1 of 33

Optimal Supersaturated Designs for �Lasso Sign Recovery

Kade Young1 Jon Stallrich1, David Edwards2,

Byran Smucker3, Maria Weese4

​

1 Department of Statistics, North Carolina State University

2 Department of Statistical Sciences and Operations Research, Virginia Commonwealth University

3 Department of Statistics, Miami University of Ohio

4 Department of Information Systems and Analytics, Miami University of Ohio

​

2 of 33

Screening experiments and models

  •  

1

 

3 of 33

Screening experiments and models

  •  

2

 

 

 

 

Active Set

Inactive Set

Sign Vector

4 of 33

Design and Analysis Approaches

  •  

3

5 of 33

Heuristic Orthogonality Measures

  •  

4

6 of 33

Related Work

  •  

5

7 of 33

Lasso Estimator Notation

  •  

6

 

8 of 33

Lasso Sign Recovery

  •  

7

 

9 of 33

Lasso Sign Recovery Criterion

  •  

8

10 of 33

Lasso Sign Recovery Criterion

  •  

9

11 of 33

Relaxing Assumptions

  •  

10

 

12 of 33

Comparing Designs

  •  

11

13 of 33

Optimizing Relaxed Criteria

  •  

12

14 of 33

Approximate Criteria

  •  

13

 

N(0,1) PDF

N(0,1) CDF

15 of 33

Approximate Criteria

  •  

14

 

16 of 33

Approximate Criteria: Demo

  •  

15

 

 

Theorem Inequality:

Theorem Inequality:

17 of 33

Design Construction

  •  

16

18 of 33

Heuristic-Initiated Lasso Sieve (H.I.L.S)

  •  

17

19 of 33

 

  •  

18

 

20 of 33

 

  •  

19

21 of 33

 

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

20

22 of 33

 

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

21

23 of 33

Overview of Contributions

  •  

22

Thank you!

24 of 33

References

  • Bickel, P. J., Ritov, Y., and Tsybakov, A. B. (2009), “Simultaneous analysis of Lasso and Dantzig selector,” The Annals of Statistics, 37, 1705–1732.
  • Booth, K. H. V. and Cox, D. R. (1962) “ Some systematic supersaturated designs,” Technometrics, 4, 489-495.
  • Candes, Emmanuel, and Terence Tao. "The Dantzig selector: Statistical estimation when p is much larger than n." The Annals of Statistics 35.6 (2007): 2313-2351.
  • Cao, Y., Smucker, B. J., and Robinson, T. J. (2017), “A hybrid elitist pareto-based coordinate exchange algorithm for constructing multi-criteria optimal experimental designs,” Statistics and Computing, 27, 423-437.
  • Cheng, C.-S., Das, A., Singh, R., and Tsai, P.-W. (2018), “E(s2)-and UE (s2)-optimal supersaturated designs,” Journal of Statistical Planning and Inference, 196, 105–114
  • Deng, X., Lin, C. D., and Qian, P Z. G. (2013), “The Lasso with Nearly Orthogonal Latin Hypercube Designs,” preprint.
  • Dragulji ́c, D., Woods, D. C., Dean, A. M., Lewis, S. M., and Vine, A.-J. E. (2014), “Screening Strategies in the Presence of Interactions,” Technometrics, 56, 1–16
  • Hastie, T., Tibshirani, R., and Wainwright, M. 2019. “Statistical learning with sparsity, the lasso and generalizations.” Chapman and Hall/CRC

​

23

25 of 33

References

  • Huang, Y., Kong, X., and Ai, M. (2020), “Optimal designs in sparse linear models,” Metrika, 83, 255–273.
  • Jones, B. and Majumdar, D. (2014), “Optimal supersaturated designs,” Journal of the American Statistical Association, 109, 1592–1600.
  • Lin, D. K. J. (1993), “A new class of supersaturated designs,” Technometrics, 35, 28-31
  • Marley, C. J. and Woods, D. C. (2010), “A Comparison of design and model selection methods for supersaturated experiments,” Computational Statistics and Data Analysis, 54, 3158–3167
  • Phoa, Frederick KH, Yu-Hui Pan, and Hongquan Xu. "Analysis of supersaturated designs via the Dantzig selector." Journal of Statistical Planning and Inference 139.7 (2009): 2362-2372.
  • Singh, R. and Stufken, J. (2022), “Selection of two-level supersaturated designs for main effects models,” Technometrics, 1–20
  • Weese, M. L., Smucker, B. J., and Edwards, D. J. (2015), “Searching for Powerful Supersaturated Designs,” Journal of Quality Technology, 47, 66–84
  • Weese, Maria L., David J. Edwards, and Byran J. Smucker. "A criterion for constructing powerful supersaturated designs when effect directions are known." Journal of Quality Technology 49.3 (2017): 265-277.

​

​

24

26 of 33

References

  • Weese, M. L., Stallrich, J. W., Smucker, B. J., and Edwards, D. J. (2021), “Strategies for Supersaturated Screening: Group Orthogonal and Constrained Var (s) Designs,” Technometrics, 63, 443–455.
  • Xing, D. (2015), “Lasso-Optimal Supersaturated Design and Analysis For Factor Screening in Simulation Experiments,” Ph.D. thesis, Purdue University.

​

​

25

27 of 33

Lasso Sign Recovery Events

  •  

 

 

Sign recovery

Inactive Set Recovery

 

 

 

 

26

28 of 33

 

  •  

27

 

 

 

29 of 33

 

  •  

28

 

 

 

 

30 of 33

 

  •  

29

 

 

 

 

 

 

31 of 33

Comparing Designs

  •  

30

32 of 33

 

  •  

31

33 of 33

 

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

​

32