Incremental Subgradient Methods for Convex Nondifferentiable Optimization

1
Incremental Subgradient Methods for Convex
Nondifferentiable Optimization
5th Ballarat Workshop on Global and Nonsmooth
Optimization 28 Nov. 2006
Angelia Nedich

• Dept. of Industrial and Enterprise Systems
  Engineering
• University of Illinois at Urbana-Champaign

2
Outline

• Large Convex Optimization
• Subgradient Methods
• Incremental Subgradient Method
• Convergence Results
• Randomized Subgradient Method
• Partially Asynchronous Incremental Method
• Convergence Rate Comparison

3
Convex Problem with Large Number of Component
Functions

• The functions Fi are convex and possibly
  non-differentiable
• The constraint set C is nonempty, closed, convex,
  and simple for projection
• F denotes the optimal value
• C denotes the set of optimal solutions

4
Agent-Task Scheduling Problem

• The number of tasks can be very large
• Relaxing the agent time constraints yields

• Issue
• The dual function q is non-differentiable
• The gradient-based methods do not work

5
Subgradient
6
Basic Subgradient Property

• Property ensuring that subgradient methods work
• Subgradient descent property
• At each iteration, for a range of stepsize
  values
• - Either function value decreases
• - Or the iterate distance from the optimal set
  decreases

7
Classical Subgradient Method

• Requires computing the subgradients of all
  components Fi at the current iterate
• Not suitable for some real-time operations (m
  large)
• Large volume of incoming data to the processing
  node
• Issues with the link capacities and congestion
• Large delays between updates

8
Incremental Subgradient Method

• Admits decentralized computations
• Suitable for real-time computations

• Intermediate updates along subgradients of
  functions Fi
• y0k xk
• Cycle through Fi
• yik ?C yi-1,k ak gik (yi-1,k)
• xk1 ymk
• gik(yi-1,k) is a subgradient of Fi at yi-1,k

9
Stepsize Rules

• Constant stepsize
• Diminishing stepsize
• Adaptive stepsize (F known )
• Adaptive stepsize (F estimated )

10
Convergence Results

• Proposition Let xk be the sequence generated
  by the incremental method.
• For a constant stepsize

B is the norm bound on subgradients of Fi for all
i
Motivated by a result in Rubinov and Demyanovs
book on Approximate Methods for Optimization
Problems

• For a diminishing stepsize

• For an adaptive stepsize (F known), when C is
  nonempty

11
Proof Idea
Let xk be the sequence generated by the
incremental method. Then for any feasible vector
y and any k, we have
Thus for any xk and a feasible vector y with
lower cost than xk, the next iterate xk1 is
closer to y than xk, provided the stepsize ak is
sufficiently small
12
Randomized Subgradient Methods
Pick randomly an index w from 1,,m
13
Convergence Results

• Proposition Let xk be the sequence generated
  by the randomized method.
• For a constant stepsize

• When C is nonempty
• For a diminishing stepsize with

• For an adaptive stepsize (F known)

Here
B is the norm bound on subgradients of Fi for all
i
14
Convergence Rate Comparison

• Convergence rate results for constant stepsize
• Assumption The optimal solution set C is
  nonempty
• For any given error level egt0, we have

B is a bound on the subgradient norms
15
Partially Asynchronous

• Estimates xk communicated with delays
• Processing nodes sends the estimates in slotted
  times tk

D is an upper bound on the communication delay
Convergence for a stepsize of the form
16
Conclusions

• A new subgradient method for large scale convex
  optimization
• Convergence and rate of convergence analysis
• Extensions to partially asynchronous settings
• Ongoing work on fully asynchronous implementations

17
References

• Nedich and Bertsekas Incremental subgradient
  methods for nondifferentiable optimization SIAM
  J. Optimization
• Nedich and Bertsekas Convergence rate of
  incremental subgradient algorithms Stochastic
  Optimization Algorithms and Applications
• Nedich, Bertsekas, and Borkar Distributed
  asynchronous incremental subgradient methods
  Inherently parallel algorithms in feasibility and
  optimization and their applications