Finding the optimal model complexity that minimizes the generalization error (GE) is a key issue of machine learning. For the conventional supervised learning, this task typically involves the bias-variance tradeoff: lowering the bias by making the model more complex entails an increase in the variance. Meanwhile, little has been studied about whether the same tradeoff exists for unsupervised learning. In this study, we propose that unsupervised learning generally exhibits a two-component tradeoff of the GE, namely the model error and the data error -- using a more complex model reduces the model error at the cost of the data error, with the data error playing a more significant role for a smaller training dataset. This is corroborated by training the restricted Boltzmann machine to generate the configurations of the two-dimensional Ising model at a given temperature and the totally asymmetric simple exclusion process with given entry and exit rates. Our results also indicate that the optimal model tends to be more complex when the data to be learned are more complex.
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This paper studies the high-dimensional quantile regression problem under the transfer learning framework, where possibly related source datasets are available to make improvements on the estimation or prediction based solely on the target data. In the oracle case with known transferable sources, a smoothed two-step transfer learning algorithm based on convolution smoothing is proposed and the L1/L2 estimation error bounds of the corresponding estimator are also established. To avoid including non-informative sources, we propose to select the transferable sources adaptively and establish its selection consistency under regular conditions. Monte Carlo simulations as well as an empirical analysis of gene expression data demonstrate the effectiveness of the proposed procedure.
We provide a new information-theoretic generalization error bound that is exactly tight (i.e., matching even the constant) for the canonical quadratic Gaussian mean estimation problem. Despite considerable existing efforts in deriving information-theoretic generalization error bounds, applying them to this simple setting where sample average is used as the estimate of the mean value of Gaussian data has not yielded satisfying results. In fact, most existing bounds are order-wise loose in this setting, which has raised concerns about the fundamental capability of information-theoretic bounds in reasoning the generalization behavior for machine learning. The proposed new bound adopts the individual-sample-based approach proposed by Bu et al., but also has several key new ingredients. Firstly, instead of applying the change of measure inequality on the loss function, we apply it to the generalization error function itself; secondly, the bound is derived in a conditional manner; lastly, a reference distribution, which bears a certain similarity to the prior distribution in the Bayesian setting, is introduced. The combination of these components produces a general KL-divergence-based generalization error bound. We further show that although the conditional bounding and the reference distribution can make the bound exactly tight, removing them does not significantly degrade the bound, which leads to a mutual-information-based bound that is also asymptotically tight in this setting.
Theoretical studies on transfer learning or domain adaptation have so far focused on situations with a known hypothesis class or model; however in practice, some amount of model selection is usually involved, often appearing under the umbrella term of hyperparameter-tuning: for example, one may think of the problem of tuning for the right neural network architecture towards a target task, while leveraging data from a related source task. Now, in addition to the usual tradeoffs on approximation vs estimation errors involved in model selection, this problem brings in a new complexity term, namely, the transfer distance between source and target distributions, which is known to vary with the choice of hypothesis class. We present a first study of this problem, focusing on classification; in particular, the analysis reveals some remarkable phenomena: adaptive rates, i.e., those achievable with no distributional information, can be arbitrarily slower than oracle rates, i.e., when given knowledge on distances.
Data valuation is a powerful framework for providing statistical insights into which data are beneficial or detrimental to model training. Many Shapley-based data valuation methods have shown promising results in various downstream tasks, however, they are well known to be computationally challenging as it requires training a large number of models. As a result, it has been recognized as infeasible to apply to large datasets. To address this issue, we propose Data-OOB, a new data valuation method for a bagging model that utilizes the out-of-bag estimate. The proposed method is computationally efficient and can scale to millions of data by reusing trained weak learners. Specifically, Data-OOB takes less than 2.25 hours on a single CPU processor when there are $10^6$ samples to evaluate and the input dimension is 100. Furthermore, Data-OOB has solid theoretical interpretations in that it identifies the same important data point as the infinitesimal jackknife influence function when two different points are compared. We conduct comprehensive experiments using 12 classification datasets, each with thousands of sample sizes. We demonstrate that the proposed method significantly outperforms existing state-of-the-art data valuation methods in identifying mislabeled data and finding a set of helpful (or harmful) data points, highlighting the potential for applying data values in real-world applications.
Inverse reinforcement learning (IRL) methods assume that the expert data is generated by an agent optimizing some reward function. However, in many settings, the agent may optimize a reward function subject to some constraints, where the constraints induce behaviors that may be otherwise difficult to express with just a reward function. We consider the setting where the reward function is given, and the constraints are unknown, and propose a method that is able to recover these constraints satisfactorily from the expert data. While previous work has focused on recovering hard constraints, our method can recover cumulative soft constraints that the agent satisfies on average per episode. In IRL fashion, our method solves this problem by adjusting the constraint function iteratively through a constrained optimization procedure, until the agent behavior matches the expert behavior. We demonstrate our approach on synthetic environments, robotics environments and real world highway driving scenarios.
The rapid recent progress in machine learning (ML) has raised a number of scientific questions that challenge the longstanding dogma of the field. One of the most important riddles is the good empirical generalization of overparameterized models. Overparameterized models are excessively complex with respect to the size of the training dataset, which results in them perfectly fitting (i.e., interpolating) the training data, which is usually noisy. Such interpolation of noisy data is traditionally associated with detrimental overfitting, and yet a wide range of interpolating models -- from simple linear models to deep neural networks -- have recently been observed to generalize extremely well on fresh test data. Indeed, the recently discovered double descent phenomenon has revealed that highly overparameterized models often improve over the best underparameterized model in test performance. Understanding learning in this overparameterized regime requires new theory and foundational empirical studies, even for the simplest case of the linear model. The underpinnings of this understanding have been laid in very recent analyses of overparameterized linear regression and related statistical learning tasks, which resulted in precise analytic characterizations of double descent. This paper provides a succinct overview of this emerging theory of overparameterized ML (henceforth abbreviated as TOPML) that explains these recent findings through a statistical signal processing perspective. We emphasize the unique aspects that define the TOPML research area as a subfield of modern ML theory and outline interesting open questions that remain.
This book develops an effective theory approach to understanding deep neural networks of practical relevance. Beginning from a first-principles component-level picture of networks, we explain how to determine an accurate description of the output of trained networks by solving layer-to-layer iteration equations and nonlinear learning dynamics. A main result is that the predictions of networks are described by nearly-Gaussian distributions, with the depth-to-width aspect ratio of the network controlling the deviations from the infinite-width Gaussian description. We explain how these effectively-deep networks learn nontrivial representations from training and more broadly analyze the mechanism of representation learning for nonlinear models. From a nearly-kernel-methods perspective, we find that the dependence of such models' predictions on the underlying learning algorithm can be expressed in a simple and universal way. To obtain these results, we develop the notion of representation group flow (RG flow) to characterize the propagation of signals through the network. By tuning networks to criticality, we give a practical solution to the exploding and vanishing gradient problem. We further explain how RG flow leads to near-universal behavior and lets us categorize networks built from different activation functions into universality classes. Altogether, we show that the depth-to-width ratio governs the effective model complexity of the ensemble of trained networks. By using information-theoretic techniques, we estimate the optimal aspect ratio at which we expect the network to be practically most useful and show how residual connections can be used to push this scale to arbitrary depths. With these tools, we can learn in detail about the inductive bias of architectures, hyperparameters, and optimizers.
Minimizing cross-entropy over the softmax scores of a linear map composed with a high-capacity encoder is arguably the most popular choice for training neural networks on supervised learning tasks. However, recent works show that one can directly optimize the encoder instead, to obtain equally (or even more) discriminative representations via a supervised variant of a contrastive objective. In this work, we address the question whether there are fundamental differences in the sought-for representation geometry in the output space of the encoder at minimal loss. Specifically, we prove, under mild assumptions, that both losses attain their minimum once the representations of each class collapse to the vertices of a regular simplex, inscribed in a hypersphere. We provide empirical evidence that this configuration is attained in practice and that reaching a close-to-optimal state typically indicates good generalization performance. Yet, the two losses show remarkably different optimization behavior. The number of iterations required to perfectly fit to data scales superlinearly with the amount of randomly flipped labels for the supervised contrastive loss. This is in contrast to the approximately linear scaling previously reported for networks trained with cross-entropy.
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