Partial Function Extension with Applications Umang Bhaskar Gunjan

Partial Function Extension with Applications Umang Bhaskar Gunjan Kumar Tata Institute of Fundamental Research

2 Partial Function Extension Given points values from domain D at the given points and family F of functions, (i. e. , extends the given partial function)

3 Example: Convex Functions Family F : convex functions Partial function: 2 Not Extendible 1 0 2 4

4 Example: Convex Functions Family F : convex functions Partial function: 2 1 0 2 4

5 Example: Convex Functions Family F : convex functions Partial function: 2 Extendible 1 0 2 4

6 Example: Monotone Functions Family F : monotone functions Partial function: Not Extendible

7 Example: Monotone Functions Family F : monotone functions Partial function:

8 Example: Monotone Functions Family F : monotone functions Partial function: Extendible

9 Partial Function Extension Given partial function and family F of functions,

1 0 Approximate Extension Given partial function and family F of functions,

1 1 Application I: Lower Bounds for PAC Learning

1 2 Application I: Lower Bounds for PAC Learning

1 3 Application I: Lower Bounds for PAC Learning Algorithm

1 4 Application I: Lower Bounds for PAC Learning

1 5 Application II: Property Testing

1 6 Application II: Property Testing Algorithm

1 7 Application II: Property Testing Algorithm

1 8 Relevant Families F • Subadditive functions (complement-free)

1 9 Relevant Families F (complement-free) • Subadditive functions 3 3 2 2 1 1 0 1 2 3 4

2 0 Relevant Families F • Subadditive functions (complement-free) • Submodular functions or (diminishing marginal returns)

2 1 Relevant Families F • Subadditive functions (complement-free) • Submodular functions or (diminishing marginal returns) 3 3 2 2 1 1 0 1 2 3 4

2 2 Relevant Families F • Subadditive functions (complement-free) • Submodular functions or (diminishing marginal returns) • Convex functions

2 3 Previous Work • Subadditive functions: [Balcan et al. , 2012] • Submodular functions: [Balcan, Harvey, 2011] [Seshadhri, Vondrak, 2014] • Convex functions: partial function extension studied in convex analysis

2 4 Our Results Partial Fn Extn: Learning: Testing: co. NP-complete

2 5 Our Results Partial Fn Extn: Learning: cannot be learned within any factor

2 6 Our Results Partial Fn Extn: Learning: cannot be learned within any factor 3 2 1 Partial Fn Extn: 0 1 2

2 7 Our Results Partial Fn Extn: Learning: cannot be learned within any factor ? 3 2 1 Partial Fn Extn: 0 1 2

2 8 This Talk Subadditive functions (extension, learning, testing) Convex functions (extension)

2 9 This Talk Subadditive functions (extension, learning, testing) Convex functions (extension)

3 0 Subadditive Functions (complement-free) • subadditive functions 3 3 2 2 1 1 0 1 2 3 4

3 1 Subadditive Functions: Extension Theorem: Subadditive extension is co. NP-hard Lemma:

3 2 Subadditive Functions: Extension Theorem: Subadditive extension is co. NP-hard Lemma: (lower bound in theorem from set cover, proof skipped) not extendible

3 3 Subadditive Functions: Learning Theorem:

3 4 Subadditive Functions: Learning Lemma: Proof:

3 5 Subadditive Functions: Learning Lemma: Theorem: [Dyachkov et al. , 2014]

3 6 Subadditive Functions: Testing Theorem: Recall: Lemma:

3 7 Subadditive Functions: Testing Theorem: Tester:

3 8 Subadditive Functions: Testing Theorem: Tester: Accept

3 9 Subadditive Functions: Testing Theorem: Tester: Accept

4 0 This Talk Subadditive functions (extension, learning, testing) Convex functions (extension)

4 1 Our Results Given partial function ? 3 2 1 0 1 2

4 2 3 2 1 0 1 2

4 3 3 2 1 0 1 2

4 4 3 2 1 0 Then: 1 2

4 5

4 6

4 7

4 8 Further Results This talk: • Subadditive functions • Submodular functions • Convex functions Further work on: • XOS, matroid rank, coverage functions • Further work on submodular functions

4 9 Open Questions • Hardness of submodular extension? • Is submodular extension in NP / co. NP? • Better testers for subadditive functions? • Further applications for partial function extension? Thank You!