Score estimation has recently emerged as a key modern statistical challenge, due to its pivotal role in generative modelling via diffusion models. Moreover, it is an essential ingredient in a new approach to linear regression via convex $M$-estimation, where the corresponding error densities are projected onto the log-concave class. Motivated by these applications, we study the minimax risk of score estimation with respect to squared $L^2(P_0)$-loss, where $P_0$ denotes an underlying log-concave distribution on $\mathbb{R}$. Such distributions have decreasing score functions, but on its own, this shape constraint is insufficient to guarantee a finite minimax risk. We therefore define subclasses of log-concave densities that capture two fundamental aspects of the estimation problem. First, we establish the crucial impact of tail behaviour on score estimation by determining the minimax rate over a class of log-concave densities whose score function exhibits controlled growth relative to the quantile levels. Second, we explore the interplay between smoothness and log-concavity by considering the class of log-concave densities with a scale restriction and a $(β,L)$-Hölder assumption on the log-density for some $β\in [1,2]$. We show that the minimax risk over this latter class is of order $L^{2/(2β+1)}n^{-β/(2β+1)}$ up to poly-logarithmic factors, where $n$ denotes the sample size. When $β< 2$, this rate is faster than could be obtained under either the shape constraint or the smoothness assumption alone. Our upper bounds are attained by a locally adaptive, multiscale estimator constructed from a uniform confidence band for the score function. This study highlights intriguing differences between the score estimation and density estimation problems over this shape-constrained class.

The construction of confidence intervals for the mean of a bounded random variable is a classical problem in statistics with numerous applications in machine learning and virtually all scientific fields. In particular, obtaining the tightest possible confidence intervals is vital every time the sampling of the random variables is expensive. The current state-of-the-art method to construct confidence intervals is by using betting algorithms. This is a very successful approach for deriving optimal confidence sequences, even matching the rate of law of iterated logarithms. However, in the fixed horizon setting, these approaches are either sub-optimal or based on heuristic solutions with strong empirical performance but without a finite-time guarantee. Hence, no betting-based algorithm guaranteeing the optimal $\mathcal{O}(\sqrt{\frac{σ^2\log\frac1δ}{n}})$ width of the confidence intervals are known. This work bridges this gap. We propose a betting-based algorithm to compute confidence intervals that empirically outperforms the competitors. Our betting strategy uses the optimal strategy in every step (in a certain sense), whereas the standard betting methods choose a constant strategy in advance. Leveraging this fact results in strict improvements even for classical concentration inequalities, such as the ones of Hoeffding or Bernstein. Moreover, we also prove that the width of our confidence intervals is optimal up to an $1+o(1)$ factor diminishing with $n$. The code is available at https://github.com/vvoracek/STaR-bets-confidence-interval.

We study the adversarial linear bandits problem and present a unified algorithmic framework that bridges Follow-the-Regularized-Leader (FTRL) and Follow-the-Perturbed-Leader (FTPL) methods, extending the known connection between them from the full-information setting. Within this framework, we introduce self-concordant perturbations, a family of probability distributions that mirror the role of self-concordant barriers previously employed in the FTRL-based SCRiBLe algorithm. Using this idea, we design a novel FTPL-based algorithm that combines self-concordant regularization with efficient stochastic exploration. Our approach achieves a regret of $O(d\sqrt{n \ln n})$ on both the $d$-dimensional hypercube and the Euclidean ball. On the Euclidean ball, this matches the rate attained by existing self-concordant FTRL methods. For the hypercube, this represents a $\sqrt{d}$ improvement over these methods and matches the optimal bound up to logarithmic factors.

In this section we first give a game-theoretic perspective of our method when applied to the statistical setting (Subsection A.1). A formal description is provided in Algorithm 4. Algorithm 4 Boosting with OCO It is described in this section. Player A's goal is to minimize the payoff, while player B's goal is to maximize it. There are several ways to circumvent it. If players A and B play according to Algorithm 5, then player B's average strategy's depend on the sequence of q's, they are also random variables, as well as p.

And if they are too close to each other, it is also quite clear that spectral methods will not work. However, we highlight these statements in Figure 1. Then we will proceed to prove technical statements made in Section 1. Let us first consider a preliminary remark on the connectivity of the random graph. Then one has that P ( 9 an isolated vertex i, 1 i N)! 0 as N!1 .

We study the properties of the Neural Tangent Kernel (NTK) $\overset{\scriptscriptstyle\infty}{K} : \mathbb{R}^{m_0} \times \mathbb{R}^{m_0} \to \mathbb{R}^{m_l \times m_l}$ corresponding to infinitely wide $l$-layer Multilayer Perceptrons (MLPs) taking inputs from $\mathbb{R}^{m_0}$ to outputs in $\mathbb{R}^{m_l}$ equipped with activation functions $\phi(s) = a s + b \vert s \vert$ for some $a,b \in \mathbb{R}$ and initialized at the Edge Of Chaos (EOC). We find that the entries $\overset{\scriptscriptstyle\infty}{K}(x_1,x_2)$ can be approximated by the inverses of the cosine distances of the activations corresponding to $x_1$ and $x_2$ increasingly better as the depth $l$ increases. By quantifying these inverse cosine distances and the spectrum of the matrix containing them, we obtain tight spectral bounds for the NTK matrix $\overset{\scriptscriptstyle\infty}{K} = [\frac{1}{n} \overset{\scriptscriptstyle\infty}{K}(x_{i_1},x_{i_2}) : i_1, i_2 \in [1:n]]$ over a dataset $\{x_1,\cdots,x_n\} \subset \mathbb{R}^{m_0}$, transferred from the inverse cosine distance matrix via our approximation result. Our results show that $\Delta_\phi = \frac{b^2}{a^2+b^2}$ determines the rate at which the condition number of the NTK matrix converges to its limit as depth increases, implying in particular that the absolute value ($\Delta_\phi=1$) is better than the ReLU ($\Delta_\phi=\frac{1}{2}$) in this regard.

We analyse preference inference, through consistency, for general preference languages based on lexicographic models. We identify a property, which we call strong compositionality, that applies for many natural kinds of preference statement, and that allows a greedy algorithm for determining consistency of a set of preference statements. We also consider different natural definitions of optimality, and their relations to each other, for general preference languages based on lexicographic models. Based on our framework, we show that testing consistency, and thus inference, is polynomial for a specific preference language LpqT, which allows strict and non-strict statements, comparisons between outcomes and between partial tuples, both ceteris paribus and strong statements, and their combination. Computing different kinds of optimal sets is also shown to be polynomial; this is backed up by our experimental results.

This paper explores the expressive power of deep neural networks for a diverse range of activation functions. An activation function set $\mathscr{A}$ is defined to encompass the majority of commonly used activation functions, such as $\mathtt{ReLU}$, $\mathtt{LeakyReLU}$, $\mathtt{ReLU}^2$, $\mathtt{ELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, $\mathtt{Mish}$, $\mathtt{Sigmoid}$, $\mathtt{Tanh}$, $\mathtt{Arctan}$, $\mathtt{Softsign}$, $\mathtt{dSiLU}$, and $\mathtt{SRS}$. We demonstrate that for any activation function $\varrho\in \mathscr{A}$, a $\mathtt{ReLU}$ network of width $N$ and depth $L$ can be approximated to arbitrary precision by a $\varrho$-activated network of width $3N$ and depth $2L$ on any bounded set. This finding enables the extension of most approximation results achieved with $\mathtt{ReLU}$ networks to a wide variety of other activation functions, albeit with slightly increased constants. Significantly, we establish that the (width,$\,$depth) scaling factors that appeared in the previous result can be further reduced from $(3,2)$ to $(1,1)$ if $\varrho$ falls within a specific subset of $\mathscr{A}$. This subset includes activation functions such as $\mathtt{ELU}$, $\mathtt{SELU}$, $\mathtt{Softplus}$, $\mathtt{GELU}$, $\mathtt{SiLU}$, $\mathtt{Swish}$, and $\mathtt{Mish}$.

The Wasserstein distance, rooted in optimal transport (OT) theory, is a popular discrepancy measure between probability distributions with various applications to statistics and machine learning. Despite their rich structure and demonstrated utility, Wasserstein distances are sensitive to outliers in the considered distributions, which hinders applicability in practice. Inspired by the Huber contamination model, we propose a new outlier-robust Wasserstein distance $\mathsf{W}_p^\varepsilon$ which allows for $\varepsilon$ outlier mass to be removed from each contaminated distribution. Our formulation amounts to a highly regular optimization problem that lends itself better for analysis compared to previously considered frameworks. Leveraging this, we conduct a thorough theoretical study of $\mathsf{W}_p^\varepsilon$, encompassing characterization of optimal perturbations, regularity, duality, and statistical estimation and robustness results. In particular, by decoupling the optimization variables, we arrive at a simple dual form for $\mathsf{W}_p^\varepsilon$ that can be implemented via an elementary modification to standard, duality-based OT solvers. We illustrate the benefits of our framework via applications to generative modeling with contaminated datasets.