Faster Convergence with Multiway Preferences
Aadirupa Saha Vitaly Feldman Tomer Koren Yishay Mansour
We address the problem of convex optimization with preference feedback, where
the goal is to minimize a convex function given a weaker form of comparison
queries. Each query consists of two points and the dueling feedback returns a
(noisy) single-bit binary comparison of the function values of the two queried
points. Here we consider the sign-function-based comparison feedback model and
analyze the convergence rates with batched and multiway (argmin of a set
queried points) comparisons. Our main goal is to understand the improved
convergence rates owing to parallelization in sign-feedback-based optimization
problems. Our work is the first to study the problem of convex optimization
with multiway preferences and analyze the optimal convergence rates. Our first
contribution lies in designing efficient algorithms with a convergence rate of
$\smash{\widetilde O}(\frac{d}{\min\{m,d\} \epsilon})$ for $m$-batched
preference feedback where the learner can query $m$-pairs in parallel. We next
study a $m$-multiway comparison (`battling') feedback, where the learner can
get to see the argmin feedback of $m$-subset of queried points and show a
convergence rate of $\smash{\widetilde O}(\frac{d}{ \min\{\log m,d\}\epsilon
})$. We show further improved convergence rates with an additional assumption
of strong convexity. Finally, we also study the convergence lower bounds for
batched preferences and multiway feedback optimization showing the optimality
of our convergence rates w.r.t. $m$.