Double Double Descent: On Generalization Errors in Transfer Learning between Linear Regression Tasks

We study the transfer learning process between two linear regression problems. An important and timely special case is when the regressors are overparameterized and perfectly interpolate their training data. We examine a parameter transfer mechanism whereby a subset of the parameters of the target task solution are constrained to the values learned for a related source task. We analytically characterize the generalization error of the target task in terms of the salient factors in the transfer learning architecture, i.e., the number of examples available, the number of (free) parameters in each of the tasks, the number of parameters transferred from the source to target task, and the correlation between the two tasks. Our non-asymptotic analysis shows that the generalization error of the target task follows a two-dimensional double descent trend (with respect to the number of free parameters in each of the tasks) that is controlled by the transfer learning factors. Our analysis points to specific cases where the transfer of parameters is beneficial.

MomentumRNN: Integrating Momentum into Recurrent Neural Networks

Designing deep neural networks is an art that often involves an expensive search over candidate architectures. To overcome this for recurrent neural nets (RNNs), we establish a connection between the hidden state dynamics in an RNN and gradient descent (GD). We then integrate momentum into this framework and propose a new family of RNNs, called {\em MomentumRNNs}. We theoretically prove and numerically demonstrate that MomentumRNNs alleviate the vanishing gradient issue in training RNNs. We study the momentum long-short term memory (MomentumLSTM) and verify its advantages in convergence speed and accuracy over its LSTM counterpart across a variety of benchmarks, with little compromise in computational or memory efficiency. We also demonstrate that MomentumRNN is applicable to many types of recurrent cells, including those in the state-of-the-art orthogonal RNNs. Finally, we show that other advanced momentum-based optimization methods, such as Adam and Nesterov accelerated gradients with a restart, can be easily incorporated into the MomentumRNN framework for designing new recurrent cells with even better performance. The code is available at \url{https://github.com/minhtannguyen/MomentumRNN}.

Attention Word Embedding

Word embedding models learn semantically rich vector representations of words and are widely used to initialize natural processing language (NLP) models. The popular continuous bag-of-words (CBOW) model of word2vec learns a vector embedding by masking a given word in a sentence and then using the other words as a context to predict it. A limitation of CBOW is that it equally weights the context words when making a prediction, which is inefficient, since some words have higher predictive value than others. We tackle this inefficiency by introducing the Attention Word Embedding (AWE) model, which integrates the attention mechanism into the CBOW model. We also propose AWE-S, which incorporates subword information. We demonstrate that AWE and AWE-S outperform the state-of-the-art word embedding models both on a variety of word similarity datasets and when used for initialization of NLP models.

qDKT: Question-centric Deep Knowledge Tracing

Knowledge tracing (KT) models, e.g., the deep knowledge tracing (DKT) model, track an individual learner's acquisition of skills over time by examining the learner's performance on questions related to those skills. A practical limitation in most existing KT models is that all questions nested under a particular skill are treated as equivalent observations of a learner's ability, which is an inaccurate assumption in real-world educational scenarios. To overcome this limitation we introduce qDKT, a variant of DKT that models every learner's success probability on individual questions over time. First, qDKT incorporates graph Laplacian regularization to smooth predictions under each skill, which is particularly useful when the number of questions in the dataset is big. Second, qDKT uses an initialization scheme inspired by the fastText algorithm, which has found success in a variety of language modeling tasks. Our experiments on several real-world datasets show that qDKT achieves state-of-art performance on predicting learner outcomes. Because of this, qDKT can serve as a simple, yet tough-to-beat, baseline for new question-centric KT models.

Deep Learning Techniques for Inverse Problems in Imaging

Recent work in machine learning shows that deep neural networks can be used to solve a wide variety of inverse problems arising in computational imaging. We explore the central prevailing themes of this emerging area and present a taxonomy that can be used to categorize different problems and reconstruction methods. Our taxonomy is organized along two central axes: (1) whether or not a forward model is known and to what extent it is used in training and testing, and (2) whether or not the learning is supervised or unsupervised, i.e., whether or not the training relies on access to matched ground truth image and measurement pairs. We also discuss the trade-offs associated with these different reconstruction approaches, caveats and common failure modes, plus open problems and avenues for future work.

Subspace Fitting Meets Regression: The Effects of Supervision and Orthonormality Constraints on Double Descent of Generalization Errors

We study the linear subspace fitting problem in the overparameterized setting, where the estimated subspace can perfectly interpolate the training examples. Our scope includes the least-squares solutions to subspace fitting tasks with varying levels of supervision in the training data (i.e., the proportion of input-output examples of the desired low-dimensional mapping) and orthonormality of the vectors defining the learned operator. This flexible family of problems connects standard, unsupervised subspace fitting that enforces strict orthonormality with a corresponding regression task that is fully supervised and does not constrain the linear operator structure. This class of problems is defined over a supervision-orthonormality plane, where each coordinate induces a problem instance with a unique pair of supervision level and softness of orthonormality constraints. We explore this plane and show that the generalization errors of the corresponding subspace fitting problems follow double descent trends as the settings become more supervised and less orthonormally constrained.

Scheduled Restart Momentum for Accelerated Stochastic Gradient Descent

Stochastic gradient descent (SGD) with constant momentum and its variants such as Adam are the optimization algorithms of choice for training deep neural networks (DNNs). Since DNN training is incredibly computationally expensive, there is great interest in speeding up convergence. Nesterov accelerated gradient (NAG) improves the convergence rate of gradient descent (GD) for convex optimization using a specially designed momentum; however, it accumulates error when an inexact gradient is used (such as in SGD), slowing convergence at best and diverging at worst. In this paper, we propose Scheduled Restart SGD (SRSGD), a new NAG-style scheme for training DNNs. SRSGD replaces the constant momentum in SGD by the increasing momentum in NAG but stabilizes the iterations by resetting the momentum to zero according to a schedule. Using a variety of models and benchmarks for image classification, we demonstrate that, in training DNNs, SRSGD significantly improves convergence and generalization; for instance in training ResNet200 for ImageNet classification, SRSGD achieves an error rate of 20.93% vs. the benchmark of 22.13%. These improvements become more significant as the network grows deeper. Furthermore, on both CIFAR and ImageNet, SRSGD reaches similar or even better error rates with fewer training epochs compared to the SGD baseline. We provide code for SRSGD at https://github.com/minhtannguyen/SRSGD.

InfoCNF: An Efficient Conditional Continuous Normalizing Flow with Adaptive Solvers

Continuous Normalizing Flows (CNFs) have emerged as promising deep generative models for a wide range of tasks thanks to their invertibility and exact likelihood estimation. However, conditioning CNFs on signals of interest for conditional image generation and downstream predictive tasks is inefficient due to the high-dimensional latent code generated by the model, which needs to be of the same size as the input data. In this paper, we propose InfoCNF, an efficient conditional CNF that partitions the latent space into a class-specific supervised code and an unsupervised code that shared among all classes for efficient use of labeled information. Since the partitioning strategy (slightly) increases the number of function evaluations (NFEs), InfoCNF also employs gating networks to learn the error tolerances of its ordinary differential equation (ODE) solvers for better speed and performance. We show empirically that InfoCNF improves the test accuracy over the baseline while yielding comparable likelihood scores and reducing the NFEs on CIFAR10. Furthermore, applying the same partitioning strategy in InfoCNF on time-series data helps improve extrapolation performance.

The Implicit Regularization of Ordinary Least Squares Ensembles

Ensemble methods that average over a collection of independent predictors that are each limited to a subsampling of both the examples and features of the training data command a significant presence in machine learning, such as the ever-popular random forest, yet the nature of the subsampling effect, particularly of the features, is not well understood. We study the case of an ensemble of linear predictors, where each individual predictor is fit using ordinary least squares on a random submatrix of the data matrix. We show that, under standard Gaussianity assumptions, when the number of features selected for each predictor is optimally tuned, the asymptotic risk of a large ensemble is equal to the asymptotic ridge regression risk, which is known to be optimal among linear predictors in this setting. In addition to eliciting this implicit regularization that results from subsampling, we also connect this ensemble to the dropout technique used in training deep (neural) networks, another strategy that has been shown to have a ridge-like regularizing effect.

Drawing early-bird tickets: Towards more efficient training of deep networks

(Frankle & Carbin, 2019) shows that there exist winning tickets (small but critical subnetworks) for dense, randomly initialized networks, that can be trained alone to achieve comparable accuracies to the latter in a similar number of iterations. However, the identification of these winning tickets still requires the costly train-prune-retrain process, limiting their practical benefits. In this paper, we discover for the first time that the winning tickets can be identified at the very early training stage, which we term as early-bird (EB) tickets, via low-cost training schemes (e.g., early stopping and low-precision training) at large learning rates. Our finding of EB tickets is consistent with recently reported observations that the key connectivity patterns of neural networks emerge early. Furthermore, we propose a mask distance metric that can be used to identify EB tickets with low computational overhead, without needing to know the true winning tickets that emerge after the full training. Finally, we leverage the existence of EB tickets and the proposed mask distance to develop efficient training methods, which are achieved by first identifying EB tickets via low-cost schemes, and then continuing to train merely the EB tickets towards the target accuracy. Experiments based on various deep networks and datasets validate: 1) the existence of EB tickets, and the effectiveness of mask distance in efficiently identifying them; and 2) that the proposed efficient training via EB tickets can achieve up to 4.7x energy savings while maintaining comparable or even better accuracy, demonstrating a promising and easily adopted method for tackling cost-prohibitive deep network training.