Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Distributed Online Optimization with Stochastic Agent Availability

Juliette Achddou, Nicolò Cesa-Bianchi, Hao Qiu

TL;DR

It is shown that the notion of regret (average-case over the agents) is essentially equivalent to the standard notion of regret (worst-case over agents), implying that the authors' bounds are not significantly improvable when $p=1$.

Abstract

Motivated by practical federated learning settings where clients may not be always available, we investigate a variant of distributed online optimization where agents are active with a known probability $p$ at each time step, and communication between neighboring agents can only take place if they are both active. We introduce a distributed variant of the FTRL algorithm and analyze its network regret, defined through the average of the instantaneous regret of the active agents. Our analysis shows that, for any connected communication graph $G$ over $N$ agents, the expected network regret of our FTRL variant after $T$ steps is at most of order $(κ/p^2)\min\big\{\sqrt{N},N^{1/4}/\sqrt{p}\big\}\sqrt{T}$, where $κ$ is the condition number of the Laplacian of $G$. We then show that similar regret bounds also hold with high probability. Moreover, we show that our notion of regret (average-case over the agents) is essentially equivalent to the standard notion of regret (worst-case over agents), implying that our bounds are not significantly improvable when $p=1$. Our theoretical results are supported by experiments on synthetic datasets.

Distributed Online Optimization with Stochastic Agent Availability

TL;DR

It is shown that the notion of regret (average-case over the agents) is essentially equivalent to the standard notion of regret (worst-case over agents), implying that the authors' bounds are not significantly improvable when .

Abstract

Motivated by practical federated learning settings where clients may not be always available, we investigate a variant of distributed online optimization where agents are active with a known probability at each time step, and communication between neighboring agents can only take place if they are both active. We introduce a distributed variant of the FTRL algorithm and analyze its network regret, defined through the average of the instantaneous regret of the active agents. Our analysis shows that, for any connected communication graph over agents, the expected network regret of our FTRL variant after steps is at most of order , where is the condition number of the Laplacian of . We then show that similar regret bounds also hold with high probability. Moreover, we show that our notion of regret (average-case over the agents) is essentially equivalent to the standard notion of regret (worst-case over agents), implying that our bounds are not significantly improvable when . Our theoretical results are supported by experiments on synthetic datasets.

Paper Structure

This paper contains 16 sections, 7 theorems, 22 equations, 5 figures, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Main contributions.
  3. Technical challenges.
  4. SETTING AND NOTATION
  5. THE Gossip-FTRL ALGORITHM
  6. UPPER BOUNDS
  7. Comparison with previous bounds.
  8. Lower bound on activation probabilities.
  9. Non stationary activation distributions.
  10. LOWER BOUND
  11. THE GOSSIP MATRIX
  12. A general bound on $\rho$.
  13. The $p$-uniform case.
  14. RANDOM EDGES
  15. EXPERIMENTS
  16. ...and 1 more sections

Key Result

Theorem 1

Assume each agent runs an instance of Gossip-FTRL with learning rate $\eta > 0$ and i.i.d gossip matrices $W_t$. Then, the expected network regret can be bounded by where $\rho = \sqrt{\lambda_2(\mathbb{E}[W_1 W_1^{\top}])}$. In the $p$-uniform case, we have If, in addition, ${ \eta = \frac{(D/L) \sqrt{\mu}}{2\sqrt{ 2p \min(p N, \sqrt N)T}} }$, then

Figures (5)

  • Figure 1: Empirical estimate $\hat{\rho}$ compared to $\rho$ and the upper bound $\hat{\rho}_{\text{up}}$\ref{['eq:up_bound_sym']} for $b = 1/\lambda_1(G)$ plotted as a function of $p \in [0,1]$.
  • Figure 2: Growth over $T=1000$ steps of the network regret of Gossip-FTRL and DOGD on a clique and on a grid for $N=36$ and for different choices of $p,q$.
  • Figure 3: Network regret of Gossip-FTRL and DOGD after $T=1000$ steps on a grid with $N=36$.
  • Figure 4: Plot of $(R^{\mathrm{net}}_T)^{-1/2}$ for Gossip-FTRL on a clique for $p \in [0,1]$ and $T=1000$.
  • Figure 5: Network regret of Gossip-FTRL after $T=1000$ steps when $p=0.5$, $q=1$, and $G$ is made up by two cliques joined by a varying number of random edges.

Theorems & Definitions (10)

  • Remark 1
  • Remark 2
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Theorem 5
  • Corollary 1
  • Corollary 2