[Paper Review] Dimension-Free Bounds on Chasing Convex Functions

C. J. Argue, Anupam Gupta+1 more|arXiv (Cornell University)|May 28, 2020

Optimization and Search Problems28 references4 citations

TL;DR

This paper presents dimension-free competitive algorithms for chasing convex functions in high-dimensional spaces by exploiting structural properties: for $κ$-well-conditioned functions, it achieves $O(\sqrt{\kappa})$-competitiveness independent of dimension; for $k$-dimensional affine subspaces, it gives $O(\min(k, \sqrt{k\log T}))$-competitiveness. The approach uses dimension reduction via affine isometries and leverages existing convex body chasing algorithms in low-dimensional subspaces.

ABSTRACT

We consider the problem of chasing convex functions, where functions arrive over time. The player takes actions after seeing the function, and the goal is to achieve a small function cost for these actions, as well as a small cost for moving between actions. While the general problem requires a polynomial dependence on the dimension, we show how to get dimension-independent bounds for well-behaved functions. In particular, we consider the case where the convex functions are $κ$-well-conditioned, and give an algorithm that achieves an $O(\sqrt κ)$-competitiveness. Moreover, when the functions are supported on $k$-dimensional affine subspaces--e.g., when the function are the indicators of some affine subspaces--we get $O(\min(k, \sqrt{k \log T}))$-competitive algorithms for request sequences of length $T$. We also show some lower bounds, that well-conditioned functions require $Ω(κ^{1/3})$-competitiveness, and $k$-dimensional functions require $Ω(\sqrt{k})$-competitiveness.

Motivation & Objective

• To overcome the curse of dimensionality in convex function chasing (CFC) by deriving dimension-independent competitive ratios.
• To study CFC under structural assumptions: well-conditioned functions and functions supported on low-dimensional affine subspaces.
• To design online algorithms that achieve competitive ratios independent of the ambient dimension $d$, while maintaining strong performance guarantees.
• To establish tight lower bounds for both settings to validate the optimality of the proposed algorithms.

Proposed method

• Propose a novel algorithm, COBD, for well-conditioned functions using a potential function and gradient-based updates to balance movement and function costs.
• Use affine isometries to embed high-dimensional $k$-dimensional convex bodies into a $2k+1$-dimensional subspace, preserving distances and costs.
• Reduce the problem of chasing $k$-dimensional affine subspaces to convex body chasing in $O(k)$ dimensions, leveraging existing $g(d)$-competitive algorithms.
• Apply dimension-reduction theorems to transform the original high-dimensional problem into a low-dimensional one with equivalent cost structure.
• Use the fact that $\|R_t(x_t) - R_{t-1}(x_{t-1})\| = \|x_t - x_{t-1}\|$ to maintain cost equivalence under transformation.
• Combine competitive algorithms for low-dimensional spaces with the transformed problem to derive overall competitive ratios.

Experimental results

Research questions

• RQ1 Can we achieve dimension-free competitive ratios for convex function chasing when functions are well-conditioned?
• RQ2 What is the optimal competitive ratio for chasing $k$-dimensional affine subspaces in $\mathbb{R}^d$?
• RQ3 Can we reduce high-dimensional convex function chasing to low-dimensional instances without losing competitiveness?
• RQ4 What are the tight lower bounds for well-conditioned and low-dimensional function chasing problems?
• RQ5 Is the $O(\sqrt{\kappa})$ competitive ratio for well-conditioned functions optimal up to constants?

Key findings

• An $O(\sqrt{\kappa})$-competitive algorithm is achieved for constrained CFC with $\kappa$-well-conditioned functions, independent of the ambient dimension $d$.
• For functions supported on $k$-dimensional affine subspaces, an $O(\min(k, \sqrt{k\log T}))$-competitive algorithm is achieved, again independent of $d$.
• A dimension-reduction technique via affine isometries preserves distances and allows reduction of $k$-dimensional chasing to $O(k)$-dimensional chasing.
• The lower bound for $k$-dimensional functions is $\Omega(\sqrt{k})$, showing that the $O(\sqrt{k\log T})$ bound is tight up to logarithmic factors.
• The lower bound for well-conditioned functions is $\Omega(\kappa^{1/3})$, indicating that the $O(\sqrt{\kappa})$ bound is not tight but represents a significant improvement over polynomial dependence on $\kappa$.
• The COBD algorithm is shown to be tight up to constants, with a matching $\Omega(\sqrt{\kappa})$ lower bound in the unconstrained setting.