### How to Scale Up Kernel Methods to Be As Good As Deep Neural Nets

The computational complexity of kernel methods has often been a major barrier for applying them to large-scale learning problems. We argue that this barrier can be effectively overcome. In particular, we develop methods to scale up kernel models to successfully tackle large-scale learning problems that are so far only approachable by deep learning architectures. Based on the seminal work by Rahimi and Recht on approximating kernel functions with features derived from random projections, we advance the state-of-the-art by proposing methods that can efficiently train models with hundreds of millions of parameters, and learn optimal representations from multiple kernels. We conduct extensive empirical studies on problems from image recognition and automatic speech recognition, and show that the performance of our kernel models matches that of well-engineered deep neural nets (DNNs). To the best of our knowledge, this is the first time that a direct comparison between these two methods on large-scale problems is reported. Our kernel methods have several appealing properties: training with convex optimization, cost for training a single model comparable to DNNs, and significantly reduced total cost due to fewer hyperparameters to tune for model selection. Our contrastive study between these two very different but equally competitive models sheds light on fundamental questions such as how to learn good representations.

### Kernel Approximation Methods for Speech Recognition

We study large-scale kernel methods for acoustic modeling in speech recognition and compare their performance to deep neural networks (DNNs). We perform experiments on four speech recognition datasets, including the TIMIT and Broadcast News benchmark tasks, and compare these two types of models on frame-level performance metrics (accuracy, cross-entropy), as well as on recognition metrics (word/character error rate). In order to scale kernel methods to these large datasets, we use the random Fourier feature method of Rahimi and Recht (2007). We propose two novel techniques for improving the performance of kernel acoustic models. First, in order to reduce the number of random features required by kernel models, we propose a simple but effective method for feature selection. The method is able to explore a large number of non-linear features while maintaining a compact model more efficiently than existing approaches. Second, we present a number of frame-level metrics which correlate very strongly with recognition performance when computed on the heldout set; we take advantage of these correlations by monitoring these metrics during training in order to decide when to stop learning. This technique can noticeably improve the recognition performance of both DNN and kernel models, while narrowing the gap between them. Additionally, we show that the linear bottleneck method of Sainath et al. (2013) improves the performance of our kernel models significantly, in addition to speeding up training and making the models more compact. Together, these three methods dramatically improve the performance of kernel acoustic models, making their performance comparable to DNNs on the tasks we explored.

### A Comparison between Deep Neural Nets and Kernel Acoustic Models for Speech Recognition

We study large-scale kernel methods for acoustic modeling and compare to DNNs on performance metrics related to both acoustic modeling and recognition. Measuring perplexity and frame-level classification accuracy, kernel-based acoustic models are as effective as their DNN counterparts. However, on token-error-rates DNN models can be significantly better. We have discovered that this might be attributed to DNN's unique strength in reducing both the perplexity and the entropy of the predicted posterior probabilities. Motivated by our findings, we propose a new technique, entropy regularized perplexity, for model selection.

### Building DNN Acoustic Models for Large Vocabulary Speech Recognition

Deep neural networks (DNNs) are now a central component of nearly all state-of-the-art speech recognition systems. Building neural network acoustic models requires several design decisions including network architecture, size, and training loss function. This paper offers an empirical investigation on which aspects of DNN acoustic model design are most important for speech recognition system performance. We report DNN classifier performance and final speech recognizer word error rates, and compare DNNs using several metrics to quantify factors influencing differences in task performance. Our first set of experiments use the standard Switchboard benchmark corpus, which contains approximately 300 hours of conversational telephone speech. We compare standard DNNs to convolutional networks, and present the first experiments using locally-connected, untied neural networks for acoustic modeling. We additionally build systems on a corpus of 2,100 hours of training data by combining the Switchboard and Fisher corpora. This larger corpus allows us to more thoroughly examine performance of large DNN models -- with up to ten times more parameters than those typically used in speech recognition systems. Our results suggest that a relatively simple DNN architecture and optimization technique produces strong results. These findings, along with previous work, help establish a set of best practices for building DNN hybrid speech recognition systems with maximum likelihood training. Our experiments in DNN optimization additionally serve as a case study for training DNNs with discriminative loss functions for speech tasks, as well as DNN classifiers more generally.

### Gaussian Quadrature for Kernel Features

Kernel methods have recently attracted resurgent interest, showing performance competitive with deep neural networks in tasks such as speech recognition. The random Fourier features map is a technique commonly used to scale up kernel machines, but employing the randomized feature map means that $O(\epsilon^{-2})$ samples are required to achieve an approximation error of at most $\epsilon$. We investigate some alternative schemes for constructing feature maps that are deterministic, rather than random, by approximating the kernel in the frequency domain using Gaussian quadrature. We show that deterministic feature maps can be constructed, for any $\gamma > 0$, to achieve error $\epsilon$ with $O(e^{e^\gamma} + \epsilon^{-1/\gamma})$ samples as $\epsilon$ goes to 0. Our method works particularly well with sparse ANOVA kernels, which are inspired by the convolutional layer of CNNs. We validate our methods on datasets in different domains, such as MNIST and TIMIT, showing that deterministic features are faster to generate and achieve accuracy comparable to the state-of-the-art kernel methods based on random Fourier features.