arXiv subjects

Keith Frankston

Publications and source records attributed to Keith Frankston.

7 recordsLinked to original sources

arXiv2026

Accelerating Monte-Carlo Tree Search with Optimized Posterior Policies

We introduce a recursive AlphaZero-style Monte--Carlo tree search algorithm, "RMCTS". The advantage of RMCTS over AlphaZero's MCTS-UCB is speed. In RMCTS, the search tree is explored in a breadth-first manner, so that network inferences naturally occur in large batches. This significantly reduces the GPU latency cost. We find that RMCTS is often more than 40 times faster than MCTS-UCB when searching a single root state, and about 3 times faster when searching a large batch of root states. The recursion in RMCTS is based on computing optimized posterior policies at each game state in the search tree, starting from the leaves and working back up to the root. Here we use the posterior policy explored in "Monte--Carlo tree search as regularized policy optimization" (Grill, et al.) Their posterior policy is the unique policy which maximizes the expected reward given estimated action rewards minus a penalty for diverging from the prior policy. The tree explored by RMCTS is not defined in an adaptive manner, as it is in MCTS-UCB. Instead, the RMCTS tree is defined by following prior network policies at each node. This is a disadvantage, but the speedup advantage is more significant, and in practice we find that RMCTS-trained networks match the quality of MCTS-UCB-trained networks in roughly one-third of the training time. We include timing and quality comparisons of RMCTS vs. MCTS-UCB for three games: Connect-4, Dots-and-Boxes, and Othello.

cs.AI↗

arXiv2025

On Independent Spanning Trees in Random and Pseudorandom Graphs

In 1989, Zehavi and Itai conjectured that every $k$-connected graph contains $k$ independent spanning trees rooted at any prescribed vertex $r$. That is, for each vertex $v$, the unique $r$-$v$ paths within these $k$ spanning trees are internally disjoint. This fundamental problem has received much attention, in part motivated by its applications to network reliability, but despite that has only been resolved for $k \le 4$ and certain restricted graph families. We establish the conjecture for almost all graphs of essentially any relevant density. Specifically, we prove that there exists a constant $C > 1$ such that, with high probability, the random graph $G(n,p)$ contains $\delta(G)$ independent spanning trees rooted at any vertex whenever $C \log n/n \leq p < 0.99$. Since the lower bound on $p$ coincides (up to the constant $C$) with the connectivity threshold of $G(n,p)$, this result is essentially optimal. In addition, we show that $(n,d,\lambda)$-graphs with fairly mild bounds on the spectral ratio $d/\lambda$ contain $(1-o(1))d$ independent spanning trees rooted at each vertex, thereby settling the conjecture asymptotically for random $d$-regular graphs as well.

math.CO↗

arXiv2024

Proving Norine's Conjecture holds for $n=7$ via SAT solvers

We say a red/blue edge-coloring of the $n$-dimensional cube graph, $Q_n$, is antipodal if all pairs of antipodal edges have different colors. Norine conjectured that in such a coloring there must exist a pair of antipodal vertices connected by a monochromatic path. Previous work has proven this conjecture for $n\le 6$. Using SAT solvers we verify that the conjecture holds for $n = 7$.

math.CO↗

arXiv2024

Efficient Unbiased Sparsification

An unbiased $m$-sparsification of a vector $p\in \mathbb{R}^n$ is a random vector $Q\in \mathbb{R}^n$ with mean $p$ that has at most $m<n$ nonzero coordinates. Unbiased sparsification compresses the original vector without introducing bias; it arises in various contexts, such as in federated learning and sampling sparse probability distributions. Ideally, unbiased sparsification should also minimize the expected value of a divergence function $\mathsf{Div}(Q,p)$ that measures how far away $Q$ is from the original $p$. If $Q$ is optimal in this sense, then we call it efficient. Our main results describe efficient unbiased sparsifications for divergences that are either permutation-invariant or additively separable. Surprisingly, the characterization for permutation-invariant divergences is robust to the choice of divergence function, in the sense that our class of optimal $Q$ for squared Euclidean distance coincides with our class of optimal $Q$ for Kullback-Leibler divergence, or indeed any of a wide variety of divergences.

cs.IT↗

arXiv2021

On a problem of M. Talagrand

We address a special case of a conjecture of M. Talagrand relating two notions of "threshold" for an increasing family $\mathcal F$ of subsets of a finite set $V$. The full conjecture implies equivalence of the "Fractional Expectation-Threshold Conjecture," due to Talagrand and recently proved by the authors and B. Narayanan, and the (stronger) "Expectation-Threshold Conjecture" of the second author and G. Kalai. The conjecture under discussion here says there is a fixed $L$ such that if, for a given $\mathcal F$, $p\in [0,1]$ admits $λ:2^V \rightarrow \mathbb R^+$ with \[ \mbox{$\sum_{S\subseteq F}λ_S\ge 1 ~~\forall F\in \mathcal F$} \] and \[ \mbox{$\sum_Sλ_Sp^{|S|} \le 1/2$} \] (a.k.a. $\mathcal F$ is weakly $p$-small), then $p/L$ admits such a $λ$ taking values in $\{0,1\}$ ($\mathcal F$ is $(p/L)$-small). Talagrand showed this when $λ$ is supported on singletons and suggested, as a more challenging test case, proving it when $λ$ is supported on pairs. The present work provides such a proof.

math.CO↗

arXiv2019

Thresholds versus fractional expectation-thresholds

Proving a conjecture of Talagrand, a fractional version of the 'expectation-threshold' conjecture of Kalai and the second author, we show for any increasing family $F$ on a finite set $X$ that $p_c (F) =O( q_f (F) \log \ell(F))$, where $p_c(F)$ and $q_f(F)$ are the threshold and 'fractional expectation-threshold' of $F$, and $\ell(F)$ is the largest size of a minimal member of $F$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson--Kahn--Vu), bounded-degree spanning trees (Montgomery), and bounded-degree spanning graphs (new). We also resolve (and vastly extend) the 'axial' version of the random multi-dimensional assignment problem (earlier considered by Martin--Mézard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdős--Rado 'Sunflower Conjecture'.

math.CO↗

arXiv2017

On regular 3-wise intersecting families

Ellis and the third author showed, verifying a conjecture of Frankl, that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ admitting a transitive automorphism group has cardinality $o(2^n)$, while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ that is regular and increasing has cardinality $o(2^n)$.

math.CO↗