Fixed-point computation

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Short description: Computing the fixed point of a function

Fixed-point computation refers to the process of computing an exact or approximate fixed point of a given function.[1] In its most common form, we are given a function f that satisfies the condition to the Brouwer fixed-point theorem, that is: f is continuous and maps the unit d-cube to itself. The Brouwer fixed-point theorem guarantees that f has a fixed point, but the proof is not constructive. Various algorithms have been devised for computing an approximate fixed point. Such algorithms are used in economics for computing a market equilibrium, in game theory for computing a Nash equilibrium, and in dynamic system analysis.

Definitions

an example function with three fixed points
The graph of an example function with three fixed points

The unit interval is denoted by [math]\displaystyle{ E := [0, 1] }[/math], and the unit d-dimensional cube is denoted by Ed. A continuous function f is defined on Ed (from Ed to itself). Often, it is assumed that f is not only continuous but also Lipschitz continuous, that is, for some constant L, [math]\displaystyle{ |f(x)-f(y)| \leq L\cdot |x-y| }[/math] for all x,y in Ed.

A fixed point of f is a point x in Ed such that f(x) = x. By the Brouwer fixed-point theorem, any continuous function from Ed to itself has a fixed point. But for general functions, it is impossible to compute a fixed point precisely, since it can be an arbitrary real number. Fixed-point computation algorithms look for approximate fixed points. There are several criteria for an approximate fixed point. Several common criteria are:[2]

  • The residual criterion: given an approximation parameter [math]\displaystyle{ \varepsilon\gt 0 }[/math] , An ε-residual fixed-point of f is a point x in Ed such that [math]\displaystyle{ |f(x)-x|\leq \varepsilon }[/math], where here |.| denotes the maximum norm. That is, all d coordinates of the difference [math]\displaystyle{ f(x)-x }[/math] should be at most ε.[3](p4)
  • The absolute criterion: given an approximation parameter [math]\displaystyle{ \delta\gt 0 }[/math], A δ-absolute fixed-point of f is a point x in Ed such that [math]\displaystyle{ |x-x_0|\leq \delta }[/math], where [math]\displaystyle{ x_0 }[/math] is any fixed-point of f.
  • The relative criterion: given an approximation parameter [math]\displaystyle{ \delta\gt 0 }[/math], A δ-relative fixed-point of f is a point x in Ed such that [math]\displaystyle{ |x-x_0|/|x_0|\leq \delta }[/math], where [math]\displaystyle{ x_0 }[/math] is any fixed-point of f.

For Lipschitz-continuous functions, the absolute criterion is stronger than the residual criterion: If f is Lipschitz-continuous with constant L, then [math]\displaystyle{ |x-x_0|\leq \delta }[/math] implies [math]\displaystyle{ |f(x)-f(x_0)|\leq L\cdot \delta }[/math]. Since [math]\displaystyle{ x_0 }[/math] is a fixed-point of f, this implies [math]\displaystyle{ |f(x)-x_0|\leq L\cdot \delta }[/math], so [math]\displaystyle{ |f(x)-x|\leq (1+L)\cdot \delta }[/math]. Therefore, a δ-absolute fixed-point is also an ε-residual fixed-point with [math]\displaystyle{ \varepsilon = (1+L)\cdot \delta }[/math].

The most basic step of a fixed-point computation algorithm is a value query: given any x in Ed, the[sentence fragment]

The function f is accessible via evaluation queries: for any x, the algorithm can evaluate f(x). The run-time complexity of an algorithm is usually given by the number of required evaluations.

Contractive functions

A Lipschitz-continuous function with constant L is called contractive if L < 1; it is called weakly-contractive if L ≤ 1.Every contractive function satisfying Brouwer's condisions has a unique fixed point. Moreover, fixed-point computation for contractive functions is easier than for general functions.

computing a fixed point using function iteration
Computing a fixed point using function iteration

The first algorithm for fixed-point computation was the fixed-point iteration algorithm of Banach. Banach's fixed-point theorem implies that, when fixed-point iteration is applied to a contraction mapping, the error after t iterations is in [math]\displaystyle{ O(L^t) }[/math]. Therefore, the number of evaluations required for a δ-relative fixed-point is approximately [math]\displaystyle{ \log_L(\delta) = \log(\delta)/\log(L) = \log(1/\delta)/\log(1/L) }[/math]. Sikorski and Wozniakowski[4] showed that Banach's algorithm is optimal when the dimension is large. Specifically, when [math]\displaystyle{ d\geq \log(1/\delta)/\log(1/L) }[/math], the number of required evaluations of any algorithm for δ-relative fixed-point is larger than 50% the number of evaluations required by the iteration algorithm. Note that when L approaches 1, the number of evaluations approaches infinity. In fact, no finite algorithm can compute a δ-absolute fixed point for all functions with L=1.[5]

When L < 1 and d = 1, the optimal algorithm is the Fixed Point Envelope (FPE) algorithm of Sikorski and Wozniakowski.[4] It finds a δ-relative fixed point using [math]\displaystyle{ O(\log(1/\delta) + \log \log(1/(1-L))) }[/math] queries, and a δ-absolute fixed point using [math]\displaystyle{ O(\log(1/\delta)) }[/math] queries. This is much faster than the fixed-point iteration algorithm.[6]

When d>1 but not too large, and L ≤ 1, the optimal algorithm is the interior-ellipsoid algorithm (based on the ellipsoid method).[7] It finds an ε-residual fixed-point is using [math]\displaystyle{ O(d\cdot \log(1/\varepsilon)) }[/math] evaluations. When L<1, it finds a δ-absolute fixed point using [math]\displaystyle{ O(d\cdot [\log(1/\delta) + \log(1/(1-L))]) }[/math] evaluations.

Shellman and Sikorski[8] presented an algorithm called BEFix (Bisection Envelope Fixed-point) for computing an ε-residual fixed-point of a two-dimensional function with L ≤ 1, using only [math]\displaystyle{ 2 \lceil\log_2(1/\varepsilon)\rceil+1 }[/math] queries. They later[9] presented an improvement called BEDFix (Bisection Envelope Deep-cut Fixed-point), with the same worst-case guarantee but better empirical performance. When L<1, BEDFix can also compute a δ-absolute fixed-point using [math]\displaystyle{ O(\log(1/\varepsilon)+\log(1/(1-L))) }[/math] queries.

Shellman and Sikorski[2] presented an algorithm called PFix for computing an ε-residual fixed-point of a d-dimensional function with L ≤ 1, using [math]\displaystyle{ O(\log^d(1/\varepsilon)) }[/math] queries. When L < 1, PFix can be executed with [math]\displaystyle{ \varepsilon = (1-L)\cdot \delta }[/math], and in that case, it computes a δ-absolute fixed-point, using [math]\displaystyle{ O(\log^d(1/[(1-L)\delta])) }[/math] queries. It is more efficient than the iteration algorithm when L is close to 1. The algorithm is recursive: it handles a d-dimensional function by recursive calls on (d-1)-dimensional functions.

Algorithms for differentiable functions

When the function f is differentiable, and the algorithm can evaluate its derivative (not only f itself), the Newton method can be used and it is much faster.[10][11]

General functions

For functions with Lipschitz constant L > 1, computing a fixed-point is much harder.

One dimension

For a 1-dimensional function (d = 1), a δ-absolute fixed-point can be found using [math]\displaystyle{ O(\log(1/\delta)) }[/math] queries using the bisection method: start with the interval [math]\displaystyle{ E := [0, 1] }[/math]; at each iteration, let x be the center of the current interval, and compute f(x); if f(x) > x then recurse on the sub-interval to the right of x; otherwise, recurse on the interval to the left of x. Note that the current interval always contains a fixed point, so after [math]\displaystyle{ O(\log(1/\delta)) }[/math] queries, any point in the remaining interval is a δ-absolute fixed-point of f. Setting [math]\displaystyle{ \delta := \varepsilon/(L+1) }[/math], where L is the Lipschitz constant, gives an ε-residual fixed-point, using [math]\displaystyle{ O(\log(L/\varepsilon) = \log(L) + \log(1/\varepsilon)) }[/math] queries.[3]

Two or more dimensions

For functions in two or more dimensions, the problem is much more challenging. Shellman and Sikorski[2] proved that, for any integers d ≥ 2 and L > 1, finding a δ-absolute fixed-point of d-dimensional L-Lipschitz functions might require infinitely-many evaluations. The proof idea is as follows. For any integer T > 1, and for any sequence of T of evaluation queries (possibly adaptive), one can construct two functions that are Lipschitz-continuous with constant L, and yield the same answer to all these queries, but one of them has a unique fixed-point at (x, 0) and the other has a unique fixed-point at (x, 1). Any algorithm using T evaluations cannot differentiate between these functions, so cannot find a δ-absolute fixed-point. This is true for any finite integer T.

Several algorithms based on function evaluations have been developed for finding an ε-residual fixed-point

  • The first algorithm to approximate a fixed point of a general function was developed by Herbert Scarf in 1967.[12][13] Scarf's algorithm finds an ε-residual fixed-point by finding a fully-labelled "primitive set", in a construction similar to Sperner's lemma.
  • A later algorithm by Harold Kuhn[14] used simplices and simplicial partitions instead of primitive sets.
  • Developing the simplicial approach further, Orin Harrison Merrill[15] presented the restart algorithm.
  • B. Curtis Eaves[16] presented the homotopy algorithm. The algorithm works by starting with an affine function that approximates f, and deforming it towards f, while following the fixed point. A book by Michael Todd[1] surveys various algorithms developed until 1976.
  • David Gale[17] showed that computing a fixed point of an n-dimensional function (on the unit d-dimensional cube) is equivalent to deciding who is the winner in a d-dimensional game of Hex (a game with d players, each of whom needs to connect two opposite faces of a d-cube). Given the desired accuracy ε
    • Construct a Hex board of size kd, where [math]\displaystyle{ k \gt 1/\varepsilon }[/math]. Each vertex z corresponds to a point z/k in the unit n-cube.
    • Compute the difference f(z/k) - z/k; note that the difference is an n-vector.
    • Label the vertex z by a label in 1, ..., d, denoting the largest coordinate in the difference vector.
    • The resulting labeling corresponds to a possible play of the d-dimensional Hex game among d players. This game must have a winner, and Gale presents an algorithm for constructing the winning path.
    • In the winning path, there must be a point in which fi(z/k) - z/k is positive, and an adjacent point in which fi(z/k) - z/k is negative. This means that there is a fixed point of f between these two points.

In the worst case, the number of function evaluations required by all these algorithms is exponential in the binary representation of the accuracy, that is, in [math]\displaystyle{ \Omega(1/\varepsilon) }[/math].

Query complexity

Hirsch, Papadimitriou and Vavasis proved that[3] any algorithm based on function evaluations, that finds an ε-residual fixed-point of f, requires [math]\displaystyle{ \Omega(L'/\varepsilon) }[/math] function evaluations, where [math]\displaystyle{ L' }[/math] is the Lipschitz constant of the function [math]\displaystyle{ f(x)-x }[/math] (note that [math]\displaystyle{ L-1 \leq L' \leq L+1 }[/math]). More precisely:

  • For a 2-dimensional function (d=2), they prove a tight bound [math]\displaystyle{ \Theta(L'/\varepsilon) }[/math].
  • For any d ≥ 3, finding an ε-residual fixed-point of a d-dimensional function requires [math]\displaystyle{ \Omega((L'/\varepsilon)^{d-2}) }[/math] queries and [math]\displaystyle{ O((L'/\varepsilon)^{d}) }[/math] queries.

The latter result leaves a gap in the exponent. Chen and Deng[18] closed the gap. They proved that, for any d ≥ 2 and [math]\displaystyle{ 1/\varepsilon \gt 4 d }[/math] and [math]\displaystyle{ L'/\varepsilon \gt 192 d^3 }[/math], the number of queries required for computing an ε-residual fixed-point is in [math]\displaystyle{ \Theta((L'/\varepsilon)^{d-1}) }[/math].

Discrete fixed-point computation

A discrete function is a function defined on a subset of [math]\displaystyle{ \mathbb{Z}^d }[/math] (the d-dimensional integer grid). There are several discrete fixed-point theorems, stating conditions under which a discrete function has a fixed point. For example, the Iimura-Murota-Tamura theorem states that (in particular) if f is a function from a rectangle subset of [math]\displaystyle{ \mathbb{Z}^d }[/math] to itself, and f is hypercubic direction-preserving, then f has a fixed point.

Let f be a direction-preserving function from the integer cube [math]\displaystyle{ \{1, \dots, n\}^d }[/math] to itself. Chen and Deng[18] prove that, for any d ≥ 2 and n > 48d, computing such a fixed point requires [math]\displaystyle{ \Theta(n^{d-1}) }[/math] function evaluations.

Chen and Deng[19] define a different discrete-fixed-point problem, which they call 2D-BROUWER. It considers a discrete function f on [math]\displaystyle{ \{0,\dots, n\}^2 }[/math] such that, for every x on the grid, f(x) - x is either (0, 1) or (1, 0) or (-1, -1). The goal is to find a square in the grid, in which all three labels occur. The function f must map the square [math]\displaystyle{ \{0,\dots, n\}^2 }[/math]to itself, so it must map the lines x = 0 and y = 0 to either (0, 1) or (1, 0); the line x = n to either (-1, -1) or (0, 1); and the line y = n to either (-1, -1) or (1,0). The problem can be reduced to 2D-SPERNER (computing a fully-labeled triangle in a triangulation satisfying the conditions to Sperner's lemma), and therefore it is PPAD-complete. This implies that computing an approximate fixed-point is PPAD-complete even for very simple functions.

Relation between fixed-point computation and root-finding algorithms

Given a function g from Ed to R, a root of g is a point x in Ed such that g(x)=0. An ε-root of g is a point x in Ed such that [math]\displaystyle{ g(x)\leq \varepsilon }[/math].

Fixed-point computation is a special case of root-finding: given a function f on Ed, define [math]\displaystyle{ g(x) := |f(x)-x| }[/math]. Clearly, x is a fixed-point of f if and only if x is a root of g, and x is an ε-residual fixed-point of f if and only if x is an ε-root of g. Therefore, any root-finding algorithm (an algorithm that computes an approximate root of a function) can be used to find an approximate fixed-point.

The opposite is not true: finding an approximate root of a general function may be harder than finding an approximate fixed point. In particular, Sikorski[20] proved that finding an ε-root requires [math]\displaystyle{ \Omega(1/\varepsilon^d) }[/math] function evaluations. This gives an exponential lower bound even for a one-dimensional function (in contrast, an ε-residual fixed-point of a one-dimensional function can be found using [math]\displaystyle{ O(\log(1/\varepsilon)) }[/math] queries using the bisection method). Here is a proof sketch.[3](p35) Construct a function g that is slightly larger than ε everywhere in Ed except in some small cube around some point x0, where x0 is the unique root of g. If g is Lipschitz continuous with constant L, then the cube around x0 can have a side-length of [math]\displaystyle{ \varepsilon/L }[/math]. Any algorithm that finds an ε-root of g must check a set of cubes that covers the entire Ed; the number of such cubes is at least [math]\displaystyle{ (L/\varepsilon)^d }[/math].

However, there are classes of functions for which finding an approximate root is equivalent to finding an approximate fixed point. One example[18] is the class of functions g such that [math]\displaystyle{ g(x)+x }[/math] maps Ed to itself (that is: [math]\displaystyle{ g(x)+x }[/math] is in Ed for all x in Ed). This is because, for every such function, the function [math]\displaystyle{ f(x) := g(x)+x }[/math] satisfies the conditions to Brouwer's fixed-point theorem. Clearly, x is a fixed-point of f if and only if x is a root of g, and x is an ε-residual fixed-point of f if and only if x is an ε-root of g. Chen and Deng[18] show that the discrete variants of these problems are computationally equivalent: both problems require [math]\displaystyle{ \Theta(n^{d-1}) }[/math] function evaluations.

Communication complexity

Roughgarden and Weinstein[21] studied the communication complexity of computing an approximate fixed-point. In their model, there are two agents: one of them knows a function f and the other knows a function g. Both functions are Lipschitz continuous and satisfy Brouwer's conditions. The goal is to compute an approximate fixed point of the composite function [math]\displaystyle{ g\circ f }[/math]. They show that the deterministic communication complexity is in [math]\displaystyle{ \Omega(2^d) }[/math].

References

  1. ↑ 1.0 1.1 The Computation of Fixed Points and Applications. Lecture Notes in Economics and Mathematical Systems. 124. 1976. doi:10.1007/978-3-642-50327-6. ISBN 978-3-540-07685-8. [page needed]
  2. ↑ 2.0 2.1 2.2 Shellman, Spencer; Sikorski, K. (December 2003). "A recursive algorithm for the infinity-norm fixed point problem". Journal of Complexity 19 (6): 799–834. doi:10.1016/j.jco.2003.06.001. 
  3. ↑ 3.0 3.1 3.2 3.3 Hirsch, Michael D; Papadimitriou, Christos H; Vavasis, Stephen A (December 1989). "Exponential lower bounds for finding Brouwer fix points". Journal of Complexity 5 (4): 379–416. doi:10.1016/0885-064X(89)90017-4. 
  4. ↑ 4.0 4.1 Sikorski, K; Woźniakowski, H (December 1987). "Complexity of fixed points, I". Journal of Complexity 3 (4): 388–405. doi:10.1016/0885-064X(87)90008-2. 
  5. ↑ Sikorski, Krzysztof A. (2001). Optimal Solution of Nonlinear Equations. Oxford University Press. ISBN 978-0-19-510690-9. [page needed]
  6. ↑ Sikorski, K. (1989). "Fast Algorithms for the Computation of Fixed Points". Robustness in Identification and Control. pp. 49–58. doi:10.1007/978-1-4615-9552-6_4. ISBN 978-1-4615-9554-0. 
  7. ↑ Huang, Z; Khachiyan, L; Sikorski, K (June 1999). "Approximating Fixed Points of Weakly Contracting Mappings". Journal of Complexity 15 (2): 200–213. doi:10.1006/jcom.1999.0504. 
  8. ↑ Shellman, Spencer; Sikorski, K. (June 2002). "A Two-Dimensional Bisection Envelope Algorithm for Fixed Points". Journal of Complexity 18 (2): 641–659. doi:10.1006/jcom.2001.0625. 
  9. ↑ Shellman, Spencer; Sikorski, K. (September 2003). "Algorithm 825: A deep-cut bisection envelope algorithm for fixed points". ACM Transactions on Mathematical Software 29 (3): 309–325. doi:10.1145/838250.838255. 
  10. ↑ Kellogg, R. B.; Li, T. Y.; Yorke, J. (September 1976). "A Constructive Proof of the Brouwer Fixed-Point Theorem and Computational Results". SIAM Journal on Numerical Analysis 13 (4): 473–483. doi:10.1137/0713041. 
  11. ↑ Smale, Steve (July 1976). "A convergent process of price adjustment and global newton methods". Journal of Mathematical Economics 3 (2): 107–120. doi:10.1016/0304-4068(76)90019-7. 
  12. ↑ Scarf, Herbert (September 1967). "The Approximation of Fixed Points of a Continuous Mapping". SIAM Journal on Applied Mathematics 15 (5): 1328–1343. doi:10.1137/0115116. 
  13. ↑ H. Scarf found the first algorithmic proof: Hazewinkel, Michiel, ed. (2001), "Brouwer theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page .
  14. ↑ Kuhn, Harold W. (1968). "Simplicial Approximation of Fixed Points". Proceedings of the National Academy of Sciences of the United States of America 61 (4): 1238–1242. doi:10.1073/pnas.61.4.1238. PMID 16591723. 
  15. ↑ Merrill, Orin Harrison (1972). Applications and Extensions of an Algorithm that Computes Fixed Points of Certain Upper Semi-continuous Point to Set Mappings (Thesis). OCLC 570461463. NAID 10006142329.
  16. ↑ Eaves, B. Curtis (December 1972). "Homotopies for computation of fixed points". Mathematical Programming 3-3 (1): 1–22. doi:10.1007/BF01584975. 
  17. ↑ Gale, David (1979). "The Game of Hex and Brouwer Fixed-Point Theorem". The American Mathematical Monthly 86 (10): 818–827. doi:10.2307/2320146. 
  18. ↑ 18.0 18.1 18.2 18.3 Chen, Xi; Deng, Xiaotie (2005). "On algorithms for discrete and approximate brouwer fixed points". Proceedings of the thirty-seventh annual ACM symposium on Theory of computing. pp. 323–330. doi:10.1145/1060590.1060638. ISBN 1581139608. 
  19. ↑ Chen, Xi; Deng, Xiaotie (October 2009). "On the complexity of 2D discrete fixed point problem". Theoretical Computer Science 410 (44): 4448–4456. doi:10.1016/j.tcs.2009.07.052. 
  20. ↑ Sikorski, K. (June 1984). "Optimal solution of nonlinear equations satisfying a Lipschitz condition". Numerische Mathematik 43 (2): 225–240. doi:10.1007/BF01390124. 
  21. ↑ Roughgarden, Tim; Weinstein, Omri (2016). "On the Communication Complexity of Approximate Fixed Points". 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS). pp. 229–238. doi:10.1109/FOCS.2016.32. ISBN 978-1-5090-3933-3. 

Further reading



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