We provide a complete theory of optimal universal rates for binary classification in the agnostic setting. This extends the realizable-case theory of Bousquet, Hanneke, Moran, van Handel, and Yehudayoff (2021) by removing the realizability assumption on the distribution. We identify a fundamental tetrachotomy of optimal rates: for every concept class, the optimal universal rate of convergence of the excess error rate is one of $e^{-n}$, $e^{-o(n)}$, $o(n^{-1/2})$, or arbitrarily slow. We further identify simple combinatorial structures which determine which of these categories any given concept class falls into.

Eq. (4) and the fact that σ () is 1-Lipschitz on [ 1, +1], we get the expression b n null σ (δ) + σ ( δ) 2 null 1 2 p Recalling Eq. (3), we get that by fixing s = Considering this constraint in Eq. (6), we see that for any choice of To continue, it will be convenient to get rid of the absolute value in the displayed expression above. Considering Eq. (8), this is Fortunately, the Rademacher complexity of such composed classes was analyzed in Golowich et al. [2017] for a Rademacher complexity of H . 15 To complete the proof, we need to employ a standard upper bound on Upper bounding this by ϵ, solving for m and simplifying a bit, the result follows. By Markov's inequality, it follows that with probability at least Thm. 3 implies that a certain dataset A.5 Proofs of Thm. 4 and Thm. 5 In what follows, given a vector u Lemma 4. Given a vector Upper bounding this by ϵ and solving for m, the result follows. We now utilize equation (4.20) in Ledoux and Talagrand [1991], which implies We now turn to prove the theorem. Cauchy-Schwartz and Jensen's inequalities, this in turn can be upper bounded as follows: E The proof follows from a covering number argument.

Convolutional Neural Networks (CNNs) have revolutionized the field of computer vision, achieving remarkable success in tasks such as image classification(Krizhevsky et al., 2012), object detection (Ren et al., 2016), and segmentation (Long et al., 2015). Their effectiveness can be partly attributed to their translation invariant architecture, enabling CNNs to recognize objects regardless of their position in an image. However, while CNNs are effective at capturing translation symmetries, there has been a growing interest in incorporating additional structure into neural networks to handle a wider range of transformations. These architectures aim to combine the flexibility of learning with the robustness of structure-preserving features (see, e.g., (Hinton and Wang, 2011; Lee et al., 2015)). GCNNs were first introduced by Cohen and Welling (2016a) to improve statistical efficiency and enhance geometric reasoning. Since then equivariant network structures have evolved to support equivariance on Euclidean groups(Bekkers et al., 2018; Bekkers, 2019; Weiler et al., 2018), compact groups(Kondor and Trivedi, 2018) and Riemannian manifolds (Weiler et al., 2021). More recent architectures have even been generalized beyondother types ofsymmetrygroups(Zhdanov et al., 2024; Dehmamy et al., 2021; Smets et al., 2023)

In 1984, Valiant [ 7 ] introduced the Probably Approximately Correct (PAC) learning framework for boolean function classes. Blumer et al. [ 2] extended this model in 1989 by introducing the VC dimension as a tool to characterize the learnability of PAC. The VC dimension was based on the work of Vapnik and Chervonenkis in 1971 [8 ], who introduced a tool called the growth function to characterize the shattering property. Researchers have since determined the VC dimension for specific classes, and efforts have been made to develop an algorithm that can calculate the VC dimension for any concept class. In 1991, Linial, Mansour, and Rivest [4] presented an algorithm for computing the VC dimension in the discrete setting, assuming that both the concept class and domain set were finite. However, no attempts had been made to design an algorithm that could compute the VC dimension in the general setting.Therefore, our work focuses on developing a method to approximately compute the VC dimension without constraints on the concept classes or their domain set. Our approach is based on our finding that the Empirical Risk Minimization (ERM) learning paradigm can be used as a new tool to characterize the shattering property of a concept class.

In the 1950s I spent a significant chunk of my pocket money buying a transistor. It was a small metal cylinder (about 5mm in diameter and 7mm deep) with three wires protruding from its base. I needed it for a little radio I was building, and buying it was a big deal for a lad living in rural Ireland. My baffled parents couldn't understand why this gizmo their son was holding between finger and thumb could be interesting; and, to be honest, you couldn't blame them. The A13 processor that powers the iPhone that I used to find a photograph of that first transistor has 8.5 billion of them etched on to a sliver of silicon no bigger than a fingernail – a "chip".

AI has the potential to benefit the field of design in several ways, one of which is the ability for us to train AI to perform many of the tedious tasks that are otherwise repetitive, allowing us to concentrate on the strategic & creative areas of design! As well as this, AI enables large steps in customer experience, enabling the automated study of user behaviours and preferences to create unique experiences for every individual based on their data -- This would be a huge boost to client engagement and satisfaction! Another point of value that we could gain here is within research & testing, as through machine learning we may be able to better understand our consumer's preferences -- Paired within swift testing & iterative design, this allows our concepts to better suit our users, driving better adoption.

We study combinatorial complexity of certain classes of products of intervals in $\mathbb{R}^d$, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the Vapnik-Chervonenkis dimension of the set of balls in $\ell_\infty^d$ -- which denotes $\R^d$ equipped with the sup norm -- equals $\lfloor (3d+1)/2\rfloor$.

Every ad tech vendor claims they have built Artificial Intelligence (AI) into their solution. Machine learning holds incredible promise, but how much do you really know about it? Get clarity on some of AI's most misunderstood terms like unsupervised learning, deep learning, and neural networks. Learn from the experts at Forrester so you can more accurately assess the AI expertise of vendors and their solutions, and avoid pitfalls that have befallen other companies.

It's a question most of us have asked at one point in our lives -is Santa real? Today's children aren't looking to their parents for an answer, but are turning to Google and the search engine is shattering the shattering the illusion. A report found that 1.1 million children learn online that Saint Nick is a fictitious character, as the first article in the search says'as adults we know Santa Claus isn't real.' When searching'Is Santa real' the first article that is displayed comes from Quartz, which provides parents with advice on how to answer the question . And the introductory sentence of the article reads: 'As adults we know Santa Claus isn't real.'