Learning how to effectively control unknown dynamical systems is crucial for intelligent autonomous systems. This task becomes a significant challenge when the underlying dynamics are changing with time. Motivated by this challenge, this paper considers the problem of controlling an unknown Markov jump linear system (MJS) to optimize a quadratic objective. By taking a model-based perspective, we consider identification-based adaptive control for MJSs. We first provide a system identification algorithm for MJS to learn the dynamics in each mode as well as the Markov transition matrix, underlying the evolution of the mode switches, from a single trajectory of the system states, inputs, and modes. Through mixing-time arguments, sample complexity of this algorithm is shown to be $\mathcal{O}(1/\sqrt{T})$. We then propose an adaptive control scheme that performs system identification together with certainty equivalent control to adapt the controllers in an episodic fashion. Combining our sample complexity results with recent perturbation results for certainty equivalent control, we prove that when the episode lengths are appropriately chosen, the proposed adaptive control scheme achieves $\mathcal{O}(\sqrt{T})$ regret, which can be improved to $\mathcal{O}(polylog(T))$ with partial knowledge of the system. Our proof strategy introduces innovations to handle Markovian jumps and a weaker notion of stability common in MJSs. Our analysis provides insights into system theoretic quantities that affect learning accuracy and control performance. Numerical simulations are presented to further reinforce these insights.

Neural Architecture Search (NAS) is a popular method for automatically designing optimized architectures for high-performance deep learning. In this approach, it is common to use bilevel optimization where one optimizes the model weights over the training data (lower-level problem) and various hyperparameters such as the configuration of the architecture over the validation data (upper-level problem). This paper explores the statistical aspects of such problems with train-validation splits. In practice, the lower-level problem is often overparameterized and can easily achieve zero loss. Thus, a-priori it seems impossible to distinguish the right hyperparameters based on training loss alone which motivates a better understanding of the role of train-validation split. To this aim this work establishes the following results. (1) We show that refined properties of the validation loss such as risk and hyper-gradients are indicative of those of the true test loss. This reveals that the upper-level problem helps select the most generalizable model and prevent overfitting with a near-minimal validation sample size. Importantly, this is established for continuous search spaces which are highly relevant for popular differentiable search schemes. (2) We establish generalization bounds for NAS problems with an emphasis on an activation search problem. When optimized with gradient-descent, we show that the train-validation procedure returns the best (model, architecture) pair even if all architectures can perfectly fit the training data to achieve zero error. (3) Finally, we highlight rigorous connections between NAS, multiple kernel learning, and low-rank matrix learning. The latter leads to novel algorithmic insights where the solution of the upper problem can be accurately learned via efficient spectral methods to achieve near-minimal risk.

Label-imbalanced and group-sensitive classification seeks to appropriately modify standard training algorithms to optimize relevant metrics such as balanced error and/or equal opportunity. For label imbalances, recent works have proposed a logit-adjusted loss modification to standard empirical risk minimization. We show that this might be ineffective in general and, in particular so, in the overparameterized regime where training continues in the zero training-error regime. Specifically for binary linear classification of a separable dataset, we show that the modified loss converges to the max-margin SVM classifier despite the logit adjustment. Instead, we propose a more general vector-scaling loss that directly relates to the cost-sensitive SVM (CS-SVM), thus favoring larger margin to the minority class. Through an insightful sharp asymptotic analysis for a Gaussian-mixtures data model, we demonstrate the efficacy of CS-SVM in balancing the errors of the minority/majority classes. Our analysis also leads to a simple strategy for optimally tuning the involved margin-ratio parameter. Then, we show how our results extend naturally to binary classification with sensitive groups, thus treating the two common types of imbalances (label/group) in a unifying way. We corroborate our theoretical findings with numerical experiments on both synthetic and real-world datasets.

Constructing good representations is critical for learning complex tasks in a sample efficient manner. In the context of meta-learning, representations can be constructed from common patterns of previously seen tasks so that a future task can be learned quickly. While recent works show the benefit of subspace-based representations, such results are limited to linear-regression tasks. This work explores a more general class of nonlinear tasks with applications ranging from binary classification, generalized linear models and neural nets. We prove that subspace-based representations can be learned in a sample-efficient manner and provably benefit future tasks in terms of sample complexity. Numerical results verify the theoretical predictions in classification and neural-network regression tasks.

Deep networks are typically trained with many more parameters than the size of the training dataset. Recent empirical evidence indicates that the practice of overparameterization not only benefits training large models, but also assists - perhaps counterintuitively - building lightweight models. Specifically, it suggests that overparameterization benefits model pruning / sparsification. This paper sheds light on these empirical findings by theoretically characterizing the high-dimensional asymptotics of model pruning in the overparameterized regime. The theory presented addresses the following core question: "should one train a small model from the beginning, or first train a large model and then prune?". We analytically identify regimes in which, even if the location of the most informative features is known, we are better off fitting a large model and then pruning rather than simply training with the known informative features. This leads to a new double descent in the training of sparse models: growing the original model, while preserving the target sparsity, improves the test accuracy as one moves beyond the overparameterization threshold. Our analysis further reveals the benefit of retraining by relating it to feature correlations. We find that the above phenomena are already present in linear and random-features models. Our technical approach advances the toolset of high-dimensional analysis and precisely characterizes the asymptotic distribution of over-parameterized least-squares. The intuition gained by analytically studying simpler models is numerically verified on neural networks.

Contemporary machine learning applications often involve classification tasks with many classes. Despite their extensive use, a precise understanding of the statistical properties and behavior of classification algorithms is still missing, especially in modern regimes where the number of classes is rather large. In this paper, we take a step in this direction by providing the first asymptotically precise analysis of linear multiclass classification. Our theoretical analysis allows us to precisely characterize how the test error varies over different training algorithms, data distributions, problem dimensions as well as number of classes, inter/intra class correlations and class priors. Specifically, our analysis reveals that the classification accuracy is highly distribution-dependent with different algorithms achieving optimal performance for different data distributions and/or training/features sizes. Unlike linear regression/binary classification, the test error in multiclass classification relies on intricate functions of the trained model (e.g., correlation between some of the trained weights) whose asymptotic behavior is difficult to characterize. This challenge is already present in simple classifiers, such as those minimizing a square loss. Our novel theoretical techniques allow us to overcome some of these challenges. The insights gained may pave the way for a precise understanding of other classification algorithms beyond those studied in this paper.

Paraphrasing is expressing the meaning of an input sentence in different wording while maintaining fluency (i.e., grammatical and syntactical correctness). Most existing work on paraphrasing use supervised models that are limited to specific domains (e.g., image captions). Such models can neither be straightforwardly transferred to other domains nor generalize well, and creating labeled training data for new domains is expensive and laborious. The need for paraphrasing across different domains and the scarcity of labeled training data in many such domains call for exploring unsupervised paraphrase generation methods. We propose Progressive Unsupervised Paraphrasing (PUP): a novel unsupervised paraphrase generation method based on deep reinforcement learning (DRL). PUP uses a variational autoencoder (trained using a non-parallel corpus) to generate a seed paraphrase that warm-starts the DRL model. Then, PUP progressively tunes the seed paraphrase guided by our novel reward function which combines semantic adequacy, language fluency, and expression diversity measures to quantify the quality of the generated paraphrases in each iteration without needing parallel sentences. Our extensive experimental evaluation shows that PUP outperforms unsupervised state-of-the-art paraphrasing techniques in terms of both automatic metrics and user studies on four real datasets. We also show that PUP outperforms domain-adapted supervised algorithms on several datasets. Our evaluation also shows that PUP achieves a great trade-off between semantic similarity and diversity of expression.

Self-training is a classical approach in semi-supervised learning which is successfully applied to a variety of machine learning problems. Self-training algorithm generates pseudo-labels for the unlabeled examples and progressively refines these pseudo-labels which hopefully coincides with the actual labels. This work provides theoretical insights into self-training algorithm with a focus on linear classifiers. We first investigate Gaussian mixture models and provide a sharp non-asymptotic finite-sample characterization of the self-training iterations. Our analysis reveals the provable benefits of rejecting samples with low confidence and demonstrates that self-training iterations gracefully improve the model accuracy even if they do get stuck in sub-optimal fixed points. We then demonstrate that regularization and class margin (i.e. separation) is provably important for the success and lack of regularization may prevent self-training from identifying the core features in the data. Finally, we discuss statistical aspects of empirical risk minimization with self-training for general distributions. We show how a purely unsupervised notion of generalization based on self-training based clustering can be formalized based on cluster margin. We then establish a connection between self-training based semi-supervision and the more general problem of learning with heterogenous data and weak supervision.

Model pruning is an essential procedure for building compact and computationally-efficient machine learning models. A key feature of a good pruning algorithm is that it accurately quantifies the relative importance of the model weights. While model pruning has a rich history, we still don't have a full grasp of the pruning mechanics even for relatively simple problems involving linear models or shallow neural nets. In this work, we provide a principled exploration of pruning by building on a natural notion of importance. For linear models, we show that this notion of importance is captured by covariance scaling which connects to the well-known Hessian-based pruning. We then derive asymptotic formulas that allow us to precisely compare the performance of different pruning methods. For neural networks, we demonstrate that the importance can be at odds with larger magnitudes and proper initialization is critical for magnitude-based pruning. Specifically, we identify settings in which weights become more important despite becoming smaller, which in turn leads to a catastrophic failure of magnitude-based pruning. Our results also elucidate that implicit regularization in the form of Hessian structure has a catalytic role in identifying the important weights, which dictate the pruning performance.

We consider the problem of finding clusters in an unweighted graph, when the graph is partially observed. We analyze two programs, one which works for dense graphs and one which works for both sparse and dense graphs, but requires some a priori knowledge of the total cluster size, that are based on the convex optimization approach for low-rank matrix recovery using nuclear norm minimization. For the commonly used Stochastic Block Model, we obtain \emph{explicit} bounds on the parameters of the problem (size and sparsity of clusters, the amount of observed data) and the regularization parameter characterize the success and failure of the programs. We corroborate our theoretical findings through extensive simulations. We also run our algorithm on a real data set obtained from crowdsourcing an image classification task on the Amazon Mechanical Turk, and observe significant performance improvement over traditional methods such as k-means.