Stochastic variance reduction

(Stochastic) variance reduction is an algorithmic approach to minimizing functions that can be decomposed into finite sums. By exploiting the finite sum structure, variance reduction techniques are able to achieve convergence rates that are impossible to achieve with methods that treat the objective as an infinite sum, as in the classical Stochastic approximation setting.

Variance reduction approaches are widely used for training machine learning models such as logistic regression and support vector machines[1] as these problems have finite-sum structure and uniform conditioning that make them ideal candidates for variance reduction.

Finite sum objectives

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A function   is considered to have finite sum structure if it can be decomposed into a summation or average:

 

where the function value and derivative of each   can be queried independently. Although variance reduction methods can be applied for any positive   and any   structure, their favorable theoretical and practical properties arise when   is large compared to the condition number of each  , and when the   have similar (but not necessarily identical) Lipschitz smoothness and strong convexity constants.

The finite sum structure should be contrasted with the stochastic approximation setting which deals with functions of the form   which is the expected value of a function depending on a random variable  . Any finite sum problem can be optimized using a stochastic approximation algorithm by using  .

Rapid Convergence

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Stochastic variance reduced methods without acceleration are able to find a minima of   within accuracy  , i.e.   in a number of steps of the order:

 

The number of steps depends only logarithmically on the level of accuracy required, in contrast to the stochastic approximation framework, where the number of steps   required grows proportionally to the accuracy required. Stochastic variance reduction methods converge almost as fast as the gradient descent method's   rate, despite using only a stochastic gradient, at a   lower cost than gradient descent.

Accelerated methods in the stochastic variance reduction framework achieve even faster convergence rates, requiring only

 

steps to reach   accuracy, potentially   faster than non-accelerated methods. Lower complexity bounds.[2] for the finite sum class establish that this rate is the fastest possible for smooth strongly convex problems.

Approaches

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Variance reduction approaches fall within 3 main categories: table averaging methods, full-gradient snapshot methods and dual methods. Each category contains methods designed for dealing with convex, non-smooth, and non-convex problems, each differing in hyper-parameter settings and other algorithmic details.

SAGA

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In the SAGA method,[3] the prototypical table averaging approach, a table of size   is maintained that contains the last gradient witnessed for each   term, which we denote  . At each step, an index   is sampled, and a new gradient   is computed. The iterate   is updated with:

 

and afterwards table entry   is updated with  .

SAGA is among the most popular of the variance reduction methods due to its simplicity, easily adaptable theory, and excellent performance. It is the successor of the SAG method,[4] improving on its flexibility and performance.

SVRG

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The stochastic variance reduced gradient method (SVRG),[5] the prototypical snapshot method, uses a similar update except instead of using the average of a table it instead uses a full-gradient that is reevaluated at a snapshot point   at regular intervals of   iterations. The update becomes:

 

This approach requires two stochastic gradient evaluations per step, one to compute   and one to compute   where-as table averaging approaches need only one.

Despite the high computational cost, SVRG is popular as its simple convergence theory is highly adaptable to new optimization settings. It also has lower storage requirements than tabular averaging approaches, which make it applicable in many settings where tabular methods can not be used.

SDCA

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Exploiting the dual representation of the objective leads to another variance reduction approach that is particularly suited to finite-sums where each term has a structure that makes computing the convex conjugate   or its proximal operator tractable. The standard SDCA method[6] considers finite sums that have additional structure compared to generic finite sum setting:

 

where each   is 1 dimensional and each   is a data point associated with  . SDCA solves the dual problem:

 

by a stochastic coordinate ascent procedure, where at each step the objective is optimized with respect to a randomly chosen coordinate  , leaving all other coordinates the same. An approximate primal solution   can be recovered from the   values:

 .

This method obtains similar theoretical rates of convergence to other stochastic variance reduced methods, while avoiding the need to specify a step-size parameter. It is fast in practice when   is large, but significantly slower than the other approaches when   is small.

Accelerated approaches

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Accelerated variance reduction methods are built upon the standard methods above. The earliest approaches make use of proximal operators to accelerate convergence, either approximately or exactly. Direct acceleration approaches have also been developed[7]

Catalyst acceleration

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The catalyst framework[8] uses any of the standard methods above as an inner optimizer to approximately solve a proximal operator:

 

after which it uses an extrapolation step to determine the next  :

 

The catalyst method's flexibility and simplicity make it a popular baseline approach. It doesn't achieve the optimal rate of convergence among accelerated methods, it is potentially slower by up to a log factor in the hyper-parameters.

Point-SAGA

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Proximal operations may also be applied directly to the   terms to yield an accelerated method. The Point-SAGA method[9] replaces the gradient operations in SAGA with proximal operator evaluations, result in a simple, direct acceleration method:

 

with the table update   performed after each step. Here   is defined as the proximal operator for the  th term:

 

Unlike other known accelerated methods, Point-SAGA requires only a single iterate sequence   to be maintained between steps, and it has the advantage of only having a single tunable parameter  . It obtains the optimal accelerated rate of convergence for strongly convex finite-sum minimization without additional log factors.

See also

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References

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  1. ^ "sklearn.linear_model.LogisticRegression". Scikit Learn. Retrieved Feb 26, 2022.
  2. ^ Lan, Guanghui; Zhou, Yi (2018). "An optimal randomized incremental gradient method". Mathematical Programming: Series A and B. 171 (1–2): 167–215. arXiv:1507.02000. doi:10.1007/s10107-017-1173-0. S2CID 9143586.
  3. ^ Defazio, Aaron; Bach, Francis; Lacoste-Julien, Simon (2014). "SAGA: A Fast Incremental Gradient Method With Support for Non-Strongly Convex Composite Objectives". Neural Information Processing Systems. arXiv:1407.0202.
  4. ^ Schmidt, Mark; Le Roux, Nicolas; Bach, Francis (2017). "Minimizing finite sums with the stochastic average gradient". Mathematical Programming. 162. arXiv:1309.2388.
  5. ^ Johnson, Rie; Zhang, Tong (2013). "Accelerating Stochastic Gradient Descent using Predictive Variance Reduction" (PDF). Neural Information Processing Systems.
  6. ^ Shalev-Shwartz, Shai; Zhang, Tong (2013). "Stochastic Dual Coordinate Ascent Methods for Regularized Loss Minimization" (PDF). Journal of Machine Learning Research. 14.
  7. ^ Lan, Guanghui; Zhou, Yi (2018). "An optimal randomized incremental gradient method". Mathematical Programming: Series A and B. 171 (1–2): 167–215. arXiv:1507.02000. doi:10.1007/s10107-017-1173-0. S2CID 9143586.
  8. ^ Lin, Hongzhou; Mairal, Julien; Harchaoui, Zaid (2016). "Catalyst Acceleration for First-order Convex Optimization: from Theory to Practice". Journal of Machine Learning Research. 18. arXiv:1712.05654.
  9. ^ Defazio, Aaron (2016). "A Simple Practical Accelerated Method for Finite Sums". Neural Information Processing Systems. arXiv:1602.02442.