Contributions An efficient algorithm, CGP-UCB, for the contextual GP bandit problem Flexibly combining kernels over contexts and actions Generic approach for deriving regret bounds for composite kernel functions Evaluate CGP-UCB on automated vaccine design and sensor management Contextual Bandits [cf., Auer ’02; Langford & Zhang ’08] Play a game for T rounds: Receive context zt ∈ Z Choose an action st ∈ S Receive a payoff yt = f (st , zt ) + t (f unknown). Cumulative regret for context specific action Incur contextual regret rt = sups0∈S f (s0, zt ) − f (st , zt ) PTAfter T rounds, the cumulative contextual regret is RT = t=1 rt . Context-specific best action is a demanding benchmark. Gaussian Processes (GP) Model payoff function using GPs: f ∼ GP(µ, k) • observations yT = [y1 . . . yT ]T at inputs AT = {x1, . . . , xT } yt = f (xt ) + t with i.i.d. Gaussian noise t ∼ N(0, σ 2) Posterior distribution over f is a GP with mean covariance variance where kT (x) = [k(x1, x) . . . k(xT , x)]T and KT is the kernel matrix. GP-UCB [Srinivas, Krause, Kakade, Seeger ICML 2010]

Context free upper confidence bound algorithm (GP-UCB) At round t, GP-UCB picks action st = xt such that with appropriate βt . Trades exploration (high σ) and exploitation (high µ). with appropriate βt . Trades exploration (high Maximum information gain bounds regret The (context-free) regret RT of GP-UCB is bounded by O∗( T βT γT ), where γT is defined as the maximum information gain: quantifies the reduction in uncertainty about f achieved by revealing yA. Bounds for Kernels Bounds on γT exist for linear, squared exponential and Mat´ern kernels. Contextual Upper Confidence Bound Algorithm (CGP-UCB) where µt−1(·) and σt−1(·) are the posterior mean and standard deviation of the GP over the joint set X = S × Z conditioned on the observations (s1, z1, y1), . . . , (st−1, zt−1, yt−1). Bounds on Contextual Regret Then√for appropriate choices of βt , the contextual regret of CGP-UCB is bounded by O∗( T γT βT ) w.h.p. Precisely, onpLet δ ∈ (0, 1). Suppose one of the following assumptions holds X is finite, f is sampled from a known GP prior with known noise variance σ 2, dX is compact and convex, ⊆ [0, r ] , d ∈ N, r > 0. Suppose f is sampled from a known GP prior with known noise variance σ 2, and that k(x, x0) has smooth derivatives, X is arbitrary; ||f ||k ≤ B. The noise variables t form an arbitrary martingale difference sequence (meaning that E[εt | ε1, . . . , εt−1] = 0 for all t ∈ N), uniformly bounded by σ. where C1 = 8/ log(1 + σ −2). Composite Kernels

Product of squared exponential kernel and linear kernel Additive combination of payoff that smoothly depends on context, and exhibits clusters of actions. Product kernel • k = kS ⊗ kZ , where (kS ⊗ kZ )((s, z), (s0, z0)) = kZ (z, z0)kS (s, s0) • Two context-action pairs are similar (large correlation) if the contexts are similar and actions are similar Additive kernel • (kS ⊕ kZ )((s, z), (s0, z0)) = kZ (z, z0) + kS (s, s0) • Generative model: first sample a function fS (s, z) that is constant along z, and varies along s with regularity as expressed by ks; then sample a function fz(s, z), which varies along z and is constant along s; Bounds for Composite Kernels Maximum information gain for a GP with kernel k on set V Product kernel Let kZ be a kernel function on Z with rank at most d . Then Additive kernel Let kS and kZ be kernel functions on S and Z respectively. Then

Task Discover peptide sequences binding to MHC molecules Context Features encoding the MHC alleles Action Choose a stimulus (the vaccine) s ∈ S that maximizes an observed response (binding affinity). Kernels Use a finite inter-task covariance kernel KZ with rank mZ to model the similarity of different experiments, and a Gaussian kernel kS (s, s0) to model the experimental parameters. Learning to Monitor Sensor Networks

Temperature data from a network of 46 sensors at Intel Research. Time (h) CGP-UCB using average temperature CGP-UCB using minimum temperature Task Given a sensor network, monitor maximum temperatures in building Context Time of day Action Pick 5 sensors to activate Kernels Joint spatio-temporal covariance function using the Mat´ern kernel