Subgradient method

Let ${\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} }$ be a convex function with domain ${\displaystyle \mathbb {R} ^{n}.}$ A classical subgradient method iterates $${\displaystyle x^{(k+1)}=x^{(k)}-\alpha _{k}g^{(k)}\ }$$ where ${\displaystyle g^{(k)}}$ denotes any subgradient of ${\displaystyle f\ }$ at ${\displaystyle x^{(k)},\ }$ and ${\displaystyle x^{(k)}}$ is the ${\displaystyle k^{th}}$ iterate of ${\displaystyle x.}$ If ${\displaystyle f\ }$ is differentiable, then its only subgradient is the gradient vector ${\displaystyle \nabla f}$ itself. It may happen that ${\displaystyle -g^{(k)}}$ is not a descent direction for ${\displaystyle f\ }$ at ${\displaystyle x^{(k)}.}$ We therefore maintain a list ${\displaystyle f_{\rm {best}}\ }$ that keeps track of the lowest objective function value found so far, i.e. $${\displaystyle f_{\rm {best}}^{(k)}=\min\{f_{\rm {best}}^{(k-1)},f(x^{(k)})\}.}$$

Many different types of step-size rules are used by subgradient methods. This article notes five classical step-size rules for which convergence proofs are known:

• Constant step size, ${\displaystyle \alpha _{k}=\alpha .}$
• Constant step length, ${\displaystyle \alpha _{k}=\gamma /\lVert g^{(k)}\rVert _{2},}$ which gives ${\displaystyle \lVert x^{(k+1)}-x^{(k)}\rVert _{2}=\gamma .}$
• Square summable but not summable step size, i.e. any step sizes satisfying $${\displaystyle \alpha _{k}\geq 0,\qquad \sum _{k=1}^{\infty }\alpha _{k}^{2}<\infty ,\qquad \sum _{k=1}^{\infty }\alpha _{k}=\infty .}$$
• Nonsummable diminishing, i.e. any step sizes satisfying $${\displaystyle \alpha _{k}\geq 0,\qquad \lim _{k\to \infty }\alpha _{k}=0,\qquad \sum _{k=1}^{\infty }\alpha _{k}=\infty .}$$
• Nonsummable diminishing step lengths, i.e. ${\displaystyle \alpha _{k}=\gamma _{k}/\lVert g^{(k)}\rVert _{2},}$ where $${\displaystyle \gamma _{k}\geq 0,\qquad \lim _{k\to \infty }\gamma _{k}=0,\qquad \sum _{k=1}^{\infty }\gamma _{k}=\infty .}$$

For all five rules, the step-sizes are determined "off-line", before the method is iterated; the step-sizes do not depend on preceding iterations. This "off-line" property of subgradient methods differs from the "on-line" step-size rules used for descent methods for differentiable functions: Many methods for minimizing differentiable functions satisfy Wolfe's sufficient conditions for convergence, where step-sizes typically depend on the current point and the current search-direction. An extensive discussion of stepsize rules for subgradient methods, including incremental versions, is given in the books by Bertsekas^[1] and by Bertsekas, Nedic, and Ozdaglar.^[2]

For constant step-length and scaled subgradients having Euclidean norm equal to one, the subgradient method converges to an arbitrarily close approximation to the minimum value, that is

${\displaystyle \lim _{k\to \infty }f_{\rm {best}}^{(k)}-f^{*}<\epsilon }$ by a result of Shor.^[3]

These classical subgradient methods have poor performance and are no longer recommended for general use.^[4][5] However, they are still used widely in specialized applications because they are simple and they can be easily adapted to take advantage of the special structure of the problem at hand.

During the 1970s, Claude Lemaréchal and Phil Wolfe proposed "bundle methods" of descent for problems of convex minimization.^[6] The meaning of the term "bundle methods" has changed significantly since that time. Modern versions and full convergence analysis were provided by Kiwiel. ^[7] Contemporary bundle-methods often use "level control" rules for choosing step-sizes, developing techniques from the "subgradient-projection" method of Boris T. Polyak (1969). However, there are problems on which bundle methods offer little advantage over subgradient-projection methods.^[4][5]

One extension of the subgradient method is the projected subgradient method, which solves the constrained optimization problem

minimize ${\displaystyle f(x)\ }$ subject to $${\displaystyle x\in {\mathcal {C}}}$$

where ${\displaystyle {\mathcal {C}}}$ is a convex set. The projected subgradient method uses the iteration $${\displaystyle x^{(k+1)}=P\left(x^{(k)}-\alpha _{k}g^{(k)}\right)}$$ where ${\displaystyle P}$ is projection on ${\displaystyle {\mathcal {C}}}$ and ${\displaystyle g^{(k)}}$ is any subgradient of ${\displaystyle f\ }$ at ${\displaystyle x^{(k)}.}$

The subgradient method can be extended to solve the inequality constrained problem

minimize ${\displaystyle f_{0}(x)\ }$ subject to $${\displaystyle f_{i}(x)\leq 0,\quad i=1,\ldots ,m}$$

where ${\displaystyle f_{i}}$ are convex. The algorithm takes the same form as the unconstrained case $${\displaystyle x^{(k+1)}=x^{(k)}-\alpha _{k}g^{(k)}\ }$$ where ${\displaystyle \alpha _{k}>0}$ is a step size, and ${\displaystyle g^{(k)}}$ is a subgradient of the objective or one of the constraint functions at ${\displaystyle x.\ }$ Take $${\displaystyle g^{(k)}={\begin{cases}\partial f_{0}(x)&{\text{ if }}f_{i}(x)\leq 0\;\forall i=1\dots m\\\partial f_{j}(x)&{\text{ for some }}j{\text{ such that }}f_{j}(x)>0\end{cases}}}$$ where ${\displaystyle \partial f}$ denotes the subdifferential of ${\displaystyle f.\ }$ If the current point is feasible, the algorithm uses an objective subgradient; if the current point is infeasible, the algorithm chooses a subgradient of any violated constraint.

• Stochastic gradient descent – Optimization algorithm