We introduce a mapping between Maximum Entropy Reinforcement Learning (MaxEnt RL) and Markovian processes conditioned on rare events. In the long time limit, this mapping allows us to derive analytical expressions for the optimal policy, dynamics and initial state distributions for the general case of stochastic dynamics in MaxEnt RL. We find that soft-$\mathcal{Q}$ functions in MaxEnt RL can be obtained from the Perron-Frobenius eigenvalue and the corresponding left eigenvector of a regular, non-negative matrix derived from the underlying Markov Decision Process (MDP). The results derived lead to novel algorithms for model-based and model-free MaxEnt RL, which we validate by numerical simulations. The mapping established in this work opens further avenues for the application of novel analytical and computational approaches to problems in MaxEnt RL. We make our code available at: https://github.com/argearriojas/maxent-rl-mdp-scripts

We tackle sequential learning under label noise in applications where a human supervisor can be queried to relabel suspicious examples. Existing approaches are flawed, in that they only relabel incoming examples that look ``suspicious'' to the model. As a consequence, those mislabeled examples that elude (or don't undergo) this cleaning step end up tainting the training data and the model with no further chance of being cleaned. We propose Cincer, a novel approach that cleans both new and past data by identifying pairs of mutually incompatible examples. Whenever it detects a suspicious example, Cincer identifies a counter-example in the training set that -- according to the model -- is maximally incompatible with the suspicious example, and asks the annotator to relabel either or both examples, resolving this possible inconsistency. The counter-examples are chosen to be maximally incompatible, so to serve as explanations of the model' suspicion, and highly influential, so to convey as much information as possible if relabeled. Cincer achieves this by leveraging an efficient and robust approximation of influence functions based on the Fisher information matrix (FIM). Our extensive empirical evaluation shows that clarifying the reasons behind the model's suspicions by cleaning the counter-examples helps acquiring substantially better data and models, especially when paired with our FIM approximation.

A large body of recent work has begun to explore the potential of parametrized quantum circuits (PQCs) as machine learning models, within the framework of hybrid quantum-classical optimization. In particular, theoretical guarantees on the out-of-sample performance of such models, in terms of generalization bounds, have emerged. However, none of these generalization bounds depend explicitly on how the classical input data is encoded into the PQC. We derive generalization bounds for PQC-based models that depend explicitly on the strategy used for data-encoding. These imply bounds on the performance of trained PQC-based models on unseen data. Moreover, our results facilitate the selection of optimal data-encoding strategies via structural risk minimization, a mathematically rigorous framework for model selection. We obtain our generalization bounds by bounding the complexity of PQC-based models as measured by the Rademacher complexity and the metric entropy, two complexity measures from statistical learning theory. To achieve this, we rely on a representation of PQC-based models via trigonometric functions. Our generalization bounds emphasize the importance of well-considered data-encoding strategies for PQC-based models.

Although Shapley Values (SV) are widely used in explainable AI, they can be poorly understood and estimated, which implies that their analysis may lead to spurious inferences and explanations. As a starting point, we remind an invariance principle for SV and derive the correct approach for computing the SV of categorical variables that are particularly sensitive to the encoding used. In the case of tree-based models, we introduce two estimators of Shapley Values that exploit efficiently the tree structure and are more accurate than state-of-the-art methods. For interpreting additive explanations, we recommend to filter the non-influential variables and to compute the Shapley Values only for groups of influential variables. For this purpose, we use the concept of "Same Decision Probability" (SDP) that evaluates the robustness of a prediction when some variables are missing. This prior selection procedure produces sparse additive explanations easier to visualize and analyse. Simulations and comparisons are performed with state-of-the-art algorithm, and show the practical gain of our approach.

Neural network compression techniques have become increasingly popular as they can drastically reduce the storage and computation requirements for very large networks. Recent empirical studies have illustrated that even simple pruning strategies can be surprisingly effective, and several theoretical studies have shown that compressible networks (in specific senses) should achieve a low generalization error. Yet, a theoretical characterization of the underlying cause that makes the networks amenable to such simple compression schemes is still missing. In this study, we address this fundamental question and reveal that the dynamics of the training algorithm has a key role in obtaining such compressible networks. Focusing our attention on stochastic gradient descent (SGD), our main contribution is to link compressibility to two recently established properties of SGD: (i) as the network size goes to infinity, the system can converge to a mean-field limit, where the network weights behave independently, (ii) for a large step-size/batch-size ratio, the SGD iterates can converge to a heavy-tailed stationary distribution. In the case where these two phenomena occur simultaneously, we prove that the networks are guaranteed to be '$\ell_p$-compressible', and the compression errors of different pruning techniques (magnitude, singular value, or node pruning) become arbitrarily small as the network size increases. We further prove generalization bounds adapted to our theoretical framework, which indeed confirm that the generalization error will be lower for more compressible networks. Our theory and numerical study on various neural networks show that large step-size/batch-size ratios introduce heavy-tails, which, in combination with overparametrization, result in compressibility.

Generalised linear models for multi-class classification problems are one of the fundamental building blocks of modern machine learning tasks. In this manuscript, we characterise the learning of a mixture of $K$ Gaussians with generic means and covariances via empirical risk minimisation (ERM) with any convex loss and regularisation. In particular, we prove exact asymptotics characterising the ERM estimator in high-dimensions, extending several previous results about Gaussian mixture classification in the literature. We exemplify our result in two tasks of interest in statistical learning: a) classification for a mixture with sparse means, where we study the efficiency of $\ell_1$ penalty with respect to $\ell_2$; b) max-margin multi-class classification, where we characterise the phase transition on the existence of the multi-class logistic maximum likelihood estimator for $K>2$. Finally, we discuss how our theory can be applied beyond the scope of synthetic data, showing that in different cases Gaussian mixtures capture closely the learning curve of classification tasks in real data sets.

It has been hypothesized that quantum computers may lend themselves well to applications in machine learning. In the present work, we analyze function classes defined via quantum kernels. Quantum computers offer the possibility to efficiently compute inner products of exponentially large density operators that are classically hard to compute. However, having an exponentially large feature space renders the problem of generalization hard. Furthermore, being able to evaluate inner products in high dimensional spaces efficiently by itself does not guarantee a quantum advantage, as already classically tractable kernels can correspond to high- or infinite-dimensional reproducing kernel Hilbert spaces (RKHS). We analyze the spectral properties of quantum kernels and find that we can expect an advantage if their RKHS is low dimensional and contains functions that are hard to compute classically. If the target function is known to lie in this class, this implies a quantum advantage, as the quantum computer can encode this inductive bias, whereas there is no classically efficient way to constrain the function class in the same way. However, we show that finding suitable quantum kernels is not easy because the kernel evaluation might require exponentially many measurements. In conclusion, our message is a somewhat sobering one: we conjecture that quantum machine learning models can offer speed-ups only if we manage to encode knowledge about the problem at hand into quantum circuits, while encoding the same bias into a classical model would be hard. These situations may plausibly occur when learning on data generated by a quantum process, however, they appear to be harder to come by for classical datasets.

We analyze a class of stochastic gradient algorithms with momentum on a high-dimensional random least squares problem. Our framework, inspired by random matrix theory, provides an exact (deterministic) characterization for the sequence of loss values produced by these algorithms which is expressed only in terms of the eigenvalues of the Hessian. This leads to simple expressions for nearly-optimal hyperparameters, a description of the limiting neighborhood, and average-case complexity. As a consequence, we show that (small-batch) stochastic heavy-ball momentum with a fixed momentum parameter provides no actual performance improvement over SGD when step sizes are adjusted correctly. For contrast, in the non-strongly convex setting, it is possible to get a large improvement over SGD using momentum. By introducing hyperparameters that depend on the number of samples, we propose a new algorithm sDANA (stochastic dimension adjusted Nesterov acceleration) which obtains an asymptotically optimal average-case complexity while remaining linearly convergent in the strongly convex setting without adjusting parameters.

Out-of-distribution generalization is one of the key challenges when transferring a model from the lab to the real world. Existing efforts mostly focus on building invariant features among source and target domains. Based on invariant features, a high-performing classifier on source domains could hopefully behave equally well on a target domain. In other words, the invariant features are \emph{transferable}. However, in practice, there are no perfectly transferable features, and some algorithms seem to learn ''more transferable'' features than others. How can we understand and quantify such \emph{transferability}? In this paper, we formally define transferability that one can quantify and compute in domain generalization. We point out the difference and connection with common discrepancy measures between domains, such as total variation and Wasserstein distance. We then prove that our transferability can be estimated with enough samples and give a new upper bound for the target error based on our transferability. Empirically, we evaluate the transferability of the feature embeddings learned by existing algorithms for domain generalization. Surprisingly, we find that many algorithms are not quite learning transferable features, although few could still survive. In light of this, we propose a new algorithm for learning transferable features and test it over various benchmark datasets, including RotatedMNIST, PACS, Office-Home and WILDS-FMoW. Experimental results show that the proposed algorithm achieves consistent improvement over many state-of-the-art algorithms, corroborating our theoretical findings.

In decentralized optimization, nodes of a communication network privately possess a local objective function, and communicate using gossip-based methods in order to minimize the average of these per-node objectives. While synchronous algorithms can be heavily slowed down by a few nodes and edges in the graph (the straggler problem), their asynchronous counterparts lack from a sharp analysis taking into account heterogeneous delays in the communication network. In this paper, we propose a novel continuous-time framework to analyze asynchronous algorithms, which does not require to define a global ordering of the events, and allows to finely characterize the time complexity in the presence of (heterogeneous) delays. Using this framework, we describe a fully asynchronous decentralized algorithm to minimize the sum of smooth and strongly convex functions. Our algorithm (DCDM, Delayed Coordinate Dual Method), based on delayed randomized gossip communications and local computational updates, achieves an asynchronous speed-up: the rate of convergence is tightly characterized in terms of the eigengap of the graph weighted by local delays only, instead of the global worst-case delays as in previous analyses.