Noisy Zeroth-Order Optimization for Non-smooth Saddle Point Problems

Abstract

This paper investigates zeroth-order methods for non-smooth convex-concave saddle point problems (with r-growth condition for duality gap). We assume that a black-box gradient-free oracle returns an inexact function value corrupted by an adversarial noise. In this work we prove that the standard zeroth-order version of the mirror descent method is optimal in terms of the oracle calls complexity and the maximum admissible noise.

The research was supported by Russian Science Foundation (project No. 21-71-30005).

A Auxiliary Results

This appendix presents auxiliary results to prove Theorem 1 from Sect. 3.

Lemma 1

Let vector $\boldsymbol{e}$ be a random unit vector from the Euclidean unit sphere $\{\boldsymbol{e}:\Vert \boldsymbol{e}\Vert _2=1\}$. Then it holds for all $r \in \mathbb R^d$

$$\begin{aligned} \mathbb E_{ \boldsymbol{e}}\left[ \left| \langle \boldsymbol{e}, r \rangle \right| \right] \le {\Vert r\Vert _2}/{\sqrt{d}}. \end{aligned}$$

Lemma 2

Let f(z) be $M_2$-Lipschitz continuous. Then for $f^\tau (z)$ from (4), it holds

$$ \sup _{z\in \mathcal {Z}}|f^\tau (z) - f(z)|\le \tau M_2. $$

Lemma 3

Function $f^\tau (z)$ is differentiable with the following gradient

$$ \nabla f^\tau (z) = \mathbb E_{{\boldsymbol{e}}}\left[ \frac{d}{\tau } f(z+\tau {\boldsymbol{e}})\boldsymbol{e}\right] . $$

Lemma 4

For $g(z,\xi ,\boldsymbol{e})$ from (3) and $f^\tau (z)$ from (4), the following holds

• 1. under Assumption 2

  $$\begin{aligned} \mathbb E_{\xi , \boldsymbol{e}}\left[ \langle g(z,\xi ,\boldsymbol{e}),r\rangle \right] \ge \langle \nabla f^\tau (z),r\rangle - {d\varDelta }{\tau ^{-1}} \mathbb E_{\boldsymbol{e}} \left[ \left| \langle \boldsymbol{e}, r \rangle \right| \right] , \end{aligned}$$

• 2. under Assumption 3

  $$\begin{aligned} \mathbb E_{\xi , \boldsymbol{e}}\left[ \langle g(z,\xi ,\boldsymbol{e}),r\rangle \right] \ge \langle \nabla f^\tau (z),r\rangle - d M_{2,\delta } \mathbb E_{\boldsymbol{e}} \left[ \left| \langle \boldsymbol{e}, r \rangle \right| \right] , \end{aligned}$$

Lemma 5

[24, Lemma 9]. For any function $f(\boldsymbol{e})$ which is M-Lipschitz w.r.t. the $\ell _2$-norm, it holds that if $\boldsymbol{e}$ is uniformly distributed on the Euclidean unit sphere, then

$$ \sqrt{\mathbb E\left[ (f(\boldsymbol{e}) - \mathbb Ef(\boldsymbol{e}))^4 \right] } \le cM_2^2/d $$

for some numerical constant c.

Lemma 6

For $g(z,\xi ,\boldsymbol{e})$ from (3), the following holds under Assumption 1

• 1. and Assumption 2

  $$\begin{aligned} \mathbb E_{\xi ,\boldsymbol{e}}\left[ \Vert g(z,\xi ,\boldsymbol{e})\Vert ^2_q\right] \le c a^2_q dM_2^2 + {d^2 a_q^2\varDelta ^2}/{\tau ^2}, \end{aligned}$$

• 2. and Assumption 3

  $$\begin{aligned} \mathbb E_{\xi ,\boldsymbol{e}}\left[ \Vert g(z,\xi ,\boldsymbol{e})\Vert ^2_q\right] \le c a^2_q d (M_2^2+M_{2,\delta }^2), \end{aligned}$$

where c is some numerical constant and $\sqrt{\mathbb E\left[ \Vert \boldsymbol{e}\Vert _q^4\right] } \le a_q^2$.