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Sparse Euclidean projections onto convex sets Volkan Cevher Laboratory for Information and Inference Systems – LIONS / EPFL joint work with Stephen Becker Anastasios Kyrillidis ISMP’12 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A A

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On the “sparse” model Sparse vector only K out of N coordinates nonzero sorted index support:

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On the “sparse” model Sparse vector only K out of N coordinates nonzero Many important applications<>model-of-choice regression (linear / nonlinear), classification, density estimation, principal component analysis, compressive sensing, spectrum estimation, denoising, deblurring, data mining, sketching, demixing / deconvolution, geophysics, medical imaging, compression, source localization, speech, function learning… Great theoretical backup explaining empirical success –provably effective in circumventing the ill-posed nature of inverse problems –provably essential in containing the generalization errors in learning –provably necessary in keeping transaction costs low in portfolio design –…

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On the “structured sparse” model Structured sparse vector only certain K out of N coordinates nonzero Many important applications<>model-of-choice regression (linear / nonlinear), classification, density estimation, principal component analysis, compressive sensing, spectrum estimation, denoising, deblurring, data mining, sketching, demixing / deconvolution, geophysics, medical imaging, compression, source localization, speech, function learning… sorted index

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On the “structured sparse” model Structured sparse vector only certain K out of N coordinates nonzero Many important applications<>model-of-choice regression (linear / nonlinear), classification, density estimation, principal component analysis, compressive sensing, spectrum estimation, denoising, deblurring, data mining, sketching, demixing / deconvolution, geophysics, medical imaging, compression, source localization, speech, function learning… sorted index

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Applications require Optimization with the sparse models data fidelity term; mostly convex, smooth. typically: Difficulties: non-smoothness, non-convexity, and large dimension sparsity-based regularizer/indicator; maybe convex and non-smooth; or non-convex...

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Goal: Non-convexConvex Encoding Ecombinatorialconvex relaxation Example Optimization with the sparse models

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Goal: Sparsityinherently discrete non-convex encoder+simpler description of the model and its structured variations +projections with combinatorial algorithms with optimal time and space bounds -difficult to analyze / no-scale convex encoder-harder description of the model & struct. +efficient/general convex projections +easier to analyze / known scale Optimization with the sparse models Non-convexConvex Encoding Ecombinatorialconvex relaxation Example

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Goal: Modus operandi:convex > non-convex * key lessons:ell1 encodes sparsity extremely effectively + convexity (a rare condition) is powerful Optimization with the sparse models * for simple sparsity models Non-convexConvex Encoding Ecombinatorialconvex relaxation Example

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Goal: Modus operandi:convex > non-convex* * for simple sparsity models CLASH/NP: the power of the two frameworks for quadratic convex problems Optimization with the sparse models [Kyrillidis and C, 2011] Non-convexConvex Encoding Ecombinatorialconvex relaxation Example sparsity & structure + geometry

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A preview CLASH Structured Sparsity Norm Constraints [Kyrillidis, Puy, and C, 2012]

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A Real Clash of Sparsity Models

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If ell1 is already taken, what do we do? Density estimation Examples: sparse mixture/kernel models with small sample size, nonparametric density estimation, aggregation of density estimators Markowitz Portfolio optimization Examples: minimum variance portfolio with no-short positions minimum variance portfolio with short positions Many other applications boosting classifiers, NNMF, ED, etc… [Bunea, Tsybakov, and Wegkamp, 2010; Willet and Nowak, 2007] [Brodie et al., 2008] C encodes constraints f 1 : convex sparsity is still required!

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If ell1 is already taken, what do we do? Density estimation Markowitz Portfolio optimization Many other applications boosting classifiers, NNMF, ED, etc… Sparsity is still required! [Bunea, Tsybakov, and Wegkamp, 2010; Willet and Nowak, 2007] [Brodie et al., 2008] C encodes constraints sparsity-based f 2 and the constraint f 3 can conflict with each other!

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If ell1 is already taken, what do we do? Density estimation Markowitz Portfolio optimization Many other applications boosting classifiers, NNMF, ED, etc… Sparsity is still required! [Bunea, Tsybakov, and Wegkamp, 2010; Willet and Nowak, 2007] [Brodie et al., 2008] C encodes constraints sparsity-based f 2 and the constraint f 3 can conflict with each other!

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If ell1 is already taken, what do we do? Density estimation Markowitz Portfolio optimization Many other applications boosting classifiers, NNMF, ED, etc… Sparsity is still required! [Bunea, Tsybakov, and Wegkamp, 2010; Willet and Nowak, 2007] [Brodie et al., 2008] C encodes constraints sparsity-based f 2 can bias the data-fidelity

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If ell1 is already taken, what do we do? Density estimation Markowitz Portfolio optimization Many other applications boosting classifiers, NNMF, ED, etc… Sparsity is still required! [Bunea, Tsybakov, and Wegkamp, 2010; Willet and Nowak, 2007] [Brodie et al., 2008] C encodes constraints sparsity-based f 2 can bias the data-fidelity 1. de-biased solutions are not necessarily sparse stationary points of f 1 + the constraint C 2. it may not sparsify the solution enough!

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Objective: Problems where ell1 is already taken and beyond…

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Objective: in general, NP-Hard! Problems where ell1 is already taken and beyond…

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Objective: Motivation:convergence of the forward-backward algorithm for non-convex sets −obtain stationary points of convex optimization objective* Problems where ell1 is already taken and beyond… [Attouch et al. 2010]*with Lipschitz gradient

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Objective: Motivation:convergence of the forward-backward algorithm for non-convex sets −obtain stationary points of convex optimization objective* Key actors for hire: −convex projector −sparse projector Problems where ell1 is already taken and beyond… [Attouch et al. 2010]*with Lipschitz gradient

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Examples −ell1 ball −simplex −positive-simplex combination of the two Convex projectors* soft thresholding *I will abuse the term and use it also for prox operators. coordinate-bias

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Examples support of the solution <> modular approximation problem Sparse projectors [Kyrillidis and C, 2011]

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Examples support of the solution <> modular approximation problem where Sparse projectors [Kyrillidis and C, 2011]

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Examples support of the solution <> modular approximation problem where support of the solution <> integer linear program Sparse projectors [Kyrillidis and C, 2011] : support indicator variables

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Examples support of the solution <> modular approximation problem support of the solution <> integer linear program Sparse projectors [Kyrillidis and C, 2011] matroid structured sparse models 1. uniform matroid<>simple sparsity intersection with the following matroids (result is still a matroid!*) 2. partition matroid<>distributed sparsity 3. graphic matroid<>spanning tree sparsity 4. matching matroid <>graph matching sparsity *: in general, the intersection of two matroids is not a matroid. projector: Greedy basis algorithm A and b<>integral first row of A <>all 1’s first entry of b<>K when A is totally unimodular, projector: linear program clustered sparsity models 1.clustered-tree 2.clustered-line model projector: dynamic programming

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Algorithm: Sparse Euclidean projections on convex sets

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Algorithm: Sparse Euclidean projections on convex sets piece of cake.

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Algorithm: Sparse Euclidean projections on convex sets seems tough.

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Algorithm: Sparse Euclidean projections on convex sets ditto.

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Algorithm: Sparse Euclidean projections on convex sets is this what we wanted to solve???

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Algorithm: Example: Sparse Euclidean projections on convex sets

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Algorithm: Example: Sparse Euclidean projections on convex sets by inspection! Error = this is hard by inspection in high-dimensions!

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Sparse Euclidean projections on convex sets by inspection! Error = Algorithm: Example: this is hard by inspection in high-dimensions!Let’s stage the actors!

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Algorithm: Naïve approaches: Sparse Euclidean projections on convex sets Error = 5.5 so, non-convex followed by convex does not work here. 0000

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Algorithm: Naïve approaches: Sparse Euclidean projections on convex sets Error = 5.5 well, neither does convex followed by non-convex!

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Algorithm: Naïve approaches: Sparse Euclidean projections on convex sets 5 2 surprisingly, our algorithm obtains the correct solution! our algorithm Error = 4 00

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Algorithm: So, what do actually we do? Sparse Euclidean projections on convex sets &

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Algorithm: So, what do actually we do? Sparse Euclidean projections on convex sets &

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Algorithm: So, what do actually we do? Sparse Euclidean projections on convex sets &

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Algorithm: So, what do actually we do? Sparse Euclidean projections on convex sets &

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme More on the algorithm

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Intuition:online optimality More on the algorithm straightforward to establish for

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Intuition: efficiency More on the algorithm monotonicity of boundedness of bisection

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Intuition:online approximation guarantee More on the algorithm bisection

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Intuition:optimality More on the algorithm bisection +monotonicity +continuity +equivalence

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Intuition:optimality More on the algorithm For simple sparsity for p=1,2, and inf

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Intuition:optimality More on the algorithm For structured sparsity for p=2 and inf

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Intuition:sub-optimality More on the algorithm For structured sparsity

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Algorithm: Highlights: 1.optimality / online optimality / online approximation guarantees 2.efficiency of the scheme Alternative: proximal alternating minimization algorithm a general hammer for such problems with sublinear convergence rate More on the algorithm [Attouch et al. 2010]

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Kernel density learning: Experiments

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Markowitz portfolio selection: Experiments

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Simplex: Experiments

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Structured sparse projections on convex sets highlightsonline optimality / approximation guarantees efficient scheme 1-shot projections for certain combinations* needs more workcase-by-case analysis more than one-parameter projectors Upshot sparse stationary points of constrained convex objectives fixed sparse solutions (t,C(x t ))-curve reveals the difficulty of the projection conjecture monotonicity is preserved if C is convex Analysis generalizes to argmax Conclusions * 1/k problem!

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Postdoc LIONS / EPFL contact:

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