Online Passive-Aggressive Algorithms Three Decision Problems Online Setting A Unified View A Unified View (Cont.)
Romanelli, Bill, Contributing Writer has reference to this Academic Journal, PHwiki organized this Journal Online Passive-Aggressive Algorithms Shai Shalev-Shwartz joint work with Koby Crammer, Ofer Dekel & Yoram Singer The Hebrew University Jerusalem, Israel Three Decision Problems Classification Regression Uniclass Online Setting Receive instance n/a Predict target value Receive true target ; suffer loss Update hypothesis Classification Regression Uniclass
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A Unified View Define discrepancy as long as : Unified Hinge-Loss: Notion of Realizability: Classification Regression Uniclass A Unified View (Cont.) Online Convex Programming: Let be a sequence of convex functions: Let be an insensitivity parameter. For Guess a vector Get the current convex function Suffer loss Goal: minimize the cumulative loss The Passive-Aggressive Algorithm Each example defines a set of consistent hypotheses: The new vector is set to be the projection of onto Classification Regression Uniclass
Passive-Aggressive An Analytic Solution where in addition to Classification Regression Uniclass Loss Bounds Theorem: – a sequence of examples. Assumption: Then if the online algorithm is run with , the following bound holds as long as any where as long as classification in addition to regression in addition to as long as uniclass.
Loss bounds (cont.) For the case of classification we have one degree of freedom since if then as long as any There as long as e, we can set in addition to get the following bounds: Loss bounds (Cont). Classification Uniclass Proof Sketch Define: Upper bound: Lower bound: Lipschitz Condition
Proof Sketch (Cont.) Combining upper in addition to lower bounds The Unrealizable Case Main idea: downsize step size by Loss Bound Theorem: – sequence of examples. bound as long as any in addition to as long as any
Implications as long as Batch Learning Batch Setting: Input: A training set , sampled i.i.d according to an unknown distribution D. Output: A hypothesis parameterized by Goal: Minimize Online Setting: Input: A sequence of examples Output: A sequence of hypotheses Goal: Minimize Implications as long as Batch Learning (Cont.) Convergence: Let be a fixed training set in addition to let be the vector obtained by PA after epochs. Then, as long as any Large margin as long as classification: For all we have: , which implies that the margin attained by PA as long as classification is at least half the optimal margin Derived Generalization Properties Average hypothesis: Let be the average hypothesis. Then, with high probability we have
A Multiplicative Version Assumption: Multiplicative update: Loss bound: Summary Unified view of three decision problems New algorithms as long as prediction with hinge loss Competitive loss bounds as long as hinge loss Unrealizable Case: Algorithms & Analysis Multiplicative Algorithms Batch Learning Implications Future Work & Extensions: Updates using general Bregman projections Applications of PA to other decision problems Related Work Projections Onto Convex Sets (POCS), e.g.: Y. Censor in addition to S.A. Zenios, Parallel Optimization H.H. Bauschke in addition to J.M. Borwein, On Projection Algorithms as long as Solving Convex Feasibility Problems Online Learning, e.g.: M. Herbster, Learning additive models online with fast evaluating kernels
Romanelli, Bill Contributing Writer
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