mathrm{kl}$-UCB algorithm of KLUCBJournal. The algorithm takes as input: the window size $w$, a non-decreasing function $f$, the horizon $T$ and the corruptions functions $g_{1},\dots,g_{K}$. We assume that the horizon T is known; an unknown $T$ can be handled using the doubling trick (besson:hal-01736357). We use $d(x,y)$ to denote the Kullback–Leibler divergence between two Bernoulli distributions with mean $x$ and $y$. We also use a shorthand of $x\wedge y$ to denote $\min(x,y)$.

At each time time step $t$, the algorithm computes an $\mathrm{Index}_{a}(t)$, which is an upper-confidence bound on $\mu_{a,t}$ built from a confidence interval on $\lambda_{a,t}$ based on the KL-divergence. The quantity $N_{a}(t,w)$ denotes the number of times arm $a$ was chosen in the last $w$ time steps until time $t$. Correspondingly, $\hat{\lambda}_{a}(t,w)$ denotes the empirical mean of the feedback observed from arm $a$ in the last $w$ time steps until time $t$: $\hat{\lambda}_{a}(t,w)\coloneqq\frac{1}{N_{a}(t,w)}\sum_{s=\min\{1,t-w+1\}}^{t}F_{s}\cdot\mathds{1}_{(\hat{a}_{s}=a)}$.

Theorem 1 gives an upper bound on the regret of SW-KLUCB-CF. A more explicit bound is proved in the Appendix.

The lower bound on regret in terms $T$ for classical non-stationary stochastic bandits is $\Omega(\sqrt{T})$ (GarivierSW). Theorem 1 matches the lower bound up to logarithmic factors, so SW-KLUCB-CF has near-optimal regret guarantees in terms of the time horizon $T$. The best known regret upper bounds for classical non-stationary stochastic bandits (e.g., pmlr-v99-auer19a) also feature logarithmic terms besides the lower bound, hence our regret bound is in line with the best known results for analogous problems. Moreover, the bound in Theorem 1 also matches the best known regret bound in terms of $L_{T}$ for classical non-stationary stochastic bandits which is $O\sqrt{L_{T}}$.

We can use SW-KLUCB-CF on non-stationary stochastic corrupts bandits where the corruption is done via randomized response. The following corollary bounds the resulting regret.

First, let us formally define local differential privacy.

As done in pmlr-v83-gajane18a, a straightforward approach to achieve local differential privacy using corrupt bandits is to employ a corruption scheme on the user feedback. This is similar to how randomized response is used in data collection by DBLP:conf/edbt/0009WH16.

When corruption is done by randomized response, local differential privacy requires that $\max_{1\leq a\leq K}{\Big{(}\frac{p_{00}(a)}{1-p_{11}(a)},\frac{p_{11}(a)}{1-p_{00}(a)}\Big{)}}\leq e^{\epsilon}$. From Corollary 1, we can see that to achieve lower regret, $p_{00}(a)+p_{11}$(a) is to be maximized for all $a\in A$. Using DBLP:conf/edbt/0009WH16, we can state that, in order to achieve $\epsilon$-local differential privacy while maximizing $p_{00}(a)+p_{11}(a)$,

As it turns out, this is equivalent to the staircase mechanism for local privacy which is the optimal local differential privacy mechanism for low privacy regime (JMLR:v17:15-135, Theorem 14). The trade-off between utility and privacy is controlled by $\epsilon$.

Here, we provide a proof outline for Theorem 1. Please refer to the Appendix for the complete proof.

We start by bounding the expected number of times a suboptimal arm (i.e., an arm other than the optimal arm at the time of selection) is pulled by the algorithm till horizon $T$. Recall that, at any time step $t$, SW-KLUCB-CF pulls an arm maximizing an index defined as

We further decompose the computation of index as follows,

where,

The interval $[\ell_{a}(t),u_{a}(t)]$ is a KL-based confidence interval on the mean feedback $\lambda_{a,t}$ of arm $a$ at time $t$. This is in contrast to $\mathrm{kl}$-UCB (KLUCBJournal) where a confidence interval is placed on the mean reward. Furthermore, This differs from $\mathrm{kl}$-UCB-CF (pmlr-v83-gajane18a) where the mean feedback of an arm remains the same for all the time steps and $f$ does not feature $w$.

In our analysis, we use the fact that when an arm $a$ is picked at time $t+1$ by SW-KLUCB-CF, one of the following is true: Either the mean feedback of the optimal arm $a_{*,t}$ with mean reward $\mu_{*,t}$ is outside its confidence interval (i.e., $g_{a_{*,t}}(\mu_{*,t})<\ell_{a_{*,t}}(t)$ or $g_{a_{*,t}}(\mu_{*,t})>u_{a_{*,t}}(t)$) which is unlikely. Or, the mean feedback of the optimal arm is where it should be, and then the fact that arm $a$ is selected indicates that the confidence interval on $\lambda_{a}$ cannot be too small as either $(u_{a}(t)\geq g_{a}(\mu_{*,t}))$ or $(\ell_{a}(t)\leq g_{a}(\mu_{*,t}))$. The previous statement follows from considering various cases depending on whether the corruption functions $g_{a}$ and $g_{a_{*,t}}$ are increasing or decreasing. We then need to control the two terms in the decomposition of the expected number of draws of arm $a$. The term regarding the “unlikely” event, is bounded using the same technique as in the $\mathrm{kl}$-UCB analysis, however with some added challenges due to the use of a sliding window. In particular, the analysis of a typical upper confidence bound algorithm for bandits relies on the fact that the confidence interval for any arm is always non-increasing, however this is not true while using a sliding window. To control the second term, depending on the monotonicity of the corruption functions $g_{a}$ and $g_{a_{*,t}}$, we need to meticulously adapt the arguments in KLUCBJournal to control the number of draws of a suboptimal arm, as can be seen in the Appendix.

In this work, we proposed the setting of non-stationary stochastic corrupt bandits for preserving privacy while still maintaining high utility in sequential decision making in a changing environment. We devised an algorithm called SW-KLUCB-CF and proved its regret upper bound which is near-optimal in the number of time steps and matches the best known bound for analogous problems in terms of the number of time steps and the number of changes. Moreover, we provided an optimal corruption scheme to be used with our algorithm in order to attain the dual goal of achieving high utility while maintaining the desired level of privacy.

Interesting directions for future work include:

1. 1.

   Complete an empirical evaluation of the proposed algorithm on simulated as well as real-life data.

2. 2.

   Characterize the changes in the environment by a variation budget (as done in NIPS2014_903ce922 for classical bandits) instead of the number of changes.

3. 3.

   Incorporate contextual information in the learning process.

4. 4.

   Propose a Bayesian algorithm for non-stationary stochastic corrupt bandits.

5. 5.

   Propose a (near-)optimal differentially private algorithm which does not need to know the number of changes.