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Register a new userQuantifying the Benefit of Using Differentiable Learning over Tangent Kernels
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We study the relative power of learning with gradient descent on differentiable models, such as neural networks, versus using the corresponding tangent kernels. We show that under certain conditions, gradient descent achieves small error only if a related tangent kernel method achieves a non-trivial advantage over random guessing (a.k.a. weak learning), though this advantage might be very small even when gradient descent can achieve arbitrarily high accuracy. Complementing this, we show that without these conditions, gradient descent can in fact learn with small error even when no kernel method, in particular using the tangent kernel, can achieve a non-trivial advantage over random guessing.
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Recent theoretical work has shown that massively overparameterized neural networks are equivalent to kernel regressors that use Neural Tangent Kernels(NTK). Experiments show that these kernel methods perform similarly to real neural networks. Here we show that NTK for fully connected networks is closely related to the standard Laplace kernel. We show theoretically that for normalized data on the hypersphere both kernels have the same eigenfunctions and their eigenvalues decay polynomially at the same rate, implying that their Reproducing Kernel Hilbert Spaces (RKHS) include the same sets of functions. This means that both kernels give rise to classes of functions with the same smoothness properties. The two kernels differ for data off the hypersphere, but experiments indicate that when data is properly normalized these differences are not significant. Finally, we provide experiments on real data comparing NTK and the Laplace kernel, along with a larger class of{gamma}-exponential kernels. We show that these perform almost identically. Our results suggest that much insight about neural networks can be obtained from analysis of the well-known Laplace kernel, which has a simple closed-form.
The Neural Tangent Kernel (NTK) characterizes the behavior of infinitely wide neural nets trained under least squares loss by gradient descent. However, despite its importance, the super-quadratic runtime of kernel methods limits the use of NTK in large-scale learning tasks. To accelerate kernel machines with NTK, we propose a near input sparsity time algorithm that maps the input data to a randomized low-dimensional feature space so that the inner product of the transformed data approximates their NTK evaluation. Our transformation works by sketching the polynomial expansions of arc-cosine kernels. Furthermore, we propose a feature map for approximating the convolutional counterpart of the NTK, which can transform any image using a runtime that is only linear in the number of pixels. We show that in standard large-scale regression and classification tasks a linear regressor trained on our features outperforms trained Neural Nets and Nystrom approximation of NTK kernel.
Calibration of neural networks is a topical problem that is becoming increasingly important for real-world use of neural networks. The problem is especially noticeable when using modern neural networks, for which there is significant difference between the model confidence and the confidence it should have. Various strategies have been successfully proposed, yet there is more space for improvements. We propose a novel approach that introduces a differentiable metric for expected calibration error and successfully uses it as an objective for meta-learning, achieving competitive results with state-of-the-art approaches. Our approach presents a new direction of using meta-learning to directly optimize model calibration, which we believe will inspire further work in this promising and new direction.
We study the power of learning via mini-batch stochastic gradient descent (SGD) on the population loss, and batch Gradient Descent (GD) on the empirical loss, of a differentiable model or neural network, and ask what learning problems can be learnt using these paradigms. We show that SGD and GD can always simulate learning with statistical queries (SQ), but their ability to go beyond that depends on the precision $rho$ of the gradient calculations relative to the minibatch size $b$ (for SGD) and sample size $m$ (for GD). With fine enough precision relative to minibatch size, namely when $b rho$ is small enough, SGD can go beyond SQ learning and simulate any sample-based learning algorithm and thus its learning power is equivalent to that of PAC learning; this extends prior work that achieved this result for $b=1$. Similarly, with fine enough precision relative to the sample size $m$, GD can also simulate any sample-based learning algorithm based on $m$ samples. In particular, with polynomially many bits of precision (i.e. when $rho$ is exponentially small), SGD and GD can both simulate PAC learning regardless of the mini-batch size. On the other hand, when $b rho^2$ is large enough, the power of SGD is equivalent to that of SQ learning.
Bayesian structure learning allows inferring Bayesian network structure from data while reasoning about the epistemic uncertainty -- a key element towards enabling active causal discovery and designing interventions in real world systems. In this work, we propose a general, fully differentiable framework for Bayesian structure learning (DiBS) that operates in the continuous space of a latent probabilistic graph representation. Contrary to existing work, DiBS is agnostic to the form of the local conditional distributions and allows for joint posterior inference of both the graph structure and the conditional distribution parameters. This makes DiBS directly applicable to posterior inference of nonstandard Bayesian network models, e.g., with nonlinear dependencies encoded by neural networks. Building on recent advances in variational inference, we use DiBS to devise an efficient general purpose method for approximating posteriors over structural models. In evaluations on simulated and real-world data, our method significantly outperforms related approaches to joint posterior inference.
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