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**Projection-free Online Learning**

Dan Garber Elad Hazan

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**Super-linear operations are infeasible!**

Matrix completion Super-linear operations are infeasible!

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**Online convex optimization**

Incurred loss x2 f1 f1(x1) x1 f2 f2(x2) fT(xT) linear (convex) bounded cost functions Total loss = t ft(xt) Regret = t ft(xt) minx* t ft(x*) Matrix completion: set = low rank matrices, Xij = prediction user i movie j functions: f(X) = |X * Eij - ±1|^2

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**Online gradient descent**

yt+1 The algorithm: move in the direction of the vector -ct (gradient of the current cost function) ct xt+1 xt yt+1 = xt - ct and project back to the convex set Thm [Zinkevich]: if = 1/ t then this alg attains worst case regret of t ft(xt) - t ft(x*) = O( T)

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**Computational efficiency?**

Gradient step: linear time Projection step: quadratic program !! Online mirror descent: general convex program The convex decision set K: In general O(m½ n3 ) Simplex / Euclidean ball / cube – linear time Flow polytope – conic opt. O(m½ n3) PSD cone (matrix completion) – Cholesky decomposition O(n3)

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**Projections out of the question!**

Matrix completion Projections out of the question!

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**Computationally difficult learning problems**

Matrix completion K = SDP cone Cholesky decomposition Online routing K = flow polytope conic optimization over flow polytope Rotations K = rotation matrices Matroids K = matroid polytope

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**Results part 1 (Hazan + Kale, ICML’12)**

Projection-less stochastic/online algorithms with regret bounds: Projections <-> Linear optimization parameter free (no learning rate) sparse predictions Stochastic Adversarial Smooth √T T¾ Non-smooth T3/4

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**Linear opt. vs. Projections**

Matrix completion K = SDP cone Cholesky decomposition largest singular vector Online routing K = flow polytope conic optimization over flow polytope shortest path computation Rotations K = rotation matrices convex opt. Wahba’s alg.

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**The Frank-Wolfe algorithm**

vt+1 xt+1 xt

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**The Frank-Wolfe algorithm (conditional grad.)**

vt+1 xt+1 xt Thm[ FW ’56]: rate of convergence = 1/Ct (C = smoothness) [Clarkson ‘06] – refined analysis [Hazan ‘07] - SDP [Jaggi ‘11] – generalization

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**The Frank-Wolfe algorithm**

vt+1 xt+1 xt At iteration t – convex comb. of <= t vertices = ((t,K)-sparse No learning rate. Convex combination with 1/t (indep. Of diameter, gradients etc.)

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**Online Frank-Wolfe – wrong approach**

example: [S. Bubek] K = interval [-1,1]

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**Online Conditional Gradient (OCG)**

xt xt+1 vt+1

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**Projections <-> Linear optimization **

parameter free (no learning rate) sparse predictions But can we get the optimal root(T) rate?? Barrier: existing projection-free algs were not linearly converging (poly-time) Stochastic Adversarial Smooth √T T¾ Non-smooth T2/3

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**New poly-time projection free alg [Garber, Hazan 2013]**

New algorithm with convergence ~ e-t/n rate CS: “poly time” Nemirovski: “linear rate” Only linear optimization calls on the original polytope! (constantly many per iteration)

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**Linearly converging Frank-Wolfe**

vt+1 xt+1 xt Assume optimum is within Euclidean distance r: Thm[ easy ]: rate of convergence = e-t But useless: under a ball-intersection constraint – quadratic optimization equivalent to projection

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**Inherent problem with balls**

x* xt No intersection -> radius is bounded by distance to boundary With intersection -> hard problem equivalent to projection

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Polytopes are OK! Can find a significantly smaller polytope (radius proportional to Euclidean distance to OPT) that: Contains x* Does not intersect original polytope same shape

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**Implications for online optimization**

Projections <-> Linear optimization parameter free (no learning rate) sparse predictions Optimal rate Stochastic Adversarial Convex √T Strongly convex log(T)

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**More research / open questions**

Projection free alg – for many problems linear step time vs. cubic or more For main ML problems today – projection-free is the only feasible optimization method Completely poly-time (log dependence on smoothness / strong convexity / diameter) Can we attain poly-time optimization using only gradient information? Thank you!

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