Scientific discovery via symbolic regression is often viewed as statistically and computationally intractable because the hypothesis space of expressions grows combinatorially with depth. This paper revisits the statistical side through the lens of PAC learning, focusing on compositional function trees built from a finite vocabulary of smooth operators (e.g., $\{+,\times,\sin,\exp\}$ and affine maps). We prove that the relevant generalization quantity, Rademacher complexity, hence the excess risk, does not necessarily blow up exponentially with the number of distinct symbolic structures, but is controlled by (i) the depth $d$ and (ii) the Lipschitz constants of the base operators along the composed computation graph. Concretely, under mild Lipschitz conditions on operators and bounded affine leaves, a finite-union bound over a vocabulary of size $K=|\mathcal{H}_{\mathrm{base}}|$ together with Maurer-type vector contraction yields $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{d}) \leq (Kb\sqrt{2}L)^{d-1}\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})$ with arity bound $b$; corresponding high-probability risk bounds scale as $\mathcal{O}(L^{d}/\sqrt{n})$ when $K,b=O(1)$ and $\mathfrak{R}_n(\mathcal{H}_{\mathrm{comp}}^{1})=O(n^{-1/2})$. We complement the theory with a modular codebase that trains differentiable operator trees (not MLPs) on synthetic "physics-like" targets of controlled depth and shows that the empirical generalization gap correlates positively with the predicted complexity term $(\widehat{L}^{d})/\sqrt{n}$.

Binary classification from positive-only samples is a variant of PAC learning in which the learner receives i.i.d. samples from the positive region of an unknown target concept, but is evaluated under the original distribution (which places mass on both positive and negative regions). This model dates back to Natarajan [1987, STOC], and the characterization of improper learning is well-known -- it even appears in textbooks. The characterization of proper positive-only learning, however, has long remained open. In this work, we revisit and settle this question: a concept class is properly learnable from positive-only samples if and only if it has finite VC dimension and satisfies a new combinatorial condition, which we call uniform exterior separability. Together with several separation results, this characterization reveals a surprisingly rich landscape that differs sharply from standard PAC learning: proper and improper learning are separated, randomized and deterministic proper learning are separated, there are classes for which no ERM is a learner, and finite VC dimension does not suffice even for non-uniform learning. Along the way, we introduce new combinatorial dimensions that we believe can be of broader interest in learning theory.

We study learning in prophet inequalities with i.i.d. rewards drawn from an exponential-type parametric family with an unknown parameter $θ$, a class that includes exponential, Pareto, and bounded-support power-family distributions. We first characterize the optimal full-information asymptotic competitive ratio for this family. In the unbounded-support case, the limit is $ {\left(θ/({θ-c_+})\right)^{c_+/θ}}/ {Γ(1-c_+/θ)},$ while in the bounded-support case, the limit is $1$. We then propose a confidence-based dynamic-programming policy for online learning. By exploiting the explicit parametric structure, the policy achieves the same optimal asymptotic competitive ratio using only online observations, without external offline samples. We further derive distribution-specific convergence rates for canonical examples. Finally, numerical experiments on synthetic instances illustrate the performance of our algorithm.

Data augmentation is a simple and model-agnostic approach for exploiting known invariances in learning problems. Given a group acting on the input space, one augments the training set with transformed copies of each sample. Because it exploits symmetries without modifying the underlying learning algorithm, data augmentation can be applied broadly across learning methods. However, this universality comes at a computational cost: when the group is large, full group-sized augmentation quickly becomes computationally infeasible. This raises a fundamental question: Can partial data augmentation achieve the same statistical benefits as full augmentation in terms of generalization and sample complexity? We develop a general framework for investigating this question using Fourier analysis and the representation theory of finite groups. We show that, for a broad class of classical learning problems, partial data augmentation based on a randomly sampled subset of group elements achieves the same minimax rates as full augmentation, up to an approximation error that vanishes as the subset size increases. Our results provide a theoretical explanation for why partial augmentation can retain the statistical benefits of full augmentation despite enforcing symmetry only approximately, and shed light on a recently raised question in learning with symmetries: whether statistically optimal learning under general group invariances can be achieved using computationally scalable methods. Moreover, we prove a complementary impossibility result: enforcing exact invariance via data augmentation requires averaging over the entire group, and cannot be achieved by any strict subset when the hypothesis space is sufficiently expressive. Together, these results provide a unified perspective on full and partial data augmentation, as well as exact and approximate symmetry enforcement.

A central question in the theory of machine learning concerns the identification of classes of data distributions for which one can provide computationally efficient learning algorithms with provable statistical learning guarantees. Indeed, in the context of probably approximately correct (PAC) learning, there has been much interest in exploring intermediate PAC learning models that, unlike the realizable PAC learning setting, allow for some stochasticity in the labels, and unlike the fully agnostic PAC learning setting, also admit computationally efficient learning algorithms with finite sample complexity bounds. Some examples of such models include random classification noise (RCN), probabilistic concepts, Massart noise, and generalized linear models (GLMs); in general, most of this work has focused on binary classification problems. In this paper, we study what we call realizablestatistic models (RSMs), wherein we allow stochastic labels but assume that some vector-valued statistic of the conditional label distribution comes from some known function class. RSMs are a flexible class of models that interpolate between the realizable and fully agnostic settings, and that also recover several previously studied models as special cases.

We revisit the problem of private online learning, in which a learner receives a sequence of T data points and has to respond at each time-step a hypothesis. It is required that the entire stream of output hypotheses should satisfy differential privacy. Prior work of Golowich and Livni [2021] established that every concept class H with finite Littlestone dimension d is privately online learnable in the realizable setting. In particular, they proposed an algorithm that achieves an Od(logT) mistake bound against an oblivious adversary. However, their approach yields a suboptimal Od( T) bound against an adaptive adversary. In this work, we present a new algorithm with a mistake bound of Od(logT)against an adaptive adversary, closing this gap. We further investigate the problem in the agnostic setting, which is more general than the realizable setting as it does not impose any assumptions on the data. We give an algorithm that obtains a sublinear regret of Od( T) for generic Littlestone classes, demonstrating that they are also privately online learnable in the agnostic setting.

We revisit the classical model of nonuniform PAC learning, introduced by Benedek and Itai [1994], where generalization guarantees may depend on the target concept (but not on the marginal distribution). In this work, we study a complementary variant, which we call marginal-nonuniform learning. In this setting, guarantees may depend on the marginal distribution over the domain, but must hold uniformly over all concepts. This captures the intuition that some data distributions are inherently easier to learn from than others, allowing for a flexible, distributionsensitive view of learnability. Our main result is a complete characterization of the achievable learning rates in this model, revealing a trichotomy: exponential rates of the form e n arise precisely when the hypothesis class is finite; linear rates of the form d/n are achievable when a recently introduced combinatorial parameter, the VC-eluder dimension d, is finite; and arbitrarily slow rates may occur when d = . Additionally, in the original (concept-)nonuniform model, we show that for all learnable classes linear rates are achievable. We conclude by situating marginal-nonuniform learning within the landscape of universal learning, and by discussing its relationship to other distribution-dependent learning paradigms.

Inspired by recent work on learning with distribution shift, we give a general outlier removal algorithm called iterative polynomial filtering and show a number of striking applications for supervised learning with contamination: (1) We show that any function class that can be approximated by low-degree polynomials with respect to a hypercontractive distribution can be efficiently learned under bounded contamination (also known as nasty noise). This is a surprising resolution to a longstanding gap between the complexity of agnostic learning and learning with contamination, as it was widely believed that low-degree approximators only implied tolerance to label noise.

According to Algorithmic Information Theory (AIT) - Intelligent representations compress data into the shortest possible program that can reconstruct its content, exhibiting low Kolmogorov Complexity (KC). In contrast, most visual representation learning systems use fixed-length representations for all inputs, ignoring variations in complexity or familiarity. Recent adaptive tokenization methods address this by allocating variable-length representations but typically require test-time search over multiple encodings to find the most predictive one. Inspired by Kolmogorov Complexity principles, we propose a single-pass adaptive tokenizer, KARL, which predicts the appropriate number of tokens for an image in a single forward pass, halting once its approximate KC is reached. The token count serves as a proxy for the minimum description length. KARL's training procedure closely resembles the Upside-Down Reinforcement Learning paradigm, as it learns to conditionally predict token halting based on a desired reconstruction quality. KARL matches the performance of recent adaptive tokenizers while operating in a single pass.

We study the problem of multiclass learning with bandit feedback in both the i.i.d.