Task Assignment in a Human-Autonomous

Transcription

1 Addison Bohannon Applied Math, Statistics, & Scientific Computation Vernon Lawhern Army Research Laboratory Advisors: Brian Sadler Army Research Laboratory December 9, 2015

2 Outline

3 Human- OSD Autonomy Research Pilot Initiative, Army Research Laboratory

4 Iterative Assignment Statement 1 Assignment How to optimally assign homogeneous binary classification tasks amongst diverse agents? 2 Joint Classification How to dynamically combine multiple labels from noisy agents without supervised knowledge?

5 Generalized Assignment (GAP) Morales and Romeijn [2004]; Kundakcioglu and Alizamir [2008] 1 2 Z = max x v ji x ji (1) i I j J c ji x ji b j, j J i I x ji = 1, i I j J 3 x ji {0, 1} 4 c ji, b ji Z + 5 v ji = g(r j, s j ) 0 n number of tasks m number of agents x ji assignment of task i to agent j v ji assignment value of task i to agent j c ji assignment cost of task i to agent j b j budget for agent j r j reliability of agent j s i classification confidence of task i

6 Algorithm Fisher [2004]; Morales and Romeijn [2004]; Kundakcioglu and Alizamir [2008] Pseudo-code: Branch and Bound Data: v, c, b Result: x, Z Z = Z 0, queue = p 0 ; while queue do 1. Select p queue 2. Branch on p 3. for j = 1,..., m do Bound p j 4. if Z j > Z then if x j is feasible then x = x j, Z = Z j else add p j to queue

7 Lagrangian Relaxation Fisher [2004]; Fisher et al. [1986]; Boyd and Vandenberghe [2004] We relax the semi-assignment constraint, [2], in (1): L a (λ) = max v ji x ji + λ i 1 x i I j J i I j J 1 c ji x ji b j, j J i I 2 x ji {0, 1} 3 c ji, b ji Z + x ji (2) which yields m distinct 0-1 knapsack problems for fixed λ: ( ) L a j (λ) = max (v ji λ i ) x ji, j J, s.t. [1, 2, 3], (3) x and the dual problem provides a bounding function: Z Da = min λ i I L a (λ) = min λ j J L a j (λ) + i I λ i Z. (4)

8 Sub-gradient Method Fisher [2004]; Boyd and Vandenberghe [2004] Although Z Da is not everywhere differentiable, a sub-gradient descent method can be implemented. A subgradient of a function, f at t 0 is a vector, ν, such that f (t) f (t 0 ) + ν(t t 0 ), t. (5) g k = (1 x j1,..., 1 x jn ) is a sub-gradient of j j L a (λ k ) = max v ji x ji + λ k i 1 x i I j J i I j J x ji at λ k, and we can use the following iterative step for the sub-gradient descent algorithm: λ k+1 i = λ k i α k gi k. (6)

9 Sub-gradient Method Fisher [2004]; Boyd and Vandenberghe [2004] Algorithm: Subgradient Method Data: v j = (v j1,..., v jn ) T, c j = (c j1,..., c jn ) T, b j, λ 0 Result: x, Z k = 0; while convergence condition is not met do for j = 1,..., m do [x j, Z j ] = knapsack(v j λ k, c j, b j ); for i = 1,..., n do λ k+1 i = λ k i α k (1 j J k = k + 1, Z = j Z j + i x ji ); λ k i ;

10 0-1 Knapsack Otten The 0-1 knapsack problem, ( ) L a ( ) j (λ) = max vji λ i xji, j J x i I ( ) = max v x ji x ji 1 c ji x ji b j, i I 2 x ji {0, 1} i I 3 c ji, b ji Z + has a pseudo-polynomial time dynamic programming algorithm. (7)

11 0-1 Knapsack Otten Algorithm: 0-1 Knapsack Data: v j = (vj1,..., vjn) T, c j = (c j1,..., c jn ) T, b j Result: x j = (x j1,..., x jn ) T, Z j M = {0} n b j, S = {0} n b j, x j = {0} n ; for i = 1,..., n do for l = 1,..., b j do M(i, l) = max(m(i 1, j), M(i 1, j c j (i)) + vj (i)); if M(i 1, j c j (i)) + vj (i)) > M(i 1, j) then S(i, l) = 1; for i = n,..., 1 do if S(i, K ) then x j (i) = 1, K = K c j (i); Z j = M(n, b j ) ;

12 0-1 Knapsack Let v j = (4, 3, 2, 1), c j = (2, 1, 3, 1), and b j = 5. M = S =

13 Validation Randomized problems of sizes from Fisher et al. [1986] Compare against MATLAB integer programming application NP-hard problem (no analytical solution) Compare target function values, Z MATLAB uses (with plane cutting techniques, integer relaxation) Implementation MATLAB R2015b Personal Laptop (8GB, Intel Core i7-4510u, 2.6 GHz, Windows 10-64)

14 Validation Set-up size derived from Fisher et al. [1986] Randomized v, c, b to facilitate feasible problems

15 of Methods

16 Figure: Target function values, Z, for each method compared against target function value of MATLAB solution. Relative error reported to account for problem size. of Methods

17 Time of Methods F subgradient = 60.29, F multiplier = , F MATLAB = 49.03

18 Time of Methods

19 Time of Methods

20 Tightness of Bound

21 Schedule (with Milestones*) - AMSC 663 Develop Assignment Module (15 OCT - 4 DEC) Implement branch and bound algorithm (6 NOV)* Validate branch and bound algorithm (25 NOV)* Implement greedy search algorithm Mid-year Review (14 DEC)*

22 Schedule (with Milestones*) - AMSC 664 Build (25 JAN - 26 FEB) Build agent classes Develop message-passing framework Integrate all components into a system (26 FEB)* Test (26 FEB - 15 APR) Testing (1 APR)* Performance analysis of test results Conclusion (15 APR - 1 MAY) Final Presentation and (6 MAY)*

23 Deliverables Software (fusion module, assignment module, agent classes) Execution script Data Office Object Database Office Object RSVP Database Analysis Performance analysis of test results Implications for human-autonomous systems

24 Greedy Method Efficient implementation of the algorithm requires a feasible solution to provide a lower bound for solutions: Z feasible Z Z Da. (8) A tight lower bound requires fewer problems to be enqueued. Pseudo-code: Greedy Search Data: v, c, b Result: x, Z x = bound(v, c, b); if x is feasible then Z = v T x, return; else I 0 = {i I x ji = 1}; j J for i I 0 do x ji = 0 j J; 1. sort(v ji ) 2. assign x ji j J, i I 0 ; if x is feasible then Z = v T x, return;

25 Multiplier Adjustment Method Fisher et al. [1986] 1. Find (x 0, Z 0 ) to (3) s.t. x ji 1 j 2. if x 0 is feasible then return; 3. while Z k < Z k 1 and x k is not feasible do for i {i I x ji = 0} and j J do j Calculate, δ ji, least decrease in λ i for x ji = 1 for i {i I j x k ji λ i = λ i min 2 δ ji ; if possible then Find (x k, Z k ) to (3) s.t. j = 0 and min 2 δ ji > 0} do x k ji 1; continue;

26 Time of Methods

27 Stephen Boyd and Lieven Vandenberghe. Convex optimization. Cambridge university press, Marshall L. Fisher, R. Jaikumar, and Luk N. Van Wassenhove. A Multiplier Adjustment Method for the Generalized Assignment. Management Science, 32(9): , September Marshall L. Fisher. The Lagrangian Relaxation Method for Solving Integer Programming s. Management Science, 50(12_supplement): , December O. Erhun Kundakcioglu and Saed Alizamir. Generalized assignment problem Generalized Assignment. In Christodoulos A. Floudas and Panos M. Pardalos, editors, Encyclopedia of Optimization, pages Springer US, Dolores Romero Morales and H. Edwin Romeijn. The Generalized Assignment and Extensions. In Ding-Zhu Du and Panos M. Pardalos, editors, Handbook of Combinatorial Optimization, pages Springer US, Ralph Otten. Lecture 13: The knapsack problem.