Learning curves plot the expected error of a learning algorithm as a function of the number of labeled samples it receives from a target distribution. They are widely used as a measure of an algorithm’s performance, but classic PAC learning theory cannot explain their behavior. As observed by Antos and Lugosi (1996, 1998), the classic ‘No Free Lunch’ lower bounds only trace the upper envelope above all learning curves of specific target distributions. For a concept class with VC dimension $d$ the classic bound decays like $d/n$, yet it is possible that the learning curve for \emph{every} specific distribution decays exponentially. In this case, for each $n$ there exists a different ‘hard’ distribution requiring $d/n$ samples. Antos and Lugosi asked which concept classes admit a ‘strong minimax lower bound’ – a lower bound of $d’/n$ that holds for a fixed distribution for infinitely many $n$.We solve this problem in a principled manner, by introducing a combinatorial dimension called VCL that characterizes the best $d’$ for which $d’/n$ is a strong minimax lower bound. Conceptually, the VCL dimension determines the asymptotic rate of decay of the minimax learning curve, which we call the ‘distribution-free trail’ of the class. Our characterization strengthens the lower bounds of Bousquet, Hanneke, Moran, van Handel, and Yehudayoff (2021), and it refines their analysis of learning curves, by showing that for classes with finite VCL the learning rate can be decomposed into a linear component that depends only on the hypothesis class and a faster (e.g., exponential) component that depends also on the target distribution. As a corollary, we recover the lower bound of Antos and Lugosi (1996, 1998) for half-spaces in $\mathbb{R}^d$.Finally, to provide another viewpoint on our work and how it compares to traditional PAC learning bounds, we also present an alternative formulation of our results in a language that is closer to the PAC setting.

Convolutional Neural Networks (CNNs) are the go-to model for computer vision. Recently, attention-based networks, such as the Vision Transformer, have also become popular. In this paper we show that while convolutions and attention are both sufficient for good performance, neither of them are necessary. We present MLP-Mixer, an architecture based exclusively on multi-layer perceptrons (MLPs). MLP-Mixer contains two types of layers: one with MLPs applied independently to image patches (i.e. "mixing" the per-location features), and one with MLPs applied across patches (i.e. "mixing" spatial information). When trained on large datasets, or with modern regularization schemes, MLP-Mixer attains competitive scores on image classification benchmarks, with pre-training and inference cost comparable to state-of-the-art models. We hope that these results spark further research beyond the realms of well established CNNs and Transformers.

We study deep neural networks (DNNs) trained on natural image data with entirely random labels. Despite its popularity in the literature, where it is often used to study memorization, generalization, and other phenomena, little is known about what DNNs learn in this setting. In this paper, we show analytically for convolutional and fully connected networks that an alignment between the principal components of network parameters and data takes place when training with random labels. We study this alignment effect by investigating neural networks pre-trained on randomly labelled image data and subsequently fine-tuned on disjoint datasets with random or real labels. We show how this alignment produces a positive transfer: networks pre-trained with random labels train faster downstream compared to training from scratch even after accounting for simple effects, such as weight scaling. We analyze how competing effects, such as specialization at later layers, may hide the positive transfer. These effects are studied in several network architectures, including VGG16 and ResNet18, on CIFAR10 and ImageNet.

In semi-supervised classification, one is given access both to labeled and unlabeled data. As unlabeled data is typically cheaper to acquire than labeled data, this setup becomes advantageous as soon as one can exploit the unlabeled data in order to produce a better classifier than with labeled data alone. However, the conditions under which such an improvement is possible are not fully understood yet. Our analysis focuses on improvements in the \emph{minimax} learning rate in terms of the number of labeled examples (with the number of unlabeled examples being allowed to depend on the number of labeled ones). We argue that for such improvements to be realistic and indisputable, certain specific conditions should be satisfied and previous analyses have failed to meet those conditions. We then demonstrate examples where these conditions can be met, in particular showing rate changes from $1/\sqrt{\ell}$ to $e^{-c\ell}$ and from $1/\sqrt{\ell}$ to $1/\ell$. These results improve our understanding of what is and isn’t possible in semi-supervised learning.

The estimation of an f-divergence between two probability distributions based on samples is a fundamental problem in statistics and machine learning. Most works study this problem under very weak assumptions, in which case it is provably hard. We consider the case of stronger structural assumptions that are commonly satisfied in modern machine learning, including representation learning and generative modelling with autoencoder architectures. Under these assumptions we propose and study an estimator that can be easily implemented, works well in high dimensions, and enjoys faster rates of convergence. We verify the behavior of our estimator empirically in both synthetic and real-data experiments, and discuss its direct implications for total correlation, entropy, and mutual information estimation.

We lay theoretical foundations for new database release mechanisms that allow third-parties to construct consistent estimators of population statistics, while ensuring that the privacy of each individual contributing to the database is protected. The proposed framework rests on two main ideas. First, releasing (an estimate of) the kernel mean embedding of the data generating random variable instead of the database itself still allows third-parties to construct consistent estimators of a wide class of population statistics. Second, the algorithm can satisfy the definition of differential privacy by basing the released kernel mean embedding on entirely synthetic data points, while controlling accuracy through the metric available in a Reproducing Kernel Hilbert Space. We describe two instantiations of the proposed framework, suitable under different scenarios, and prove theoretical results guaranteeing differential privacy of the resulting algorithms and the consistency of estimators constructed from their outputs.

Generative Adversarial Networks (GAN) are an effective method for training generative models of complex data such as natural images. However, they are notoriously hard to train and can suffer from the problem of missing modes where the model is not able to produce examples in certain regions of the space. We propose an iterative procedure, called AdaGAN, where at every step we add a new component into a mixture model by running a GAN algorithm on a re-weighted sample. This is inspired by boosting algorithms, where many potentially weak individual predictors are greedily aggregated to form a strong composite predictor. We prove analytically that such an incremental procedure leads to convergence to the true distribution in a finite number of steps if each step is optimal, and convergence at an exponential rate otherwise. We also illustrate experimentally that this procedure addresses the problem of missing modes.

In this paper, we study the minimax estimation of the Bochner integral \[ \mu_k(P) := \int_\mathcal{X} k(\cdot,x)\, dP(x), \] also called the kernel mean embedding, based on random samples drawn i.i.d. from $P$, where $k:\mathcal{X}\times\mathcal{X}\rightarrow \mathbb{R}$ is a positive definite kernel. Various estimators (including the empirical estimator), $\hat{\theta}_n$ of $\mu_k(P)$ are studied in the literature wherein all of them satisfy $\|\hat{\theta}_n-\mu_k(P)\|_{\mathcal{H}_k}=O_P(n^{-1/2})$ with $\mathcal{H}_k$ being the reproducing kernel Hilbert space induced by $k$. The main contribution of the paper is in showing that the above mentioned rate of $n^{-1/2}$ is minimax in $\|\cdot\|_{\mathcal{H}_k}$ and $\|\cdot\|_{L^2(\mathbb{R}^d)}$-norms over the class of discrete measures and the class of measures that has an infinitely differentiable density, with $k$ being a continuous translation- invariant kernel on $\mathbb{R}^d$. The interesting aspect of this result is that the minimax rate is independent of the smoothness of the kernel and the density of $P$ (if it exists).

We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function f, consistent estimators of the mean embedding of a random variable X lead to consistent estimators of the mean embedding of f(X). For Matern kernels and sufficiently smooth functions we also provide rates of convergence. Our results extend to functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are independent, it is sufficient to have separate estimators of the mean embeddings of their marginal distributions. In either case, our results cover both mean embeddings based on i.i.d. samples as well as "reduced set" expansions in terms of dependent expansion points. The latter serves as a justification for using such expansions to limit memory resources when applying the approach as a basis for probabilistic programming.

Maximum Mean Discrepancy (MMD) is a distance on the space of probability measures which has found numerous applications in machine learning and nonparametric testing. This distance is based on the notion of embedding probabilities in a reproducing kernel Hilbert space. In this paper, we present the first known lower bounds for the estimation of MMD based on finite samples. Our lower bounds hold for any radial universal kernel on $\R^d$ and match the existing upper bounds up to constants that depend only on the properties of the kernel. Using these lower bounds, we establish the minimax rate optimality of the empirical estimator and its $U$-statistic variant, which are usually employed in applications.

We pose causal inference as the problem of learning to classify probability distributions. In particular, we assume access to a collection {(S_i,l_i)}_i=1^n, where each S_i is a sample drawn from the probability distribution of X_i \times Y_i, and l_i is a binary label indicating whether “X_i \to Y_i” or “X_i ←Y_i”. Given these data, we build a causal inference rule in two steps. First, we featurize each S_i using the kernel mean embedding associated with some characteristic kernel. Second, we train a binary classifier on such embeddings to distinguish between causal directions. We present generalization bounds showing the statistical consistency and learning rates of the proposed approach, and provide a simple implementation that achieves state-of-the-art cause-effect inference. Furthermore, we extend our ideas to infer causal relationships between more than two variables.

We show two novel concentration inequalities for suprema of empirical processes when sampling without replacement, which both take the variance of the functions into account. While these inequalities may potentially have broad applications in learning theory in general, we exemplify their significance by studying the transductive setting of learning theory. For which we provide the first excess risk bounds based on the localized complexity of the hypothesis class, which can yield fast rates of convergence also in the transductive learning setting. We give a preliminary analysis of the localized complexities for the prominent case of kernel classes.

We present PAC-Bayes-Empirical-Bernstein inequality. The inequality is based on combination of PAC-Bayesian bounding technique with Empirical Bernstein bound. It allows to take advantage of small empirical variance and is especially useful in regression. We show that when the empirical variance is significantly smaller than the empirical loss PAC-Bayes-Empirical-Bernstein inequality is significantly tighter than PAC-Bayes-kl inequality of Seeger (2002) and otherwise it is comparable. PAC-Bayes-Empirical-Bernstein inequality is an interesting example of application of PAC-Bayesian bounding technique to self-bounding functions. We provide empirical comparison of PAC-Bayes-Empirical-Bernstein inequality with PAC-Bayes-kl inequality on a synthetic example and several UCI datasets.