Final version Visualization in Multiobjective Optimization Bogdan Filipič 1 Tea Tušar 2;1 GECCO Tutorial, Denver, July 20, 2016 Tutorial slides are available at http://dis.ijs.si/tea/research.htm 1Department of Intelligent Systems Jožef Stefan Institute Ljubljana, Slovenia 2DOPHIN Group Inria Lille – Nord Europe Villeneuve d’Ascq, France Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for third-party components of this work must be honored. For all other uses, contact the Owner/Author. Copyright is held by the owner/author(s). GECCO’16 Companion, July 20-24, 2016, Denver, CO, USA ACM 978-1-4503-4323-7/16/07. 2 http://dx.doi.org/10.1145/2908961.2926994 Contents Introduction Visualizing approximation sets Visualizing EAF values and differences Introduction Summary References 3 Introduction Introduction Multiobjective optimization problem Visualization in multiobjective optimization Minimize Useful for different purposes [13] f: X ! F • Analysis of solutions and solution sets f:(x ;:::; x ) 7! (f (x ;:::; x );:::; f (x ;:::; x )) 1 n 1 1 n m 1 n • Decision support in interactive optimization • Analysis of algorithm performance • X is an n-dimensional decision space ⊆ Rm ≥ • F is an m-dimensional objective space (m 2) Visualizing solution sets in the decision space • Problem-specific ! Conflicting objectives a set of optimal solutions • If X ⊆ Rm, any method for visualizing multidimensional • Pareto set in the decision space solutions can be used • Pareto front in the objective space • Not the focus of this tutorial 4 5 Introduction Introduction Visualization can be hard even in 2-D Stochastic optimization algorithms Visualizing solution sets in the objective space • Single run ! single approximation set • Interested in sets of mutually nondominated solutions called approximation sets • Multiple runs ! multiple approximation sets • Different from ordinary multidimensional solution sets Single run Three runs Ten runs • The focus of this tutorial 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 Challenges 0.6 0.6 0.6 0.4 0.4 0.4 • High dimension and large number of solutions 0.2 0.2 0.2 0 0 0 • Limitations of computing and displaying technologies 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 0 0.2 0.4 0.6 0.8 1 1.2 Run 1 Run 3 Run 5 Run 7 Run 9 • Cognitive limitations Run 2 Run 4 Run 6 Run 8 Run 10 Visualization of the Empirical Attainment Function (EAF) can be used 6 in such cases 7 Introduction This tutorial is not about • Visualization for decision making purposes [26] • Visualization in the decision space • General multidimensional visualization methods not previously used on approximation sets Visualizing approximation sets This tutorial covers • Visualization in the objective space • Visualization of separate approximation sets [1] • Visualization of EAF values and differences in EAF values [2] 8 Methodology Benchmark approximation sets Two different sets that can be instantiated in any dimension [1] Comparing visualization methods • Linear with a uniform distribution of solutions • No existing methodology for comparing visualization methods • Spherical with a nonuniform distribution of solutions (more at • Propose benchmark approximation sets (analog to benchmark the corners and less at the center) problems in multiobjective optimization) • Sets are intertwined • Visualize the sets using different methods Size of each set • Observe which set properties are distinguishable after visualization • 2-D: 50 solutions • 3-D: 500 solutions • 4-D: only 300 solutions since most methods cannot handle more 9 10 Benchmark approximation sets Visualizing approximation sets Desired properties of visualization methods 1 Linear Linear Spherical • Preservation of the Spherical 1 0.8 • Dominance relation 0.8 • Front shape 0.6 1 0.6 f3 0.8 • Objective range f 2 0.4 2 0.6 f 0.4 • Distribution of solutions 0.4 0.2 0.2 0 • Robustness 0.2 0.2 • Handling of large sets 0.4 0.6 • Simultaneous visualization of multiple sets 0 f 0.8 0 0.2 0.4 0.6 0.8 1 1 1 • Scalability in number of objectives f1 • Simplicity 11 12 Visualizing approximation sets General methods • Scatter plot matrix • Bubble chart Existing methods • Radial coordinate visualization [16, 36] Showing only methods previously used in multiobjective • Parallel coordinates [17] optimization • Heatmaps [29] • General methods • Sammon mapping [30, 33] • Specific methods – designed for visualizing approximation sets • Neuroscale [24, 10] • Self-organizing maps [18, 27] Demonstration on 4-D benchmark approximation sets • Principal component analysis [39] • Isomap [31, 21] 13 14 Scatter plot matrix Scatter plot matrix 1 1 1 2 3 4 Most often f 0.5 f 0.5 f 0.5 • Scatter plot in a 2-D space 0 0 0 0 0.5 1 0 0.5 1 0 0.5 1 • Matrix of all possible combinations f1 f1 f1 1 1 ! m(m−1) • m objectives 2 different combinations 3 4 f 0.5 f 0.5 Alternatively 0 0 0 0.5 1 0 0.5 1 f f • Scatter plot in a 3-D space 2 2 1 ! m(m−1)(m−2) • m objectives 6 different combinations 4 Linear f 0.5 Spherical 0 0 0.5 1 f3 15 16 Scatter plot matrix Bubble chart Linear 1 1 Spherical f4 0.5 f4 0.5 0 0 0.5 0.5 4-D objective space f 1 f 1 2 0.5 1 0.5 1 0 1 0 f3 f3 • Similar to a 3-D scatter plot 1 1 • Fourth objective visualized with point size f4 0.5 f3 0.5 0 0 5-D objective space 0.5 0.5 f 1 f 1 1 0.5 1 0.5 1 0 1 0 • Fifth objective visualized with colors f2 f2 Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 ≈ 3 ≈ 3 ≈ 3 5 3 17 18 Bubble chart Radial coordinate visualization 1 0.8 Linear Also called RadViz Spherical 0.6 • Inspired from physics f3 f1 0.4 • Objectives treated as anchors, 0.2 equally spaced around the 0 circumference of a unit circle 0.2 • Solutions attached to anchors with f2 f4 0.4 ‘springs’ f1 0.6 1 0.8 0.8 • Spring stiffness proportional to the 0.6 0.4 1 0.2 objective value f2 f3 • Solution placed where the spring Preservation of the forces are in equilibrium Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 ≈ 3 ≈ 3 ≈ 3 5 3 19 20 Radial coordinate visualization Parallel coordinates f 1 Linear Spherical • m objectives ! m parallel axes • Solution represented as a polyline with vertices on the axes • Position of each vertex corresponds to that objective value • No loss of information f3 f2 1.0 0.8 0.6 0.4 0.2 f4 0.0 f1 f2 f3 f4 Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 5 5 ≈ 3 ≈ 3 3 3 21 22 Parallel coordinates Heatmaps Linear Spherical 1 1 0.8 0.8 0.6 0.6 • m objectives ! m columns 0.4 0.4 • One solution per row 0.2 0.2 • Each cell colored according to objective value 0 0 • No loss of information f1 f2 f3 f4 f1 f2 f3 f4 Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization ≈ 5 3 ≈ 3 5 5 3 3 23 24 Heatmaps Sammon mapping Linear Spherical 1.0 0.8 0.9 0.7 0.8 • A non-linear mapping 0.6 0.7 • Aims to preserve distances between solutions 0.6 0.5 d∗ x x 0.5 0.4 • ij distance between solutions i and j in the objective space • d distance between solutions x and x in the visualized space 0.4 0.3 ij i j 0.3 0.2 • Stress function to be minimized 0.2 0.1 0.1 X X S = (d∗ − d )2 0.0 0.0 ij ij f1 f2 f3 f4 f1 f2 f3 f4 i j>i • Minimization by gradient descent or other (iterative) methods Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 5 3 5 3 5 5 3 3 25 26 Sammon mapping Sammon mapping 0.8 Linear Spherical 0.4 0.8 0.4 0 Third 0 coordinate -0.4 0.8 Second coordinate 0.4 -0.8 -0.8 0 -0.4 -0.4 Second 0 -0.4 coordinate First 0.4 -0.8 coordinate 0.8 Linear Spherical -0.8 -0.8 -0.4 0 0.4 0.8 Preservation of the Handling of Simultaneous First coordinate dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 5 5 3 ≈ ≈ 3 3 5 27 28 Neuroscale Neuroscale 1.2 0.8 0.4 • A non-linear mapping • Aims to minimize the same stress function as Sammon mapping 0 • Uses a radial basis function neural network to model the -0.4 projection Second coordinate -0.8 Linear Spherical -1.2 -1.2 -0.8 -0.4 0 0.4 0.8 1.2 First coordinate 29 30 Neuroscale Self-organizing maps Linear Spherical 0.8 • Self-organizing maps (SOMs) are neural networks 0.4 • Nearby solutions are mapped to nearby neurons in the SOM Third 0 coordinate • A SOM can be visualized using the unified distance matrix -0.4 0.8 • Distance between adjacent neurons is denoted with color 0.4 -0.8 -0.8 0 • Similar neurons ! light color -0.4 Second 0 -0.4 0.4 coordinate • Different neurons (cluster boundaries) ! dark color First 0.8 -0.8 coordinate 1.2 Preservation of the Handling of Simultaneous dominance front shape objective distribution Robustness Scalability Simplicity relation range of solutions large sets visualization 5 5 5 5 ≈ ≈ 3 3 5 31 32 Self-organizing maps Principal component analysis Linear Spherical • Principal components are linear combinations of objectives that