Differentially Private High Dimensional Bandits
Apurv Shukla
TL;DR
This work addresses high-dimensional stochastic contextual linear bandits with a sparse parameter and joint differential privacy by introducing PrivateLASSO, a privacy-preserving LASSO-bandit algorithm. The method combines a sparse hard-thresholding privacy mechanism with an episodic thresholding rule to identify the support of $\theta$, and then applies private $\ell_2$ regression on the identified support via a tree-based aggregation framework, controlled by the privacy budget. The authors prove $(\varepsilon,\delta)$-DP guarantees and derive regret bounds: $\mathcal{R}(T)=\mathcal{O}\left(\frac{s_0^{3/2}\log^3 T}{\varepsilon}\right)$ under a margin condition and $\mathcal{O}\left(\frac{s_0^{3/2}\sqrt{T}\log^2 T}{\varepsilon}\right)$ without it, along with minimax lower bounds that characterize the privacy-accuracy trade-offs. The results demonstrate the feasibility of private learning in high-dimensional contextual bandits with sparse structure and clarify how privacy parameters shape learning performance in practice.
Abstract
We consider a high-dimensional stochastic contextual linear bandit problem when the parameter vector is $s_{0}$-sparse and the decision maker is subject to privacy constraints under both central and local models of differential privacy. We present PrivateLASSO, a differentially private LASSO bandit algorithm. PrivateLASSO is based on two sub-routines: (i) a sparse hard-thresholding-based privacy mechanism and (ii) an episodic thresholding rule for identifying the support of the parameter $θ$. We prove minimax private lower bounds and establish privacy and utility guarantees for PrivateLASSO for the central model under standard assumptions.