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Decision Making in Changing Environments: Robustness, Query-Based Learning, and Differential Privacy

Fan Chen, Alexander Rakhlin

TL;DR

A framework that provides an interpolation between the stochastic and adversarial settings of decision making, and establishes strong connections between the DEC's behavior, the SQ dimension, local minimax complexity, learnability, and joint differential privacy is proposed.

Abstract

We study the problem of interactive decision making in which the underlying environment changes over time subject to given constraints. We propose a framework, which we call \textit{hybrid Decision Making with Structured Observations} (hybrid DMSO), that provides an interpolation between the stochastic and adversarial settings of decision making. Within this framework, we can analyze local differentially private (LDP) decision making, query-based learning (in particular, SQ learning), and robust and smooth decision making under the same umbrella, deriving upper and lower bounds based on variants of the Decision-Estimation Coefficient (DEC). We further establish strong connections between the DEC's behavior, the SQ dimension, local minimax complexity, learnability, and joint differential privacy. To showcase the framework's power, we provide new results for contextual bandits under the LDP constraint.

Decision Making in Changing Environments: Robustness, Query-Based Learning, and Differential Privacy

TL;DR

A framework that provides an interpolation between the stochastic and adversarial settings of decision making, and establishes strong connections between the DEC's behavior, the SQ dimension, local minimax complexity, learnability, and joint differential privacy is proposed.

Abstract

We study the problem of interactive decision making in which the underlying environment changes over time subject to given constraints. We propose a framework, which we call \textit{hybrid Decision Making with Structured Observations} (hybrid DMSO), that provides an interpolation between the stochastic and adversarial settings of decision making. Within this framework, we can analyze local differentially private (LDP) decision making, query-based learning (in particular, SQ learning), and robust and smooth decision making under the same umbrella, deriving upper and lower bounds based on variants of the Decision-Estimation Coefficient (DEC). We further establish strong connections between the DEC's behavior, the SQ dimension, local minimax complexity, learnability, and joint differential privacy. To showcase the framework's power, we provide new results for contextual bandits under the LDP constraint.
Paper Structure (222 sections, 100 theorems, 614 equations, 5 algorithms)

This paper contains 222 sections, 100 theorems, 614 equations, 5 algorithms.

Table of Contents

  1. Introduction
  2. Contributions
  3. Related work
  4. Decision-Estimation Coefficient Framework
  5. Exploration-by-Optimization
  6. Local differential privacy
  7. The role of interaction in LDP learning
  8. Statistical Queries
  9. Robust statistics
  10. Overview of Results
  11. Hybrid DMSO
  12. Learning objective
  13. PAC hybrid DEC and guarantees
  14. Query-based learning
  15. SQ DEC
  16. ...and 207 more sections

Key Result

Theorem 1

Let $T\geq 1$, and $L$ be metric-like. Under mild growth assumption, the following holds: where $\inf_{\mathsf{Alg}}$ is taken over all $T$-round algorithms $\mathsf{Alg}$, $\sup_{\mathsf{Env}}$ is taken over all environments $\mathsf{Env}$ constrained by $\mathscr{P}$, $\uline{\varepsilon}(T)\asymp \frac{1}{\sqrt{T}}$, $\bar{\varepsilon}(T)\asymp \sqrt{\frac{\log|\mathscr{P}|}{T}}$, and

Theorems & Definitions (134)

  • Definition 1: Metric-based loss function
  • Theorem 1: PAC lower and upper bounds; Informal
  • Theorem 2: SQ DEC lower and upper bounds; Informal
  • Definition 2: Differentially private channels
  • Theorem 3: Private PAC-DEC lower and upper bounds; Informal
  • Theorem 4: Robust risk bounds; Informal
  • Theorem 5: Regret lower and upper bounds; Informal
  • Theorem 6: Regret bounds against smooth adversaries; Informal
  • Definition 3: Moderate decay
  • Definition 4
  • ...and 124 more