OpenRaiser/Intern-Atlas huggingface dataset:
Fundamental Limits and Tradeoffs in Invariant Representation Learning
Venue: Journal JMLR 2022
Year: 2022

A wide range of machine learning applications such as privacy-preserving learning, algorithmic fairness, and domain adaptation/generalization among others, involve learning invariant representations of the data that aim to achieve two competing goals: (a) maximize information or accuracy with respect to a target response, and (b) maximize invariance or independence with respect to a set of protected features (e.g., for fairness, privacy, etc). Despite their wide applicability, theoretical understanding of the optimal tradeoffs -- with respect to accuracy, and invariance -- achievable by invariant representations is still severely lacking. In this paper, we provide an information theoretic analysis of such tradeoffs under both classification and regression settings. More precisely, we provide a geometric characterization of the accuracy and invariance achievable by any representation of the data; we term this feasible region the information plane. We provide an inner bound for this feasible region for the classification case, and an exact characterization for the regression case, which allows us to either bound or exactly characterize the Pareto optimal frontier between accuracy and invariance. Although our contributions are mainly theoretical, a key practical application of our results is in certifying the potential sub-optimality of any given representation learning algorithm for either classification or regression tasks. Our results shed new light on the fundamental interplay between accuracy and invariance, and may be useful in guiding the design of future representation learning algorithms.
The Separation Capacity of Random Neural Networks
Venue: Journal JMLR 2022
Year: 2022

Neural networks with random weights appear in a variety of machine learning applications, most prominently as the initialization of many deep learning algorithms and as a computationally cheap alternative to fully learned neural networks. In the present article, we enhance the theoretical understanding of random neural networks by addressing the following data separation problem: under what conditions can a random neural network make two classes $\mathcal{X}^-, \mathcal{X}^+ \subset \mathbb{R}^d$ (with positive distance) linearly separable? We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. Crucially, the number of required neurons is explicitly linked to geometric properties of the underlying sets $\mathcal{X}^-, \mathcal{X}^+$ and their mutual arrangement. This instance-specific viewpoint allows us to overcome the usual curse of dimensionality (exponential width of the layers) in non-pathological situations where the data carries low-complexity structure. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity (based on a localized version of Gaussian mean width), which leads to sound and informative separation guarantees. We connect our result with related lines of work on approximation, memorization, and generalization.
ktrain: A Low-Code Library for Augmented Machine Learning
Venue: Journal JMLR 2022
Year: 2022

We present ktrain, a low-code Python library that makes machine learning more accessible and easier to apply. As a wrapper to TensorFlow and many other libraries (e.g., transformers, scikit-learn, stellargraph), it is designed to make sophisticated, state-of-the-art machine learning models simple to build, train, inspect, and apply by both beginners and experienced practitioners. Featuring modules that support text data (e.g., text classification, sequence tagging, open-domain question-answering), vision data (e.g., image classification), graph data (e.g., node classification, link prediction), and tabular data, ktrain presents a simple unified interface enabling one to quickly solve a wide range of tasks in as little as three or four "commands" or lines of code.
Tianshou: A Highly Modularized Deep Reinforcement Learning Library
Venue: Journal JMLR 2022
Year: 2022

In this paper, we present Tianshou, a highly modularized Python library for deep reinforcement learning (DRL) that uses PyTorch as its backend. Tianshou intends to be research-friendly by providing a flexible and reliable infrastructure of DRL algorithms. It supports online and offline training with more than 20 classic algorithms through a unified interface. To facilitate related research and prove Tianshou's reliability, we have released Tianshou's benchmark of MuJoCo environments, covering eight classic algorithms with state-of-the-art performance. We open-sourced Tianshou at https://github.com/thu-ml/tianshou/.
Provable Tensor-Train Format Tensor Completion by Riemannian Optimization
Venue: Journal JMLR 2022
Year: 2022

The tensor train (TT) format enjoys appealing advantages in handling structural high-order tensors. The recent decade has witnessed the wide applications of TT-format tensors from diverse disciplines, among which tensor completion has drawn considerable attention. Numerous fast algorithms, including the Riemannian gradient descent (RGrad), have been proposed for the TT-format tensor completion. However, the theoretical guarantees of these algorithms are largely missing or sub-optimal, partly due to the complicated and recursive algebraic operations in TT-format decomposition. Moreover, existing results established for the tensors of other formats, for example, Tucker and CP, are inapplicable because the algorithms treating TT-format tensors are substantially different and more involved. In this paper, we provide, to our best knowledge, the first theoretical guarantees of the convergence of RGrad algorithm for TT-format tensor completion, under a nearly optimal sample size condition. The RGrad algorithm converges linearly with a constant contraction rate that is free of tensor condition number without the necessity of re-conditioning. We also propose a novel approach, referred to as the sequential second-order moment method, to attain a warm initialization under a similar sample size requirement. As a byproduct, our result even significantly refines the prior investigation of RGrad algorithm for matrix completion. Lastly, statistically (near) optimal rate is derived for RGrad algorithm if the observed entries consist of random sub-Gaussian noise. Numerical experiments confirm our theoretical discovery and showcase the computational speedup gained by the TT-format decomposition.
solo-learn: A Library of Self-supervised Methods for Visual Representation Learning
Venue: Journal JMLR 2022
Year: 2022

This paper presents solo-learn, a library of self-supervised methods for visual representation learning. Implemented in Python, using Pytorch and Pytorch lightning, the library fits both research and industry needs by featuring distributed training pipelines with mixed-precision, faster data loading via Nvidia DALI, online linear evaluation for better prototyping, and many additional training tricks. Our goal is to provide an easy-to-use library comprising a large amount of Self-supervised Learning (SSL) methods, that can be easily extended and fine-tuned by the community. solo-learn opens up avenues for exploiting large-budget SSL solutions on inexpensive smaller infrastructures and seeks to democratize SSL by making it accessible to all. The source code is available at https://github.com/vturrisi/solo-learn.
Darts: User-Friendly Modern Machine Learning for Time Series
Venue: Journal JMLR 2022
Year: 2022

We present Darts, a Python machine learning library for time series, with a focus on forecasting. Darts offers a variety of models, from classics such as ARIMA to state-of-the-art deep neural networks. The emphasis of the library is on offering modern machine learning functionalities, such as supporting multidimensional series, meta-learning on multiple series, training on large datasets, incorporating external data, ensembling models, and providing a rich support for probabilistic forecasting. At the same time, great care goes into the API design to make it user-friendly and easy to use. For instance, all models can be used using fit()/predict(), similar to scikit-learn.
Fairness-Aware PAC Learning from Corrupted Data
Venue: Journal JMLR 2022
Year: 2022

Addressing fairness concerns about machine learning models is a crucial step towards their long-term adoption in real-world automated systems. While many approaches have been developed for training fair models from data, little is known about the robustness of these methods to data corruption. In this work we consider fairness-aware learning under worst-case data manipulations. We show that an adversary can in some situations force any learner to return an overly biased classifier, regardless of the sample size and with or without degrading accuracy, and that the strength of the excess bias increases for learning problems with underrepresented protected groups in the data. We also prove that our hardness results are tight up to constant factors. To this end, we study two natural learning algorithms that optimize for both accuracy and fairness and show that these algorithms enjoy guarantees that are order-optimal in terms of the corruption ratio and the protected groups frequencies in the large data limit.
tntorch: Tensor Network Learning with PyTorch
Venue: Journal JMLR 2022
Year: 2022

We present tntorch, a tensor learning framework that supports multiple decompositions (including Candecomp/Parafac, Tucker, and Tensor Train) under a unified interface. With our library, the user can learn and handle low-rank tensors with automatic differentiation, seamless GPU support, and the convenience of PyTorch's API. Besides decomposition algorithms, tntorch implements differentiable tensor algebra, rank truncation, cross-approximation, batch processing, comprehensive tensor arithmetics, and more.
Ranking and Tuning Pre-trained Models: A New Paradigm for Exploiting Model Hubs
Venue: Journal JMLR 2022
Year: 2022

Model hubs with many pre-trained models (PTMs) have become a cornerstone of deep learning. Although built at a high cost, they remain \emph{under-exploited} -- practitioners usually pick one PTM from the provided model hub by popularity and then fine-tune the PTM to solve the target task. This naïve but common practice poses two obstacles to full exploitation of pre-trained model hubs: first, the PTM selection by popularity has no optimality guarantee, and second, only one PTM is used while the remaining PTMs are ignored. An alternative might be to consider all possible combinations of PTMs and extensively fine-tune each combination, but this would not only be prohibitive computationally but may also lead to statistical over-fitting. In this paper, we propose a new paradigm for exploiting model hubs that is intermediate between these extremes. The paradigm is characterized by two aspects: (1) We use an evidence maximization procedure to estimate the maximum value of label evidence given features extracted by pre-trained models. This procedure can rank all the PTMs in a model hub for various types of PTMs and tasks \emph{before fine-tuning}. (2) The best ranked PTM can either be fine-tuned and deployed if we have no preference for the model's architecture or the target PTM can be tuned by the top $K$ ranked PTMs via a Bayesian procedure that we propose. This procedure, which we refer to as \emph{B-Tuning}, not only improves upon specialized methods designed for tuning homogeneous PTMs, but also applies to the challenging problem of tuning heterogeneous PTMs where it yields a new level of benchmark performance.
Toward Understanding Convolutional Neural Networks from Volterra Convolution Perspective
Venue: Journal JMLR 2022
Year: 2022

We make an attempt to understanding convolutional neural network by exploring the relationship between (deep) convolutional neural networks and Volterra convolutions. We propose a novel approach to explain and study the overall characteristics of neural networks without being disturbed by the horribly complex architectures. Specifically, we attempt to convert the basic structures of a convolutional neural network (CNN) and their combinations to the form of Volterra convolutions. The results show that most of convolutional neural networks can be approximated in the form of Volterra convolution, where the approximated proxy kernels preserve the characteristics of the original network. Analyzing these proxy kernels may give valuable insight about the original network. Base on this setup, we presented methods to approximating the order-zero and order-one proxy kernels, and verified the correctness and effectiveness of our results.
Topologically penalized regression on manifolds
Venue: Journal JMLR 2022
Year: 2022

We study a regression problem on a compact manifold M. In order to take advantage of the underlying geometry and topology of the data, the regression task is performed on the basis of the first several eigenfunctions of the Laplace-Beltrami operator of the manifold, that are regularized with topological penalties. The proposed penalties are based on the topology of the sub-level sets of either the eigenfunctions or the estimated function. The overall approach is shown to yield promising and competitive performance on various applications to both synthetic and real data sets. We also provide theoretical guarantees on the regression function estimates, on both its prediction error and its smoothness (in a topological sense). Taken together, these results support the relevance of our approach in the case where the targeted function is ''topologically smooth''.
On the Approximation of Cooperative Heterogeneous Multi-Agent Reinforcement Learning (MARL) using Mean Field Control (MFC)
Venue: Journal JMLR 2022
Year: 2022

Mean field control (MFC) is an effective way to mitigate the curse of dimensionality of cooperative multi-agent reinforcement learning (MARL) problems. This work considers a collection of $N_{\mathrm{pop}}$ heterogeneous agents that can be segregated into $K$ classes such that the $k$-th class contains $N_k$ homogeneous agents. We aim to prove approximation guarantees of the MARL problem for this heterogeneous system by its corresponding MFC problem. We consider three scenarios where the reward and transition dynamics of all agents are respectively taken to be functions of $(1)$ joint state and action distributions across all classes, $(2)$ individual distributions of each class, and $(3)$ marginal distributions of the entire population. We show that, in these cases, the $K$-class MARL problem can be approximated by MFC with errors given as $e_1=\mathcal{O}(\frac{\sqrt{|\mathcal{X}|}+\sqrt{|\mathcal{U}|}}{N_{\mathrm{pop}}}\sum_{k}\sqrt{N_k})$, $e_2=\mathcal{O}(\left[\sqrt{|\mathcal{X}|}+\sqrt{|\mathcal{U}|}\right]\sum_{k}\frac{1}{\sqrt{N_k}})$ and $e_3=\mathcal{O}\left(\left[\sqrt{|\mathcal{X}|}+\sqrt{|\mathcal{U}|}\right]\left[\frac{A}{N_{\mathrm{pop}}}\sum_{k\in[K]}\sqrt{N_k}+\frac{B}{\sqrt{N_{\mathrm{pop}}}}\right]\right)$, respectively, where $A, B$ are some constants and $|\mathcal{X}|,|\mathcal{U}|$ are the sizes of state and action spaces of each agent. Finally, we design a Natural Policy Gradient (NPG) based algorithm that, in the three cases stated above, can converge to an optimal MARL policy within $\mathcal{O}(e_j)$ error with a sample complexity of $\mathcal{O}(e_j^{-3})$, $j\in\{1,2,3\}$, respectively.
Mean-field Analysis of Piecewise Linear Solutions for Wide ReLU Networks
Venue: Journal JMLR 2022
Year: 2022

Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the "knot" points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.
Policy Gradient and Actor-Critic Learning in Continuous Time and Space: Theory and Algorithms
Venue: Journal JMLR 2022
Year: 2022

We study policy gradient (PG) for reinforcement learning in continuous time and space under the regularized exploratory formulation developed by Wang et al. (2020). We represent the gradient of the value function with respect to a given parameterized stochastic policy as the expected integration of an auxiliary running reward function that can be evaluated using samples and the current value function. This effectively turns PG into a policy evaluation (PE) problem, enabling us to apply the martingale approach recently developed by Jia and Zhou (2021) for PE to solve our PG problem. Based on this analysis, we propose two types of the actor-critic algorithms for RL, where we learn and update value functions and policies simultaneously and alternatingly. The first type is based directly on the aforementioned representation which involves future trajectories and hence is offline. The second type, designed for online learning, employs the first-order condition of the policy gradient and turns it into martingale orthogonality conditions. These conditions are then incorporated using stochastic approximation when updating policies. Finally, we demonstrate the algorithms by simulations in two concrete examples.
Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements
Venue: Journal JMLR 2022
Year: 2022

Tensors, which provide a powerful and flexible model for representing multi-attribute data and multi-way interactions, play an indispensable role in modern data science across various fields in science and engineering. A fundamental task is to faithfully recover the tensor from highly incomplete measurements in a statistically and computationally efficient manner. Harnessing the low-rank structure of tensors in the Tucker decomposition, this paper develops a scaled gradient descent (ScaledGD) algorithm to directly recover the tensor factors with tailored spectral initializations, and shows that it provably converges at a linear rate independent of the condition number of the ground truth tensor for two canonical problems -- tensor completion and tensor regression -- as soon as the sample size is above the order of $n^{3/2}$ ignoring other parameter dependencies, where $n$ is the dimension of the tensor. This leads to an extremely scalable approach to low-rank tensor estimation compared with prior art, which suffers from at least one of the following drawbacks: extreme sensitivity to ill-conditioning, high per-iteration costs in terms of memory and computation, or poor sample complexity guarantees. To the best of our knowledge, ScaledGD is the first algorithm that achieves near-optimal statistical and computational complexities simultaneously for low-rank tensor completion with the Tucker decomposition. Our algorithm highlights the power of appropriate preconditioning in accelerating nonconvex statistical estimation, where the iteration-varying preconditioners promote desirable invariance properties of the trajectory with respect to the underlying symmetry in low-rank tensor factorization.
Deep Network Approximation: Achieving Arbitrary Accuracy with Fixed Number of Neurons
Venue: Journal JMLR 2022
Year: 2022

This paper develops simple feed-forward neural networks that achieve the universal approximation property for all continuous functions with a fixed finite number of neurons. These neural networks are simple because they are designed with a simple, computable, and continuous activation function $σ$ leveraging a triangular-wave function and the softsign function. We first prove that $σ$-activated networks with width $36d(2d+1)$ and depth $11$ can approximate any continuous function on a $d$-dimensional hypercube within an arbitrarily small error. Hence, for supervised learning and its related regression problems, the hypothesis space generated by these networks with a size not smaller than $36d(2d+1)\times 11$ is dense in the continuous function space $C([a,b]^d)$ and therefore dense in the Lebesgue spaces $L^p([a,b]^d)$ for $p\in [1,\infty)$. Furthermore, we show that classification functions arising from image and signal classification are in the hypothesis space generated by $σ$-activated networks with width $36d(2d+1)$ and depth $12$ when there exist pairwise disjoint bounded closed subsets of $\mathbb{R}^d$ such that the samples of the same class are located in the same subset. Finally, we use numerical experimentation to show that replacing the rectified linear unit (ReLU) activation function by ours would improve the experiment results.
Inherent Tradeoffs in Learning Fair Representations
Venue: Journal JMLR 2022
Year: 2022

Real-world applications of machine learning tools in high-stakes domains are often regulated to be fair, in the sense that the predicted target should satisfy some quantitative notion of parity with respect to a protected attribute. However, the exact tradeoff between fairness and accuracy is not entirely clear, even for the basic paradigm of classification problems. In this paper, we characterize an inherent tradeoff between statistical parity and accuracy in the classification setting by providing a lower bound on the sum of group-wise errors of any fair classifiers. Our impossibility theorem could be interpreted as a certain uncertainty principle in fairness: if the base rates differ among groups, then any fair classifier satisfying statistical parity has to incur a large error on at least one of the groups. We further extend this result to give a lower bound on the joint error of any (approximately) fair classifiers, from the perspective of learning fair representations. To show that our lower bound is tight, assuming oracle access to Bayes (potentially unfair) classifiers, we also construct an algorithm that returns a randomized classifier that is both optimal (in terms of accuracy) and fair. Interestingly, when the protected attribute can take more than two values, an extension of this lower bound does not admit an analytic solution. Nevertheless, in this case, we show that the lower bound can be efficiently computed by solving a linear program, which we term as the TV-Barycenter problem, a barycenter problem under the TV-distance. On the upside, we prove that if the group-wise Bayes optimal classifiers are close, then learning fair representations leads to an alternative notion of fairness, known as the accuracy parity, which states that the error rates are close between groups. Finally, we also conduct experiments on real-world datasets to confirm our theoretical findings.