# Fast Monte-Carlo Algorithms for Matrix Multiplication

Approximate computation and implicit regularization in large-scale data analysis Michael W. Mahoney Stanford University Sept 2011 (For more info, see: http://cs.stanford.edu/people/mmahoney) Motivating observation Theory of NP-completeness is a very useful theory captures qualitative notion of fast, provides an qualitative guidance to how algorithms perform in practice, etc. LP, simplex, ellipsoid, etc. - the exception that proves the rule Theory of Approximation Algorithms is NOT analogously useful

(at least for many machine learning and data analysis problems) bounds very weak; cant get constants; dependence on parameters not qualitatively right; does not provide qualitative guidance w.r.t. practice; usually want a vector/graph achieving optimum, bu dont care about the particular vector/graph; etc. Start with the conclusions Modern theory of approximation algorithms is often NOT a useful theory for many large-scale data analysis problem Approximation algorithms and heuristics often implicitly perform regularization, leading to more robust or better solutions Can characterize the regularization properties implicit in worst-case approximation algorithms Take-home message: Solutions of approximation algorithms

dont need to be something we settle for, since they can be better than the solution to the original intractable problem Algorithmic vs. Statistical Perspectives Lambert (2000) Computer Scientists Data: are a record of everything that happened. Goal: process the data to find interesting patterns and associations. Methodology: Develop approximation algorithms under different models of data access since the goal is typically computationally hard. Statisticians (and Natural Scientists)

Data: are a particular random instantiation of an underlying process describing unobserved patterns in the world. Goal: is to extract information about the world from noisy data. Methodology: Make inferences (perhaps about unseen events) by positing a model that describes the random variability of the data around the deterministic model. Statistical regularization (1 of 2) Regularization in statistics, ML, and data analysis arose in integral equation theory to solve ill-posed problems computes a better or more robust solution, so better inference involves making (explicitly or implicitly) assumptions

about the data provides a trade-off between solution quality versus solution niceness often, heuristic approximation have regularization properties as a side effect Statistical regularization (2 of 2) Usually implemented in 2 steps: add a norm constraint (or geometric capacity control function) g(x) to objective function f(x) solve the modified optimization problem x = argminx f(x) + g(x)

Often, this is a harder problem, e.g., L1-regularized L2regression x = argminx ||Ax-b||2 + ||x||1 Two main results Big question: Can we formalize the notion that/when approximate computation can implicitly lead to better or more regular solutions than exact computation? Approximate first nontrivial eigenvector of Laplacian Three random-walk-based procedures (heat kernel, PageRank, truncated lazy random walk) are implicitly solving a regularized optimization exactly! Spectral versus flow-based approximation

algorithms for graph partitioning Theory suggests each should regularize in different ways, and empirical results agree! Approximating the top eigenvector Basic idea: Given a Laplacian matrix A, Power method starts with v0, and iteratively computes vt+1 = Avt / ||Avt||2 . Then, vt = i it vi -> v1 . If we truncate after (say) 3 or 10 iterations, still have some mixing from other eigen-directions What objective does the exact eigenvector optimize? Rayleigh quotient R(A,x) = xTAx /xTx, for a vector x.

But can also express this as an SDP, for a SPSD matrix X. (We will put regularization on this SDP!) Views of approximate spectral methods Three common procedures (L=Laplacian, and M=r.w. matrix): Heat Kernel: PageRank: q-step Lazy Random Walk: Ques: Do these approximation procedures exactly optimizing some regularized objective? Two versions of spectral partitioning

VP: R-VP: Two versions of spectral partitioning VP: SDP : R-VP: RSDP: A simple theorem

Mahoney and Orecchia (2010) Modification of the usual SDP form of spectral to have regularization (but, on the matrix X, not the vector x). Three simple corollaries FH(X) = Tr(X log X) - Tr(X) (i.e., generalized entropy) gives scaled Heat Kernel matrix, with t = FD(X) = -logdet(X) (i.e., Log-determinant)

gives scaled PageRank matrix, with t ~ Fp(X) = (1/p)||X||pp (i.e., matrix p-norm, for p>1) gives Truncated Lazy Random Walk, with ~ Answer: These approximation procedures compute regularized versions of the Fiedler vector Graph partitioning A family of combinatorial optimization problems - want to partition a graphs nodes into two sets s.t.: Not much edge weight across the cut (cut quality) Both sides contain a lot of nodes Several standard formulations:

Graph bisection (minimum cut with 50-50 balance) -balanced bisection (minimum cut with 70-30 balance) cutsize/min{|A|,|B|}, or cutsize/(|A||B|) (expansion) cutsize/min{Vol(A),Vol(B)}, or cutsize/(Vol(A)Vol(B)) (conductance or N-Cuts) All of these formalizations of the bi-criterion are NP-hard! The lay of the land Spectral methods* - compute eigenvectors of associated matrices Local improvement - easily get trapped in local minima, but can be used to clean up other cuts Multi-resolution - view (typically space-like graphs) at multiple size scales Flow-based methods* - single-commodity or multicommodity version of max-flow-min-cut ideas

*Comes with strong underlying theory to guide heuristics. Comparison of spectral versus flow Spectral: Flow: Compute an eigenvector Compute a LP Quadratic worst-case bounds

O(log n) worst-case bounds Worst-case achieved -- on long stringy graphs Worst-case achieved -- on expanders Worst case is global Worse-case is local property property Embeds you on a line (or Kn) Embeds you in L1 Two methods -- complementary strengths and weaknesses

What we compute is determined at least as much by as the approximation algorithm as by objective function. Explicit versus implicit geometry Explicitlyimposed geometry Traditional regularization uses explicit norm constraint to make sure solution vector is small and not-toocomplex

Implicitly-imposed geometry Approximation algorithms implicitly embed the data in a nice metric/geometric place and then round the solution. (X,d) d(x,y) x y f

(X,d ) f(x) f(y) Regularized and non-regularized communities Conductance of bounding cut Local Spectral Diameter of the cluster

Connected Disconnected Local Spectral (blue) gives tighter and more well-rounded sets. Lower is good Metis+MQI - a Flow-based method (red) gives sets with better conductance. External/internal conductance

Conclusions Modern theory of approximation algorithms is often NOT a useful theory for many large-scale data analysis problem Approximation algorithms and heuristics often implicitly perform regularization, leading to more robust or better solutions Can characterize the regularization properties implicit in worst-case approximation algorithms Take-home message: Solutions of approximation algorithms dont need to be something we settle for, since they can be better than the solution to the original intractable problem

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