Carmosino et al. (2016) demonstrated that natural proofs of circuit lower bounds for $\Lambda$ imply efficient algorithms for learning $\Lambda$-circuits, but only over \textit{the uniform distribution}, with \textit{membership queries}, and provided $\AC^0[p] \subseteq \Lambda$. We consider whether this implication can be generalized to $\Lambda \not\supseteq \AC^0[p]$, and to learning algorithms which use only random examples and learn over arbitrary example distributions (Valiant's PAC-learning model). We first observe that, if, for any circuit class $\Lambda$, there is an implication from natural proofs for $\Lambda$ to PAC-learning for $\Lambda$, then standard assumptions from lattice-based cryptography do not hold. In particular, we observe that depth-2 majority circuits are a (conditional) counter example to the implication, since Nisan (1993) gave a natural proof, but Klivans and Sherstov (2009) showed hardness of PAC-learning under lattice-based assumptions. We thus ask: what learning algorithms can we reasonably expect to follow from Nisan's natural proofs? Our main result is that all natural proofs arising from a type of communication complexity argument, including Nisan's, imply PAC-learning algorithms in a new \textit{distributional} variant (i.e., an ``average-case'' relaxation) of Valiant's PAC model. Our distributional PAC model is stronger than the average-case prediction model of Blum et al. (1993) and the heuristic PAC model of Nanashima (2021), and has several important properties which make it of independent interest, such as being \textit{boosting-friendly}. The main applications of our result are new distributional PAC-learning algorithms for depth-2 majority circuits, polytopes and DNFs over natural target distributions, as well as the nonexistence of encoded-input weak PRFs that can be evaluated by depth-2 majority circuits.

We present new upper and lower bounds on the number of learner mistakes in the `transductive' online learning setting of Ben-David, Kushilevitz and Mansour (1997). This setting is similar to standard online learning, except that the adversary fixes a sequence of instances $x_1,\dots,x_n$ to be labeled at the start of the game, and this sequence is known to the learner. Qualitatively, we prove a trichotomy, stating that the minimal number of mistakes made by the learner as $n$ grows can take only one of precisely three possible values: $n$, $\Theta\left(\log (n)\right)$, or $\Theta(1)$. Furthermore, this behavior is determined by a combination of the VC dimension and the Littlestone dimension. Quantitatively, we show a variety of bounds relating the number of mistakes to well-known combinatorial dimensions. In particular, we improve the known lower bound on the constant in the $\Theta(1)$ case from $\Omega\left(\sqrt{\log(d)}\right)$ to $\Omega(\log(d))$ where $d$ is the Littlestone dimension. Finally, we extend our results to cover multiclass classification and the agnostic setting.

Virtually all machine learning tasks are characterized using some form of loss function, and "good performance" is typically stated in terms of a sufficiently small average loss, taken over the random draw of test data. While optimizing for performance on average is intuitive, convenient to analyze in theory, and easy to implement in practice, such a choice brings about trade-offs. In this work, we survey and introduce a wide variety of non-traditional criteria used to design and evaluate machine learning algorithms, place the classical paradigm within the proper historical context, and propose a view of learning problems which emphasizes the question of "what makes for a desirable loss distribution?" in place of tacit use of the expected loss.

We study optimal teaching of Linear Behavior Cloning (LBC) learners. In this setup, the teacher can select which states to demonstrate to an LBC learner. The learner maintains a version space of infinite linear hypotheses consistent with the demonstration. The goal of the teacher is to teach a realizable target policy to the learner using minimum number of state demonstrations. This number is known as the Teaching Dimension(TD). We present a teaching algorithm called ``Teach using Iterative Elimination(TIE)" that achieves instance optimal TD. However, we also show that finding optimal teaching set computationally is NP-hard. We further provide an approximation algorithm that guarantees an approximation ratio of $\log(|A|-1)$ on the teaching dimension. Finally, we provide experimental results to validate the efficiency and effectiveness of our algorithm.

Machine Learning (ML) and Algorithmic Information Theory (AIT) look at Complexity from different points of view. We explore the interface between AIT and Kernel Methods (that are prevalent in ML) by adopting an AIT perspective on the problem of learning kernels from data, in kernel ridge regression, through the method of Sparse Kernel Flows. In particular, by looking at the differences and commonalities between Minimal Description Length (MDL) and Regularization in Machine Learning (RML), we prove that the method of Sparse Kernel Flows is the natural approach to adopt to learn kernels from data. This paper shows that it is not necessary to use the statistical route to derive Sparse Kernel Flows and that one can directly work with code-lengths and complexities that are concepts that show up in AIT.

We discuss the problem of learning a deterministic finite automaton (DFA) from a confidence oracle. That is, we are given access to an oracle $Q$ with incomplete knowledge of some target language $L$ over an alphabet $\Sigma$; the oracle maps a string $x\in\Sigma^*$ to a score in the interval $[-1,1]$ indicating its confidence that the string is in the language. The interpretation is that the sign of the score signifies whether $x\in L$, while the magnitude $|Q(x)|$ represents the oracle's confidence. Our goal is to learn a DFA representation of the oracle that preserves the information that it is confident in. The learned DFA should closely match the oracle wherever it is highly confident, but it need not do this when the oracle is less sure of itself.

State-of-the-art NLP methods achieve human-like performance on many tasks, but make errors nevertheless. Characterizing these errors in easily interpretable terms gives insight into whether a classifier is prone to making systematic errors, but also gives a way to act and improve the classifier. We propose to discover those patterns of tokens that distinguish correct and erroneous predictions as to obtain global and interpretable descriptions for arbitrary NLP classifiers. We formulate the problem of finding a succinct and non-redundant set of such patterns in terms of the Minimum Description Length principle. Through an extensive set of experiments, we show that our method, Premise, performs well in practice. Unlike existing solutions, it recovers ground truth, even on highly imbalanced data over large vocabularies. In VQA and NER case studies, we confirm that it gives clear and actionable insight into the systematic errors made by NLP classifiers.

We derive and study time-uniform confidence spheres - termed confidence sphere sequences (CSSs) - which contain the mean of random vectors with high probability simultaneously across all sample sizes. Inspired by the original work of Catoni and Giulini, we unify and extend their analysis to cover both the sequential setting and to handle a variety of distributional assumptions. More concretely, our results include an empirical-Bernstein CSS for bounded random vectors (resulting in a novel empirical-Bernstein confidence interval), a CSS for sub-$\psi$ random vectors, and a CSS for heavy-tailed random vectors based on a sequentially valid Catoni-Giulini estimator. Finally, we provide a version of our empirical-Bernstein CSS that is robust to contamination by Huber noise.

In this paper we start with a simple question, how is it possible that humans can recognize different movements over skin with only a prior visual experience of them? Or in general, what is the representation of spatial sequences that are invariant to scale, rotation, and translation across different modalities? To answer, we rethink the mathematical representation of spatial sequences, argue against the minimum description length principle, and focus on the movements of attention. We advance the idea that spatial sequences must be represented on different levels of abstraction, this adds redundancy but is necessary for recognition and generalization. To address the open question of how these abstractions are formed we propose two hypotheses: the first invites exploring selectionism learning, instead of finding parameters in some models; the second proposes to find new data structures, not neural network architectures, to efficiently store and operate over redundant features to be further selected. Movements of attention are central to human cognition and lessons should be applied to new better learning algorithms.

Associative memory architectures are designed for memorization but also offer, through their retrieval method, a form of generalization to unseen inputs: stored memories can be seen as prototypes from this point of view. Focusing on Modern Hopfield Networks (MHN), we show that a large memorization capacity undermines the generalization opportunity. We offer a solution to better optimize this tradeoff. It relies on Minimum Description Length (MDL) to determine during training which memories to store, as well as how many of them.