The unit ℓ2-spherein d-dimensions that is centered at the origin is denoted bySd 1. Additionally, given a pair of symmetric matricesA,B Rd, we write A B if and only if x (A B)x 0, x Rd. More linear algebra facts appear in AppendixE. Let V P be a subset of distributions indexed by the points in the hypercubeEd = { 1,1}d. For a number of facts from probability and statistics (both related and unrelated to exponential families),wereferthereadertoAppendixF.

In this setting, unlike basic setting, objective and constraints are not linear. We focus on a single state-action pairs,a, stage h, and objectivem. Similarly, in constrained settings, its estimated resource consumptions are underestimates of the true resource consumptions. B.5 BoundingtheBellmanerror We now provide an upper bound on the Bellman error which arises in the RHS of the regret decomposition(Proposition3.3). When neither failure events occur (probability 1 2δ), Proposition 3.3 upper bounds either of reward or consumption regret by In this section, we prove the main guarantee for the convex-concave setting.

The variance was computed empirically overk = 100 normally distributed samplesw. As can be seen, the variance of the kernel tends to zero only when both widths increase. The hyperkernel used corresponds to the infinite width limit ofthesame architecture. As can be seen in Figure 1, whenf is wide and kept fixed, there is a clear improvement in test performance as the width ofg increases, for every learning rate in which the networks provide non-trivial performance. Whenf is wide and kept fixed, a deeperg incurs slower training and lower overall test performance.

Inthispaper,weanalyze the theoretical properties of the IS estimator by deriving a novel anticoncentration bound that formalizes the intuition behind itsundesired behavior. Then, we propose anew class of IS transformations, based on the notion of power mean. To the best of our knowledge, the resulting estimator is the first to achieve, under certainconditions, twokeyproperties: (i)itdisplays asubgaussianconcentration rate; (ii) it preserves the differentiability in the target distribution.