| <Poster Width="1003" Height="1419"> |
| <Panel left="27" right="94" width="462" height="117"> |
| <Text>Contributions</Text> |
| <Text>An efficient algorithm, CGP-UCB, for the contextual GP bandit problem</Text> |
| <Text>Flexibly combining kernels over contexts and actions</Text> |
| <Text>Generic approach for deriving regret bounds for composite kernel functions</Text> |
| <Text>Evaluate CGP-UCB on automated vaccine design and sensor management</Text> |
| </Panel> |
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| <Panel left="24" right="214" width="465" height="187"> |
| <Text>Contextual Bandits [cf., Auer ’02; Langford & Zhang ’08]</Text> |
| <Text>Play a game for T rounds:</Text> |
| <Text>Receive context zt ∈ Z</Text> |
| <Text>Choose an action st ∈ S</Text> |
| <Text>Receive a payoff yt = f (st , zt ) + t (f unknown).</Text> |
| <Text>Cumulative regret for context specific action</Text> |
| <Text>Incur contextual regret rt = sups0∈S f (s0, zt ) − f (st , zt )</Text> |
| <Text>PTAfter T rounds, the cumulative contextual regret is RT =</Text> |
| <Text>t=1 rt .</Text> |
| <Text>Context-specific best action is a demanding benchmark.</Text> |
| </Panel> |
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| <Panel left="24" right="406" width="464" height="190"> |
| <Text>Gaussian Processes (GP)</Text> |
| <Text>Model payoff function using GPs: f ∼ GP(µ, k)</Text> |
| <Text>• observations yT = [y1 . . . yT ]T at inputs AT = {x1, . . . , xT }</Text> |
| <Text>yt = f (xt ) + t with i.i.d. Gaussian noise t ∼ N(0, σ 2)</Text> |
| <Text>Posterior distribution over f is a GP with</Text> |
| <Text>mean</Text> |
| <Text>covariance</Text> |
| <Text>variance</Text> |
| <Text>where kT (x) = [k(x1, x) . . . k(xT , x)]T and KT is the kernel matrix.</Text> |
| </Panel> |
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| <Panel left="25" right="603" width="465" height="387"> |
| <Text>GP-UCB [Srinivas, Krause, Kakade, Seeger ICML 2010]</Text> |
| <Figure left="75" right="636" width="143" height="116" no="1" OriWidth="0" OriHeight="0 |
| " /> |
| <Figure left="295" right="633" width="149" height="118" no="2" OriWidth="0" OriHeight="0 |
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| <Text>Context free upper confidence bound algorithm (GP-UCB)</Text> |
| <Text>At round t, GP-UCB picks action st = xt such that</Text> |
| <Text>with appropriate βt . Trades exploration (high σ) and exploitation (high µ).</Text> |
| <Text>with appropriate βt . Trades exploration (high </Text> |
| <Text>Maximum information gain bounds regret</Text> |
| <Text>The (context-free) regret RT of GP-UCB is bounded by O∗( T βT γT ), where</Text> |
| <Text>γT is defined as the maximum information gain:</Text> |
| <Text>quantifies the reduction in uncertainty about f achieved by revealing yA.</Text> |
| <Text>Bounds for Kernels</Text> |
| <Text>Bounds on γT exist for linear, squared exponential and Mat´ern kernels.</Text> |
| </Panel> |
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| <Panel left="26" right="992" width="463" height="128"> |
| <Text>Contextual Upper Confidence Bound Algorithm (CGP-UCB)</Text> |
| <Text>where µt−1(·) and σt−1(·) are the posterior mean and standard deviation of the GP</Text> |
| <Text>over the joint set X = S × Z conditioned on the observations</Text> |
| <Text>(s1, z1, y1), . . . , (st−1, zt−1, yt−1).</Text> |
| </Panel> |
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| <Panel left="25" right="1127" width="464" height="274"> |
| <Text>Bounds on Contextual Regret</Text> |
| <Text>Then√for appropriate choices of βt , the contextual regret of CGP-UCB is bounded by</Text> |
| <Text>O∗( T γT βT ) w.h.p. Precisely,</Text> |
| <Text>onpLet δ ∈ (0, 1). Suppose one of the following assumptions holds</Text> |
| <Text>X is finite, f is sampled from a known GP prior with known noise variance σ 2,</Text> |
| <Text>dX is compact and convex, ⊆ [0, r ] , d ∈ N, r > 0. Suppose f is sampled from a</Text> |
| <Text>known GP prior with known noise variance σ 2, and that k(x, x0) has smooth</Text> |
| <Text>derivatives,</Text> |
| <Text>X is arbitrary; ||f ||k ≤ B. The noise variables t form an arbitrary martingale</Text> |
| <Text>difference sequence (meaning that E[εt | ε1, . . . , εt−1] = 0 for all t ∈ N),</Text> |
| <Text>uniformly bounded by σ.</Text> |
| <Text>where C1 = 8/ log(1 + σ −2).</Text> |
| </Panel> |
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| <Panel left="513" right="94" width="464" height="404"> |
| <Text>Composite Kernels</Text> |
| <Figure left="539" right="142" width="180" height="126" no="3" OriWidth="0.233526" OriHeight="0.12958 |
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| <Figure left="773" right="130" width="176" height="135" no="4" OriWidth="0.231214" OriHeight="0.136282 |
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| <Text>Product of squared exponential kernel</Text> |
| <Text>and linear kernel</Text> |
| <Text>Additive combination of payoff that</Text> |
| <Text>smoothly depends on context, and</Text> |
| <Text>exhibits clusters of actions.</Text> |
| <Text>Product kernel</Text> |
| <Text>• k = kS ⊗ kZ , where (kS ⊗ kZ )((s, z), (s0, z0)) = kZ (z, z0)kS (s, s0)</Text> |
| <Text>• Two context-action pairs are similar (large correlation) if the contexts are</Text> |
| <Text>similar and actions are similar</Text> |
| <Text>Additive kernel</Text> |
| <Text>• (kS ⊕ kZ )((s, z), (s0, z0)) = kZ (z, z0) + kS (s, s0)</Text> |
| <Text>• Generative model: first sample a function fS (s, z) that is constant along z, and</Text> |
| <Text>varies along s with regularity as expressed by ks; then sample a function fz(s, z),</Text> |
| <Text>which varies along z and is constant along s;</Text> |
| </Panel> |
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| <Panel left="511" right="501" width="467" height="216"> |
| <Text>Bounds for Composite Kernels</Text> |
| <Text>Maximum information gain for a GP with kernel k on set V</Text> |
| <Text>Product kernel</Text> |
| <Text>Let kZ be a kernel function on Z with rank at most d . Then</Text> |
| <Text>Additive kernel</Text> |
| <Text>Let kS and kZ be kernel functions on S and Z respectively. Then</Text> |
| </Panel> |
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| <Panel left="513" right="723" width="464" height="336"> |
| <Figure left="529" right="764" width="118" height="110" no="5" OriWidth="0.185549" OriHeight="0.127793 |
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| <Figure left="672" right="764" width="150" height="121" no="6" OriWidth="0.198266" OriHeight="0.127793 |
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| <Figure left="832" right="763" width="140" height="122" no="7" OriWidth="0.192486" OriHeight="0.125559 |
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| <Text>Task Discover peptide sequences binding to MHC molecules</Text> |
| <Text>Context Features encoding the MHC alleles</Text> |
| <Text>Action Choose a stimulus (the vaccine) s ∈ S that maximizes an observed response</Text> |
| <Text>(binding affinity).</Text> |
| <Text>Kernels Use a finite inter-task covariance kernel KZ with rank mZ to model the</Text> |
| <Text>similarity of different experiments, and a Gaussian kernel kS (s, s0) to model the</Text> |
| <Text>experimental parameters.</Text> |
| </Panel> |
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| <Panel left="510" right="1060" width="466" height="281"> |
| <Text>Learning to Monitor Sensor Networks</Text> |
| <Figure left="520" right="1099" width="135" height="103" no="8" OriWidth="0.195376" OriHeight="0.119303 |
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| <Figure left="670" right="1101" width="150" height="114" no="9" OriWidth="0.194798" OriHeight="0.117069 |
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| <Figure left="826" right="1102" width="147" height="114" no="10" OriWidth="0.191329" OriHeight="0.116175 |
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| <Text>Temperature data from a</Text> |
| <Text>network of 46 sensors at</Text> |
| <Text>Intel Research.</Text> |
| <Text>Time (h)</Text> |
| <Text>CGP-UCB using average</Text> |
| <Text>temperature</Text> |
| <Text>CGP-UCB using</Text> |
| <Text>minimum temperature</Text> |
| <Text>Task Given a sensor network, monitor maximum temperatures in building</Text> |
| <Text>Context Time of day</Text> |
| <Text>Action Pick 5 sensors to activate</Text> |
| <Text>Kernels Joint spatio-temporal covariance function using the Mat´ern kernel</Text> |
| </Panel> |
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| </Poster> |
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