We study the PAC property of scenario decision-making algorithms, that is, the ability to make a decision that has an arbitrarily low risk of violating an unknown safety constraint, provided sufficiently many realizations (called scenarios) of the safety constraint are sampled. Sufficient conditions for scenario decision-making algorithms to be PAC are available in the literature, such as finiteness of the VC dimension of its associated classifier and existence of a compression scheme. We study the question of whether these sufficient conditions are also necessary. We show with counterexamples that this is not the case in general. This contrasts with binary classification learning, for which the analogous conditions are sufficient and necessary. Popular scenario decision-making algorithms, such as scenario optimization, enjoy additional properties, such as stability and consistency. We show that even under these additional assumptions the above conclusions hold. Finally, we derive a necessary condition for scenario decision-making algorithms to be PAC, inspired by the VC dimension and the so-called no-free-lunch theorem.

Bandits with knapsacks (BwK) is an influential model of sequential decision-making under uncertainty that incorporates resource consumption constraints. In each round, the decision-maker observes an outcome consisting of a reward and a vector of nonnegative resource consumptions, and the budget of each resource is decremented by its consumption. In this paper we introduce a natural generalization of the stochastic BwK problem that allows non-monotonic resource utilization. In each round, the decision-maker observes an outcome consisting of a reward and a vector of resource drifts that can be positive, negative or zero, and the budget of each resource is incremented by its drift. Our main result is a Markov decision process (MDP) policy that has constant regret against a linear programming (LP) relaxation when the decision-maker knows the true outcome distributions.

We provide a full characterization of the concept classes that are optimistically universally online learnable with $\{0, 1\}$ labels. The notion of optimistically universal online learning was defined in [Hanneke, 2021] in order to understand learnability under minimal assumptions. In this paper, following the philosophy behind that work, we investigate two questions, namely, for every concept class: (1) What are the minimal assumptions on the data process admitting online learnability? (2) Is there a learning algorithm which succeeds under every data process satisfying the minimal assumptions? Such an algorithm is said to be optimistically universal for the given concept class. We resolve both of these questions for all concept classes, and moreover, as part of our solution, we design general learning algorithms for each case. Finally, we extend these algorithms and results to the agnostic case, showing an equivalence between the minimal assumptions on the data process for learnability in the agnostic and realizable cases, for every concept class, as well as the equivalence of optimistically universal learnability.

In 2020, OpenAI proposed the first type of Scaling Laws, describing the relationships between model performance and parameters, data, and compute. In 2024, OpenAI proposed the second type of Scaling Laws, describing the relationship between model inference performance and inference computation. In this paper, we analyze LLM training and inference processes from the perspective of lossless compression using conditional Kolmogorov complexity, and unify these two types of Scaling Laws. We find that both types of Scaling Laws improve approximation of conditional Kolmogorov complexity by increasing execution steps $t$. The first type of Scaling Laws increases $t$ by increasing model parameters $y$. The second type of Scaling Laws increases $t$ by increasing the number of output tokens.

We consider the problem of contextual kernel bandits with stochastic contexts, where the underlying reward function belongs to a known Reproducing Kernel Hilbert Space (RKHS). We study this problem under the additional constraint of joint differential privacy, where the agents needs to ensure that the sequence of query points is differentially private with respect to both the sequence of contexts and rewards. We propose a novel algorithm that improves upon the state of the art and achieves an error rate of $\mathcal{O}\left(\sqrt{\frac{\gamma_T}{T}} + \frac{\gamma_T}{T \varepsilon}\right)$ after $T$ queries for a large class of kernel families, where $\gamma_T$ represents the effective dimensionality of the kernel and $\varepsilon > 0$ is the privacy parameter. Our results are based on a novel estimator for the reward function that simultaneously enjoys high utility along with a low-sensitivity to observed rewards and contexts, which is crucial to obtain an order optimal learning performance with improved dependence on the privacy parameter.

We consider coresets for $k$-clustering problems, where the goal is to assign points to centers minimizing powers of distances. A popular example is the $k$-median objective $\sum_{p}\min_{c\in C}dist(p,C)$. Given a point set $P$, a coreset $\Omega$ is a small weighted subset that approximates the cost of $P$ for all candidate solutions $C$ up to a $(1\pm\varepsilon )$ multiplicative factor. In this paper, we give a sharp VC-dimension based analysis for coreset construction. As a consequence, we obtain improved $k$-median coreset bounds for the following metrics: Coresets of size $\tilde{O}\left(k\varepsilon^{-2}\right)$ for shortest path metrics in planar graphs, improving over the bounds $\tilde{O}\left(k\varepsilon^{-6}\right)$ by [Cohen-Addad, Saulpic, Schwiegelshohn, STOC'21] and $\tilde{O}\left(k^2\varepsilon^{-4}\right)$ by [Braverman, Jiang, Krauthgamer, Wu, SODA'21]. Coresets of size $\tilde{O}\left(kd\ell\varepsilon^{-2}\log m\right)$ for clustering $d$-dimensional polygonal curves of length at most $m$ with curves of length at most $\ell$ with respect to Frechet metrics, improving over the bounds $\tilde{O}\left(k^3d\ell\varepsilon^{-3}\log m\right)$ by [Braverman, Cohen-Addad, Jiang, Krauthgamer, Schwiegelshohn, Toftrup, and Wu, FOCS'22] and $\tilde{O}\left(k^2d\ell\varepsilon^{-2}\log m \log |P|\right)$ by [Conradi, Kolbe, Psarros, Rohde, SoCG'24].

Monotone learning refers to learning processes in which expected performance consistently improves as more training data is introduced. Non-monotone behavior of machine learning has been the topic of a series of recent works, with various proposals that ensure monotonicity by applying transformations or wrappers on learning algorithms. In this work, from a different perspective, we tackle the topic of monotone learning within the framework of Probably Approximately Correct (PAC) learning theory. Following the mechanism that estimates sample complexity of a PAC-learnable problem, we derive a performance lower bound for that problem, and prove the monotonicity of that bound as the sample sizes increase. By calculating the lower bound distribution, we are able to prove that given a PAC-learnable problem with a hypothesis space that is either of finite size or of finite VC dimension, any learning algorithm based on Empirical Risk Minimization (ERM) is monotone if training samples are independent and identically distributed (i.i.d.). We further carry out an experiment on two concrete machine learning problems, one of which has a finite hypothesis set, and the other of finite VC dimension, and compared the experimental data for the empirical risk distributions with the estimated theoretical bound. The results of the comparison have confirmed the monotonicity of learning for the two PAC-learnable problems.

Two seminal papers--Alon, Livni, Malliaris, Moran (STOC 2019) and Bun, Livni, and Moran (FOCS 2020)--established the equivalence between online learnability and globally stable PAC learnability in binary classification. However, Chase, Chornomaz, Moran, and Yehudayoff (STOC 2024) recently showed that this equivalence does not hold in the agnostic setting. Specifically, they proved that in the agnostic setting, only finite hypothesis classes are globally stable learnable. Therefore, agnostic global stability is too restrictive to capture interesting hypothesis classes. To address this limitation, Chase \emph{et al.} introduced two relaxations of agnostic global stability. In this paper, we characterize the classes that are learnable under their proposed relaxed conditions, resolving the two open problems raised in their work. First, we prove that in the setting where the stability parameter can depend on the excess error (the gap between the learner's error and the best achievable error by the hypothesis class), agnostic stability is fully characterized by the Littlestone dimension. Consequently, as in the realizable case, this form of learnability is equivalent to online learnability. As part of the proof of this theorem, we strengthen the celebrated result of Bun et al. by showing that classes with infinite Littlestone dimension are not stably PAC learnable, even if we allow the stability parameter to depend on the excess error. For the second relaxation proposed by Chase et al., we prove that only finite hypothesis classes are globally stable learnable even if we restrict the agnostic setting to distributions with small population loss.

In this work, we study the weighted empirical risk minimization (weighted ERM) schema, in which an additional data-dependent weight function is incorporated when the empirical risk function is being minimized. We show that under a general ``balanceable" Bernstein condition, one can design a weighted ERM estimator to achieve superior performance in certain sub-regions over the one obtained from standard ERM, and the superiority manifests itself through a data-dependent constant term in the error bound. These sub-regions correspond to large-margin ones in classification settings and low-variance ones in heteroscedastic regression settings, respectively. Our findings are supported by evidence from synthetic data experiments.

The learner's ability to generate a hypothesis that closely approximates the target function is crucial in machine learning. Achieving this requires sufficient data; however, unauthorized access by an eavesdropping learner can lead to security risks. Thus, it is important to ensure the performance of the "authorized" learner by limiting the quality of the training data accessible to eavesdroppers. Unlike previous studies focusing on encryption or access controls, we provide a theorem to ensure superior learning outcomes exclusively for the authorized learner with quantum label encoding. In this context, we use the probably-approximately-correct (PAC) learning framework and introduce the concept of learning probability to quantitatively assess learner performance. Our theorem allows the condition that, given a training dataset, an authorized learner is guaranteed to achieve a certain quality of learning outcome, while eavesdroppers are not. Notably, this condition can be constructed based only on the authorized-learning-only measurable quantities of the training data, i.e., its size and noise degree. We validate our theoretical proofs and predictions through convolutional neural networks (CNNs) image classification learning.