Stochastic variance reduction

Rapid Convergence

Stochastic variance reduced methods without acceleration are able to find a minima of ${\displaystyle f}$ within accuracy ${\displaystyle ϵ>}$, i.e. ${\displaystyle f(x)-f(x_{\ast })\le ϵ}$ in a number of steps of the order:

${\displaystyle O\left(\left(\frac{L}{\mu }+n\right)\log \left(\frac{1}{ϵ}\right)\right).}$

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 ${\displaystyle O(L/(\mu ϵ))}$ required grows proportionally to the accuracy required. Stochastic variance reduction methods converge almost as fast as the gradient descent method's ${\displaystyle O((L/\mu )\log (1/ϵ))}$ rate, despite using only a stochastic gradient, at a ${\displaystyle 1/n}$ lower cost than gradient descent.

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

${\displaystyle O\left(\left(\sqrt{\frac{nL}{\mu }}+n\right)\log \left(\frac{1}{ϵ}\right)\right)}$

steps to reach ${\displaystyle ϵ}$ accuracy, potentially ${\displaystyle \sqrt{n}}$ faster than non-accelerated methods. Lower complexity bounds. for the finite sum class establish that this rate is the fastest possible for smooth strongly convex problems.

Approaches

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

In the SAGA method, the prototypical table averaging approach, a table of size ${\displaystyle n}$ is maintained that contains the last gradient witnessed for each ${\displaystyle f_i}$ term, which we denote ${\displaystyle g_i}$. At each step, an index ${\displaystyle i}$ is sampled, and a new gradient ${\displaystyle \mathrm{\nabla }{f}_i(x_k)}$ is computed. The iterate ${\displaystyle x_k}$ is updated with:

${\displaystyle x_{k+1}=x_k-\gamma \left[\mathrm{\nabla }{f}_i(x_k)-g_i+\frac{1}{n}\sum _{i=1}^n{g}_i\right],}$

and afterwards table entry ${\displaystyle i}$ is updated with ${\displaystyle g_i=\mathrm{\nabla }{f}_i(x_k)}$.

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, improving on its flexibility and performance.

SVRG

The stochastic variance reduced gradient method (SVRG), 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 ${\displaystyle \tilde{x}}$ at regular intervals of ${\displaystyle m\ge n}$ iterations. The update becomes:

${\displaystyle x_{k+1}=x_k-\gamma [\mathrm{\nabla }{f}_i(x_k)-\mathrm{\nabla }{f}_i(\tilde{x})+\mathrm{\nabla }f(\tilde{x})],}$

This approach requires two stochastic gradient evaluations per step, one to compute ${\displaystyle \mathrm{\nabla }{f}_i(x_k)}$ and one to compute ${\displaystyle \mathrm{\nabla }{f}_i(\tilde{x}),}$ 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

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 ${\displaystyle f_i^{\ast },}$ or its proximal operator tractable. The standard SDCA method considers finite sums that have additional structure compared to generic finite sum setting:

${\displaystyle f(x)=\frac{1}{n}\sum _{i=1}^n{f}_i(x^T{v}_i)+\frac{\lambda }{2}\Vert x{\Vert }^2,}$

where each ${\displaystyle f_i}$ is 1 dimensional and each ${\displaystyle v_i}$ is a data point associated with ${\displaystyle f_i}$. SDCA solves the dual problem:

${\displaystyle \underset{\alpha \in {\mathbb{R}}^n}{max}-\frac{1}{n}\sum _{i=1}^n{f}_i^{\ast }(-{\alpha }_i)-\frac{\lambda }{2}{\Vert \frac{1}{\lambda n}\sum _{i=1}^n{\alpha }_i{v}_i\Vert }^2,}$

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

${\displaystyle x=\frac{1}{\lambda n}\sum _{i=1}^n{\alpha }_i{v}_i}$.

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 ${\displaystyle \lambda }$ is large, but significantly slower than the other approaches when ${\displaystyle \lambda }$ is small.

Accelerated approaches

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

Catalyst acceleration

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

${\displaystyle x_k\approx {\text{argmin}}_x\left\{f(x)+\frac{\kappa }{2}\Vert x-y_{k-1}{\Vert }^2\right\}}$

after which it uses an extrapolation step to determine the next ${\displaystyle y}$:

${\displaystyle y_k=x_k+{\beta }_k(x_k-x_{k-1})}$

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

Proximal operations may also be applied directly to the ${\displaystyle f_i}$ terms to yield an accelerated method. The Point-SAGA method replaces the gradient operations in SAGA with proximal operator evaluations, result in a simple, direct acceleration method:

${\displaystyle x_{k+1}={\text{prox}}_j^{\gamma }\left(z_k\triangleq x_k+\gamma \left[g_j-\frac{1}{n}\sum _{i=1}^n{g}_i\right]\right),}$

with the table update ${\displaystyle g_j=\frac{1}{\gamma }(z_k-x_{k+1})}$ performed after each step. Here ${\displaystyle {\text{prox}}_j^{\gamma }}$ is defined as the proximal operator for the ${\displaystyle j}$th term:

${\displaystyle {\text{prox}}_j^{\gamma }(y)={\text{argmin}}_x\left\{f_j(x)+\frac{1}{2\gamma }\Vert x-y{\Vert }^2\right\}.}$

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