Let M be an n × n random matrix with entries in { 0 , 1 } , where each row is independently and uniformly sampled from the set of all vectors in { 0 , 1 } n containing exactly d ones, with d = p n for some fixed constant p ∈ ( 0 , 1 / 2 ] . A recent result of Tran states that the smallest singular value s n ( M ) is bounded below by c p n − 1 / 2 with high probability. In this note, we establish a complementary upper bound for s n ( M ) , proving that ∀ ε > 0 P ( s n ( M ) ≤ d ε 2 n ) ≥ 1 − C p ( ε + 1 d ) , where C p is a positive constant depending only on p . This result confirms that the least singular value s n ( M ) of dense random combinatorial matrices is typically of the order n − 1 / 2 .

We provide the first useful and rigorous analysis of ensemble sampling for the stochastic linear bandit setting. In particular, we show that, under standard assumptions, for a $d$-dimensional stochastic linear bandit with an interaction horizon $T$, ensemble sampling with an ensemble of size of order $\smash{d \log T}$ incurs regret at most of the order $\smash{(d \log T)^{5/2} \sqrt{T}}$. Ours is the first result in any structured setting not to require the size of the ensemble to scale linearly with $T$---which defeats the purpose of ensemble sampling---while obtaining near $\smash{\sqrt{T}}$ order regret. Our result is also the first to allow for infinite action sets.

We improve some upper bounds for minimal dispersion on the cube and torus. Our new ingredient is an improvement of a probabilistic lemma used to obtain upper bounds for dispersion in several previous works. Our new lemma combines a random and non-random choice of points in the cube. This leads to better upper bounds for the minimal dispersion.