Simultaneous Optimization Ashish Goel University of Southern California Joint work with Deborah Estrin, UCLA (Concave Costs) Adam Meyerson, Stanford/CMU (Convex Costs, Concave Utilities) http://cs.usc.edu/~agoel 4/25/2019 agoel@cs.usc.edu

Algorithms/Theory of Computation at USC Len Adleman Tim Chen Ashish Goel Ming-Deh Huang Mike Waterman Applications/Variations: Arbib, Desbrun, Schaal, Sukhatme, Tavare… 4/25/2019 agoel@cs.usc.edu

My Long Term Research Agenda Foundations of computer science (computation/interaction/information) Combinatorial optimization Approximation algorithms Discrete Applied Probability Find interesting connections Operations research; queueing theory; stochastic processes; functional analysis; game theory Find interesting application domains Communication Networks Self-Assembly 4/25/2019 agoel@cs.usc.edu

Simultaneous Optimization Approximately minimize the cost of a system simultaneously for a large family of cost functions Concave cost functions correspond to economies of scale Convex cost functions measure congestion on machines, on agents, or in networks Maximizing utility/profit without knowing the profit function Allocated bandwidths to customers in a network Concave profit functions correspond to the “law” of diminishing returns 4/25/2019 agoel@cs.usc.edu

Example of Concave Utilities: Revenue Maximization in Networks Let x = hx1,x2, …,xNi denote the bandwidth allocated to N users in a communication network These bandwidths must satisfy some linear capacity constraints, say Ax · C Let U(x) denote the total revenue (or the utility) that the network operator can derive from the system Goal: Maximize U(x), subject to Ax · C x can be thought of as “wealth” in any resource allocation problem 4/25/2019 agoel@cs.usc.edu

The Utility Function U Standard assumptions: U is concave (law of diminishing returns) U is non-decreasing (more bandwidth can’t harm) U(0) = 0 We will also assume that U is symmetric in x1,x2,…,xN The corresponding optimization problem is easy to solve using Operations Research techniques (eg. interior point methods) But what if U is not known? Simultaneous optimization: Maximize U(x) simultaneously for all Utility functions U 4/25/2019 agoel@cs.usc.edu

Why simultaneous optimization? Often, the utility function is poorly understood, eg. Customer satisfaction Often, we might want to promote several different objectives, eg. Fairness Let us focus on fairness. Should we Maximize the average utility? Be Max-min fair (steal from the rich)? Maximize the average income of the bottom half of society? Minimize the variance? Do all of the above? 4/25/2019 agoel@cs.usc.edu

Fair Allocation Problem: Example How to split a pie fairly? But what if there are more constraints? Alice and Eddy want only apple pie Frank: allergic to apples Cathy: equal portions of both pies David: twice as much apple pie as lemon What is a “Fair” allocation? How do we find one? 4/25/2019 agoel@cs.usc.edu

Simultaneous optimization and Fairness Consider the following three allocations of bandwidths to two users ha,bi hb,ai h(a+b)/2,(a+b)/2i If f measures fairness, then intuitively, f(a,b) = f(b,a) · f((a+b)/2,(a+b)/2) Hence, f is a symmetric concave function f(0) = 0, and f non-decreasing are also reasonable restrictions ie. f is a utility function All the fairness measures we found in literature are equivalent to maximizing a utility function Simultaneous optimization also promotes fairness 4/25/2019 agoel@cs.usc.edu

Can we do Simultaneous Optimization? Of course not But, perhaps we can do it “approximately”? Aha!! Theoretical Computer Science! More modest goal: Find x subject to Ax · C, such that for any utility function U, U(x) is a good approximation to the maximum achievable value of U 4/25/2019 agoel@cs.usc.edu

Approximate Majorization Given an allocation y of bandwidths to users, let Pi(y) denote the sum of the i smallest components of y Let Pi* denote the largest possible value of Pi(y) Definition: x is said to be a-majorized if Pi(x) ¸ Pi*/a for all 1· i· N Variant of the notion of majorization Interpretation: the K poorest individuals in the allocation x are collectively at least 1/a times as rich as the K poorest individuals in any other feasible allocation 4/25/2019 agoel@cs.usc.edu

Why Approximate Majorization? Theorem 1: An allocation x is a-majorized if and only if U(x) is an a-approximation to the maximum possible value of U for all utility functions U ie. a-majorization results in approximate simultaneous optimization Proof invokes a classic theorem of Hardy, Littlewood, and Polya from the 1920s 4/25/2019 agoel@cs.usc.edu

Existence Theorem 2: For the bandwidth allocation problem in networks, there exists an O(log N)-majorized solution ie. Can simultaneously approximate all utility functions up to a factor O(log N) Results extend to arbitrary linear (even convex) programs, and not just the bandwidth allocation problem 4/25/2019 agoel@cs.usc.edu

Tractability Theorem 3: Given arbitrary linear constraints, we can find (in polynomial time) the smallest a such that an a-majorized solution exists Can also find the corresponding a-majorized solution This completes the study of approximate simultaneous optimization for linear programs [Goel, Meyerson; Unpublished] [Bhargava, Goel, Meyerson; short abstract in Sigmetrics ’01] 4/25/2019 agoel@cs.usc.edu

Examples of Utility Functions Min Pi(x) Sum/Average åi f(xi) where f is a uni-variate utility function Eg. Entropy, åilog (1+xi) etc. Variance is also symmetric convex Can also approximately minimize the variance  Can simultaneously approximate capitalism, communism, and many other “ism”s. 4/25/2019 agoel@cs.usc.edu

Open Problem Distributed Algorithms?? 4/25/2019 agoel@cs.usc.edu

Example of Concave Costs: Data Aggregation in Sensor Networks There is a single sink and multiple sources of information Need to construct an aggregation tree Data flows from the sources to the sink along the tree When two data streams collide, they aggregate Let f(k) denote the size of k merged streams Assume f(0) = 0, f is concave, f is non-decreasing Concavity corresponds to concavity of entropy/information Canonical Aggregation Functions The amount of aggregation might depend on nature of information f is not known in advance 4/25/2019 agoel@cs.usc.edu

Another Example of Concave Costs: Buy-at-Bulk Network Design Single source, several sinks (consumers) of information If a link serves k sinks, its cost is f(k) Economies of scale => f is concave Also, assume that f is increasing, and f(0) = 0 Goal: Construct the cheapest distribution tree Buy-at-Bulk Network Design Assume f is not known in advance Same problem as before Multicast communication: f(k) = 1 Unicast communication: f(k) = k 4/25/2019 agoel@cs.usc.edu

A Simple Algorithmic Trick Jensen’s Inequality E[f(X)] · f(E[X]) for any concave function f Hence, given a fractional/multipath solution, randomized rounding can only help 2 Prob. ½ 1 Prob. ¼ 0.5 2 sink source 0.5 Prob. ¼ 2 4/25/2019 agoel@cs.usc.edu

Notation Given graph G=(V,E) and Cost function c : E! <+ on edges Sink t Set S of K sources Cost of supporting j users on edge e is c(e)f(j), where f is an unknown canonical aggregation function Given an aggregation tree T, CT(f) = Cost of tree T for function f C*(f) = minT{CT(f)} RT(f) = CT(f)/ C*(f) 4/25/2019 agoel@cs.usc.edu

Problem Definition Deterministic Algorithms: Problem D Construct a tree T and give a bound on maxf{RT(f)} Randomized Algorithms: Two possibilities Problem R1: Bound on maxf{ET[RT(f)]} Problem R2: Bound on ET[maxf{RT(f)}] Will focus on problem R2 (R2 subsumes R1) Problem R1 does not model “simultaneous” optimization: no one tree needs to be good for all canonical functions. Problem R1 can be tackled using known techniques A solution of Problem R2 is likely to result in a solution of problem D using de-randomization techniques 4/25/2019 agoel@cs.usc.edu

Previous Work Problem is NP-Hard even when f is known Randomized O(log K log log K) approximation for problem R1 using Bartal’s tree embeddings [Bartal ’98; Awerbuch and Azar ’97] Improved to a constant factor [Guha, Meyerson, Munagala ‘01] O(log K) approximation when f is known, but can be different for different links [Meyerson, Munagala, Plotkin ’00] 4/25/2019 agoel@cs.usc.edu

Background: Bartal’s Result Randomized algorithm which takes an arbitrary metric space (V,dV) as input and constructs a tree metric (V,dT) such that dV(u,v) · dT(u,v), and E[dT(u,v)] · a dV(u,v), where a = O(log n log log n) Results in an O(log K log log K) guarantee for problem R1 No obvious way to extend to problem R2 Quite complicated 4/25/2019 agoel@cs.usc.edu

Our Results Simple Algorithm Gives a bound of 1 + log K for problem R2 Intreresting rules of thumb Can be de-randomized using pessimistic estimators and the O(1) approximation algorithm for known f Quite technical; details omitted 4/25/2019 agoel@cs.usc.edu

Our Algorithm: Hierarchical Matching Find the minimum cost matching between sources The “Matching” Step cost is measured in terms of shortest path distance For each matched pair, pick one at random and discard it The “Random Selection” Step Pretend that the demand from the discarded node is moved to the remaining node If two or more sources remain, go back to step 1 At the end, take a union of all the matchings and also connect the single remaining source to the sink 4/25/2019 agoel@cs.usc.edu

Example Demands are all 1 Sink 4/25/2019 agoel@cs.usc.edu

Example Demands are all 1 Matching 1 4/25/2019 agoel@cs.usc.edu

Example Random Selection Step Demands are all 2 4/25/2019 agoel@cs.usc.edu

Example Demands are all 2 Matching 2 4/25/2019 agoel@cs.usc.edu

Example Random Selection Step Demands are all 4 4/25/2019 agoel@cs.usc.edu

Example Demands are all 4 Matching 3 4/25/2019 agoel@cs.usc.edu

Example Random Selection Step Demands are all 8 4/25/2019 agoel@cs.usc.edu

Example: The Final Solution 1 2 1 Mi = Total cost of edges in i-th matching CT(f) = åi Mi f(2i-1) 8 4 1 1 2 4/25/2019 agoel@cs.usc.edu

Bounding the Cost: Matching Step Ci*(f) = Cost of optimal tree for function f in the residual problem after i iterations Claim:Matching cost in step i = Mi¢f(2i-1) · Ci*(f) Sink Optimal aggregation tree for six sources (t1, t2, t3, t4,t5,t6) t4 t3 t6 t5 t1 t2 4/25/2019 agoel@cs.usc.edu

Bounding the Cost: Random Selection Consider any edge e in the optimum aggregation tree for function f Let k(e) be the number of sinks which use e Focus on the “Random selection” step for one matched pair (u,v) k’(e) = total demand routed on edge e after this step For each of u and v, the demand is doubled with probability ½ and becomes 0 otherwise => E[k’(e)] = k(e) By Jensen’s inequality: E[f(k’(e)] · f[E[k’(e)] = f(k(e)) 4/25/2019 agoel@cs.usc.edu

A Bound for Problem R1 Residual cost of opt. soln. is a super-martingale E[Ci*(f)] · C*(f) Expected Matching Cost in each matching step · åiE[Ci*(f)] · C*(f) In each matching step, the number of sources goes down by half =>1+log K matching steps =>Theorem 1: E[RT(f)] · 1+log K Marginal improvement of O(log log K) over Bartal’s embeddings for this problem Not very interesting 4/25/2019 agoel@cs.usc.edu

Bound for Problem R2 Atomic aggregation functions Ai(x) = min{x,2i} Ai is a canonical aggregation function Main Idea: Suffices to study the performance of our algorithm just for the atomic functions Details complicated; Omitted Ai(x) 2i x 4/25/2019 agoel@cs.usc.edu

Open Algorithmic Problems Multicommodity version (many source-sink pairs) Preliminary Progress: Can obtain O(log n log K log log n) guarantee using Bartal’s embeddings combined with our analysis Lower Bounds? Conjecture: W(log K) Handle arbitrary demands at sinks Our algorithm yields 1 + log K + log D guarantee for problem R2 where D is the maximum demand 4/25/2019 agoel@cs.usc.edu

Open Modeling Problems Realistic models of more general aggregation functions Information cancellation One node senses a pest infestation and sends an alarm. Another node senses high pesticide levels in the atmosphere, and sends another alarm. An intermediate node might receive both pieces of information and suppress both alarms. Amount of aggregation may depend on the set of nodes being aggregated rather than just the number Concave function f(Se) as opposed to f(ke) Bartal’s algorithm still gives an O(log K log log K) guarantee for problem R1 4/25/2019 agoel@cs.usc.edu

Moral Why settle for one cost function when you can approximate them all? Argument against approximate modeling of aggregation functions Particularly useful for poorly understood or inherently multi-criteria problems “Information independent” aggregation 4/25/2019 agoel@cs.usc.edu