Bandit Convex Optimisation

TL;DR

Bandit convex optimisation studies zeroth-order convex minimisation over a convex set $K\subset\mathbb{R}^d$ using only noisy function evaluations, with regret as the primary performance metric. The book surveys a broad toolkit—cutting planes, gradient-descent with surrogate losses, barrier/self-concordant methods, exponential weights, and information-theoretic minimax duality—to derive finite-time minimax bounds under adversarial and stochastic settings, with detailed treatment of curvature, smoothing, and scaling. It connects fundamental lower bounds to a range of upper-bound results, highlighting the trade-offs between statistical efficiency and computational feasibility, and extends the theory to submodular minimisation via Lovász extensions and to linear/quadratic bandits through covering numbers and optimal designs. Overall, it provides a cohesive, theory-heavy framework for understanding zeroth-order bandits in convex settings, offering deep insights into algorithm design and fundamental limits, even as some methods remain computationally challenging in high dimensions.

Abstract

Bandit convex optimisation is a fundamental framework for studying zeroth-order convex optimisation. This book covers the many tools used for this problem, including cutting plane methods, interior point methods, continuous exponential weights, gradient descent and online Newton step. The nuances between the many assumptions and setups are explained. Although there is not much truly new here, some existing tools are applied in novel ways to obtain new algorithms. A few bounds are improved in minor ways.

Figures (21)

• Figure 1: The Minkowski and support functions.
• Figure 2: $\phi$ in dimension one.
• Figure 3: Three convex functions
• Figure 4: The smoothed surrogates for different functions and precisions. Because of convexity the surrogate function is always an upper bound on the original function. Notice how much better the approximation is for $-\log(x)$, which on the interval considered is much smoother than $|x|$.
• Figure 5: Dikin ellpsoids for a polytope and the ball using the barriers in Note \ref{['note:ftrl:logarithmic']}.

Theorems & Definitions (297)

• Definition 1.1
• Remark 1.3
• Remark 1.5
• Proposition 1.7
• proof
• Proposition 3.1
• Lemma 3.3
• proof
• Lemma 3.4
• proof