Learning-to-learn non-convex piecewise-Lipschitz functions

We analyze the meta-learning of the initialization and step-size of learning algorithms for piecewise-Lipschitz functions, a non-convex setting with applications to both machine learning and algorithms. Starting from recent regret bounds for the exponential forecaster on losses with dispersed discontinuities, we generalize them to be initialization-dependent and then use this result to propose a practical meta-learning procedure that learns both the initialization and the step-size of the algorithm from multiple online learning tasks. Asymptotically, we guarantee that the average regret across tasks scales with a natural notion of task-similarity that measures the amount of overlap between near-optimal regions of different tasks. Finally, we instantiate the method and its guarantee in two important settings: robust meta-learning and multi-task data-driven algorithm design.

Sample Complexity of Tree Search Configuration: Cutting Planes and Beyond

Cutting-plane methods have enabled remarkable successes in integer programming over the last few decades. State-of-the-art solvers integrate a myriad of cutting-plane techniques to speed up the underlying tree-search algorithm used to find optimal solutions. In this paper we prove the first guarantees for learning high-performing cut-selection policies tailored to the instance distribution at hand using samples. We first bound the sample complexity of learning cutting planes from the canonical family of Chvatal-Gomory cuts. Our bounds handle any number of waves of any number of cuts and are fine tuned to the magnitudes of the constraint coefficients. Next, we prove sample complexity bounds for more sophisticated cut selection policies that use a combination of scoring rules to choose from a family of cuts. Finally, beyond the realm of cutting planes for integer programming, we develop a general abstraction of tree search that captures key components such as node selection and variable selection. For this abstraction, we bound the sample complexity of learning a good policy for building the search tree.

Data driven algorithms for limited labeled data learning

We consider a novel data driven approach for designing learning algorithms that can effectively learn with only a small number of labeled examples. This is crucial for modern machine learning applications where labels are scarce or expensive to obtain. We focus on graph-based techniques, where the unlabeled examples are connected in a graph under the implicit assumption that similar nodes likely have similar labels. Over the past decades, several elegant graph-based semi-supervised and active learning algorithms for how to infer the labels of the unlabeled examples given the graph and a few labeled examples have been proposed. However, the problem of how to create the graph (which impacts the practical usefulness of these methods significantly) has been relegated to domain-specific art and heuristics and no general principles have been proposed. In this work we present a novel data driven approach for learning the graph and provide strong formal guarantees in both the distributional and online learning formalizations. We show how to leverage problem instances coming from an underlying problem domain to learn the graph hyperparameters from commonly used parametric families of graphs that perform well on new instances coming from the same domain. We obtain low regret and efficient algorithms in the online setting, and generalization guarantees in the distributional setting. We also show how to combine several very different similarity metrics and learn multiple hyperparameters, providing general techniques to apply to large classes of problems. We expect some of the tools and techniques we develop along the way to be of interest beyond semi-supervised and active learning, for data driven algorithms for combinatorial problems more generally.

Generalization in portfolio-based algorithm selection

Portfolio-based algorithm selection has seen tremendous practical success over the past two decades. This algorithm configuration procedure works by first selecting a portfolio of diverse algorithm parameter settings, and then, on a given problem instance, using an algorithm selector to choose a parameter setting from the portfolio with strong predicted performance. Oftentimes, both the portfolio and the algorithm selector are chosen using a training set of typical problem instances from the application domain at hand. In this paper, we provide the first provable guarantees for portfolio-based algorithm selection. We analyze how large the training set should be to ensure that the resulting algorithm selectors average performance over the training set is close to its future (expected) performance. This involves analyzing three key reasons why these two quantities may diverge: 1) the learning-theoretic complexity of the algorithm selector, 2) the size of the portfolio, and 3) the learning-theoretic complexity of the algorithms performance as a function of its parameters. We introduce an end-to-end learning-theoretic analysis of the portfolio construction and algorithm selection together. We prove that if the portfolio is large, overfitting is inevitable, even with an extremely simple algorithm selector. With experiments, we illustrate a tradeoff exposed by our theoretical analysis: as we increase the portfolio size, we can hope to include a well-suited parameter setting for every possible problem instance, but it becomes impossible to avoid overfitting.

Scalable and Provably Accurate Algorithms for Differentially Private Distributed Decision Tree Learning

This paper introduces the first provably accurate algorithms for differentially private, top-down decision tree learning in the distributed setting (Balcan et al., 2012). We propose DP-TopDown, a general privacy preserving decision tree learning algorithm, and present two distributed implementations. Our first method NoisyCounts naturally extends the single machine algorithm by using the Laplace mechanism. Our second method LocalRNM significantly reduces communication and added noise by performing local optimization at each data holder. We provide the first utility guarantees for differentially private top-down decision tree learning in both the single machine and distributed settings. These guarantees show that the error of the privately-learned decision tree quickly goes to zero provided that the dataset is sufficiently large. Our extensive experiments on real datasets illustrate the trade-offs of privacy, accuracy and generalization when learning private decision trees in the distributed setting.

Data-driven Algorithm Design

Data driven algorithm design is an important aspect of modern data science and algorithm design. Rather than using off the shelf algorithms that only have worst case performance guarantees, practitioners often optimize over large families of parametrized algorithms and tune the parameters of these algorithms using a training set of problem instances from their domain to determine a configuration with high expected performance over future instances. However, most of this work comes with no performance guarantees. The challenge is that for many combinatorial problems of significant importance including partitioning, subset selection, and alignment problems, a small tweak to the parameters can cause a cascade of changes in the algorithms behavior, so the algorithms performance is a discontinuous function of its parameters. In this chapter, we survey recent work that helps put data-driven combinatorial algorithm design on firm foundations. We provide strong computational and statistical performance guarantees, both for the batch and online scenarios where a collection of typical problem instances from the given application are presented either all at once or in an online fashion, respectively.

On the Power of Abstention and Data-Driven Decision Making for Adversarial Robustness

We formally define a feature-space attack where the adversary can perturb datapoints by arbitrary amounts but in restricted directions. By restricting the attack to a small random subspace, our model provides a clean abstraction for non-Lipschitz networks which map small input movements to large feature movements. We prove that classifiers with the ability to abstain are provably more powerful than those that cannot in this setting. Specifically, we show that no matter how well-behaved the natural data is, any classifier that cannot abstain will be defeated by such an adversary. However, by allowing abstention, we give a parameterized algorithm with provably good performance against such an adversary when classes are reasonably well-separated in feature space and the dimension of the feature space is high. We further use a data-driven method to set our algorithm parameters to optimize over the accuracy vs. abstention trade-off with strong theoretical guarantees. Our theory has direct applications to the technique of contrastive learning, where we empirically demonstrate the ability of our algorithms to obtain high robust accuracy with only small amounts of abstention in both supervised and self-supervised settings. Our results provide a first formal abstention-based gap, and a first provable optimization for the induced trade-off in an adversarial defense setting.

Noise in Classification

This chapter considers the computational and statistical aspects of learning linear thresholds in presence of noise. When there is no noise, several algorithms exist that efficiently learn near-optimal linear thresholds using a small amount of data. However, even a small amount of adversarial noise makes this problem notoriously hard in the worst-case. We discuss approaches for dealing with these negative results by exploiting natural assumptions on the data-generating process.

Refined bounds for algorithm configuration: The knife-edge of dual class approximability

Automating algorithm configuration is growing increasingly necessary as algorithms come with more and more tunable parameters. It is common to tune parameters using machine learning, optimizing performance metrics such as runtime and solution quality. The training set consists of problem instances from the specific domain at hand. We investigate a fundamental question about these techniques: how large should the training set be to ensure that a parameters average empirical performance over the training set is close to its expected, future performance? We answer this question for algorithm configuration problems that exhibit a widely-applicable structure: the algorithms performance as a function of its parameters can be approximated by a simple function. We show that if this approximation holds under the L-infinity norm, we can provide strong sample complexity bounds. On the flip side, if the approximation holds only under the L-p norm for p smaller than infinity, it is not possible to provide meaningful sample complexity bounds in the worst case. We empirically evaluate our bounds in the context of integer programming, one of the most powerful tools in computer science. Via experiments, we obtain sample complexity bounds that are up to 700 times smaller than the previously best-known bounds.

Geometry-Aware Gradient Algorithms for Neural Architecture Search

Recent state-of-the-art methods for neural architecture search (NAS) exploit gradient-based optimization by relaxing the problem into continuous optimization over architectures and shared-weights, a noisy process that remains poorly understood. We argue for the study of single-level empirical risk minimization to understand NAS with weight-sharing, reducing the design of NAS methods to devising optimizers and regularizers that can quickly obtain high-quality solutions to this problem. Invoking the theory of mirror descent, we present a geometry-aware framework that exploits the underlying structure of this optimization to return sparse architectural parameters, leading to simple yet novel algorithms that enjoy fast convergence guarantees and achieve state-of-the-art accuracy on the latest NAS benchmarks in computer vision. Notably, we exceed the best published results for both CIFAR and ImageNet on both the DARTS search space and NAS-Bench201; on the latter we achieve near-oracle-optimal performance on CIFAR-10 and CIFAR-100. Together, our theory and experiments demonstrate a principled way to co-design optimizers and continuous relaxations of discrete NAS search spaces.