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Delayed Feedback in Online Non-Convex Optimization: A Non-Stationary Approach with Applications

Felipe Lara, Cristian Vega

TL;DR

This work studies non-convex delayed-noise online optimization problems by evaluating dynamic regret in the non-stationary setting when the loss functions are quasar-convex and provides new examples of non-convex functions that are quasar-convex.

Abstract

We study non-convex delayed-noise online optimization problems by evaluating dynamic regret in the non-stationary setting when the loss functions are quasar-convex. In particular, we consider scenarios involving quasar-convex functions either with a Lipschitz gradient or weakly smooth and, for each case, we ensure bounded dynamic regret in terms of cumulative path variation achieving sub-linear regret rates. Furthermore, we illustrate the flexibility of our framework by applying it to both theoretical settings such as zeroth-order (bandit) and also to practical applications with quadratic fractional functions. Moreover, we provide new examples of non-convex functions that are quasar-convex by proving that the class of differentiable strongly quasiconvex functions (Polyak 1966) are strongly quasar-convex on convex compact sets. Finally, several numerical experiments validate our theoretical findings, illustrating the effectiveness of our approach.

Delayed Feedback in Online Non-Convex Optimization: A Non-Stationary Approach with Applications

TL;DR

This work studies non-convex delayed-noise online optimization problems by evaluating dynamic regret in the non-stationary setting when the loss functions are quasar-convex and provides new examples of non-convex functions that are quasar-convex.

Abstract

We study non-convex delayed-noise online optimization problems by evaluating dynamic regret in the non-stationary setting when the loss functions are quasar-convex. In particular, we consider scenarios involving quasar-convex functions either with a Lipschitz gradient or weakly smooth and, for each case, we ensure bounded dynamic regret in terms of cumulative path variation achieving sub-linear regret rates. Furthermore, we illustrate the flexibility of our framework by applying it to both theoretical settings such as zeroth-order (bandit) and also to practical applications with quadratic fractional functions. Moreover, we provide new examples of non-convex functions that are quasar-convex by proving that the class of differentiable strongly quasiconvex functions (Polyak 1966) are strongly quasar-convex on convex compact sets. Finally, several numerical experiments validate our theoretical findings, illustrating the effectiveness of our approach.

Paper Structure

This paper contains 17 sections, 14 theorems, 73 equations, 7 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. State of the Art and Related Work
  3. Preliminaries
  4. Main Results
  5. DOGD for Quasar-Convex Functions with Non-Stationary Regret
  6. Extension to the Bandit Setting with Delays
  7. Examples and Applications
  8. New Examples of Quasar-Convex Functions
  9. Other Non-Convex Examples
  10. Numerical Experiments
  11. Conclusions and Future Work
  12. Declarations
  13. Availability of supporting data
  14. Author Contributions
  15. Competing Interests
  16. ...and 2 more sections

Key Result

Lemma 3.1

(lara2022strongly) Let $\mathcal{X} \subseteq \mathbb{R}^{p}$ be a closed and convex set and $f\colon \mathcal{X} \rightarrow \mathbb{R}$ be a lsc and strongly qua-si-con-vex function on $\mathcal{X}$ with modulus $\gamma> 0$. Then $\mathop{\mathrm{\text{\rm argmin}}}\limits\limits_{ \mathcal{X}}\,

Figures (7)

  • Figure 1: An illustration of the non-convex function $f(x)$ described in Example \ref{['ex1']}. On the left-hand side, a 3D plot of the function $f(x)$ is displayed, while on the right-hand side, an arbitrary segment that does not contain the minimizer is shown, highlighting the non-convexity of $f$.
  • Figure 2: Average regret of Algorithm \ref{['DOGD-SC']} on the Lipschitz continuous function defined in \ref{['ex:function1']} withh $d \in \{1, 5, 10, 20\}$.
  • Figure 3: Average regret of Algorithm \ref{['DOGD-SC']} on the Lipschitz continuous function defined in \ref{['ex:function1']} in a high-delay setting with $d \in \{20, 50, 100, 150, 200\}.$
  • Figure 4: Average regret of Algorithm \ref{['DOGD-SC']} on the weakly smooth function defined in \ref{['ex:f2']}.
  • Figure 5: Average regret of Algorithm \ref{['DOGD-SC']} on the weakly smooth function defined in \ref{['ex:f2']} for differents $V_{T}$.
  • ...and 2 more figures

Theorems & Definitions (36)

  • Lemma 3.1
  • Lemma 3.2
  • Remark 3.3
  • Definition 3.4
  • Lemma 3.5
  • Proposition 3.6
  • proof
  • Example 3.7
  • Lemma 3.8
  • proof
  • ...and 26 more