Improved regret bounds in tabular RL under local differential privacy

AI summary

Overview

Research area: Theoretical reinforcement learning and privacy-preserving sequential decision-making — specifically regret minimization in tabular episodic RL and contextual bandits under local differential privacy (LDP).

Technical level: Advanced. The paper is a theory contribution built on minimax regret analysis, information-theoretic lower bounds, and two well-known algorithm families (EXP3 for bandits, UCBVI for episodic RL). The abstract is stated entirely in terms of asymptotic bounds rather than experiments.

Scope (one sentence): The paper derives upper and lower bounds on minimax regret for tabular RL and contextual bandits when every piece of user feedback must be ε-locally differentially private, and shows these bounds improve on previously known ones.

What This Paper Is About

A learner repeatedly faces a user: it sends a policy, the user runs that policy for some number of steps, and instead of reporting what actually happened, the user sends back a randomized message that satisfies ε-local differential privacy — meaning the message is sanitized on the user's own device before the learner ever sees it. The learner's goal is to minimize regret (the gap between its accumulated reward and that of the best fixed policy in hindsight) despite never receiving clean data.

The paper asks how much this privacy constraint actually costs in regret, and what the best achievable rates are. It answers this in two settings: a single-step setting, which reduces to an LDP contextual bandit, and a multi-step setting where each episode has a horizon of H steps.

Key Contributions

1. A minimax regret characterization for the LDP contextual bandit. For H = 1 — which the paper frames as a contextual bandit with S contexts, A arms, and horizon K — it proves the minimax regret is $\min\big(K, \frac{\sqrt{\max(S,, e^\varepsilon+1), S A K (e^\varepsilon+1)}}{e^\varepsilon-1}\big)$, up to constant factors and a $\sqrt{\log A}$ factor.

2. A matching algorithm for the bandit case, obtained by adapting EXP3 and combining it with a unary encoding mechanism for the privatized feedback.

3. Upper and lower bounds for the multi-step case (H > 1, small ε): a lower bound of $\Omega(H S \sqrt{AK}/\varepsilon)$ and an upper bound of $\tilde{O}(H^2 S^2 \sqrt{AK}/\varepsilon)$, the latter achieved by an algorithm based on UCBVI plus unary encoding.

4. Quantitative improvement over prior work for small ε: the new lower bound improves on the previously known one by a factor of $\sqrt{S}$, and the new upper bound improves on the previously known one by a factor of $H\sqrt{A}$.

Main Findings

• The bandit case is settled up to small factors. The paper's upper and lower bounds for H = 1 agree up to constants and a $\sqrt{\log A}$ term, so the minimax regret for LDP contextual bandits is pinned down at that resolution. The abstract does not give the proof technique behind the lower bound.

• Privacy cost is explicit in the formula. Reading the stated bound, the dependence on ε is governed by the factor $\sqrt{(e^\varepsilon+1)}/(e^\varepsilon-1)$ together with the $\max(S, e^\varepsilon+1)$ term: for small ε (strong privacy) the rate degrades roughly like $S\sqrt{AK}/\varepsilon$, while for large ε the bound approaches the familiar non-private $\sqrt{SAK}$ scaling. The abstract itself does not spell out this limiting behaviour; it follows from the formula as written.

• The multi-step case leaves a gap. The stated lower bound $\Omega(HS\sqrt{AK}/\varepsilon)$ and upper bound $\tilde{O}(H^2S^2\sqrt{AK}/\varepsilon)$ differ by a factor of about $H S$. The abstract presents them as separate results and does not claim they match.

• Both regimes use the same privacy primitive. Unary encoding appears in both the EXP3-based bandit algorithm and the UCBVI-based multi-step algorithm, suggesting the encoding mechanism is the reusable ingredient while the learning algorithm is swapped per setting.

• Improvements are stated only for small ε. The factors $\sqrt{S}$ (lower bound) and $H\sqrt{A}$ (upper bound) are described as improvements "for small ε"; the abstract makes no claim about the large-ε regime for H > 1.

Methodology in Plain English

The threat model is non-interactive local privacy: the decision-maker sends out a policy, the user executes it for H steps, and returns exactly one message per episode. Because that message is randomized before it leaves the user, the learner must plan around noisy, deliberately corrupted observations — there is no trusted central curator and no follow-up queries.

The technical programme has two halves. For upper bounds, the author takes a standard no-privacy algorithm — EXP3 when there is only one decision per episode, UCBVI when the policy plays out over H steps — and pairs it with a unary encoding mechanism that turns the user's observation into a randomized bit-vector representation, so that any single message carries only limited information about the underlying state-action data. Regret is then analyzed for the combined procedure. For lower bounds, the paper establishes limits on what any ε-LDP algorithm can achieve; the abstract states the resulting rates but does not describe the argument used to derive them.

The comparison to prior work is made purely at the level of these rates: the new bounds are tighter by a stated polynomial factor in the problem parameters.

Why This Matters

Impact on research. The paper sharpens the minimax theory of privacy-constrained reinforcement learning. Prior work had loose upper and lower bounds for the multi-step LDP setting; tightening both and closing the H = 1 case up to logarithmic factors gives the community a cleaner target for future work, and shows which problem parameters (here S and H) account for the remaining uncertainty.

Real-world applications — settings where data cannot be centralized because the raw signal lives on a user's device:

• On-device personalization, such as keyboards and phone assistants that adapt from typing or usage traces without transmitting them.
• Mobile health and telemedicine, where a phone or wearable runs a suggested routine and reports only a privatized summary of the outcome.
• Recommendation and content ranking, where a provider proposes a policy to a user and receives a noisy signal about engagement rather than an exact click or dwell record.
• Smart metering and household energy management, where fine-grained consumption data is sensitive and only sanitized feedback can leave the home.

Industry relevance. Any product that wants sequential decision-making with formal, auditable per-user privacy guarantees — rather than relying on a trusted aggregator — needs exactly this kind of rate analysis. The results tell practitioners what accuracy to expect as a function of the privacy budget ε and the size of the state and action spaces, which is the input to deciding whether an LDP design is viable at all.

Future Directions

• Close the H·S gap for multi-step RL. The upper and lower bounds for H > 1 do not match; determining whether the truth lies nearer $\tilde{O}(HS\sqrt{AK}/\varepsilon)$ or higher is the most direct open question the abstract implies.
• Extend beyond the small-ε regime. The multi-step bounds are stated for small ε only; the large-ε behaviour of the H > 1 setting is not addressed.
• Relax the model assumptions. The setting is time-homogeneous, tabular, and non-interactive with a single message per episode. Time-varying dynamics, function approximation, or multi-round interaction with users would each change the problem substantially.
• Push from rates to practice. The abstract reports no experiments; empirical behaviour of the unary-encoding-plus-EXP3/UCBVI algorithms at realistic values of S, A, and ε remains untested in what is described here.

Target Audience

Theory researchers and graduate students working on reinforcement learning, bandits, or differential privacy who want the current state of minimax regret bounds under local privacy constraints. It is also useful for privacy engineers and applied scientists who need to reason about the accuracy cost of on-device privatization, provided they are comfortable reading regret expressions. Readers looking for algorithms with empirical validation or implementation guidance will not find it in this abstract, which is purely a rates-and-bounds contribution.

Authors’ abstract

We study regret minimization for time-homogeneous episodic tabular reinforcement learning with $S$ states and $A$ actions under non-interactive $ε$-local differential privacy (LDP). In each episode $k \in [K]$, the decision-maker sends a policy to a user, the user executes the policy for $H$ steps, and sends an $ε$-locally differentially private message back to the decision-maker. When $H=1$, the problem reduces to an LDP contextual bandit with $S$ contexts, $A$ arms and horizon $K$. We prove that the minimax regret is $\min\bigg(K, \frac{ \sqrt{\max( S, \exp(ε) + 1) S A K (\exp(ε) + 1)}}{\exp(ε) - 1}\bigg)$, up to constants and a $\sqrt{\log(A)}$ factor. The upper bound is achieved by an adaptation of EXP3 and a unary encoding mechanism. When $H > 1$, and for small $ε$, we show a lower bound in $Ω(HS\frac{\sqrt{A K}}ε)$ and an upper bound in $\tilde{O}(H^2 S^2 \frac{\sqrt{ AK}}ε)$. The upper bound is achieved by an algorithm based on UCBVI and a unary encoding mechanism. For small $ε$, these bounds improve the previously known lower bound by a factor $\sqrt{S}$ and the previously known upper bound by a factor of $H \sqrt{A}$.