Tuning Algorithmic and Architectural Hyperparameters in Graph-Based Semi-Supervised Learning with Provable Guarantees

Graph-based semi-supervised learning is a powerful paradigm in machine learning for modeling and exploiting the underlying graph structure that captures the relationship between labeled and unlabeled data. A large number of classical as well as modern deep learning based algorithms have been proposed for this problem, often having tunable hyperparameters. We initiate a formal study of tuning algorithm hyperparameters from parameterized algorithm families for this problem. We obtain novel $O(\log n)$ pseudo-dimension upper bounds for hyperparameter selection in three classical label propagation-based algorithm families, where $n$ is the number of nodes, implying bounds on the amount of data needed for learning provably good parameters. We further provide matching $\Omega(\log n)$ pseudo-dimension lower bounds, thus asymptotically characterizing the learning-theoretic complexity of the parameter tuning problem. We extend our study to selecting architectural hyperparameters in modern graph neural networks. We bound the Rademacher complexity for tuning the self-loop weighting in recently proposed Simplified Graph Convolution (SGC) networks. We further propose a tunable architecture that interpolates graph convolutional neural networks (GCN) and graph attention networks (GAT) in every layer, and provide Rademacher complexity bounds for tuning the interpolation coefficient.

Sample complexity of data-driven tuning of model hyperparameters in neural networks with structured parameter-dependent dual function

Modern machine learning algorithms, especially deep learning based techniques, typically involve careful hyperparameter tuning to achieve the best performance. Despite the surge of intense interest in practical techniques like Bayesian optimization and random search based approaches to automating this laborious and compute-intensive task, the fundamental learning theoretic complexity of tuning hyperparameters for deep neural networks is poorly understood. Inspired by this glaring gap, we initiate the formal study of hyperparameter tuning complexity in deep learning through a recently introduced data driven setting. We assume that we have a series of deep learning tasks, and we have to tune hyperparameters to do well on average over the distribution of tasks. A major difficulty is that the utility function as a function of the hyperparameter is very volatile and furthermore, it is given implicitly by an optimization problem over the model parameters. This is unlike previous work in data driven design, where one can typically explicitly model the algorithmic behavior as a function of the hyperparameters. To tackle this challenge, we introduce a new technique to characterize the discontinuities and oscillations of the utility function on any fixed problem instance as we vary the hyperparameter, our analysis relies on subtle concepts including tools from differential/algebraic geometry and constrained optimization. This can be used to show that the learning theoretic complexity of the corresponding family of utility functions is bounded. We instantiate our results and provide sample complexity bounds for concrete applications tuning a hyperparameter that interpolates neural activation functions and setting the kernel parameter in graph neural networks.

Offline-to-online hyperparameter transfer for stochastic bandits

Classic algorithms for stochastic bandits typically use hyperparameters that govern their critical properties such as the trade-off between exploration and exploitation. Tuning these hyperparameters is a problem of great practical significance. However, this is a challenging problem and in certain cases is information theoretically impossible. To address this challenge, we consider a practically relevant transfer learning setting where one has access to offline data collected from several bandit problems (tasks) coming from an unknown distribution over the tasks. Our aim is to use this offline data to set the hyperparameters for a new task drawn from the unknown distribution. We provide bounds on the inter-task (number of tasks) and intra-task (number of arm pulls for each task) sample complexity for learning near-optimal hyperparameters on unseen tasks drawn from the distribution. Our results apply to several classic algorithms, including tuning the exploration parameters in UCB and LinUCB and the noise parameter in GP-UCB. Our experiments indicate the significance and effectiveness of the transfer of hyperparameters from offline problems in online learning with stochastic bandit feedback.

Provable Hyperparameter Tuning for Structured Pfaffian Settings

Data-driven algorithm design automatically adapts algorithms to specific application domains, achieving better performance. In the context of parameterized algorithms, this approach involves tuning the algorithm parameters using problem instances drawn from the problem distribution of the target application domain. While empirical evidence supports the effectiveness of data-driven algorithm design, providing theoretical guarantees for several parameterized families remains challenging. This is due to the intricate behaviors of their corresponding utility functions, which typically admit piece-wise and discontinuity structures. In this work, we present refined frameworks for providing learning guarantees for parameterized data-driven algorithm design problems in both distributional and online learning settings. For the distributional learning setting, we introduce the Pfaffian GJ framework, an extension of the classical GJ framework, capable of providing learning guarantees for function classes for which the computation involves Pfaffian functions. Unlike the GJ framework, which is limited to function classes with computation characterized by rational functions, our proposed framework can deal with function classes involving Pfaffian functions, which are much more general and widely applicable. We then show that for many parameterized algorithms of interest, their utility function possesses a refined piece-wise structure, which automatically translates to learning guarantees using our proposed framework. For the online learning setting, we provide a new tool for verifying dispersion property of a sequence of loss functions. This sufficient condition allows no-regret learning for sequences of piece-wise structured loss functions where the piece-wise structure involves Pfaffian transition boundaries.

Subsidy design for better social outcomes

Overcoming the impact of selfish behavior of rational players in multiagent systems is a fundamental problem in game theory. Without any intervention from a central agent, strategic users take actions in order to maximize their personal utility, which can lead to extremely inefficient overall system performance, often indicated by a high Price of Anarchy. Recent work (Lin et al. 2021) investigated and formalized yet another undesirable behavior of rational agents, that of avoiding freely available information about the game for selfish reasons, leading to worse social outcomes. A central planner can significantly mitigate these issues by injecting a subsidy to reduce certain costs associated with the system and obtain net gains in the system performance. Crucially, the planner needs to determine how to allocate this subsidy effectively. We formally show that designing subsidies that perfectly optimize the social good, in terms of minimizing the Price of Anarchy or preventing the information avoidance behavior, is computationally hard under standard complexity theoretic assumptions. On the positive side, we show that we can learn provably good values of subsidy in repeated games coming from the same domain. This data-driven subsidy design approach avoids solving computationally hard problems for unseen games by learning over polynomially many games. We also show that optimal subsidy can be learned with no-regret given an online sequence of games, under mild assumptions on the cost matrix. Our study focuses on two distinct games: a Bayesian extension of the well-studied fair cost-sharing game, and a component maintenance game with engineering applications.

Efficiently Learning the Graph for Semi-supervised Learning

Computational efficiency is a major bottleneck in using classic graph-based approaches for semi-supervised learning on datasets with a large number of unlabeled examples. Known techniques to improve efficiency typically involve an approximation of the graph regularization objective, but suffer two major drawbacks - first the graph is assumed to be known or constructed with heuristic hyperparameter values, second they do not provide a principled approximation guarantee for learning over the full unlabeled dataset. Building on recent work on learning graphs for semi-supervised learning from multiple datasets for problems from the same domain, and leveraging techniques for fast approximations for solving linear systems in the graph Laplacian matrix, we propose algorithms that overcome both the above limitations. We show a formal separation in the learning-theoretic complexity of sparse and dense graph families. We further show how to approximately learn the best graphs from the sparse families efficiently using the conjugate gradient method. Our approach can also be used to learn the graph efficiently online with sub-linear regret, under mild smoothness assumptions. Our online learning results are stated generally, and may be useful for approximate and efficient parameter tuning in other problems. We implement our approach and demonstrate significant ($\sim$10-100x) speedups over prior work on semi-supervised learning with learned graphs on benchmark datasets.

Reliable Learning for Test-time Attacks and Distribution Shift

Machine learning algorithms are often used in environments which are not captured accurately even by the most carefully obtained training data, either due to the possibility of `adversarial' test-time attacks, or on account of `natural' distribution shift. For test-time attacks, we introduce and analyze a novel robust reliability guarantee, which requires a learner to output predictions along with a reliability radius $\eta$, with the meaning that its prediction is guaranteed to be correct as long as the adversary has not perturbed the test point farther than a distance $\eta$. We provide learners that are optimal in the sense that they always output the best possible reliability radius on any test point, and we characterize the reliable region, i.e. the set of points where a given reliability radius is attainable. We additionally analyze reliable learners under distribution shift, where the test points may come from an arbitrary distribution Q different from the training distribution P. For both cases, we bound the probability mass of the reliable region for several interesting examples, for linear separators under nearly log-concave and s-concave distributions, as well as for smooth boundary classifiers under smooth probability distributions.

Provably tuning the ElasticNet across instances

An important unresolved challenge in the theory of regularization is to set the regularization coefficients of popular techniques like the ElasticNet with general provable guarantees. We consider the problem of tuning the regularization parameters of Ridge regression, LASSO, and the ElasticNet across multiple problem instances, a setting that encompasses both cross-validation and multi-task hyperparameter optimization. We obtain a novel structural result for the ElasticNet which characterizes the loss as a function of the tuning parameters as a piecewise-rational function with algebraic boundaries. We use this to bound the structural complexity of the regularized loss functions and show generalization guarantees for tuning the ElasticNet regression coefficients in the statistical setting. We also consider the more challenging online learning setting, where we show vanishing average expected regret relative to the optimal parameter pair. We further extend our results to tuning classification algorithms obtained by thresholding regression fits regularized by Ridge, LASSO, or ElasticNet. Our results are the first general learning-theoretic guarantees for this important class of problems that avoid strong assumptions on the data distribution. Furthermore, our guarantees hold for both validation and popular information criterion objectives.

Faster algorithms for learning to link, align sequences, and price two-part tariffs

Data-driven algorithm configuration is a promising, learning-based approach for beyond worst-case analysis of algorithms with tunable parameters. An important open problem is the design of efficient data-driven algorithms for algorithm families with more than one parameter. In this work we provide algorithms for efficient (output-polynomial) multidimensional parameter tuning, i.e. for families with a small constant number of parameters, for three very different combinatorial problems -- linkage-based clustering, dynamic programming for sequence alignment, and auction design for two-part tariff schemes. We extend the single-parameter clustering algorithm of Balcan et al. 2020 arXiv:1907.00533 to multiple parameters and to the sequence alignment problem by proposing an execution graph which compactly represents all the states the algorithm could attain for all possible parameter values. A key problem-specific challenge is to efficiently compute how the partition of the parameter space (into regions with unique algorithmic states) changes with a single algorithmic step. We give algorithms which improve on the runtime of previously best known results for linkage-based clustering, sequence alignment and two-part tariff pricing.

Robustly-reliable learners under poisoning attacks

Data poisoning attacks, in which an adversary corrupts a training set with the goal of inducing specific desired mistakes, have raised substantial concern: even just the possibility of such an attack can make a user no longer trust the results of a learning system. In this work, we show how to achieve strong robustness guarantees in the face of such attacks across multiple axes. We provide robustly-reliable predictions, in which the predicted label is guaranteed to be correct so long as the adversary has not exceeded a given corruption budget, even in the presence of instance targeted attacks, where the adversary knows the test example in advance and aims to cause a specific failure on that example. Our guarantees are substantially stronger than those in prior approaches, which were only able to provide certificates that the prediction of the learning algorithm does not change, as opposed to certifying that the prediction is correct, as we are able to achieve in our work. Remarkably, we provide a complete characterization of learnability in this setting, in particular, nearly-tight matching upper and lower bounds on the region that can be certified, as well as efficient algorithms for computing this region given an ERM oracle. Moreover, for the case of linear separators over logconcave distributions, we provide efficient truly polynomial time algorithms (i.e., non-oracle algorithms) for such robustly-reliable predictions. We also extend these results to the active setting where the algorithm adaptively asks for labels of specific informative examples, and the difficulty is that the adversary might even be adaptive to this interaction, as well as to the agnostic learning setting where there is no perfect classifier even over the uncorrupted data.