The price of joint differential privacy in linear contextual bandits

Close the gap between regret upper and lower bounds for linear contextual bandits under joint differential privacy.

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	1	+# The price of joint differential privacy in linear contextual bandits
	2	+
	3	+Contextual bandits choose recommendations or treatments for a stream of users, and both a user's context and their response are personal data. For linear contextual bandits under joint differential privacy, the best algorithm pays for privacy at the same $\sqrt{T}$ rate as for learning itself, while the best lower bound leaves room for a price that does not grow with $T$ at all. Which is right?
	4	+
	5	+## Problem
	6	+
	7	+In each round $t = 1, \dots, T$, the environment reveals a decision set $A_t \subset \mathbb{R}^d$, which may be chosen adversarially; the policy picks $a_t \in A_t$ and observes $r_t = \langle \theta^\star, a_t \rangle + \eta_t$, where $\theta^\star \in \mathbb{R}^d$ is unknown and $\eta_t$ is conditionally 1-subgaussian noise. The regret is
	8	+
	9	+$$R_T = \mathbb{E}\Big[\sum_{t=1}^{T} \max_{a \in A_t} \langle \theta^\star, a - a_t \rangle\Big].$$
	10	+
	11	+Treat round $t$ as the private data $(A_t, r_t)$ of one user. Two sequences of such pairs are $t$-neighbors if they differ only in round $t$. A policy $\pi$ is $(\varepsilon, \delta)$-jointly differentially private (JDP) if for every $t$, every pair of $t$-neighbors $S, S'$ and every set $E$ of action sequences for rounds $t+1, \dots, T$,
	12	+
	13	+$$\Pr[\pi(S) \in E] \le e^{\varepsilon} \Pr[\pi(S') \in E] + \delta .$$
	14	+
	15	+The action at round $t$ itself is not protected; protecting it as well forces linear regret [2].
	16	+
	17	+Question. What is the optimal regret of $(\varepsilon, \delta)$-JDP policies for adversarial contexts, up to logarithmic factors? In particular, is privacy almost free: is there a JDP policy whose regret is the non-private $\widetilde O(d \sqrt{T})$ plus a privacy term that grows only polylogarithmically in $T$?
	18	+
	19	+## Background
	20	+
	21	+Shariff and Sheffet introduced joint differential privacy for contextual linear bandits in 2018 and made LinUCB private by releasing noisy running sums with the tree-based mechanism [2]. For bandits without contexts the picture is now sharp. For finite-armed and linear bandits under zero-concentrated differential privacy, the extra regret due to privacy is only polylogarithmic in $T$, and upper and lower bounds match up to logarithmic factors (Azize and Basu, 2024) [4]. Azize and Basu asked at COLT 2024 whether the same holds once contexts are private and adversarial [1].
	22	+
	23	+## What is known
	24	+
	25	+- Upper bound. JDP LinUCB has regret $O\big(d\sqrt{T}\log T + d^{3/4}\sqrt{T \log(1/\delta)}/\sqrt{\varepsilon}\big)$ for adversarial contexts (Shariff and Sheffet, 2018) [2]. The privacy term grows like $\sqrt{T}$.
	26	+- Lower bound. Every JDP policy has worst-case regret $\Omega\big(\sqrt{dT\log K} + d/(\varepsilon + \delta)\big)$, where $K$ is the number of actions per round (He, Zhang and Zhang, 2022) [3]. It is proved for stochastic contexts and so also holds for adversarial ones.
	27	+- Public stochastic contexts. The phased algorithm AdaC-OFUL makes the price of privacy negligible when contexts are public and stochastic [4]. The bottleneck is estimating the Gram matrix of private, adversarial contexts.
	28	+- Generalized linear models. A 2026 algorithm for generalized linear contextual bandits under JDP with adversarial contexts again has a privacy term that grows like $\sqrt{T/\varepsilon}$, up to factors polynomial in $d$ and $\log T$ [5].
	29	+
	30	+## What would count as progress
	31	+
	32	+- A JDP policy for adversarial contexts with regret $\widetilde O(d\sqrt{T})$ plus a privacy term of order $\mathrm{poly}(d, \log T)/\varepsilon$, with a proof.
	33	+- A lower bound showing that, for some adversarial context sequences, every JDP policy pays a privacy term that grows polynomially in $T$.
	34	+- Conditions on how contexts are generated (stochastic, with margin or diversity assumptions) under which privacy is provably almost free, with matching lower bounds.
	35	+
	36	+## References
	37	+
	38	+1. A. Azize and D. Basu, Open problem: What is the complexity of joint differential privacy in linear contextual bandits?, COLT 2024, PMLR 247 (2024), 5306–5311. [PMLR](https://proceedings.mlr.press/v247/azize24a.html)
	39	+2. R. Shariff and O. Sheffet, Differentially private contextual linear bandits, NeurIPS (2018). [arXiv:1810.00068](https://arxiv.org/abs/1810.00068)
	40	+3. J. He, J. Zhang and R. Q. Zhang, A reduction from linear contextual bandits lower bounds to estimations lower bounds, ICML 2022, PMLR 162 (2022), 8660–8677. [PMLR](https://proceedings.mlr.press/v162/he22e.html)
	41	+4. A. Azize and D. Basu, Concentrated differential privacy for bandits, IEEE SaTML (2024), 78–109. [arXiv:2309.00557](https://arxiv.org/abs/2309.00557)
	42	+5. S. Sarmasarkar, Shuffle and joint differential privacy for generalized linear contextual bandits, preprint (2026). [arXiv:2602.00417](https://arxiv.org/abs/2602.00417)
	43	+
	44	+## Source
	45	+
	46	+Imported from [COLT open problems, COLT 2024](https://proceedings.mlr.press/v247/azize24a.html), curated by the Conference on Learning Theory (COLT). Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.