Related papers: Algorithms and Complexity for Functions on General…

Existing bounds on the generalization error of deep networks assume some form of smooth or bounded dependence on the input variable, falling short of investigating the mechanisms controlling such factors in practice. In this work, we…

The unconstrained minimization of a sufficiently smooth objective function $f(x)$ is considered, for which derivatives up to order $p$, $p\geq 2$, are assumed to be available. An adaptive regularization algorithm is proposed that uses…

Modern large-scale statistical models require to estimate thousands to millions of parameters. This is often accomplished by iterative algorithms such as gradient descent, projected gradient descent or their accelerated versions. What are…

In the setup of i.i.d.~observations and a real valued differentiable functional~$T$, locally asymptotic upper bounds are derived for the power of one-sided tests (simple, versus large values of~$T$)and for the confidence probability of…

Dynamic nonlinear systems exhibit distortions arising from coupled static and dynamic effects. Their intertwined nature poses major challenges for data-driven modeling. This paper presents a theoretical framework grounded in structured…

In this paper, we provide the universal first-order methods of Composite Optimization with new complexity analysis. It delivers some universal convergence guarantees, which are not linked directly to any parametric problem class. However,…

We investigate the sample complexity of networks with bounds on the magnitude of its weights. In particular, we consider the class \[ H=\left\{W_t\circ\rho\circ \ldots\circ\rho\circ W_{1} :W_1,\ldots,W_{t-1}\in M_{d, d}, W_t\in…

In a previous article the authors determined the best-known upper bound for the cardinality of the image set for several classes of functions, including planar functions. Here, we show that the upper bound cannot be tight for planar…

We provide tools to help automate the error analysis of algorithms that evaluate simple functions over the floating-point numbers. The aim is to obtain tight relative error bounds for these algorithms, expressed as a function of the unit…

Many problems of theoretical and practical interest involve finding an optimum over a family of convex functions. For instance, finding the projection on the convex functions in $H^k(\Omega)$, and optimizing functionals arising from some…

This paper is devoted to the analysis of a finite horizon discrete-time stochastic optimal control problem, in presence of constraints. We study the regularity of the value function which comes from the dynamic programming algorithm. We…

The performance of online convex optimization algorithms in a dynamic environment is often expressed in terms of the dynamic regret, which measures the decision maker's performance against a sequence of time-varying comparators. In the…

While deep learning is successful in a number of applications, it is not yet well understood theoretically. A satisfactory theoretical characterization of deep learning however, is beginning to emerge. It covers the following questions: 1)…

Within a mathematically rigorous model, we analyse the curse of dimensionality for deterministic exact similarity search in the context of popular indexing schemes: metric trees. The datasets $X$ are sampled randomly from a domain $\Omega$,…

We prove an exponential size separation between depth 2 and depth 3 neural networks (with real inputs), when approximating a $\mathcal{O}(1)$-Lipschitz target function to constant accuracy, with respect to a distribution with support in the…

We study the problem of estimating the average of a Lipschitz continuous function $f$ defined over a metric space, by querying $f$ at only a single point. More specifically, we explore the role of randomness in drawing this sample. Our goal…

Typical adversarial-training-based unsupervised domain adaptation methods are vulnerable when the source and target datasets are highly-complex or exhibit a large discrepancy between their data distributions. Recently, several…

The paper addresses parametric inequality systems described by polynomial functions in finite dimensions, where state-dependent infinite parameter sets are given by finitely many polynomial inequalities and equalities. Such systems can be…

This paper provides statistical guarantees on the accuracy of dynamical models learned from dependent data sequences. Specifically, we develop uniform error bounds that apply to quantized models and imperfect optimization algorithms…

Given values of a piecewise smooth function $f$ on a square grid within a domain $\Omega$, we look for a piecewise adaptive approximation to $f$. Standard approximation techniques achieve reduced approximation orders near the boundary of…