We investigate the phenomenon of generalization through the lens of compression. In particular, we study the complexity dynamics of neural networks to explain grokking, where networks suddenly transition from memorizing to generalizing solutions long after over-fitting the training data. To this end we introduce a new measure of intrinsic complexity for neural networks based on the theory of Kolmogorov complexity. Tracking this metric throughout network training, we find a consistent pattern in training dynamics, consisting of a rise and fall in complexity. We demonstrate that this corresponds to memorization followed by generalization. Based on insights from rate--distortion theory and the minimum description length principle, we lay out a principled approach to lossy compression of neural networks, and connect our complexity measure to explicit generalization bounds. Based on a careful analysis of information capacity in neural networks, we propose a new regularization method which encourages networks towards low-rank representations by penalizing their spectral entropy, and find that our regularizer outperforms baselines in total compression of the dataset.

We develop a unifying framework for information-theoretic lower bound in statistical estimation and interactive decision making. Classical lower bound techniques -- such as Fano's method, Le Cam's method, and Assouad's lemma -- are central to the study of minimax risk in statistical estimation, yet are insufficient to provide tight lower bounds for \emph{interactive decision making} algorithms that collect data interactively (e.g., algorithms for bandits and reinforcement learning). Recent work of Foster et al. (2021, 2023) provides minimax lower bounds for interactive decision making using seemingly different analysis techniques from the classical methods. These results -- which are proven using a complexity measure known as the \emph{Decision-Estimation Coefficient} (DEC) -- capture difficulties unique to interactive learning, yet do not recover the tightest known lower bounds for passive estimation. We propose a unified view of these distinct methodologies through a new lower bound approach called \emph{interactive Fano method}. As an application, we introduce a novel complexity measure, the \emph{Fractional Covering Number}, which facilitates the new lower bounds for interactive decision making that extend the DEC methodology by incorporating the complexity of estimation. Using the fractional covering number, we (i) provide a unified characterization of learnability for \emph{any} stochastic bandit problem, (ii) close the remaining gap between the upper and lower bounds in Foster et al. (2021, 2023) (up to polynomial factors) for any interactive decision making problem in which the underlying model class is convex.

A canonical desideratum for prediction problems is that performance guarantees should hold not just on average over the population, but also for meaningful subpopulations within the overall population. But what constitutes a meaningful subpopulation? In this work, we take the perspective that relevant subpopulations should be defined with respect to the clusters that naturally emerge from the distribution of individuals for which predictions are being made. In this view, a population refers to a mixture model whose components constitute the relevant subpopulations. We suggest two formalisms for capturing per-subgroup guarantees: first, by attributing each individual to the component from which they were most likely drawn, given their features; and second, by attributing each individual to all components in proportion to their relative likelihood of having been drawn from each component. Using online calibration as a case study, we study a multi-objective algorithm that provides guarantees for each of these formalisms by handling all plausible underlying subpopulation structures simultaneously, and achieve an $O(T^{1/2})$ rate even when the subpopulations are not well-separated. In comparison, the more natural cluster-then-predict approach that first recovers the structure of the subpopulations and then makes predictions suffers from a $O(T^{2/3})$ rate and requires the subpopulations to be separable. Along the way, we prove that providing per-subgroup calibration guarantees for underlying clusters can be easier than learning the clusters: separation between median subgroup features is required for the latter but not the former.

The $\ell_2^2$ min-sum $k$-clustering problem is to partition an input set into clusters $C_1,\ldots,C_k$ to minimize $\sum_{i=1}^k\sum_{p,q\in C_i}\|p-q\|_2^2$. Although $\ell_2^2$ min-sum $k$-clustering is NP-hard, it is not known whether it is NP-hard to approximate $\ell_2^2$ min-sum $k$-clustering beyond a certain factor. In this paper, we give the first hardness-of-approximation result for the $\ell_2^2$ min-sum $k$-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than $1.056$ and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. We then complement our hardness result by giving the first $(1+\varepsilon)$-coreset construction for $\ell_2^2$ min-sum $k$-clustering. Our coreset uses $\mathcal{O}\left(k^{\varepsilon^{-4}}\right)$ space and can be leveraged to achieve a polynomial-time approximation scheme with runtime $nd\cdot f(k,\varepsilon^{-1})$, where $d$ is the underlying dimension of the input dataset and $f$ is a fixed function. Finally, we consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label $i\in[k]$ for input point, thereby implicitly partitioning the input dataset into $k$ clusters that induce an approximately optimal solution, up to some amount of adversarial error $\alpha\in\left[0,\frac{1}{2}\right)$. We give a polynomial-time algorithm that outputs a $\frac{1+\gamma\alpha}{(1-\alpha)^2}$-approximation to $\ell_2^2$ min-sum $k$-clustering, for a fixed constant $\gamma>0$.

Many natural phenomena are characterized by self-similarity, for example the symmetry of human faces, or a repetitive motif of a song. Studying of such symmetries will allow us to gain deeper insights into the underlying mechanisms of complex systems. Recognizing the importance of understanding these patterns, we propose a geometrically inspired framework to study such phenomena in artificial neural networks. To this end, we introduce \emph{CantorNet}, inspired by the triadic construction of the Cantor set, which was introduced by Georg Cantor in the $19^\text{th}$ century. In mathematics, the Cantor set is a set of points lying on a single line that is self-similar and has a counter intuitive property of being an uncountably infinite null set. Similarly, we introduce CantorNet as a sandbox for studying self-similarity by means of novel topological and geometrical complexity measures. CantorNet constitutes a family of ReLU neural networks that spans the whole spectrum of possible Kolmogorov complexities, including the two opposite descriptions (linear and exponential as measured by the description length). CantorNet's decision boundaries can be arbitrarily ragged, yet are analytically known. Besides serving as a testing ground for complexity measures, our work may serve to illustrate potential pitfalls in geometry-ignorant data augmentation techniques and adversarial attacks.

This study addresses the issue of graph generation with generative models. In particular, we are concerned with graph community augmentation problem, which refers to the problem of generating unseen or unfamiliar graphs with a new community out of the probability distribution estimated with a given graph dataset. The graph community augmentation means that the generated graphs have a new community. There is a chance of discovering an unseen but important structure of graphs with a new community, for example, in a social network such as a purchaser network. Graph community augmentation may also be helpful for generalization of data mining models in a case where it is difficult to collect real graph data enough. In fact, there are many ways to generate a new community in an existing graph. It is desirable to discover a new graph with a new community beyond the given graph while we keep the structure of the original graphs to some extent for the generated graphs to be realistic. To this end, we propose an algorithm called the graph community augmentation (GCA). The key ideas of GCA are (i) to fit Gaussian mixture model (GMM) to data points in the latent space into which the nodes in the original graph are embedded, and (ii) to add data points in the new cluster in the latent space for generating a new community based on the minimum description length (MDL) principle. We empirically demonstrate the effectiveness of GCA for generating graphs with a new community structure on synthetic and real datasets.

Understanding the inner mechanisms of black-box foundation models (FMs) is essential yet challenging in artificial intelligence and its applications. Over the last decade, the long-running focus has been on their explainability, leading to the development of post-hoc explainable methods to rationalize the specific decisions already made by black-box FMs. However, these explainable methods have certain limitations in terms of faithfulness and resource requirement. Consequently, a new class of interpretable methods should be considered to unveil the underlying mechanisms of FMs in an accurate, comprehensive, heuristic, and resource-light way. This survey aims to review those interpretable methods that comply with the aforementioned principles and have been successfully applied to FMs. These methods are deeply rooted in machine learning theory, covering the analysis of generalization performance, expressive capability, and dynamic behavior. They provide a thorough interpretation of the entire workflow of FMs, ranging from the inference capability and training dynamics to their ethical implications. Ultimately, drawing upon these interpretations, this review identifies the next frontier research directions for FMs.

Delle Rose et al.~(COLT'23) introduced an effective version of the Vapnik-Chervonenkis dimension, and showed that it characterizes improper PAC learning with total computable learners. In this paper, we introduce and study a similar effectivization of the notion of Littlestone dimension. Finite effective Littlestone dimension is a necessary condition for computable online learning but is not a sufficient one -- which we already establish for classes of the effective Littlestone dimension 2. However, the effective Littlestone dimension equals the optimal mistake bound for computable learners in two special cases: a) for classes of Littlestone dimension 1 and b) when the learner receives as additional information an upper bound on the numbers to be guessed. Interestingly, finite effective Littlestone dimension also guarantees that the class consists only of computable functions.

Many NP-hard graph problems become easy for some classes of graphs, such as coloring is easy for bipartite graphs, but NP-hard in general. So we can ask question like when does a hard problem become easy? What is the minimum substructure for which the problem remains hard? We use the notion of boundary classes to study such questions. In this paper, we introduce a method for transforming the boundary class of one NP-hard graph problem into a boundary class for another problem. If $\Pi$ and $\Gamma$ are two NP-hard graph problems where $\Pi$ is reducible to $\Gamma$, we transform a boundary class of $\Pi$ into a boundary class of $\Gamma$. More formally if $\Pi$ is reducible to $\Gamma$, where the reduction is bijective and it maps hereditary classes of graphs to hereditary classes of graphs, then $X$ is a boundary class of $\Pi$ if and only if the image of $X$ under the reduction is a boundary class of $\Gamma$. This gives us a relationship between boundary classes and reducibility among several NP-hard problems. To show the strength of our main result, we apply our theorem to obtain some previously unknown boundary classes for a few graph problems namely; vertex-cover, clique, traveling-salesperson, bounded-degree-spanning-tree, subgraph-isomorphism and clique-cover.

Precision and Recall are foundational metrics in machine learning where both accurate predictions and comprehensive coverage are essential, such as in recommender systems and multi-label learning. In these tasks, balancing precision (the proportion of relevant items among those predicted) and recall (the proportion of relevant items successfully predicted) is crucial. A key challenge is that one-sided feedback--where only positive examples are observed during training--is inherent in many practical problems. For instance, in recommender systems like YouTube, training data only consists of videos that a user has actively selected, while unselected items remain unseen. Despite this lack of negative feedback in training, avoiding undesirable recommendations at test time is essential. We introduce a PAC learning framework where each hypothesis is represented by a graph, with edges indicating positive interactions, such as between users and items. This framework subsumes the classical binary and multi-class PAC learning models as well as multi-label learning with partial feedback, where only a single random correct label per example is observed, rather than all correct labels. Our work uncovers a rich statistical and algorithmic landscape, with nuanced boundaries on what can and cannot be learned. Notably, classical methods like Empirical Risk Minimization fail in this setting, even for simple hypothesis classes with only two hypotheses. To address these challenges, we develop novel algorithms that learn exclusively from positive data, effectively minimizing both precision and recall losses. Specifically, in the realizable setting, we design algorithms that achieve optimal sample complexity guarantees. In the agnostic case, we show that it is impossible to achieve additive error guarantees--as is standard in PAC learning--and instead obtain meaningful multiplicative approximations.