Fixed-point computation

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, the given function ${\displaystyle f}$ satisfies the condition to the Brouwer fixed-point theorem: that is, ${\displaystyle f}$ is continuous and maps the unit d-cube to itself. The Brouwer fixed-point theorem guarantees that ${\displaystyle f}$ 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

The graph of an example function with three fixed points

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

A fixed point of ${\displaystyle f}$ is a point ${\displaystyle x}$ in ${\displaystyle E^{d}}$ such that ${\displaystyle f(x)=x}$. By the Brouwer fixed-point theorem, any continuous function from ${\displaystyle E^{d}}$ 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 ${\displaystyle \varepsilon >0}$ , An ε-residual fixed-point of${\displaystyle f}$ is a point ${\displaystyle x}$ in ${\displaystyle E^{d}}$' such that ${\displaystyle |f(x)-x|\leq \varepsilon }$, where here ${\displaystyle |\cdot |}$ denotes the maximum norm. That is, all ${\displaystyle d}$ coordinates of the difference ${\displaystyle f(x)-x}$ should be at most ε. ^{[3] : 4}
• The absolute criterion: given an approximation parameter ${\displaystyle \delta >0}$, A δ-absolute fixed-point of${\displaystyle f}$ is a point ${\displaystyle x}$ in ${\displaystyle E^{d}}$ such that ${\displaystyle |x-x_{0}|\leq \delta }$, where ${\displaystyle x_{0}}$ is any fixed-point of ${\displaystyle f}$.
• The relative criterion: given an approximation parameter ${\displaystyle \delta >0}$, A δ-relative fixed-point of${\displaystyle f}$ is a point x in ${\displaystyle E^{d}}$ such that ${\displaystyle |x-x_{0}|/|x_{0}|\leq \delta }$, where ${\displaystyle x_{0}}$ is any fixed-point of ${\displaystyle f}$.

For Lipschitz-continuous functions, the absolute criterion is stronger than the residual criterion: If ${\displaystyle f}$ is Lipschitz-continuous with constant ${\displaystyle L}$, then ${\displaystyle |x-x_{0}|\leq \delta }$ implies ${\displaystyle |f(x)-f(x_{0})|\leq L\cdot \delta }$. Since ${\displaystyle x_{0}}$ is a fixed-point of ${\displaystyle f}$, this implies ${\displaystyle |f(x)-x_{0}|\leq L\cdot \delta }$, so ${\displaystyle |f(x)-x|\leq (1+L)\cdot \delta }$. Therefore, a δ-absolute fixed-point is also an ε-residual fixed-point with ${\displaystyle \varepsilon =(1+L)\cdot \delta }$.

The most basic step of a fixed-point computation algorithm is a value query: given any ${\displaystyle x}$ in ${\displaystyle E^{d}}$, the algorithm is provided with an oracle ${\displaystyle {\tilde {f}}}$ to ${\displaystyle f}$ that returns the value ${\displaystyle f(x)}$. The accuracy of the approximate fixed-point depends upon the error in the oracle ${\displaystyle {\tilde {f}}(x)}$.

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

Contractive functions

A Lipschitz-continuous function with constant ${\displaystyle L}$ is called contractive if ${\displaystyle L<1}$; it is called weakly-contractive if ${\displaystyle L\leq 1}$. Every contractive function satisfying Brouwer's conditions 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

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 ${\displaystyle t}$ iterations is in ${\displaystyle O(L^{t})}$. Therefore, the number of evaluations required for a ${\displaystyle \delta }$-relative fixed-point is approximately ${\displaystyle \log _{L}(\delta )=\log(\delta )/\log(L)=\log(1/\delta )/\log(1/L)}$. Sikorski and Wozniakowski ^[4] showed that Banach's algorithm is optimal when the dimension is large. Specifically, when ${\displaystyle d\geq \log(1/\delta )/\log(1/L)}$, the number of required evaluations of any algorithm for ${\displaystyle \delta }$-relative fixed-point is larger than 50% the number of evaluations required by the iteration algorithm. Note that when ${\displaystyle L}$ approaches 1, the number of evaluations approaches infinity. No finite algorithm can compute a ${\displaystyle \delta }$-absolute fixed point for all functions with ${\displaystyle L=1}$. ^[5]

When ${\displaystyle L}$ < 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 ${\displaystyle O(\log(1/\delta )+\log \log(1/(1-L)))}$ queries, and a δ-absolute fixed point using ${\displaystyle O(\log(1/\delta ))}$ queries. This is faster than the fixed-point iteration algorithm. ^[6]

When ${\displaystyle d>1}$ but not too large, and ${\displaystyle L\leq 1}$, the optimal algorithm is the interior-ellipsoid algorithm (based on the ellipsoid method). ^[7] It finds an ε-residual fixed-point using ${\displaystyle O(d\cdot \log(1/\varepsilon ))}$ evaluations. When ${\displaystyle L<1}$, it finds a ${\displaystyle \delta }$-absolute fixed point using ${\displaystyle O(d\cdot [\log(1/\delta )+\log(1/(1-L))])}$ 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 '${\displaystyle L\leq 1}$, using only ${\displaystyle 2\lceil \log _{2}(1/\varepsilon )\rceil +1}$ 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 ${\displaystyle L<1}$, BEDFix can also compute a ${\displaystyle \delta }$-absolute fixed-point using ${\displaystyle O(\log(1/\varepsilon )+\log(1/(1-L)))}$ 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 ${\displaystyle O(\log ^{d}(1/\varepsilon ))}$ queries. When ${\displaystyle L}$ < 1, PFix can be executed with ${\displaystyle \varepsilon =(1-L)\cdot \delta }$, and in that case, it computes a δ-absolute fixed-point, using ${\displaystyle O(\log ^{d}(1/[(1-L)\delta ]))}$ queries. It is more efficient than the iteration algorithm when ${\displaystyle L}$ 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 ${\displaystyle f}$ is differentiable, and the algorithm can evaluate its derivative (not only ${\displaystyle f}$ itself), the Newton method can be used and it is much faster. ^{[10] [11]}

General functions

For functions with Lipschitz constant ${\displaystyle L}$ > 1, computing a fixed-point is much harder.

One dimension

For a 1-dimensional function (d = 1), a ${\displaystyle \delta }$-absolute fixed-point can be found using ${\displaystyle O(\log(1/\delta ))}$ queries using the bisection method: start with the interval ${\displaystyle E:=[0,1]}$; at each iteration, let ${\displaystyle x}$ be the center of the current interval, and compute ${\displaystyle f(x)}$; if ${\displaystyle f(x)>x}$ then recurse on the sub-interval to the right of ${\displaystyle x}$; otherwise, recurse on the interval to the left of ${\displaystyle x}$. Note that the current interval always contains a fixed point, so after ${\displaystyle O(\log(1/\delta ))}$ queries, any point in the remaining interval is a ${\displaystyle \delta }$-absolute fixed-point of ${\displaystyle f}$ Setting ${\displaystyle \delta$ :=\varepsilon /(L+1)}, where ${\displaystyle L}$ is the Lipschitz constant, gives an ε-residual fixed-point, using ${\displaystyle O(\log(L/\varepsilon )=\log(L)+\log(1/\varepsilon ))}$ 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 ${\displaystyle L}$ > 1, finding a δ-absolute fixed-point of d-dimensional ${\displaystyle L}$-Lipschitz functions might require infinitely many evaluations. The proof idea is as follows. For any integer T > 1 and any sequence of T of evaluation queries (possibly adaptive), one can construct two functions that are Lipschitz-continuous with constant ${\displaystyle L}$, 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 labeled "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 ${\displaystyle f}$, and deforming it towards ${\displaystyle f}$ 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 ${\displaystyle k>1/\varepsilon }$. Each vertex z corresponds to a point z/k in the unit n-cube.
  • Compute the difference ${\displaystyle f}$(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 f_i(z/k) - z/k is positive, and an adjacent point in which f_i(z/k) - z/k is negative. This means that there is a fixed point of ${\displaystyle f}$ 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 ${\displaystyle \Omega (1/\varepsilon )}$.

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 ${\displaystyle \Omega (L'/\varepsilon )}$ function evaluations, where ${\displaystyle L'}$ is the Lipschitz constant of the function ${\displaystyle f(x)-x}$ (note that ${\displaystyle L-1\leq L'\leq L+1}$). More precisely:

• For a 2-dimensional function (d=2), they prove a tight bound ${\displaystyle \Theta (L'/\varepsilon )}$.
• For any d ≥ 3, finding an ε-residual fixed-point of a d-dimensional function requires ${\displaystyle \Omega ((L'/\varepsilon )^{d-2})}$ queries and ${\displaystyle O((L'/\varepsilon )^{d})}$ 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 ${\displaystyle 1/\varepsilon >4d}$ and ${\displaystyle L'/\varepsilon >192d^{3}}$, the number of queries required for computing an ε-residual fixed-point is in ${\displaystyle \Theta ((L'/\varepsilon )^{d-1})}$.

Discrete fixed-point computation

A discrete function is a function defined on a subset of ${\displaystyle \mathbb {Z} ^{d}}$ (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 ${\displaystyle f}$ is a function from a rectangle subset of ${\displaystyle \mathbb {Z} ^{d}}$ to itself, and ${\displaystyle f}$ is hypercubic direction-preserving , then ${\displaystyle f}$ has a fixed point.

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

Chen and Deng ^[19] define a different discrete-fixed-point problem, which they call 2D-BROUWER. It considers a discrete function ${\displaystyle f}$ on ${\displaystyle \{0,\dots ,n\}^{2}}$ such that, for every x on the grid, ${\displaystyle f}$(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 ${\displaystyle f}$ must map the square ${\displaystyle \{0,\dots ,n\}^{2}}$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 ${\displaystyle g}$ from ${\displaystyle E^{d}}$ to R, a root of ${\displaystyle g}$ is a point x in ${\displaystyle E^{d}}$ such that ${\displaystyle g}$(x)=0. An ε-root of g is a point x in ${\displaystyle E^{d}}$ such that ${\displaystyle g(x)\leq \varepsilon }$.

Fixed-point computation is a special case of root-finding: given a function ${\displaystyle f}$ on ${\displaystyle E^{d}}$, define ${\displaystyle g(x):=|f(x)-x|}$. X is a fixed-point of ${\displaystyle f}$ if and only if x is a root of ${\displaystyle g}$, and x is an ε-residual fixed-point of ${\displaystyle f}$ if and only if x is an ε-root of ${\displaystyle g}$. 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 ${\displaystyle \Omega (1/\varepsilon ^{d})}$ 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 ${\displaystyle O(\log(1/\varepsilon ))}$ queries using the bisection method). Here is a proof sketch. ^{[3] : 35} Construct a function ${\displaystyle g}$ that is slightly larger than ε everywhere in ${\displaystyle E^{d}}$ except in some small cube around some point x₀, where x₀ is the unique root of ${\displaystyle g}$. If ${\displaystyle g}$ is Lipschitz continuous with constant ${\displaystyle L}$, then the cube around x₀ can have a side-length of ${\displaystyle \varepsilon /L}$. Any algorithm that finds an ε-root of ${\displaystyle g}$ must check a set of cubes that covers the entire ${\displaystyle E^{d}}$; the number of such cubes is at least ${\displaystyle (L/\varepsilon )^{d}}$.

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 ${\displaystyle g}$ such that ${\displaystyle g(x)+x}$ maps ${\displaystyle E^{d}}$ to itself (that is: ${\displaystyle g(x)+x}$ is in ${\displaystyle E^{d}}$ for all x in ${\displaystyle E^{d}}$). This is because, for every such function, the function ${\displaystyle f(x):=g(x)+x}$ satisfies the conditions of Brouwer's fixed-point theorem. X is a fixed-point of ${\displaystyle f}$ if and only if x is a root of ${\displaystyle g}$, and x is an ε-residual fixed-point of ${\displaystyle f}$ if and only if x is an ε-root of ${\displaystyle g}$. Chen and Deng ^[18] show that the discrete variants of these problems are computationally equivalent: both problems require ${\displaystyle \Theta (n^{d-1})}$ 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 ${\displaystyle f}$ and the other knows a function ${\displaystyle g}$. Both functions are Lipschitz continuous and satisfy Brouwer's conditions. The goal is to compute an approximate fixed point of the composite function ${\displaystyle g\circ f}$. They show that the deterministic communication complexity is in ${\displaystyle \Omega (2^{d})}$.

References

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Further reading

• Yannakakis, Mihalis (May 2009). "Equilibria, fixed points, and complexity classes". Computer Science Review. 3 (2): 71–85. arXiv: 0802.2831 . doi:10.1016/j.cosrev.2009.03.004.