Algebraic Dynamic Programming

Transcription

1 Algebraic Dynamic Programming Unit 3.b: Products of Evaluation Algebras Robert Giegerich 1 (Lecture) Benedikt Löwes (Exercises) Faculty of Technology Bielefeld University Summer robert@techfak.uni-bielefeld.de Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

2 Motivation for a bag-of-surprises You have worked with the ADP examples Relaxed Palindromes NW alignments Nussinow RNA folding using different algebras: minimizing cost, distance maximizing base pairs, similarity counting numbers of candidates enumerating candidate trees or pretty printing So far, we have always used one algebra at a time. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

3 Algebras in combination In working with applications, often we want several types of information about the candidates in our search space. Optimal solutions under a primary and a secondary objectives (the cheapest of the largest pizzas, or vice versa) the optimal score and the prettyprint of the candidate(s) that achieve(s) it the optial score and the number of co-optial candidates enumeration of candidates in tree and prettyprint form (for debugging)... or combinations of the above Whenever the desired information can be derived via various evaluation algebras, this amounts to using several algebras simultaneously and in cooperation. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

4 Ad-hoc combinations of evaluation algebras Given two Σ-algebras A and B, holding (say) functions f A, f B and h A, h B, we can define algebra C operating on value pairs from A and B: f C ((a, b)) = some function using f A (a) and f B (b) h C ([(a, b)]) = some function on lists of pairs, implementing the desired combination of h A and h B This means coding a new algebra (using the available f A, f B ), a new proof of Bellman s Principle for f C versus h C, and new debugging. There are standard cases where we can do better...! Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

5 Algebra products We define several product operations *,,... to combine two algebras in specific ways. Instead of hand-coding algebra C and calling we directly call G(C, x) G(A B, x) with no coding and debugging effort on our side. The ADP compiler knows the definitin of A B and constructs and uses a correct implementation of C. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

6 General scheme of product operations Product algebras A b compute answer value pairs (a, b) using functions f A f B : h A h B : component wise dependent on our purpose In fact, there are at least four useful variants of h A h B. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

7 The Cartesian product ( ) Definition Let A and B be evaluation algebras over Σ. Their Cartesian product algebra C = A B is an evaluation algebra over Σ and with f C (x 1,..., x k ) = (f A (a 1,..., a k ), f B (b 1,..., b k )) if x i = (xi A, xi B ), then a i = xi A, b i = xi B, if x i A, then a i = x i = b i for each f in Σ, and with the objective function h C [(a 1, b 1 ),..., (a k, b k )] = [(l, r) l h A [a 1,..., a k ], r h B [b 1,..., b k ] ] Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

8 Meaning of the Cartesian Product The Cartesian product algebra A B optimizes independently according to A and B returns the Cartesian product of the result sets from A and B a result value (a, b) may or may not be derived from the same candidate This appears too simple to be useful but be patient. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

9 Definition of lexicographic product * Definition Let A and B be evaluation algebras over Σ. Their lexicographic product algebra A B is an evaluation algebra C over Σ with f C defined component-wise (as with ) for each f in Σ, and with the objective function h A B ([(a 1, b 1 ),..., (a k, b k )]) = [(l, r)] l set(h A [a 1,..., a k ]), r h B ([r (l, r) [(a 1, b 1 ),..., (a k, b k )], l = l]) where [] denotes a multi-set and the function set returns a multi-set in which all duplicate elements are removed. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

10 The meaning of G(A B, x) is actually lready explained in ADP definitions, but still... Intuitive explanation: Phase 1 evaluates all candidates via functions f A f B into a gigantic list [..., (a i, b i ),...] Phase 2 chooses pairs with h A based on the a i component, and Phase 3 chooses from those the final pairs, based on te b i component Thus, if h A maximizes under ordering < A and h B maximizes under ordering < B, the product maximizes under the lexicographic combination of orderinge, (< A, < B ). Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

11 How about Bellman s Principle? In general, there is a proof obligation with * A B must satisfy Bellman s principle... because the three phases are not executed separately. Instead, h A B is applied at intermediate steps. Bellman s Principle guarantees that we loose no final result along the way. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

12 Advantages over hand-coded algebra combinations Benefits of products: the ADP compiler knows the definition of * and produces automatically code for A B. the resulting functions f A B etc. maybe a little less efficient than in a handcoded product, BUT: products formed with require no coding and testing Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

13 Versatility of the lexicographic product A B has a surprising number of uses optimization under a lexicographic ordering class-wise optimization reporting candidates The latter two cases use algebras A or B that do not do optimization. More later... Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

14 Useful properties of the lexicographic product (1) Theorem (1) Selectivity: Let h A (X ) X and h B (X ) X. For each value (a, b) computed by h A h B, there is a candidate t such that (a, b) = (A(t), B(t)). I.e., each returned value pair is backed up by a candidate that evaluates to these two values. Selectivity does not hold e.g. for the counting algebra. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

15 Useful properties of the lexicographic product (2) Theorem (2) Identity: For any algebras A and B, and multiset X, (id A id B )(X ) = X. Combining two identity functions gives the identity function on pairs. (On lists, (id A id B )(X ) is a permutation of X.) Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

16 Useful elementary properties (3) Since A B is just another algebra, we can make bigger products... Theorem (3) Associativity: G(A (B C), x) = G((A B) C, x) modulo re-grouping of tuples of result values, e.g. (a, (b, c)) = ((a, b), c) = (a, b, c) Hence we can simply write G(A B C, x). Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

17 Useful elementary properties (4) Theorem (4) Lexicographic optimization: Let A and B optimize over < A and < B. Then A B satisfies Bellman s Principle and optimizes over the lexicographic ordering (< A, < B ). So, for optimizing over two orderings in lexicographic fashion, there is a general proof that Bellman s Principle is always satisfied. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

18 Interleaved product The interleaved product: Motivation Let B(k) be an algebra that computes the best k scores under some optimization scheme. Let A be a an algebra that assigns a class attribute to each candidate. With G(A B(k), x), we obtain the best k for each class. How can we get the best 3 B-values from different A-classes? We first compute B(1) for each A-class, and then choose the classes with the best k B-values. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

19 Interleaved product The interleaved product: Definition Let A be a Σ- algebra and B(k) a generic Σ-algebra, and note that B(1) is unitary. Definition The interleaved product (A/B)(k) is a generic Σ-algebra and has the functions f A/B = f A B (1) for each f Σ, and the objective function h (A/B)(k) [(a 1, b 1 ),..., (a m, b m )] = [ (l, r) (l, r) U, p V, p = r] where U = h A B(1) [(a 1, b 1 ),..., (a m, b m )] V = set(h B(k) [ v (, v) U]) (2) Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

20 Interleaved product Interleaved product (2) In Equation 2, U stores the best results in each A-class, V stores the global rank of all solutions from U, and the main set comprehension filters U to contain only the results from V. This product is useful when B(k) alone produces the k-best-scoring candidates, and A maps candidates to different classes (not making any choices). Then, A/B yields the 1-best-scoring candidates from k different classes. This situation required sophisticated hand-coding in RNA shape analysis in 2004 it is now simply written as (Shape/Mfe)(k). Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

21 Pareto product Pareto-Optimization Pareto-optimization is a general way of optimizing with two (or more) idependent criteria. The solution set one computes is the Pareto front of the search space X A B. Taking h A and h B as maximization, the Pareto front operator pf is defined as pf(x ) = {(a, b) X (a, b ) X \ {(a, b)} with a a, b b }. In words: No element of the Pareto front is dominated in both dimensions by another element of X. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

22 Pareto product The Pareto-Product Let A and B be two Σ-algebras which maximize with respect to < A and < B. Definition The Pareto-product AB is a Σ-algebra C with f C defined component-wise from f A and f B, and h C (X ) = pf(x ) with pf as defined above for < A and < B. The Pareto product is a recent development and not yet implemented in Bellman s GAP. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23

23 Pareto product Conclusion on algebra products A number of algebra product operators have been defined there may be more. it is faster and safer to use products where suitable than hand-coding a combined evalation algebra. since products yield new algebras, these can be used with further products, as in G((A/B) (C D), x), which makes them a powerful programming device on a very high level of abstraction. Understanding the definitions of products is one thing. Using products effectively takes quite some practice!! That s the topic of our next set of slides. Robert Giegerich, Benedikt Löwes ADP Lecture Summer / 23