arXiv subjects

Samet Oymak

Publications and source records attributed to Samet Oymak.

At least 91 records · Page 5Linked to original sources

arXiv2020

On the Marginal Benefit of Active Learning: Does Self-Supervision Eat Its Cake?

Active learning is the set of techniques for intelligently labeling large unlabeled datasets to reduce the labeling effort. In parallel, recent developments in self-supervised and semi-supervised learning (S4L) provide powerful techniques, based on data-augmentation, contrastive learning, and self-training, that enable superior utilization of unlabeled data which led to a significant reduction in required labeling in the standard machine learning benchmarks. A natural question is whether these paradigms can be unified to obtain superior results. To this aim, this paper provides a novel algorithmic framework integrating self-supervised pretraining, active learning, and consistency-regularized self-training. We conduct extensive experiments with our framework on CIFAR10 and CIFAR100 datasets. These experiments enable us to isolate and assess the benefits of individual components which are evaluated using state-of-the-art methods (e.g.~Core-Set, VAAL, simCLR, FixMatch). Our experiments reveal two key insights: (i) Self-supervised pre-training significantly improves semi-supervised learning, especially in the few-label regime, (ii) The benefit of active learning is undermined and subsumed by S4L techniques. Specifically, we fail to observe any additional benefit of state-of-the-art active learning algorithms when combined with state-of-the-art S4L techniques.

cs.LG↗

arXiv2020

Unsupervised Paraphrasing via Deep Reinforcement Learning

Paraphrasing is expressing the meaning of an input sentence in different wording while maintaining fluency (i.e., grammatical and syntactical correctness). Most existing work on paraphrasing use supervised models that are limited to specific domains (e.g., image captions). Such models can neither be straightforwardly transferred to other domains nor generalize well, and creating labeled training data for new domains is expensive and laborious. The need for paraphrasing across different domains and the scarcity of labeled training data in many such domains call for exploring unsupervised paraphrase generation methods. We propose Progressive Unsupervised Paraphrasing (PUP): a novel unsupervised paraphrase generation method based on deep reinforcement learning (DRL). PUP uses a variational autoencoder (trained using a non-parallel corpus) to generate a seed paraphrase that warm-starts the DRL model. Then, PUP progressively tunes the seed paraphrase guided by our novel reward function which combines semantic adequacy, language fluency, and expression diversity measures to quantify the quality of the generated paraphrases in each iteration without needing parallel sentences. Our extensive experimental evaluation shows that PUP outperforms unsupervised state-of-the-art paraphrasing techniques in terms of both automatic metrics and user studies on four real datasets. We also show that PUP outperforms domain-adapted supervised algorithms on several datasets. Our evaluation also shows that PUP achieves a great trade-off between semantic similarity and diversity of expression.

cs.CL↗

arXiv2020

Exploring Weight Importance and Hessian Bias in Model Pruning

Model pruning is an essential procedure for building compact and computationally-efficient machine learning models. A key feature of a good pruning algorithm is that it accurately quantifies the relative importance of the model weights. While model pruning has a rich history, we still don't have a full grasp of the pruning mechanics even for relatively simple problems involving linear models or shallow neural nets. In this work, we provide a principled exploration of pruning by building on a natural notion of importance. For linear models, we show that this notion of importance is captured by covariance scaling which connects to the well-known Hessian-based pruning. We then derive asymptotic formulas that allow us to precisely compare the performance of different pruning methods. For neural networks, we demonstrate that the importance can be at odds with larger magnitudes and proper initialization is critical for magnitude-based pruning. Specifically, we identify settings in which weights become more important despite becoming smaller, which in turn leads to a catastrophic failure of magnitude-based pruning. Our results also elucidate that implicit regularization in the form of Hessian structure has a catalytic role in identifying the important weights, which dictate the pruning performance.

cs.LG↗

arXiv2020

Statistical and Algorithmic Insights for Semi-supervised Learning with Self-training

Self-training is a classical approach in semi-supervised learning which is successfully applied to a variety of machine learning problems. Self-training algorithm generates pseudo-labels for the unlabeled examples and progressively refines these pseudo-labels which hopefully coincides with the actual labels. This work provides theoretical insights into self-training algorithm with a focus on linear classifiers. We first investigate Gaussian mixture models and provide a sharp non-asymptotic finite-sample characterization of the self-training iterations. Our analysis reveals the provable benefits of rejecting samples with low confidence and demonstrates that self-training iterations gracefully improve the model accuracy even if they do get stuck in sub-optimal fixed points. We then demonstrate that regularization and class margin (i.e. separation) is provably important for the success and lack of regularization may prevent self-training from identifying the core features in the data. Finally, we discuss statistical aspects of empirical risk minimization with self-training for general distributions. We show how a purely unsupervised notion of generalization based on self-training based clustering can be formalized based on cluster margin. We then establish a connection between self-training based semi-supervision and the more general problem of learning with heterogenous data and weak supervision.

cs.LG↗

arXiv2020

On the Role of Dataset Quality and Heterogeneity in Model Confidence

Safety-critical applications require machine learning models that output accurate and calibrated probabilities. While uncalibrated deep networks are known to make over-confident predictions, it is unclear how model confidence is impacted by the variations in the data, such as label noise or class size. In this paper, we investigate the role of the dataset quality by studying the impact of dataset size and the label noise on the model confidence. We theoretically explain and experimentally demonstrate that, surprisingly, label noise in the training data leads to under-confident networks, while reduced dataset size leads to over-confident models. We then study the impact of dataset heterogeneity, where data quality varies across classes, on model confidence. We demonstrate that this leads to heterogenous confidence/accuracy behavior in the test data and is poorly handled by the standard calibration algorithms. To overcome this, we propose an intuitive heterogenous calibration technique and show that the proposed approach leads to improved calibration metrics (both average and worst-case errors) on the CIFAR datasets.

cs.LG↗

arXiv2019

Generalization Guarantees for Neural Networks via Harnessing the Low-rank Structure of the Jacobian

Modern neural network architectures often generalize well despite containing many more parameters than the size of the training dataset. This paper explores the generalization capabilities of neural networks trained via gradient descent. We develop a data-dependent optimization and generalization theory which leverages the low-rank structure of the Jacobian matrix associated with the network. Our results help demystify why training and generalization is easier on clean and structured datasets and harder on noisy and unstructured datasets as well as how the network size affects the evolution of the train and test errors during training. Specifically, we use a control knob to split the Jacobian spectum into "information" and "nuisance" spaces associated with the large and small singular values. We show that over the information space learning is fast and one can quickly train a model with zero training loss that can also generalize well. Over the nuisance space training is slower and early stopping can help with generalization at the expense of some bias. We also show that the overall generalization capability of the network is controlled by how well the label vector is aligned with the information space. A key feature of our results is that even constant width neural nets can provably generalize for sufficiently nice datasets. We conduct various numerical experiments on deep networks that corroborate our theoretical findings and demonstrate that: (i) the Jacobian of typical neural networks exhibit low-rank structure with a few large singular values and many small ones leading to a low-dimensional information space, (ii) over the information space learning is fast and most of the label vector falls on this space, and (iii) label noise falls on the nuisance space and impedes optimization/generalization.

cs.LG↗

arXiv2019

Gradient Descent with Early Stopping is Provably Robust to Label Noise for Overparameterized Neural Networks

Modern neural networks are typically trained in an over-parameterized regime where the parameters of the model far exceed the size of the training data. Such neural networks in principle have the capacity to (over)fit any set of labels including pure noise. Despite this, somewhat paradoxically, neural network models trained via first-order methods continue to predict well on yet unseen test data. This paper takes a step towards demystifying this phenomena. Under a rich dataset model, we show that gradient descent is provably robust to noise/corruption on a constant fraction of the labels despite overparameterization. In particular, we prove that: (i) In the first few iterations where the updates are still in the vicinity of the initialization gradient descent only fits to the correct labels essentially ignoring the noisy labels. (ii) to start to overfit to the noisy labels network must stray rather far from from the initialization which can only occur after many more iterations. Together, these results show that gradient descent with early stopping is provably robust to label noise and shed light on the empirical robustness of deep networks as well as commonly adopted heuristics to prevent overfitting.

cs.LG↗

arXiv2019

Quickly Finding the Best Linear Model in High Dimensions

We study the problem of finding the best linear model that can minimize least-squares loss given a data-set. While this problem is trivial in the low dimensional regime, it becomes more interesting in high dimensions where the population minimizer is assumed to lie on a manifold such as sparse vectors. We propose projected gradient descent (PGD) algorithm to estimate the population minimizer in the finite sample regime. We establish linear convergence rate and data dependent estimation error bounds for PGD. Our contributions include: 1) The results are established for heavier tailed sub-exponential distributions besides sub-gaussian. 2) We directly analyze the empirical risk minimization and do not require a realizable model that connects input data and labels. 3) Our PGD algorithm is augmented to learn the bias terms which boosts the performance. The numerical experiments validate our theoretical results.

cs.LG↗

arXiv2019

Towards moderate overparameterization: global convergence guarantees for training shallow neural networks

Many modern neural network architectures are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Sufficiently overparameterized neural network architectures in principle have the capacity to fit any set of labels including random noise. However, given the highly nonconvex nature of the training landscape it is not clear what level and kind of overparameterization is required for first order methods to converge to a global optima that perfectly interpolate any labels. A number of recent theoretical works have shown that for very wide neural networks where the number of hidden units is polynomially large in the size of the training data gradient descent starting from a random initialization does indeed converge to a global optima. However, in practice much more moderate levels of overparameterization seems to be sufficient and in many cases overparameterized models seem to perfectly interpolate the training data as soon as the number of parameters exceed the size of the training data by a constant factor. Thus there is a huge gap between the existing theoretical literature and practical experiments. In this paper we take a step towards closing this gap. Focusing on shallow neural nets and smooth activations, we show that (stochastic) gradient descent when initialized at random converges at a geometric rate to a nearby global optima as soon as the square-root of the number of network parameters exceeds the size of the training data. Our results also benefit from a fast convergence rate and continue to hold for non-differentiable activations such as Rectified Linear Units (ReLUs).

cs.LG↗

arXiv2019

Non-asymptotic Identification of LTI Systems from a Single Trajectory

We consider the problem of learning a realization for a linear time-invariant (LTI) dynamical system from input/output data. Given a single input/output trajectory, we provide finite time analysis for learning the system's Markov parameters, from which a balanced realization is obtained using the classical Ho-Kalman algorithm. By proving a stability result for the Ho-Kalman algorithm and combining it with the sample complexity results for Markov parameters, we show how much data is needed to learn a balanced realization of the system up to a desired accuracy with high probability.

cs.LG↗

arXiv2018

Overparameterized Nonlinear Learning: Gradient Descent Takes the Shortest Path?

Many modern learning tasks involve fitting nonlinear models to data which are trained in an overparameterized regime where the parameters of the model exceed the size of the training dataset. Due to this overparameterization, the training loss may have infinitely many global minima and it is critical to understand the properties of the solutions found by first-order optimization schemes such as (stochastic) gradient descent starting from different initializations. In this paper we demonstrate that when the loss has certain properties over a minimally small neighborhood of the initial point, first order methods such as (stochastic) gradient descent have a few intriguing properties: (1) the iterates converge at a geometric rate to a global optima even when the loss is nonconvex, (2) among all global optima of the loss the iterates converge to one with a near minimal distance to the initial point, (3) the iterates take a near direct route from the initial point to this global optima. As part of our proof technique, we introduce a new potential function which captures the precise tradeoff between the loss function and the distance to the initial point as the iterations progress. For Stochastic Gradient Descent (SGD), we develop novel martingale techniques that guarantee SGD never leaves a small neighborhood of the initialization, even with rather large learning rates. We demonstrate the utility of our general theory for a variety of problem domains spanning low-rank matrix recovery to neural network training. Underlying our analysis are novel insights that may have implications for training and generalization of more sophisticated learning problems including those involving deep neural network architectures.

cs.LG↗

arXiv2018

Stochastic Gradient Descent Learns State Equations with Nonlinear Activations

We study discrete time dynamical systems governed by the state equation $h_{t+1}=ϕ(Ah_t+Bu_t)$. Here $A,B$ are weight matrices, $ϕ$ is an activation function, and $u_t$ is the input data. This relation is the backbone of recurrent neural networks (e.g. LSTMs) which have broad applications in sequential learning tasks. We utilize stochastic gradient descent to learn the weight matrices from a finite input/state trajectory $(u_t,h_t)_{t=0}^N$. We prove that SGD estimate linearly converges to the ground truth weights while using near-optimal sample size. Our results apply to increasing activations whose derivatives are bounded away from zero. The analysis is based on i) a novel SGD convergence result with nonlinear activations and ii) careful statistical characterization of the state vector. Numerical experiments verify the fast convergence of SGD on ReLU and leaky ReLU in consistence with our theory.

cs.LG↗

arXiv2018

Learning Compact Neural Networks with Regularization

Proper regularization is critical for speeding up training, improving generalization performance, and learning compact models that are cost efficient. We propose and analyze regularized gradient descent algorithms for learning shallow neural networks. Our framework is general and covers weight-sharing (convolutional networks), sparsity (network pruning), and low-rank constraints among others. We first introduce covering dimension to quantify the complexity of the constraint set and provide insights on the generalization properties. Then, we show that proposed algorithms become well-behaved and local linear convergence occurs once the amount of data exceeds the covering dimension. Overall, our results demonstrate that near-optimal sample complexity is sufficient for efficient learning and illustrate how regularization can be beneficial to learn over-parameterized networks.

cs.LG↗

arXiv2018

End-to-end Learning of a Convolutional Neural Network via Deep Tensor Decomposition

In this paper we study the problem of learning the weights of a deep convolutional neural network. We consider a network where convolutions are carried out over non-overlapping patches with a single kernel in each layer. We develop an algorithm for simultaneously learning all the kernels from the training data. Our approach dubbed Deep Tensor Decomposition (DeepTD) is based on a rank-1 tensor decomposition. We theoretically investigate DeepTD under a realizable model for the training data where the inputs are chosen i.i.d. from a Gaussian distribution and the labels are generated according to planted convolutional kernels. We show that DeepTD is data-efficient and provably works as soon as the sample size exceeds the total number of convolutional weights in the network. We carry out a variety of numerical experiments to investigate the effectiveness of DeepTD and verify our theoretical findings.

cs.LG↗

arXiv2017

Universality laws for randomized dimension reduction, with applications

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain geometric features of the set. The question is how large the embedding dimension must be to ensure that randomized dimension reduction succeeds with high probability. This paper studies a natural family of randomized dimension reduction maps and a large class of data sets. It proves that there is a phase transition in the success probability of the dimension reduction map as the embedding dimension increases. For a given data set, the location of the phase transition is the same for all maps in this family. Furthermore, each map has the same stability properties, as quantified through the restricted minimum singular value. These results can be viewed as new universality laws in high-dimensional stochastic geometry. Universality laws for randomized dimension reduction have many applications in applied mathematics, signal processing, and statistics. They yield design principles for numerical linear algebra algorithms, for compressed sensing measurement ensembles, and for random linear codes. Furthermore, these results have implications for the performance of statistical estimation methods under a large class of random experimental designs.

math.PR↗

arXiv2017

Learning Feature Nonlinearities with Non-Convex Regularized Binned Regression

For various applications, the relations between the dependent and independent variables are highly nonlinear. Consequently, for large scale complex problems, neural networks and regression trees are commonly preferred over linear models such as Lasso. This work proposes learning the feature nonlinearities by binning feature values and finding the best fit in each quantile using non-convex regularized linear regression. The algorithm first captures the dependence between neighboring quantiles by enforcing smoothness via piecewise-constant/linear approximation and then selects a sparse subset of good features. We prove that the proposed algorithm is statistically and computationally efficient. In particular, it achieves linear rate of convergence while requiring near-minimal number of samples. Evaluations on synthetic and real datasets demonstrate that algorithm is competitive with current state-of-the-art and accurately learns feature nonlinearities. Finally, we explore an interesting connection between the binning stage of our algorithm and sparse Johnson-Lindenstrauss matrices.

cs.LG↗

arXiv2016

Fast and Reliable Parameter Estimation from Nonlinear Observations

In this paper we study the problem of recovering a structured but unknown parameter ${\bfθ}^*$ from $n$ nonlinear observations of the form $y_i=f(\langle {\bf{x}}_i,{\bfθ}^*\rangle)$ for $i=1,2,\ldots,n$. We develop a framework for characterizing time-data tradeoffs for a variety of parameter estimation algorithms when the nonlinear function $f$ is unknown. This framework includes many popular heuristics such as projected/proximal gradient descent and stochastic schemes. For example, we show that a projected gradient descent scheme converges at a linear rate to a reliable solution with a near minimal number of samples. We provide a sharp characterization of the convergence rate of such algorithms as a function of sample size, amount of a-prior knowledge available about the parameter and a measure of the nonlinearity of the function $f$. These results provide a precise understanding of the various tradeoffs involved between statistical and computational resources as well as a-prior side information available for such nonlinear parameter estimation problems.

stat.ML↗

arXiv2016

Near-Optimal Sample Complexity Bounds for Circulant Binary Embedding

Binary embedding is the problem of mapping points from a high-dimensional space to a Hamming cube in lower dimension while preserving pairwise distances. An efficient way to accomplish this is to make use of fast embedding techniques involving Fourier transform e.g.~circulant matrices. While binary embedding has been studied extensively, theoretical results on fast binary embedding are rather limited. In this work, we build upon the recent literature to obtain significantly better dependencies on the problem parameters. A set of $N$ points in $\mathbb{R}^n$ can be properly embedded into the Hamming cube $\{\pm 1\}^k$ with $δ$ distortion, by using $k\simδ^{-3}\log N$ samples which is optimal in the number of points $N$ and compares well with the optimal distortion dependency $δ^{-2}$. Our optimal embedding result applies in the regime $\log N\lesssim n^{1/3}$. Furthermore, if the looser condition $\log N\lesssim \sqrt{n}$ holds, we show that all but an arbitrarily small fraction of the points can be optimally embedded. We believe our techniques can be useful to obtain improved guarantees for other nonlinear embedding problems.

cs.DS↗