27 research outputs found
Resolving a Conjecture on Degree of Regularity of Linear Homogeneous Equations
A linear equation is -regular, if, for every -coloring of the positive
integers, there exist positive integers of the same color which satisfy the
equation. In 2005, Fox and Radoicic conjectured that the equation , for any , has a degree of
regularity of , which would verify a conjecture of Rado from 1933. Rado's
conjecture has since been verified with a different family of equations. In
this paper, we show that Fox and Radoicic's family of equations indeed have a
degree of regularity of . We also provide a few extensions of this result.Comment: 8 page
Acyclic Subgraphs of Planar Digraphs
An acyclic set in a digraph is a set of vertices that induces an acyclic
subgraph. In 2011, Harutyunyan conjectured that every planar digraph on
vertices without directed 2-cycles possesses an acyclic set of size at least
. We prove this conjecture for digraphs where every directed cycle has
length at least 8. More generally, if is the length of the shortest
directed cycle, we show that there exists an acyclic set of size at least .Comment: 9 page
Near-Optimal No-Regret Learning in General Games
We show that Optimistic Hedge -- a common variant of
multiplicative-weights-updates with recency bias -- attains regret in multi-player general-sum games. In particular, when every player
of the game uses Optimistic Hedge to iteratively update her strategy in
response to the history of play so far, then after rounds of interaction,
each player experiences total regret that is . Our bound
improves, exponentially, the regret attainable by standard
no-regret learners in games, the regret attainable by no-regret
learners with recency bias (Syrgkanis et al., 2015), and the
bound that was recently shown for Optimistic Hedge in the special case of
two-player games (Chen & Pen, 2020). A corollary of our bound is that
Optimistic Hedge converges to coarse correlated equilibrium in general games at
a rate of .Comment: 40 page