On the ksupport and Related Norms


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1 On the ksupport and Related Norms Massimiliano Pontil Department of Computer Science Centre for Computational Statistics and Machine Learning University College London (Joint work with Andrew McDonald and Dimitris Stamos) Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
2 Plan Problem Spectral regularization ksupport norm Box norm Link to cluster norm Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
3 Problem Learn a matrix from a set of linear measurements: y i = W, X i + noise i, i = 1,..., n Method min W R d m n (y i W, X i ) 2 + λω(w ) Matrix completion: X i = e r e c Multitask learning: X i = e r x i Regularizer Ω encourages matrix structure Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
4 Spectral Regularization min W R d m n (y i W, X i ) 2 + λω(w ) Ω favors matrix structure (low rank, low variance, clustering, etc.) Choose an OInorm: Ω(W ) W = UWV, U, V orthogonal von Neumann (1937): W = g(σ(w )), with g is an SGfunction Well studied example is trace norm: g( ) = 1 Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
5 ksupport Norm [Argyriou et al. 2012] Special case of group lasso with overlap [Jacob et al., 2009] w (k) = inf v J 2 : v J = w, supp(v J ) J J k J k Includes the l 1 norm (k = 1) and l 2 norm (k = d) Unit ball of (k) is the convex hull of {card(w) k, w 2 1} k Dual norm: u,(k) = ( u i )2 Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
6 Spectral ksupport Norm ksupport norm is an SGfunction, inducing the OInorm W (k) := σ(w ) (k) Proposition. Unit ball of σ( ) (k) is the convex hull of {rank(w ) k, W F 1} Includes trace norm (k = 1) and Frobenius norm (k = d) Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
7 Matrix Completion Experiment dataset norm test error r k a ML 100k tr ρ = 50% en ks box e5 ML 1M tr ρ = 50% en ks box e6 Jester1 tr per en line ks box e5 Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
8 MTL Experiment Table: Multitask learning clustering on Lenk dataset, with simple thresholding. dataset norm test error k a Lenk fr (0.07) per task tr (0.04)   en (0.04)   ks (0.04) box (0.04) e3 cfr (0.08)   ctr (0.03)   cen (0.03)   cks (0.03) cbox (0.03) e3 Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
9 Box Norm Let Θ R d ++, bounded and convex and consider the norm: Box norm: Θ = w 2 Θ = inf θ Θ d w 2 i θ i, { a < θ i b, u 2,Θ = sup θ Θ d θ i c} Includes ksupport norm for a = 0, b = 1, c = k d θ i ui 2 Unit ball is the convex hull of { w R d : i J J k w 2 i b + i / J } wi 2 a 1 Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
10 Unit Balls Figure: Unit balls of the box norm in R 2 for k = 1, a {0.01, 0.25, 0.50}. Figure: Unit balls of the dual box norm in R 2 for k = 1, a {0.01, 0.25, 0.50}. Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
11 Cluster Norm Box norm is an SGfunction, inducing the OInorm { d W 2 Θ = σ(w ) 2 Θ = inf σ i (W ) 2 : θ (a, b] d, θ i d } θ i c Associated OInorm has been used to favour task clustering [Jacob et al. 2008]. It can be written as } W 2 Θ {tr(w = inf Σ 1 W T ) : ai Σ bi, tr Σ c Includes spectral ksupport norm for a = 0, b = 1, c = k Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
12 Interpretation of a Proposition. If c = da + k(b a), the solution of the regularization problem is given by Ŵ = ˆV + Ẑ, where ( ˆV, Ẑ) = arg min V,Z n ( 1 (y i V + Z, X i ) 2 + λ a V 2 F + 1 ) b a Z 2 (k) Parameter a balances the relative importance of the two components Cluster norm is the Moureau envelope of spectral ksupport norm: { 1 W 2 Θ = a W Z 2 F + 1 } b a Z 2 (k) min Z R d m Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
13 Computation of the Θ norm Assume w.l.o.g. w 0 with non increasing components w 2 Θ = 1 b w [1:q] c qb la w [q+1:d l] a w [l+1:d] 2 2, where q, l {0,..., d} are uniquely determined In particular: w (k) = w [1:q] k q w [q+1:d] 2 1 d where q {0,..., k 1} is determined by w q 1 k q w j > w q+1 j=q+1 Computation of norm is O(d log(d)) For ksupport improves previous O(kd) method Efficient optimization using proximalgradient methods Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
14 Extensions/Open Problems Other sets Θ allow for exact prox, e.g. Θ = {θ 1... θ d > 0}. Can give a general characterization? Online learning / stochastic optimization Kernel extensions Massimiliano Pontil (UCL) On the ksupport and Related Norms Sestri Levante, Sept / 14
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