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Register a new userA Universal Kernel for Learning Regular Languages
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Added by
Leonid (Aryeh) Kontorovich
Publication date
2007
fields
Informatics Engineering
and research's language is
English
Authors
Leonid Kontorovich
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We give a universal kernel that renders all the regular languages linearly separable. We are not able to compute this kernel efficiently and conjecture that it is intractable, but we do have an efficient $eps$-approximation.
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In structured prediction, a major challenge for models is to represent the interdependencies within their output structures. For the common case where outputs are structured as a sequence, linear-chain conditional random fields (CRFs) are a widely used model class which can learn local dependencies in output sequences. However, the CRFs Markov assumption makes it impossible for these models to capture nonlocal dependencies, and standard CRFs are unable to respect nonlocal constraints of the data (such as global arity constraints on output labels). We present a generalization of CRFs that can enforce a broad class of constraints, including nonlocal ones, by specifying the space of possible output structures as a regular language $mathcal{L}$. The resulting regular-constrained CRF (RegCCRF) has the same formal properties as a standard CRF, but assigns zero probability to all label sequences not in $mathcal{L}$. Notably, RegCCRFs can incorporate their constraints during training, while related models only enforce constraints during decoding. We prove that constrained training is never worse than constrained decoding, and show using synthetic data that it can be substantially better in practice. Additionally, we demonstrate a practical benefit on downstream tasks by incorporating a RegCCRF into a deep neural model for semantic role labeling, exceeding state-of-the-art results on a standard dataset.
We introduce a general framework for approximating regular conditional distributions (RCDs). Our approximations of these RCDs are implemented by a new class of geometric deep learning models with inputs in $mathbb{R}^d$ and outputs in the Wasserstein-$1$ space $mathcal{P}_1(mathbb{R}^D)$. We find that the models built using our framework can approximate any continuous functions from $mathbb{R}^d$ to $mathcal{P}_1(mathbb{R}^D)$ uniformly on compacts, and quantitative rates are obtained. We identify two methods for avoiding the curse of dimensionality; i.e.: the number of parameters determining the approximating neural network depends only polynomially on the involved dimension and the approximation error. The first solution describes functions in $C(mathbb{R}^d,mathcal{P}_1(mathbb{R}^D))$ which can be efficiently approximated on any compact subset of $mathbb{R}^d$. Conversely, the second approach describes sets in $mathbb{R}^d$, on which any function in $C(mathbb{R}^d,mathcal{P}_1(mathbb{R}^D))$ can be efficiently approximated. Our framework is used to obtain an affirmative answer to the open conjecture of Bishop (1994); namely: mixture density networks are universal regular conditional distributions. The predictive performance of the proposed models is evaluated against comparable learning models on various probabilistic predictions tasks in the context of ELMs, model uncertainty, and heteroscedastic regression. All the results are obtained for more general input and output spaces and thus apply to geometric deep learning contexts.
Large machine learning models achieve unprecedented performance on various tasks and have evolved as the go-to technique. However, deploying these compute and memory hungry models on resource constraint environments poses new challenges. In this work, we propose mathematically provable Representer Sketch, a concise set of count arrays that can approximate the inference procedure with simple hashing computations and aggregations. Representer Sketch builds upon the popular Representer Theorem from kernel literature, hence the name, providing a generic fundamental alternative to the problem of efficient inference that goes beyond the popular approach such as quantization, iterative pruning and knowledge distillation. A neural network function is transformed to its weighted kernel density representation, which can be very efficiently estimated with our sketching algorithm. Empirically, we show that Representer Sketch achieves up to 114x reduction in storage requirement and 59x reduction in computation complexity without any drop in accuracy.
We propose a deep learning approach for discovering kernels tailored to identifying clusters over sample data. Our neural network produces sample embeddings that are motivated by--and are at least as expressive as--spectral clustering. Our training objective, based on the Hilbert Schmidt Information Criterion, can be optimized via gradient adaptations on the Stiefel manifold, leading to significant acceleration over spectral methods relying on eigendecompositions. Finally, our trained embedding can be directly applied to out-of-sample data. We show experimentally that our approach outperforms several state-of-the-art deep clustering methods, as well as traditional approaches such as $k$-means and spectral clustering over a broad array of real-life and synthetic datasets.
Motivated by the question of which completely regular semigroups have context-free word problem, we show that for certain classes of languages $mathfrak{C}$(including context-free), every completely regular semigroup that is a union of finitely many finitely generated groups with word problem in $mathfrak{C}$ also has word problem in $mathfrak{C}$. We give an example to show that not all completely regular semigroups with context-free word problem can be so constructed.
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