Adversarial training yields robust models against a specific threat model, e.g., $L_\infty$ adversarial examples. Typically robustness does not generalize to previously unseen threat models, e.g., other $L_p$ norms, or larger perturbations. Our confidence-calibrated adversarial training (CCAT) tackles this problem by biasing the model towards low confidence predictions on adversarial examples. By allowing to reject examples with low confidence, robustness generalizes beyond the threat model employed during training. CCAT, trained only on $L_\infty$ adversarial examples, increases robustness against larger $L_\infty$, $L_2$, $L_1$ and $L_0$ attacks, adversarial frames, distal adversarial examples and corrupted examples and yields better clean accuracy compared to adversarial training. For thorough evaluation we developed novel white- and black-box attacks directly attacking CCAT by maximizing confidence. For each threat model, we use $7$ attacks with up to $50$ restarts and $5000$ iterations and report worst-case robust test error, extended to our confidence-thresholded setting, across all attacks.

Interest has been rising lately towards methods representing data in non-Euclidean spaces, e.g. hyperbolic or spherical that provide specific inductive biases useful for certain real-world data properties, e.g. scale-free, hierarchical or cyclical. However, the popular graph neural networks are currently limited in modeling data only via Euclidean geometry and associated vector space operations. Here, we bridge this gap by proposing mathematically grounded generalizations of graph convolutional networks (GCN) to (products of) constant curvature spaces. We do this by i) introducing a unified formalism permitting a differentiable interpolation between all geometries of constant curvature irrespective of their sign, ii) leveraging gyro-barycentric coordinates that generalize the classic Euclidean concept of the center of mass. Our class of models smoothly recover their Euclidean counterparts when the curvature goes to zero from either side. Empirically, we outperform Euclidean GCNs in the tasks of node classification and distortion minimization for symbolic data exhibiting non-Euclidean behavior, according to their discrete curvature.

Operator-Valued Kernels (OVKs) and associated vector-valued Reproducing Kernel Hilbert Spaces provide an elegant way to extend scalar kernel methods when the output space is a Hilbert space. Although primarily used in finite dimension for problems like multi-task regression, the ability of this framework to deal with infinite dimensional output spaces unlocks many more applications, such as functional regression, structured output prediction, and structured data representation. However, these sophisticated schemes crucially rely on the kernel trick in the output space, so that most of previous works have focused on the square norm loss function, completely neglecting robustness issues that may arise in such surrogate problems. To overcome this limitation, this paper develops a duality approach that allows to solve OVK machines for a wide range of loss functions. The infinite dimensional Lagrange multipliers are handled through a Double Representer Theorem, and algorithms for \epsilon-insensitive losses and the Huber loss are thoroughly detailed. Robustness benefits are emphasized by a theoretical stability analysis, as well as empirical improvements on structured data applications.

Stochastic gradient descent with momentum (SGDm) is one of the most popular optimization algorithms in deep learning. While there is a rich theory of SGDm for convex problems, the theory is considerably less developed in the context of deep learning where the problem is non-convex and the gradient noise might exhibit a heavy-tailed behavior, as empirically observed in recent studies. In this study, we consider a \emph{continuous-time} variant of SGDm, known as the underdamped Langevin dynamics (ULD), and investigate its asymptotic properties under heavy-tailed perturbations. Supported by recent studies from statistical physics, we argue both theoretically and empirically that the heavy-tails of such perturbations can result in a bias even when the step-size is small, in the sense that \emph{the optima of stationary distribution} of the dynamics might not match \emph{the optima of the cost function to be optimized}. As a remedy, we develop a novel framework, which we coin as \emph{fractional} ULD (FULD), and prove that FULD targets the so-called Gibbs distribution, whose optima exactly match the optima of the original cost. We observe that the Euler discretization of FULD has noteworthy algorithmic similarities with \emph{natural gradient} methods and \emph{gradient clipping}, bringing a new perspective on understanding their role in deep learning. We support our theory with experiments conducted on a synthetic model and neural networks.

A fundamental trait of intelligence is the ability to achieve goals in the face of novel circumstances, such as making decisions from new action choices. However, standard reinforcement learning assumes a fixed set of actions and requires expensive retraining when given a new action set. To make learning agents more adaptable, we introduce the problem of zero-shot generalization to new actions. We propose a two-stage framework where the agent first infers action representations from action information acquired separately from the task. A policy flexible to varying action sets is then trained with generalization objectives. We benchmark generalization on sequential tasks, such as selecting from an unseen tool-set to solve physical reasoning puzzles and stacking towers with novel 3D shapes. Videos and code are available at https://sites.google.com/view/action-generalization.

We propose inertial versions of block coordinate descent methods for solving non-convex non-smooth composite optimization problems. Our methods possess three main advantages compared to current state-of-the-art accelerated first-order methods: (1) they allow using two different extrapolation points to evaluate the gradients and to add the inertial force (we will empirically show that it is more efficient than using a single extrapolation point), (2) they allow to randomly select the block of variables to update, and (3) they do not require a restarting step. We prove the subsequential convergence of the generated sequence under mild assumptions, prove the global convergence under some additional assumptions, and provide convergence rates. We deploy the proposed methods to solve non-negative matrix factorization (NMF) and show that they compete favorably with the state-of-the-art NMF algorithms. Additional experiments on non-negative approximate canonical polyadic decomposition, also known as nonnegative tensor factorization, are also provided.

We tackle the problem disentangling the latent space of an autoencoder in order to separate labelled attribute information from other characteristic information. This then allows us to change selected attributes while preserving other information. Our method, matrix subspace projection, is much simpler than previous approaches to latent space factorisation, for example not requiring multiple discriminators or a careful weighting among their loss functions. Furthermore our new model can be applied to autoencoders as a plugin, and works across diverse domains such as images or text. We demonstrate the utility of our method for attribute manipulation in autoencoders trained across varied domains, using both human evaluation and automated methods. The quality of generation of our new model (e.g. reconstruction, conditional generation) is highly competitive to a number of strong baselines.

We propose a method for training a deterministic deep model that can find and reject out of distribution data points at test time with a single forward pass. Our approach, deterministic uncertainty quantification (DUQ), builds upon ideas of RBF networks. We scale training in these with a novel loss function and centroid updating scheme and match the accuracy of softmax models. By enforcing detectability of changes in the input using a gradient penalty, we are able to reliably detect out of distribution data. Our uncertainty quantification scales well to large datasets, and using a single model, we improve upon or match Deep Ensembles in out of distribution detection on notable difficult dataset pairs such as FashionMNIST vs. MNIST, and CIFAR-10 vs. SVHN.

We consider the problem of allocating a fixed budget of samples to a finite set of discrete distributions to learn them uniformly well (minimizing the maximum error) in terms of four common distance measures: $\ell_2^2$, $\ell_1$, $f$-divergence, and separation distance. To present a unified treatment of these distances, we first propose a general \emph{optimistic tracking algorithm} and analyze its sample allocation performance w.r.t.~an oracle. We then instantiate this algorithm for the four distance measures and derive bounds on their regret. We also show that the allocation performance of the proposed algorithm cannot, in general, be improved, by deriving lower-bounds on the expected deviation from the oracle allocation for any adaptive scheme. We verify our theoretical findings through some experiments. Finally, we show that the techniques developed in the paper can be easily extended to learn some classes of continuous distributions as well as to the related setting of minimizing the average error (in terms of the four distances) in learning a set of distributions.

With a view to bridging the gap between deep learning and symbolic AI, we present a novel end-to-end neural network architecture that learns to form propositional representations with an explicitly relational structure from raw pixel data. In order to evaluate and analyse the architecture, we introduce a family of simple visual relational reasoning tasks of varying complexity. We show that the proposed architecture, when pre-trained on a curriculum of such tasks, learns to generate reusable representations that better facilitate subsequent learning on previously unseen tasks when compared to a number of baseline architectures. The workings of a successfully trained model are visualised to shed some light on how the architecture functions.

Traditional differential privacy is independent of the data distribution. However, this is not well-matched with the modern machine learning context, where models are trained on specific data. As a result, achieving meaningful privacy guarantees in ML often excessively reduces accuracy. We propose Bayesian differential privacy (BDP), which takes into account the data distribution to provide more practical privacy guarantees. We also derive a general privacy accounting method under BDP, building upon the well-known moments accountant. Our experiments demonstrate that in-distribution samples in classic machine learning datasets, such as MNIST and CIFAR-10, enjoy significantly stronger privacy guarantees than postulated by DP, while models maintain high classification accuracy.

Deep neural networks are typically initialized with random weights, with variances chosen to facilitate signal propagation and stable gradients. It is also believed that diversity of features is an important property of these initializations. We construct a deep convolutional network with identical features by initializing almost all the weights to $0$. The architecture also enables perfect signal propagation and stable gradients, and achieves high accuracy on standard benchmarks. This indicates that random, diverse initializations are \textit{not} necessary for training neural networks. An essential element in training this network is a mechanism of symmetry breaking; we study this phenomenon and find that standard GPU operations, which are non-deterministic, can serve as a sufficient source of symmetry breaking to enable training.

Causal inference using observational data is challenging, especially in the bivariate case. Through the minimum description length principle, we link the postulate of independence between the generating mechanisms of the cause and of the effect given the cause to quantile regression. Based on this theory, we develop Bivariate Quantile Causal Discovery (bQCD), a new method to distinguish cause from effect assuming no confounding, selection bias or feedback. Because it uses multiple quantile levels instead of the conditional mean only, bQCD is adaptive not only to additive, but also to multiplicative or even location-scale generating mechanisms. To illustrate the effectiveness of our approach, we perform an extensive empirical comparison on both synthetic and real datasets. This study shows that bQCD is robust across different implementations of the method (i.e., the quantile regression), computationally efficient, and compares favorably to state-of-the-art methods.

Normalizing flows are exact-likelihood generative neural networks which approximately transform samples from a simple prior distribution to samples of the probability distribution of interest. Recent work showed that such generative models can be utilized in statistical mechanics to sample equilibrium states of many-body systems in physics and chemistry. To scale and generalize these results, it is essential that the natural symmetries in the probability density -- in physics defined by the invariances of the target potential -- are built into the flow.
We provide a theoretical sufficient criterion showing that the distribution generated by equivariant normalizing flows is invariant with respect to these symmetries by design. Furthermore, we propose building blocks for flows which preserve symmetries which are usually found in physical/chemical many-body particle systems. Using benchmark systems motivated from molecular physics, we demonstrate that those symmetry preserving flows can provide better generalization capabilities and sampling efficiency.

Recurrent neural networks (RNNs) are instrumental in modelling sequential and time-series data. Yet, when using RNNs to inform decision-making, predictions by themselves are not sufficient — we also need estimates of predictive uncertainty. Existing approaches for uncertainty quantification in RNNs are based predominantly on Bayesian methods; these are computationally prohibitive, and require major alterations to the RNN architecture and training. Capitalizing on ideas from classical jackknife resampling, we develop a frequentist alternative that: (a) does not interfere with model training or compromise its accuracy, (b) applies to any RNN architecture, and (c) provides theoretical coverage guarantees on the estimated uncertainty intervals. Our method derives predictive uncertainty from the variability of the (jackknife) sampling distribution of the RNN outputs, which is estimated by repeatedly deleting “blocks” of (temporally-correlated) training data, and collecting the predictions of the RNN re-trained on the remaining data. To avoid exhaustive re-training, we utilize influence functions to estimate the effect of removing training data blocks on the learned RNN parameters. Using data from a critical care setting, we demonstrate the utility of uncertainty quantification in sequential decision-making.

This paper presents a new Graph Neural Network (GNN) type using feature-wise linear modulation (FiLM). Many standard GNN variants propagate information along the edges of a graph by computing messages based only on the representation of the source of each edge. In GNN-FiLM, the representation of the target node of an edge is used to compute a transformation that can be applied to all incoming messages, allowing feature-wise modulation of the passed information.

Different GNN architectures are compared in extensive experiments on three tasks from the literature, using re-implementations of many baseline methods. Hyperparameters for all methods were found using extensive search, yielding somewhat surprising results: differences between state of the art models are much smaller than reported in the literature and well-known simple baselines that are often not compared to perform better than recently proposed GNN variants. Nonetheless, GNN-FiLM outperforms these methods on a regression task on molecular graphs and performs competitively on other tasks.

We present Graph Random Neural Features (GRNF), a novel embedding method from graph-structured data to real vectors based on a family of graph neural networks. The embedding naturally deals with graph isomorphism and preserves the metric structure of the graph domain, in probability. In addition to being an explicit embedding method, it also allows us to efficiently and effectively approximate graph metric distances (as well as complete kernel functions); a criterion to select the embedding dimension trading off the approximation accuracy with the computational cost is also provided. GRNF can be used within traditional processing methods or as a training-free input layer of a graph neural network. The theoretical guarantees that accompany GRNF ensure that the considered graph distance is metric, hence allowing to distinguish any pair of non-isomorphic graphs.

In complex tasks, such as those with large combinatorial action spaces, random exploration may be too inefficient to achieve meaningful learning progress. In this work, we use a curriculum of progressively growing action spaces to accelerate learning. We assume the environment is out of our control, but that the agent may set an internal curriculum by initially restricting its action space. Our approach uses off-policy reinforcement learning to estimate optimal value functions for multiple action spaces simultaneously and efficiently transfers data, value estimates, and state representations from restricted action spaces to the full task. We show the efficacy of our approach in proof-of-concept control tasks and on challenging large-scale StarCraft micromanagement tasks with large, multi-agent action spaces.

Setting regularization parameters for Lasso-type estimators is notoriously difficult, though crucial for obtaining the best accuracy. The most popular hyperparameter optimization approach is grid-search on a held-out dataset. However, grid-search requires to choose a predefined grid of parameters and scales exponentially in the number of parameters. Another class of approaches casts hyperparameter optimization as a bi-level optimization problem, typically solved by gradient descent. The key challenge for these approaches is the estimation of the gradient w.r.t. the hyperparameters. Computing that gradient via forward or backward automatic differentiation usually suffers from high memory consumption, while implicit differentiation typically involves solving a linear system which can be prohibitive and numerically unstable. In addition, implicit differentiation usually assumes smooth loss functions, which is not the case of Lasso-type problems. This work introduces an efficient implicit differentiation algorithm, without matrix inversion, tailored for Lasso-type problems. Our proposal scales to high-dimensional data by leveraging the sparsity of the solutions. Empirically, we demonstrate that the proposed method outperforms a large number of standard methods for hyperparameter optimization.

In this paper, we study inexact high-order Tensor Methods for solving convex optimization problems with composite objective. At every step of such methods, we use approximate solution of the auxiliary problem, defined by the bound for the residual in function value. We propose two dynamic strategies for choosing the inner accuracy: the first one is decreasing as $1/k^{p + 1}$, where $p \geq 1$ is the order of the method and $k$ is the iteration counter, and the second approach is using for the inner accuracy the last progress in the target objective. We show that inexact Tensor Methods with these strategies achieve the same global convergence rate as in the error-free case. For the second approach we also establish local superlinear rates (for $p \geq 2$), and propose the accelerated scheme. Lastly, we present computational results on a variety of machine learning problems for several methods and different accuracy policies.

The construction in the recent paper by Du et al. [2019] implies that searching for a near-optimal action in a bandit sometimes requires examining essentially all the actions, even if the learner is given linear features in R^d that approximate the rewards with a small uniform error. We use the Kiefer-Wolfowitz theorem to prove a positive result that by checking only a few actions, a learner can always find an action that is suboptimal with an error of at most O(ε√d) where ε is the approximation error of the features. Thus, features are useful when the approximation error is small relative to the dimensionality of the features. The idea is applied to stochastic bandits and reinforcement learning with a generative model where the learner has access to d-dimensional linear features that approximate the action-value functions for all policies to an accuracy of ε. For linear bandits, we prove a bound on the regret of order d√(n log(k)) + εn√d log(n) with k the number of actions and n the horizon. For RL we show that approximate policy iteration can learn a policy that is optimal up to an additive error of order ε√d/(1 − γ)^2 and using about d/(ε^2(1 − γ)^4) samples from the generative model. These bounds are independent of the finer details of the features. We also investigate how the structure of the feature set impacts the tradeoff between sample complexity and estimation error.

Posterior inference with an intractable likelihood is becoming an increasingly common task in scientific domains which rely on sophisticated computer simulations. Typically, these forward models do not admit tractable densities forcing practitioners to rely on approximations. This work introduces a novel approach to address the intractability of the likelihood and the marginal model. We achieve this by learning a flexible amortized estimator which approximates the likelihood-to-evidence ratio. We demonstrate that the learned ratio estimator can be embedded in \textsc{mcmc} samplers to approximate likelihood-ratios between consecutive states in the Markov chain, allowing us to draw samples from the intractable posterior. Techniques are presented to improve the numerical stability and to measure the quality of an approximation. The accuracy of our approach is demonstrated on a variety of benchmarks against well-established techniques. Scientific applications in physics show its applicability.

We study whether a neural network optimizes to the same, linearly connected minimum under different samples of SGD noise (e.g., random data order and augmentation). We find that standard vision models become stable to SGD noise in this way early in training. From then on, the outcome of optimization is determined to a linearly connected region. We use this technique to study iterative magnitude pruning (IMP), the procedure used by work on the lottery ticket hypothesis to identify subnetworks that could have trained in isolation to full accuracy. We find that these subnetworks only reach full accuracy when they are stable to SGD noise, which either occurs at initialization for small-scale settings (MNIST) or early in training for large-scale settings (ResNet-50 and Inception-v3 on ImageNet).

We investigate meta-learning procedures in the setting of stochastic linear bandits tasks. The goal is to select a learning algorithm which works well on average over a class of bandits tasks, that are sampled from a task-distribution. Inspired by recent work on learning-to-learn linear regression, we consider a class of bandit algorithms that implement a regularized version of the well-known OFUL algorithm, where the regularization is a square euclidean distance to a bias vector. We first study the benefit of the biased OFUL algorithm in terms of regret minimization. We then propose two strategies to estimate the bias within the learning-to-learn setting. We show both theoretically and experimentally, that when the number of tasks grows and the variance of the task-distribution is small, our strategies have a significant advantage over learning the tasks in isolation.

Support Vector Machines (SVMs) are among the most fundamental tools for binary classification.

In its simplest formulation, an SVM produces a hyperplane separating two classes of data using the largest possible margin to the data. The focus on maximizing the margin has been well motivated through numerous generalization bounds.

In this paper, we revisit and improve the classic generalization bounds in terms of margins. Furthermore, we complement our new generalization bound by a nearly matching lower bound, thus almost settling the generalization performance of SVMs in terms of margins.

Online recommender systems often face long delays in receiving feedback, especially when optimizing for some long-term metrics. While mitigating the effects of delays in learning is well-understood in stationary environments, the problem becomes much more challenging when the environment changes. In fact, if the timescale of the change is comparable to the delay, it is impossible to learn about the environment, since the available observations are already obsolete. However, the arising issues can be addressed if intermediate signals are available without delay, such that given those signals, the long-term behavior of the system is stationary. To model this situation, we introduce the problem of stochastic, non-stationary, delayed bandits with intermediate observations. We develop a computationally efficient algorithm based on UCRL, and prove sublinear regret guarantees for its performance. Experimental results demonstrate that our method is able to learn in non-stationary delayed environments where existing methods fail.

Online Convex Optimization (OCO) is a powerful framework for sequential prediction, portraying the natural uncertainty inherent in data-streams as though the data were generated by an almost omniscient adversary. However, this view, which is often too pessimistic for real-world data, comes with a price. The complexity of solving many important online tasks in this adversarial framework becomes much worse than that of their offline and even stochastic counterparts. In this work we consider a natural random-order version of the OCO model, in which the adversary can choose the set of loss functions, but does not get to choose the order in which they are supplied to the learner; Instead, they are observed in uniformly random order. Focusing on two important families of online tasks, one which includes online linear regression, and the other being online $k$-PCA, we show that under standard well-conditioned-data assumptions, standard online gradient descent (OGD) methods become much more efficient in the random-order model. In particular, for the first group of tasks OGD guarantees poly-logarithmic regret (this result holds even without assuming convexity of individual loss functions). In the case of online $k$-PCA, OGD guarantees sublinear regret using only a rank-$k$ SVD on each iteration and memory linear in the size of the solution.

We provide an exact analysis of a class of randomized algorithms for solving overdetermined least-squares problems. We consider first-order methods, where the gradients are pre-conditioned by an approximation of the Hessian, based on a subspace embedding of the data matrix. This class of algorithms encompasses several randomized methods among the fastest solvers for least-squares problems. We focus on two classical embeddings, namely, Gaussian projections and subsampled randomized Hadamard transforms (SRHT). Our key technical innovation is the derivation of the limiting spectral density of SRHT embeddings. Leveraging this novel result, we derive the family of normalized orthogonal polynomials of the SRHT density and we find the optimal pre-conditioned first-order method along with its rate of convergence. Our analysis of Gaussian embeddings proceeds similarly, and leverages classical random matrix theory results. In particular, we show that for a given sketch size, SRHT embeddings exhibits a faster rate of convergence than Gaussian embeddings. Then, we propose a new algorithm by optimizing the computational complexity over the choice of the sketching dimension. To our knowledge, our resulting algorithm yields the best known complexity for solving least-squares problems with no condition number dependence.

Autoregressive models (ARMs) currently hold state-of-the-art performance in likelihood-based modeling of image and audio data. Generally, neural network based ARMs are designed to allow fast inference, but sampling from these models is impractically slow. In this paper, we introduce the predictive sampling algorithm: a procedure that exploits the fast inference property of ARMs in order to speed up sampling, while keeping the model intact. We propose two variations of predictive sampling, namely sampling with ARM fixed-point iteration and learned forecasting modules. Their effectiveness is demonstrated in two settings: i) explicit likelihood modeling on binary MNIST, SVHN and CIFAR10, and ii) discrete latent modeling in an autoencoder trained on SVHN, CIFAR10 and Imagenet32. Empirically, we show considerable improvements over baselines in number of ARM inference calls and sampling speed.

Increasing the scale of reinforcement learning experiments has allowed researchers to achieve unprecedented results in both training sophisticated agents for video games, and in sim-to-real transfer for robotics. Typically such experiments rely on large distributed systems and require expensive hardware setups, limiting wider access to this exciting area of research. In this work we aim to solve this problem by optimizing the efficiency and resource utilization of reinforcement learning algorithms instead of relying on distributed computation. We present the "Sample Factory", a high-throughput training system optimized for a single-machine setting. Our architecture combines a highly efficient, asynchronous, GPU-based sampler with off-policy correction techniques, allowing us to achieve throughput higher than $10^5$ environment frames/second on non-trivial control problems in 3D without sacrificing sample efficiency. We extend Sample Factory to support self-play and population-based training and apply these techniques to train highly capable agents for a multiplayer first-person shooter game. Github: https://github.com/alex-petrenko/sample-factory

We are interested in how to design reinforcement learning agents that provably reduce the sample complexity for learning new tasks by transferring knowledge from previously-solved ones. The availability of solutions to related problems poses a fundamental trade-off: whether to seek policies that are expected to immediately achieve high (yet sub-optimal) performance in the new task or whether to seek information to quickly identify an optimal solution, potentially at the cost of poor initial behaviour. In this work, we focus on the second objective when the agent has access to a generative model of state-action pairs. First, given a set of solved tasks containing an approximation of the target one, we design an algorithm that quickly identifies an accurate solution by seeking the state-action pairs that are most informative for this purpose. We derive PAC bounds on its sample complexity which clearly demonstrate the benefits of using this kind of prior knowledge. Then, we show how to learn these approximate tasks sequentially by reducing our transfer setting to a hidden Markov model and employing spectral methods to recover its parameters. Finally, we empirically verify our theoretical findings in simple simulated domains.

In this paper, we introduce a powerful and efficient framework for direct optimization of ranking metrics. The problem is ill-posed due to the discrete structure of the loss, and to deal with that, we introduce two important techniques: a stochastic smoothing and a novel gradient estimate based on partial integration. We also address the problem of smoothing bias and present a universal solution for a proper debiasing. To guarantee the global convergence of our method, we adopt a recently proposed Stochastic Gradient Langevin Boosting algorithm. Our algorithm is implemented as a part of the CatBoost gradient boosting library and outperforms the existing approaches on several learning-to-rank datasets. In addition to ranking metrics, our framework applies to any scale-free discrete loss function.

Supervised learning requires the specification of a loss function to minimise. While the theory of admissible losses from both a computational and statistical perspective is well-developed, these offer a panoply of different choices. In practice, this choice is typically made in an \emph{ad hoc} manner. In hopes of making this procedure more principled, the problem of \emph{learning the loss function} for a downstream task (e.g., classification) has garnered recent interest. However, works in this area have been generally empirical in nature. In this paper, we revisit the {\sc SLIsotron} algorithm of Kakade et al. (2011) through a novel lens, derive a generalisation based on Bregman divergences, and show how it provides a principled procedure for learning the loss. In detail, we cast {\sc SLIsotron} as learning a loss from a family of composite square losses. By interpreting this through the lens of \emph{proper losses}, we derive a generalisation of {\sc SLIsotron} based on Bregman divergences. The resulting {\sc BregmanTron} algorithm jointly learns the loss along with the classifier. It comes equipped with a simple guarantee of convergence for the loss it learns, and its set of possible outputs comes with a guarantee of agnostic approximability of Bayes rule. Experiments indicate that the {\sc BregmanTron} significantly outperforms the {\sc SLIsotron}, and that the loss it learns can be minimized by other algorithms for different tasks, thereby opening the interesting problem of \textit{loss transfer} between domains.

We propose a modeling framework for event data and aim to answer questions such as {\it when} and {\it why} the next event would happen. Our proposed model excels in small data regime with the ability to incorporate domain knowledge in terms of logic rules. We model the dynamics of the event starts and ends via intensity function with the structures informed by a set of first-order temporal logic rules. Using the softened representation of temporal relations, and a weighted combination of logic rules, our probabilistic model can deal with uncertainty in events. Furthermore, many well-known point processes (e.g., Hawkes process, self-correcting point process) can be interpreted as special cases of our model given simple temporal logic rules. Our model, therefore, riches the family of point processes. We derive a maximum likelihood estimation procedure for our model and show that it can lead to accurate predictions when data are sparse and domain knowledge is critical.

We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number $n$ of samples and their dimension $d$ go to infinity while their ratio is fixed to $\alpha=n/d$. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances. We also illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.