Chasing Convex Bodies Optimally

Abstract

In the chasing convex bodies problem, an online player receives a request sequence of N convex sets $K_1,\dots , K_N$ contained in a normed space X of dimension d. The player starts at $x_0=0\in X$, and at time n observes the set $K_n$ and then moves to a new point $x_n\in K_n$, paying a cost $||x_n-x_{n-1}||$. The player aims to ensure the total cost exceeds the minimum possible total cost by at most a bounded factor $\alpha _d$ independent of N, despite $x_n$ being chosen without knowledge of the future sets $K_{n+1},\dots ,K_N$. The best possible $\alpha _d$ is called the competitive ratio. Finiteness of the competitive ratio for convex body chasing was proved for $d=2$ in Friedman and Linial (Discrete Comput. Geom. 9(3):293–321, 1993.) and conjectured for all d. Bubeck et al. (Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pp. 861–868, 2019) recently resolved this conjecture, proving an exponential $2^{O(d)}$ upper bound on the competitive ratio.

We give an improved algorithm achieving competitive ratio d in any normed space, which is exactly tight for $\ell ^{\infty }$. In Euclidean space, our algorithm also achieves competitive ratio $O(\sqrt {d\log N})$, nearly matching a $\sqrt {d}$ lower bound when N is subexponential in d. Our approach extends that of Bubeck et al. (Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1496–1508. SIAM, 2020.) for nested convex bodies, which is based on the classical Steiner point of a convex body. We define the functional Steiner point of a convex function and apply it to the associated work function.

Appendix: Proof of Lemma 3.7

Proof

We prove the result for all $v\in B_1^*$ where $\nabla W^*_t(v)$ exists. This includes almost all v by Alexandrov’s theorem. Moreover it ensures the conjugate point $v_t^*=\arg \min _{w\in X} W(w)-\langle v,w\rangle$ is well-defined and that $W_t$ is strictly convex at $v_t^*$ [30, Corollary 25.1.2]. We write:

$$\displaystyle \begin{aligned} W_{t+\delta}(v)=& \min_{x_s:[0,t+\delta]\to X} \left(\int_{0}^{t+\delta} (f_s(x_s)+||x^{\prime}_s||\text{d} s -\langle v,x_{t+\delta}\rangle\right)\\ =& \min_{ x_s:[t,t+\delta]\to X}\left(W_t(x_t)+\int_{t}^{t+\delta}f_s(x_s)+||x^{\prime}_s||\text{d} s - \langle v,x_{t+\delta}\rangle\right) \end{aligned} $$

For small $\delta \in (0,\varepsilon )$, we show $W^*_{t+\delta }(v)=W^*_t(v)+\delta f_t(v^*_t)+o(\delta )$. For the upper bound,

$$\displaystyle \begin{aligned} W_{t+\delta}(v_t^*)&\leq W_t(v_t^*)+\int_t^{t+\delta} f_s(v^*_t)\text{d} s\\ &= W_t(v_t^*)+\delta f_t(v^*_t)+o(\delta) \end{aligned} $$

holds by taking $x_s=v_t^*$ constant for $s\in [t,t+\delta )$ and recalling the assumption that $f_s(x)$ is continuous on $s\in [t,t+\delta )$. Since $v_t^*=\arg \min _x \big (W_t(x)-\langle x,v\rangle \big )$, the upper bound follows from

$$\displaystyle \begin{aligned} W^*_{t+\delta}(v)&\leq W_{t+\delta}(v_t^*)-\langle v,v_t^*\rangle\\ &\leq W_t(v_t^*)+\delta f_t(v^*_t)+o(\delta)-\langle v,v_t^*\rangle\\ &=W^*_t(v)+\delta f_t(v^*_t)+o(\delta). \end{aligned} $$

For the lower bound, the strict convexity of $W_t$ at $v_t^*$ implies

$$\displaystyle \begin{aligned} W_t(x)= W_t(v^*_t)+\langle v,x-v^*_t\rangle +\gamma(||x-v^*_t||) \end{aligned}$$

where $\gamma :\mathbb R^+\to \mathbb R^+$ is continuous and increasing with unique minimum $F(0)=0$. Therefore any path $x_s:[0,t+\delta ]\to X$ satisfies:

$$\displaystyle \begin{aligned} W_t(x_t)&+\int_{t}^{t+\delta}f_s(x_s)+||x^{\prime}_s||\text{d} s - \langle v,x_{t+\delta}\rangle \geq W_t(v^*_t)+\langle v,x_t-v^*_t\rangle\\ &+\gamma(||x_t-v^*_t||) + \int_t^{t+\delta} f_s(x_s)+||x^{\prime}_s||\text{d} s-\langle v,x_{t+\delta}\rangle. \end{aligned} $$

The observation $\int _t^{t+\delta } ||x^{\prime }_s||\text{d} s \geq ||x_{t+\delta }-x_t|| \geq \langle v,x_{t+\delta }-x_t\rangle$ implies

$$\displaystyle \begin{aligned} W_t(x_t)+\int_{t}^{t+\delta}f_s(x_s)+||x^{\prime}_s||\text{d} s - \langle v,x_{t+\delta}\rangle\geq & W_t(x_t)-\langle v,v^*_t\rangle+f(||x_t-v^*_t||) \\ &\quad + \int_t^{t+\delta} f_s(x_s)\text{d} s\\ \geq & W_t(v^*_t)-\langle v,v^*_t\rangle +\gamma(||x_t-v^*_t||) \\ & \quad + \int_t^{t+\delta} f_s(x_s)\text{d} s\\ \geq & W^*_t(v)+\gamma(||x_t-v^*_t||)\\ &\quad + \int_t^{t+\delta} f_s(x_s)\text{d} s. \end{aligned} $$

Because $W_{t+\delta }(v)=W_t(v)+O(\delta )$, we see that for $\delta \to 0$ small we must have $||x_t-v_t^*||=o_{\delta \to 0}(1)$ for any optimal trajectory $x_s$ witnessing the correct value $W_{t+\delta }$. Additionally,

$$\displaystyle \begin{aligned} \int_t^{t+\delta} ||x^{\prime}_s||\text{d} s + \langle v, x_t-x_{t+\delta}\rangle \geq (1-|v|)\int_t^{t+\delta} ||x^{\prime}_s||\text{d} s \geq (1-|v|)\sup_{s\in [t,t+\delta]} |x_t-x_s|. \end{aligned}$$

which similarly implies $\sup _{s\in [t,t+\delta ]}||x_t-x_s||=o(1)$ for any optimal trajectory since $||v||<1$. It follows that all optimal trajectories satisfy $\int _t^{t+\delta } f_s(x_s)\text{d} s = \delta f_t(v^*_t) +o(\delta ).$ This concludes the proof. □