stochastic gradient descent and robustness to ill … › cohen › sanservolotalks › bach.pdf–...

Stochastic gradient descentand robustness to ill-conditioning

Francis Bach

INRIA - Ecole Normale Superieure, Paris, France

ÉCOLE NORMALE

S U P É R I E U R E

Joint work with Aymeric Dieuleveut, Nicolas Flammarion,

Eric Moulines - Venezia, May 2016

“Big data” revolution?

A new scientific context

• Data everywhere: size does not (always) matter

• Science and industry

• Size and variety

• Learning from examples

– n observations in dimension p

Search engines - Advertising

Visual object recognition

Bioinformatics

• Protein: Crucial elements of cell life

• Massive data: 2 millions for humans

• Complex data

Context

Machine learning for “big data”

• Large-scale machine learning: large p, large n

– p : dimension of each observation (input)

– n : number of observations

• Examples: computer vision, bioinformatics, advertising

– Ideal running-time complexity: O(pn)

– Going back to simple methods

– Stochastic gradient methods (Robbins and Monro, 1951)

– Mixing statistics and optimization

– Using smoothness to go beyond stochastic gradient descent

Context

• Ideal running-time complexity: O(pn)

– Going back to simple methods

Context

• Ideal running-time complexity: O(pn)

• Going back to simple methods

Supervised machine learning

• Data: n observations (xi, yi) ∈ X × Y, i = 1, . . . , n, i.i.d.

• Prediction as a linear function 〈θ,Φ(x)〉 of features Φ(x) ∈ Rp

– Explicit features adapted to inputs (can be learned as well)

– Using Hilbert spaces for non-linear / non-parametric estimation

• (regularized) empirical risk minimization: find θ solution of

minθ∈Rp

yi, 〈θ,Φ(xi)〉)

+ µΩ(θ)

convex data fitting term + regularizer

Usual losses

• Regression: y ∈ R, prediction y = 〈θ,Φ(x)〉– quadratic loss 1

2(y − y)2 = 12(y − 〈θ,Φ(x)〉)2

Usual losses

• Regression: y ∈ R, prediction y = 〈θ,Φ(x)〉– quadratic loss 1

2(y − y)2 = 12(y − 〈θ,Φ(x)〉)2

• Classification : y ∈ −1, 1, prediction y = sign(〈θ,Φ(x)〉)– loss of the form ℓ(y 〈θ,Φ(x)〉)– “True” 0-1 loss: ℓ(y 〈θ,Φ(x)〉) = 1y 〈θ,Φ(x)〉<0

– Usual convex losses:

−3 −2 −1 0 1 2 3 40

0−1hingesquarelogistic

minθ∈Rp

+ µΩ(θ)

minθ∈Rp

+ µΩ(θ)

• Empirical risk: f(θ) = 1n

∑ni=1 ℓ(yi, 〈θ,Φ(xi)〉) training cost

• Expected risk: f(θ) = E(x,y)ℓ(y, 〈θ,Φ(x)〉) testing cost

• Two fundamental questions: (1) computing θ and (2) analyzing θ

Smoothness and strong convexity

• A function g : Rp → R is L-smooth if and only if it is twice

differentiable and

∀θ ∈ Rp, eigenvalues

g′′(θ)]

smooth non−smooth

• A function g : Rp → R is L-smooth if and only if it is twice

differentiable and

g′′(θ)]

• Machine learning

– with g(θ) = 1n

∑ni=1 ℓ(yi, 〈θ,Φ(xi)〉)

– Hessian ≈ covariance matrix 1n

∑ni=1Φ(xi)⊗ Φ(xi)

– Bounded data: ‖Φ(x)‖ 6 R ⇒ L = O(R2)

• A twice differentiable function g : Rp → R is µ-strongly convex if

and only if

g′′(θ)]

convexstronglyconvex

and only if

g′′(θ)]

(large µ/L) (small µ/L)

and only if

g′′(θ)]

– with g(θ) = 1n

– Data with invertible covariance matrix (low correlation/dimension)

and only if

g′′(θ)]

– with g(θ) = 1n

– Data with invertible covariance matrix (low correlation/dimension)

• Adding regularization by µ2‖θ‖2

– creates additional bias unless µ is small

Iterative methods for minimizing smooth functions

• Assumption: g convex and smooth on Rp

• Gradient descent: θt = θt−1 − γt g′(θt−1)

– O(1/t) convergence rate for convex functions

– O(e−(µ/L)t) convergence rate for strongly convex functions

• Newton method: θt = θt−1 − g′′(θt−1)−1g′(θt−1)

– O(

e−ρ2t)

convergence rate

• Newton method: θt = θt−1 − g′′(θt−1)−1g′(θt−1)

– O(

e−ρ2t)

convergence rate

• Key insights from Bottou and Bousquet (2008)

1. In machine learning, no need to optimize below statistical error

2. In machine learning, cost functions are averages

⇒ Stochastic approximation

Stochastic approximation

• Goal: Minimizing a function f defined on Rp

– given only unbiased estimates f ′n(θn) of its gradients f ′(θn) at

certain points θn ∈ Rp

Stochastic approximation

• Goal: Minimizing a function f defined on Rp

– given only unbiased estimates f ′n(θn) of its gradients f ′(θn) at

certain points θn ∈ Rp

• Machine learning - statistics

– f(θ) = Efn(θ) = E ℓ(yn, 〈θ,Φ(xn)〉) = generalization error

– Loss for a single pair of observations: fn(θ) = ℓ(yn, 〈θ,Φ(xn)〉)– Expected gradient:

f ′(θ) = Ef ′n(θ) = E

ℓ′(yn, 〈θ,Φ(xn)〉)Φ(xn)

• Beyond convex optimization: see, e.g., Benveniste et al. (2012)

Convex stochastic approximation

• Key assumption: smoothness and/or strong convexity

• Key algorithm: stochastic gradient descent (a.k.a. Robbins-Monro)

θn = θn−1 − γn f′n(θn−1)

– Polyak-Ruppert averaging: θn = 1n+1

∑nk=0 θk

– Which learning rate sequence γn? Classical setting: γn = Cn−α

• Key assumption: smoothness and/or strong convexity

• Key algorithm: stochastic gradient descent (a.k.a. Robbins-Monro)

θn = θn−1 − γn f′n(θn−1)

– Polyak-Ruppert averaging: θn = 1n+1

∑nk=0 θk

– Which learning rate sequence γn? Classical setting: γn = Cn−α

• Running-time = O(np)

– Single pass through the data

– One line of code among many

Existing analysis

• Known global minimax rates of convergence for non-smooth

problems (Nemirovsky and Yudin, 1983; Agarwal et al., 2012)

– Strongly convex: O((µn)−1)

Attained by averaged stochastic gradient descent with γn ∝ (µn)−1

– Non-strongly convex: O(n−1/2)

Attained by averaged stochastic gradient descent with γn ∝ n−1/2

Existing analysis

• Asymptotic analysis of averaging (Polyak and Juditsky, 1992;

Ruppert, 1988)

– All step sizes γn = Cn−α with α ∈ (1/2, 1) lead to O(n−1) for

smooth strongly convex problems

A single algorithm with global adaptive convergence rate for

smooth problems?

Existing analysis

• Asymptotic analysis of averaging (Polyak and Juditsky, 1992;

Ruppert, 1988)

– All step sizes γn = Cn−α with α ∈ (1/2, 1) lead to O(n−1) for

smooth strongly convex problems

• A single algorithm for smooth problems with global convergence

rate O(1/n) in all situations?

Least-mean-square algorithm

• Least-squares: f(θ) = 12E

(yn − 〈Φ(xn), θ〉)2]

with θ ∈ Rp

– SGD = least-mean-square algorithm (see, e.g., Macchi, 1995)

– usually studied without averaging and decreasing step-sizes

– with strong convexity assumption E[

Φ(xn)⊗Φ(xn)]

= H < µ · Id

Least-mean-square algorithm

• Least-squares: f(θ) = 12E

(yn − 〈Φ(xn), θ〉)2]

with θ ∈ Rp

– SGD = least-mean-square algorithm (see, e.g., Macchi, 1995)

– usually studied without averaging and decreasing step-sizes

– with strong convexity assumption E[

Φ(xn)⊗Φ(xn)]

= H < µ · Id

• New analysis for averaging and constant step-size γ = 1/(4R2)

– Assume ‖Φ(xn)‖ 6 R and |yn − 〈Φ(xn), θ∗〉| 6 σ almost surely

– No assumption regarding lowest eigenvalues of H

– Main result: Ef(θn)− f(θ∗) 64σ2p

4R2‖θ0 − θ∗‖2n

• Matches statistical lower bound (Tsybakov, 2003)

– Non-asymptotic robust version of Gyorfi and Walk (1996)

Markov chain interpretation of constant step sizes

• LMS recursion for fn(θ) =12

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)

• The sequence (θn)n is a homogeneous Markov chain

– convergence to a stationary distribution πγ

– with expectation θγdef=

θπγ(dθ)

- For least-squares, θγ = θ∗

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)

θπγ(dθ)

• For least-squares, θγ = θ∗

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)

θπγ(dθ)

yn − 〈Φ(xn), θ〉)2

θn = θn−1 − γ(

〈Φ(xn), θn−1〉 − yn)

Φ(xn)

θπγ(dθ)

– θn does not converge to θ∗ but oscillates around it

– oscillations of order√γ

• Ergodic theorem:

– Averaged iterates converge to θγ = θ∗ at rate O(1/n)

Simulations - synthetic examples

• Gaussian distributions - p = 20

0 2 4 6−5

log 10

[f(θ)

θ *)]

synthetic square

1/32R2

1/2R2n1/2

Simulations - benchmarks

• alpha (p = 500, n = 500 000), news (p = 1 300 000, n = 20 000)

0 2 4 6

−1.5

−0.5

log 10

[f(θ)

θ *)]

alpha square C=1 test

1/R2n1/2

0 2 4 6

−1.5

−0.5

alpha square C=opt test

C/R2n1/2

−0.8

−0.6

−0.4

−0.2

log 10

[f(θ)

θ *)]

news square C=1 test

1/R2n1/2

−0.8

−0.6

−0.4

−0.2

news square C=opt test

C/R2n1/2

Isn’t least-squares regression a “regression”?

• Least-squares regression

– Simpler to analyze and understand

– Explicit relationship to bias/variance trade-offs (next slides)

• Many important loss functions are not quadratic

– Beyond least-squares with online Newton steps

– Complexity of O(p) per iteration with rate O(p/n)

– See Bach and Moulines (2013) for details

Optimal bounds for least-squares?

• Least-squares: cannot beat σ2p/n (Tsybakov, 2003). Really?

• Refined analysis (Defossez and Bach, 2015)

Ef(θn)− f(θ∗) 6σ2p

‖H−1/2(θ0 − θ∗)‖22γ2n2

– In practice: bias may be larger than variance, σ2p/n pessimistic

• Refined analysis (Defossez and Bach, 2015)

Ef(θn)− f(θ∗) 6σ2p

‖H−1/2(θ0 − θ∗)‖22γ2n2

– In practice: bias may be larger than variance, σ2p/n pessimistic

• Refined assumptions with adaptivity (Dieuleveut and Bach, 2014)

Ef(θn)− f(θ∗) 6σ2γ1/α trH1/α

n1−1/α+

‖H1/2−r(θ0 − θ∗)‖22γ2rn2minr,1

– SGD is adaptive to the covariance matrix eigenvalue decay

– Leads to optimal rates for non-parametric regression

Achieving optimal bias and variance terms

• Current results with averaged SGD (ill-conditioned problems)

– Variance (starting from optimal θ∗) =σ2p

– Bias (no noise)= minR2‖θ0 − θ∗‖2

n,R4〈θ0 − θ∗,H

−1(θ0 − θ∗)〉n2

n,R4〈θ0 − θ∗,H

−1(θ0 − θ∗)〉n2

n,R4〈θ0 − θ∗,H

−1(θ0 − θ∗)〉n2

Bias Variance

Averaged gradient descent

(Bach and Moulines, 2013)R2‖θ0 − θ∗‖2

Bias Variance

Accelerated gradient descent

(Nesterov, 1983)R2‖θ0 − θ∗‖2

n2σ2p

• Acceleration is notoriously non-robust to noise (d’Aspremont,

2008; Schmidt et al., 2011)

– For non-structured noise, see Lan (2012)

Bias Variance

(Nesterov, 1983)R2‖θ0 − θ∗‖2

n2σ2p

“Between” averaging and acceleration

(Flammarion and Bach, 2015)R2‖θ0 − θ∗‖2

n1−α

Bias Variance

(Nesterov, 1983)R2‖θ0 − θ∗‖2

n2σ2p

“Between” averaging and acceleration

(Flammarion and Bach, 2015)R2‖θ0 − θ∗‖2

n1−α

Averaging and acceleration

(Dieuleveut, Flammarion, and Bach, 2016)R2‖θ0 − θ∗‖2

Conclusions

• Constant-step-size averaged stochastic gradient descent

– Reaches convergence rate O(1/n) in all regimes

– Improves on the O(1/√n) lower-bound of non-smooth problems

– Efficient online Newton step for non-quadratic problems

– Robustness to step-size selection and adaptivity

Conclusions

• Constant-step-size averaged stochastic gradient descent

– Reaches convergence rate O(1/n) in all regimes

– Improves on the O(1/√n) lower-bound of non-smooth problems

– Efficient online Newton step for non-quadratic problems

– Robustness to step-size selection and adaptivity

• Extensions and future work

– Going beyond a single pass (Le Roux, Schmidt, and Bach, 2012;

Defazio, Bach, and Lacoste-Julien, 2014)

– Proximal extensions fo non-differentiable terms

– Kernels and nonparametric estimation (Dieuleveut and Bach, 2014)

– Parallelization

– Non-convex problems

References

A. Agarwal, P. L. Bartlett, P. Ravikumar, and M. J. Wainwright. Information-theoretic lower bounds

on the oracle complexity of stochastic convex optimization. Information Theory, IEEE Transactions

on, 58(5):3235–3249, 2012.

F. Bach and E. Moulines. Non-strongly-convex smooth stochastic approximation with convergence

rate o(1/n). Technical Report 00831977, HAL, 2013.

Albert Benveniste, Michel Metivier, and Pierre Priouret. Adaptive algorithms and stochastic

approximations. Springer Publishing Company, Incorporated, 2012.

L. Bottou and O. Bousquet. The tradeoffs of large scale learning. In Adv. NIPS, 2008.

A. d’Aspremont. Smooth optimization with approximate gradient. SIAM J. Optim., 19(3):1171–1183,

Aaron Defazio, Francis Bach, and Simon Lacoste-Julien. Saga: A fast incremental gradient method

with support for non-strongly convex composite objectives. In Advances in Neural Information

Processing Systems, pages 1646–1654, 2014.

A. Defossez and F. Bach. Constant step size least-mean-square: Bias-variance trade-offs and optimal

sampling distributions. 2015.

A. Dieuleveut and F. Bach. Non-parametric Stochastic Approximation with Large Step sizes. Technical

report, ArXiv, 2014.

A. Dieuleveut, N. Flammarion, and F. Bach. Harder, better, faster, stronger convergence rates for

least-squares regression. Technical Report 1602.05419, arXiv, 2016.

N. Flammarion and F. Bach. From averaging to acceleration, there is only a step-size. arXiv preprint

arXiv:1504.01577, 2015.

L. Gyorfi and H. Walk. On the averaged stochastic approximation for linear regression. SIAM Journal

on Control and Optimization, 34(1):31–61, 1996.

G. Lan. An optimal method for stochastic composite optimization. Math. Program., 133(1-2, Ser. A):

365–397, 2012.

N. Le Roux, M. Schmidt, and F. Bach. A stochastic gradient method with an exponential convergence

rate for strongly-convex optimization with finite training sets. In Adv. NIPS, 2012.

O. Macchi. Adaptive processing: The least mean squares approach with applications in transmission.

Wiley West Sussex, 1995.

A. S. Nemirovsky and D. B. Yudin. Problem complexity and method efficiency in optimization. Wiley

& Sons, 1983.

Y. Nesterov. A method for solving a convex programming problem with rate of convergence O(1/k2).

Soviet Math. Doklady, 269(3):543–547, 1983.

B. T. Polyak and A. B. Juditsky. Acceleration of stochastic approximation by averaging. SIAM Journal

on Control and Optimization, 30(4):838–855, 1992.

H. Robbins and S. Monro. A stochastic approximation method. Ann. Math. Statistics, 22:400–407,

1951. ISSN 0003-4851.

D. Ruppert. Efficient estimations from a slowly convergent Robbins-Monro process. Technical Report

781, Cornell University Operations Research and Industrial Engineering, 1988.

M. Schmidt, N. Le Roux, and F. Bach. Convergence rates for inexact proximal-gradient method. In

Adv. NIPS, 2011.

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stochastic gradient descent and robustness to ill … › cohen › sanservolotalks › bach.pdf–...

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