Fixed Point Computation: Beating Brute Force with Smoothed Analysis

TL;DR

This work studies the computational task of finding an $\varepsilon$-approximate fixed point of a smooth map on the $n$-dimensional $\ell_2$ unit ball by formulating it as a variational-inequality problem. It introduces a path-following algorithm driven by a reference function $K(x)=(x^T x I-xx^T)F(x)$, and proves that under smoothed perturbations, the algorithm runs in time roughly $e^{O(n)} \cdot \mathrm{poly}(1/\varepsilon)$, outperforming brute-force bounds of $\big(1/\varepsilon\big)^{O(n)}$. A complementary lower-bound analysis shows that any algorithm, even in a smoothed setting, requires $2^{\Omega(n)}$ queries to achieve an $O(1)$-approximate fixed point, matching the prevailing intuition that high-dimensional fixed-point problems resist fast general-purpose methods. The approach leverages a carefully regularized Jacobian spectral-gap property, bounds on path curvature and length via Weyl-tube arguments, and a discretized continuous dynamic that can generalize to variational inequalities and Nash equilibria. Overall, the paper demonstrates how higher-order smoothness and smoothed perturbations can yield exponential improvements over brute force in fixed-point computations, while clarifying fundamental limits through a tight lower bound.

Abstract

We propose a new algorithm that finds an $\varepsilon$-approximate fixed point of a smooth function from the $n$-dimensional $\ell_2$ unit ball to itself. We use the general framework of finding approximate solutions to a variational inequality, a problem that subsumes fixed point computation and the computation of a Nash Equilibrium. The algorithm's runtime is bounded by $e^{O(n)}/\varepsilon$, under the smoothed-analysis framework. This is the first known algorithm in such a generality whose runtime is faster than $(1/\varepsilon)^{O(n)}$, which is a time that suffices for an exhaustive search. We complement this result with a lower bound of $e^{Ω(n)}$ on the query complexity for finding an $O(1)$-approximate fixed point on the unit ball, which holds even in the smoothed-analysis model, yet without the assumption that the function is smooth. Existing lower bounds are only known for the hypercube, and adapting them to the ball does not give non-trivial results even for finding $O(1/\sqrt{n})$-approximate fixed points.

Figures (2)

• Figure 1: Consider $F(x,y)=(3x+y-1,x-2y+1)$. In the left figure, the boundary of the region is the yellow circle, the path $\gamma$ is the green ellipse inside, and the fixed point $x_0$ is $(2/7,3/7)$. The right figure zooms in and shows the first few steps of the following-the-path algorithm. From the origin (the $x_0$ in the figure denotes the fixed point), it goes through a vector $v_1$ traversing along the path to $y_1$, then go by a correction step $c_1$ to $x_1$. Then, use the same steps to $y_2,x_2,x_3,$ and so on, and finally reach the output point $x_{output}$ where it is close to the fixed point.
• Figure 2: Here, a step is going from $x(0)$ to $x(t)$, and its projection is from $\gamma(s(0))$ to $\gamma(s(t))$. We measure our changes by measuring the distance from $\gamma(s(0))$ to $\gamma(s(t))$, traversing on $\gamma$.

Theorems & Definitions (82)

• Definition 1.2: $\varepsilon$-approximate solution to a variational inequality
• Remark 1.3
• theorem 1.10: Smoothed-case. Informal version of Theorem \ref{['thm:smooth analysis final']}
• Corollary 1.11: Worst-case. Informal version of Theorem \ref{['thm:algorithm final']}
• Corollary 1.12: Convergence to Nash equilibrium
• theorem 1.13: Lower bound. Informal version of Theorem \ref{['thm:lowerBoundMain']}
• Lemma 2.2
• Lemma 2.3
• Proof
• Proposition 2.4: $L_J$-smooth and $L_K$-Lipschitz