(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 as these problems have finite-sum structure and uniform conditioning that make them ideal candidates for variance reduction.
Finite sum objectives A function f {\displaystyle f} is considered to have finite sum structure if it can be decomposed into a summation or average:
f ( x ) = 1 n ∑ i = 1 n f i ( x ) , {\displaystyle f(x)={\frac {1}{n}}\sum _{i=1}^{n}f_{i}(x),}
where the function value and derivative of each f i {\displaystyle f_{i}} can be queried independently. Although variance reduction methods can be applied for any positive n {\displaystyle n} and any f i {\displaystyle f_{i}} structure, their favorable theoretical and practical properties arise when n {\displaystyle n} is large compared to the condition number of each f i {\displaystyle f_{i}} , and when the f i {\displaystyle f_{i}} 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 f ( θ ) = E ξ [ F ( θ , ξ ) ] {\textstyle f(\theta )=\operatorname {E} _{\xi }[F(\theta ,\xi )]}
which is the expected value of a function depending on a random variable ξ {\textstyle \xi } . Any finite sum problem can be optimized using a stochastic approximation algorithm by using F ( ⋅ , ξ ) = f ξ {\displaystyle F(\cdot ,\xi )=f_{\xi }} .
Rapid Convergence Stochastic variance reduced methods without acceleration are able to find a minima of f {\displaystyle f} within accuracy ϵ > {\displaystyle \epsilon >} , i.e. f ( x ) − f ( x ∗ ) ≤ ϵ {\displaystyle f(x)-f(x_{*})\leq \epsilon } in a number of steps of the order:
O ( ( L μ + n ) log ( 1 ϵ ) ) . {\displaystyle O\left(\left({\frac {L}{\mu }}+n\right)\log \left({\frac {1}{\epsilon }}\right)\right).}
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 O ( L / ( μ ϵ ) ) {\displaystyle O{\bigl (}L/(\mu \epsilon ){\bigr )}} required grows proportionally to the accuracy required. Stochastic variance reduction methods converge almost as fast as the gradient descent method's O ( ( L / μ ) log ( 1 / ϵ ) ) {\displaystyle O{\bigl (}(L/\mu )\log(1/\epsilon ){\bigr )}} rate, despite using only a stochastic gradient, at a 1 / n {\displaystyle 1/n} lower cost than gradient descent. Accelerated methods in the stochastic variance reduction framework achieve even faster convergence rates, requiring only
O ( ( n L μ + n ) log ( 1 ϵ ) ) {\displaystyle O\left(\left({\sqrt {\frac {nL}{\mu }}}+n\right)\log \left({\frac {1}{\epsilon }}\right)\right)}
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