Sample Complexity of Algorithm Configuration for Sequence Alignment - - PowerPoint PPT Presentation

Sample Complexity of Algorithm Configuration for Sequence Alignment

Travis Dick

Nina Balcan Dan DeBlasio Ellen Vitercik Carl Kingsford Tuomas Sandholm

Sequence alignment

Goal: Line up pairs of strings (DNA, RNA, protein, …) Uncover functional, structural, or evolutionary relationships

GRTCP---KPDDLPFSTVVPLKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP EVKCPFPSRPDN-GFVNYPAKPTLYYK-DKATFGCHDGY-SLDGPEEIECTKLGNWS-AMPSCKA 𝑻𝟐 = GRTCPKPDDLPFSTVVPLKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP 𝑻𝟑 = EVKCPFPSRPDNGFVNYPAKPTLYYKDKATFGCHDGYSLDGPEEIECTKLGNWSAMPSCKA

Sequence alignment algorithms

Typically optimize for alignment features: Number of matching characters, number of gaps, …

[Needleman and Wunsch ‘70; Gotoh ’82]

Standard algos solve for alignment maximizing weighted sum How to tune the feature weights?

Sequence alignment algorithms

Can sometimes access ground-truth alignment

Requires extensive manual alignments

Given set of application’s “typical” alignment problems, together with ground-truth alignments, can we learn parameters that recover ground truth?

Model

• 1. Fix a parameterized alignment optimization function
• 2. Receive sample problems from unknown distribution
• 3. Find parameter values with best performance over samples

Closest to ground truth, for example

Sequence 𝑇& Sequence 𝑇&

'

Alignment

⋯

Sequence 𝑇) Sequence 𝑇)

'

Alignment

Model

• 1. Fix a parameterized alignment optimization function
• 2. Receive sample problems from unknown distribution
• 3. Find parameter values with best performance over samples

Model studied from empirical perspective

Kim and Kececioglu ’07; Xu, Hutter, Hoos, Leyton-Brown ’08; Dai, Khalil, Zhang, Dilkina, Song ’17 …

Sequence 𝑇& Sequence 𝑇&

'

Alignment

⋯

Sequence 𝑇) Sequence 𝑇)

'

Alignment

Model

• 1. Fix a parameterized alignment optimization function
• 2. Receive sample problems from unknown distribution
• 3. Find parameter values with best performance over samples

Model studied from theoretical perspective

Gupta and Roughgarden ’16; Kleinberg, Leyton-Brown, Lucier ‘17; Weisz, György, Szepesvári ‘18 …

Sequence 𝑇& Sequence 𝑇&

'

Alignment

⋯

Sequence 𝑇) Sequence 𝑇)

'

Alignment

Questions

Focus of this talk: Will those parameters have high performance in expectation? Focus of prior work [e.g., Kim and Kececioglu ’07]: Algorithmically, how to find good parameters over training set Sequence 𝑇& Sequence 𝑇&

'

Alignment Sequence 𝑇 Sequence 𝑇′? Sequence 𝑇) Sequence 𝑇)

'

Alignment

⋯

Model

𝒠: Distribution over sequence pairs (𝑇, 𝑇') ℝ0: Set of parameters For any sequence pair (𝑇, 𝑇'): 𝑣𝝇 𝑇, 𝑇' = utility of using params 𝝇 ∈ ℝ0 to align 𝑇, 𝑇' Similarity between algorithm’s output & ground truth Generalization: Given samples 𝑇&, 𝑇&

' , … , 𝑇), 𝑇) '

~𝒠, for any 𝝇 ∈ ℝ0,

& ) ∑89& ) 𝑣𝝇 𝑇8, 𝑇8 ' − 𝔽(<,<=)~𝒠[𝑣𝝇 𝑇, 𝑇′ ] ≤?

Primary challenge: Algorithmic performance is volatile function of parameters For well-understood functions in machine learning: Close connection between function parameters and value

Similarity to ground truth 𝜍& 𝜍B

Outline

• 1. Pairwise sequence alignment algorithms
• 2. Sample complexity for pairwise alignment
• 3. Multiple-sequence alignment algorithms
• 4. Sample complexity for multiple-sequence alignments
• 5. Additional applications

Pairwise sequence alignment

Input: Two sequences 𝑇, 𝑇′ ∈ ΣD Alignment: Sequences 𝜐, 𝜐′ ∈ Σ ∪ −

∗ such that:

Deleting “−” yields 𝑇 from 𝜐 and 𝑇′ from 𝜐′ 𝑇 = A C T G 𝑇′ = G T C A 𝜐 = A – - C T G 𝜐′ = - G T C A -

Insertion/deletion (indel) Match Mismatch Gap

Pairwise sequence alignment algorithms

Standard algorithm with parameters 𝜍&, 𝜍B, 𝜍H ≥ 0: Use dynamic programming to find alignment 𝐵 maximizing: (# matches) − 𝜍& L (# mismatches) − 𝜍B L (# indels) − 𝜍H L (# gaps) 𝑇 = A C T G 𝑇′ = G T C A 𝜐 = A – - C T G 𝜐′ = - G T C A -

Insertion/deletion (indel) Match Mismatch Gap

Pairwise sequence alignment algorithms

More generally, given parameters 𝝇 ∈ ℝ0: Use dynamic programming to find alignment 𝐵 maximizing: 𝜍& L 𝑔

& 𝐵 + ⋯ + 𝜍0 L 𝑔 0 𝐵

𝑔

& 𝐵 , … , 𝑔 0 𝐵 features of alignment 𝐵 (e.g., # matches, …)

Pairwise sequence alignment algorithms

• GRTCPKPDDLPFSTVVP-LKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP

E-VKCPFPSRPDNGFVNYPAKPTLYYKDKATFGCHDGYSLDGP-EEIECTKLGNWSAMPSC-KA

Ground-truth alignment

Pairwise sequence alignment algorithms

• GRTCPKPDDLPFSTVVP-LKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP

E-VKCPFPSRPDNGFVNYPAKPTLYYKDKATFGCHDGYSLDGP-EEIECTKLGNWSAMPSC-KA

Ground-truth alignment

GRTCP---KPDDLPFSTVVPLKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP EVKCPFPSRPDN-GFVNYPAKPTLYYK-DKATFGCHDGY-SLDGPEEIECTKLGNWS-AMPSCKA

Alignment by algorithm with poorly-tuned parameters

Pairwise sequence alignment algorithms

• GRTCPKPDDLPFSTVVP-LKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP

E-VKCPFPSRPDNGFVNYPAKPTLYYKDKATFGCHDGYSLDGP-EEIECTKLGNWSAMPSC-KA

Ground-truth alignment

GRTCP---KPDDLPFSTVVPLKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP EVKCPFPSRPDN-GFVNYPAKPTLYYK-DKATFGCHDGY-SLDGPEEIECTKLGNWS-AMPSCKA

Alignment by algorithm with poorly-tuned parameters

GRTCPKPDDLPFSTV-VPLKTFYEPGEEITYSCKPGYVSRGGMRKFICPLTGLWPINTLKCTP EVKCPFPSRPDNGFVNYPAKPTLYYKDKATFGCHDGY-SLDGPEEIECTKLGNWSA-MPSCKA

Alignment by algorithm with well-tuned parameters

Outline

• 1. Pairwise sequence alignment algorithms
• 2. Sample complexity for pairwise alignment
• 3. Multiple-sequence alignment algorithms
• 4. Sample complexity for multiple-sequence alignments
• 5. Additional applications

Piecewise-constant utility functions

Theorem If for any problem 𝑦, the func 𝜍 ↦ 𝑣Q 𝑦 is piecewise constant and boundaries between pieces defined by 𝑙 hyperplanes: Pseudo-dimension of 𝑣𝝇 𝝇 ∈ ℝ0 is O 𝑒 log 𝑙 An optimal 𝝇 on 𝑃

0 YZ[ \ ]^

samples is 𝜗-optimal on 𝒠.

𝑣` 𝝇 𝜍& 𝜍B

Need to show piecewise constant utilities and bound log(𝑙)

𝑦 = (𝑇, 𝑇')

Key structural property

Lemma:

• For any sequence pair 𝑇, 𝑇' ∈ ΣD, there exists partition of ℝ0 such that:

For any region 𝑆, across all 𝝇 ∈ 𝑆, algorithm’s output is invariant

• Partition induced by 𝐮𝐩𝐮𝐛𝐦 # 𝐛𝐦𝐣𝐡𝐨𝐧𝐟𝐨𝐮𝐭

B

hyperplanes

𝜍& 𝜍B

Key structural property

Lemma:

• For any sequence pair 𝑇, 𝑇′ ∈ ΣD, there exists partition of ℝ0 such that:

For any region 𝑆, across all 𝝇 ∈ 𝑆, algorithm’s output is invariant

• Partition induced by 𝐮𝐩𝐮𝐛𝐦 # 𝐛𝐦𝐣𝐡𝐨𝐧𝐟𝐨𝐮𝐭

B

hyperplanes

Proof:

• For any pair of alignments 𝐵, 𝐵′, prefer 𝐵 over 𝐵' when

∑8 𝜍8 ⋅ 𝑔

8 𝐵 > ∑8 𝜍8 ⋅ 𝑔 8(𝐵').

• Preference for 𝐵 vs 𝐵' determined by hyperplane 𝐼pp=.
• Let ℋ = {𝐼pp= ∣ 𝐵, 𝐵′ alignments}.
• On any region 𝑆 in ℝ0 ∖ ℋ , alignment ordering fixed.
• If DP solver breaks ties reasonably, output constant.

𝜍& 𝜍B

𝐼pp=

Key structural property

Lemma:

• For any sequence pair 𝑇, 𝑇′ ∈ ΣD, there exists partition of ℝ0 such that:

For any region 𝑆, across all 𝝇 ∈ 𝑆, algorithm’s output is invariant

• Partition induced by 𝐮𝐩𝐮𝐛𝐦 # 𝐛𝐦𝐣𝐡𝐨𝐧𝐟𝐨𝐮𝐭

B

hyperplanes

Corollary:

• For fixed 𝑇, 𝑇′, algorithm’s utility is

piecewise-constant function of 𝝇

Similarity to ground truth 𝜍& 𝜍B

Key structural property

Lemma:

• For any sequence pair 𝑇, 𝑇′ ∈ ΣD, there exists partition of ℝ0 such that:

For any region 𝑆, across all 𝝇 ∈ 𝑆, algorithm’s output is invariant

• Partition induced by 𝐮𝐩𝐮𝐛𝐦 # 𝐛𝐦𝐣𝐡𝐨𝐧𝐟𝐨𝐮𝐭

B

hyperplanes

Total # alignments when 𝑇 , 𝑇' ≤ 𝑜 at most 2D𝑜BDx&

Generalization for pairwise alignment

For any sequence pair (𝑇, 𝑇'): 𝑣𝝇 𝑇, 𝑇' = utility of using params 𝝇 ∈ ℝ0 to align 𝑇, 𝑇' Similarity between algorithm’s output & ground truth Theorem Pseudo-dimension of 𝑣𝝇 | 𝝇 ∈ ℝ0 is z 𝑃 𝑒𝑜 where 𝑜 = max |𝑇| Corollary Optimal 𝝇 on sample of size z 𝑃(

0D ]^) is 𝜗-optimal for 𝒠 w.h.p.

Proof: Pseudo-dimension is 𝑃(𝑒 log 𝑙 ) where 𝑙 = 𝑃(2D𝑜BDx&)

Improvement for a special case

Special case widely used in practice: Given parameters 𝜍&, 𝜍B, 𝜍H ≥ 0, find alignment maximizing: (# matches) − 𝜍& L (# mismatches) − 𝜍B L (# indels) − 𝜍H L (# gaps) Theorem

[Gusfield, Balasubramanian, Naor ’94; Fernández-Baca, Seppäläinen, Slutzki ‘04]

• For any sequence pair 𝑇, 𝑇′, there exists partition of ℝH such that:

For any region 𝑆, across all 𝝇 ∈ 𝑆, algorithm’s output is invariant

• Partition induced by 𝑃 𝑜~ hyperplanes

Improvement from ≈ 𝑜D to 𝑜~

Improvement for a special case

Given parameters 𝜍&, 𝜍B, 𝜍H ≥ 0, find alignment maximizing: (# matches) − 𝜍& L (# mismatches) − 𝜍B L (# indels) − 𝜍H L (# gaps) Theorem Pseudo-dim of 𝑣𝝇 | 𝝇 ∈ ℝH is 𝑃 log 𝑜 where 𝑜 = max |𝑇| Corollary

• Optimal 𝝇 on sample of size z

𝑃(

YZ[ D ]^ ) is 𝜗-optimal for 𝒠 w.h.p.

𝜍& 𝜍B

vs z 𝑃(𝑒𝑜) vs z 𝑃(

0D ]^)

Outline

• 1. Pairwise sequence alignment algorithms
• 2. Sample complexity for pairwise alignment
• 3. Multiple-sequence alignment algorithms
• 4. Sample complexity for multiple-sequence alignments
• 5. Additional applications

Multiple sequence alignment

Multiple sequence alignment

Input: Collection of sequences S&, … , S• ∈ ΣD Alignment: Sequences 𝜐&, … , 𝜐‚ ∈ Σ ∪ −

∗ such that:

Deleting “−” from 𝜐8 yields 𝑇8. 𝑇& = A C T G 𝑇B = G T C A 𝑇H = C T T A 𝜐& = A – - C T G 𝜐B = - G T C A – 𝜐H = C - T T A –

Multiple sequence alignment algorithms

Given parameters 𝝇 ∈ ℝ0: Find alignment 𝐵 maximizing: 𝜍& L 𝑔

& 𝐵 + ⋯ + 𝜍0 L 𝑔 0 𝐵

𝑔

& 𝐵 , … , 𝑔 0 𝐵 features of alignment 𝐵 (e.g., # matches, …)

Dynamic programming table has 𝑜‚ entries – exp. running time! Finding min

p 𝜍& ⋅ 𝑔 & 𝐵 + ⋯ + 𝜍0 ⋅ 𝑔 0(𝐵) is NP-complete!

[Wang and Jiang, 1994, Kececioglu and Starrett, 2004]

In practice, use heuristic algorithms

Progressive multiple sequence alignment

Given a binary guide tree over sequences

e.g. obtained by clustering sequences

Use pairwise algo to align children of each node

Find pairwise alignments minimizing ∑8 𝜍8 ⋅ 𝑔

8(𝐵)

Output alignment at the root node

Algorithm parameters: 𝜍&, … , 𝜍0

𝑇& 𝑇B 𝑇H 𝜐&B 𝜐&BH

Progressive multiple sequence alignment

Outline

• 1. Pairwise sequence alignment algorithms
• 2. Sample complexity for pairwise alignment
• 3. Multiple-sequence alignment algorithms
• 4. Sample complexity for multiple-sequence alignments
• 5. Additional applications

Key structural property

Lemma:

• For any sequences 𝑇&, … , 𝑇‚ ∈ ΣD, there exists partition of ℝ0 such that:

For any region 𝑆, across all 𝝇 ∈ 𝑆, algorithm’s output is invariant

• Partition induced by 𝑙 hyperplanes with log 𝑙 = z

𝑃 𝑒…x&𝑜𝑂 𝜃 = bound on depth of guide trees

𝑇& 𝑇B 𝑇H Idea:

• Solve pairwise alignment at each node.
• Collect the hyperplanes from each node!
• Complication: prob. at internal node depends
• n children alignment.
• Include hyperplanes for every possible

problem faced at each node.

Pseudo-dim of multi-sequence alignment

Theorem Pseudo-dimension of 𝑣𝝇 | 𝝇 ∈ ℝ0 is z 𝑃 𝑒…xB𝑜𝑂

𝑜 = number of problems 𝑂 = number of sequences per problem 𝑒 = number of alignment features 𝜃 = bound on guide-tree depth.

Corollary Optimal 𝝇 on sample of size z 𝑃(

0ˆ‰^D‚ ]^

) is 𝜗-optimal for 𝒠 w.h.p. If guide trees roughly balanced, then 𝜃 = O(log(𝑜)).

Outline

• 1. Pairwise sequence alignment algorithms
• 2. Sample complexity for pairwise alignment
• 3. Multiple-sequence alignment algorithms
• 4. Sample complexity for multiple-sequence alignments
• 5. Additional applications

RNA folding

RNA assembled as a chain of bases

Denoted as sequence in {𝐵, 𝑉, 𝐷, 𝐻}∗

Often found as single strand folded into itself

Non-adjacent bases physically bound together

Given unfolded RNA strand: Infer how would naturally fold

Sheds light on function

We provide sample complexity guarantees for inferring RNA folding

Predicting TADs

Linear DNA of genome wraps into 3D structures

Influence genome function

Topologically associating domains (TADs): Contiguous segments of genome that fold into compact regions We provide sample complexity guarantees for predicting TADs

Conclusion

• Goal: Learn parameters for sequence alignment to recover

ground truth alignments

• Sample complexity for pairwise alignment.
• Sample complexity for progressive multi-sequence alignment
• Mentioned other computational biology applications