An MDL framework for sparse coding and dictionary learning

The power of sparse signal modeling with learned over-complete dictionaries has been demonstrated in a variety of applications and fields, from signal processing to statistical inference and machine learning. However, the statistical properties of these models, such as under-fitting or over-fitting given sets of data, are still not well characterized in the literature. As a result, the success of sparse modeling depends on hand-tuning critical parameters for each data and application. This work aims at addressing this by providing a practical and objective characterization of sparse models by means of the Minimum Description Length (MDL) principle -- a well established information-theoretic approach to model selection in statistical inference. The resulting framework derives a family of efficient sparse coding and dictionary learning algorithms which, by virtue of the MDL principle, are completely parameter free. Furthermore, such framework allows to incorporate additional prior information to existing models, such as Markovian dependencies, or to define completely new problem formulations, including in the matrix analysis area, in a natural way. These virtues will be demonstrated with parameter-free algorithms for the classic image denoising and classification problems, and for low-rank matrix recovery in video applications

Constraining Implicit Space with Minimum Description Length: An Unsupervised Attention Mechanism across Neural Network Layers

Inspired by the adaptation phenomenon of neuronal firing, we propose the regularity normalization (RN) as an unsupervised attention mechanism (UAM) which computes the statistical regularity in the implicit space of neural networks under the Minimum Description Length (MDL) principle. Treating the neural network optimization process as a partially observable model selection problem, UAM constrains the implicit space by a normalization factor, the universal code length. We compute this universal code incrementally across neural network layers and demonstrated the flexibility to include data priors such as top-down attention and other oracle information. Empirically, our approach outperforms existing normalization methods in tackling limited, imbalanced and non-stationary input distribution in image classification, classic control, procedurally-generated reinforcement learning, generative modeling, handwriting generation and question answering tasks with various neural network architectures. Lastly, UAM tracks dependency and critical learning stages across layers and recurrent time steps of deep networks

Evaluating Overfit and Underfit in Models of Network Community Structure

A common data mining task on networks is community detection, which seeks an unsupervised decomposition of a network into structural groups based on statistical regularities in the network's connectivity. Although many methods exist, the No Free Lunch theorem for community detection implies that each makes some kind of tradeoff, and no algorithm can be optimal on all inputs. Thus, different algorithms will over or underfit on different inputs, finding more, fewer, or just different communities than is optimal, and evaluation methods that use a metadata partition as a ground truth will produce misleading conclusions about general accuracy. Here, we present a broad evaluation of over and underfitting in community detection, comparing the behavior of 16 state-of-the-art community detection algorithms on a novel and structurally diverse corpus of 406 real-world networks. We find that (i) algorithms vary widely both in the number of communities they find and in their corresponding composition, given the same input, (ii) algorithms can be clustered into distinct high-level groups based on similarities of their outputs on real-world networks, and (iii) these differences induce wide variation in accuracy on link prediction and link description tasks. We introduce a new diagnostic for evaluating overfitting and underfitting in practice, and use it to roughly divide community detection methods into general and specialized learning algorithms. Across methods and inputs, Bayesian techniques based on the stochastic block model and a minimum description length approach to regularization represent the best general learning approach, but can be outperformed under specific circumstances. These results introduce both a theoretically principled approach to evaluate over and underfitting in models of network community structure and a realistic benchmark by which new methods may be evaluated and compared.Comment: 22 pages, 13 figures, 3 table

A Unified Theory of Dual-Process Control

Dual-process theories play a central role in both psychology and neuroscience, figuring prominently in fields ranging from executive control to reward-based learning to judgment and decision making. In each of these domains, two mechanisms appear to operate concurrently, one relatively high in computational complexity, the other relatively simple. Why is neural information processing organized in this way? We propose an answer to this question based on the notion of compression. The key insight is that dual-process structure can enhance adaptive behavior by allowing an agent to minimize the description length of its own behavior. We apply a single model based on this observation to findings from research on executive control, reward-based learning, and judgment and decision making, showing that seemingly diverse dual-process phenomena can be understood as domain-specific consequences of a single underlying set of computational principles

HyperVAE: A Minimum Description Length Variational Hyper-Encoding Network

We propose a framework called HyperVAE for encoding distributions of distributions. When a target distribution is modeled by a VAE, its neural network parameters \theta is drawn from a distribution p(\theta) which is modeled by a hyper-level VAE. We propose a variational inference using Gaussian mixture models to implicitly encode the parameters \theta into a low dimensional Gaussian distribution. Given a target distribution, we predict the posterior distribution of the latent code, then use a matrix-network decoder to generate a posterior distribution q(\theta). HyperVAE can encode the parameters \theta in full in contrast to common hyper-networks practices, which generate only the scale and bias vectors as target-network parameters. Thus HyperVAE preserves much more information about the model for each task in the latent space. We discuss HyperVAE using the minimum description length (MDL) principle and show that it helps HyperVAE to generalize. We evaluate HyperVAE in density estimation tasks, outlier detection and discovery of novel design classes, demonstrating its efficacy

Asymptotics of Discrete MDL for Online Prediction

Minimum Description Length (MDL) is an important principle for induction and prediction, with strong relations to optimal Bayesian learning. This paper deals with learning non-i.i.d. processes by means of two-part MDL, where the underlying model class is countable. We consider the online learning framework, i.e. observations come in one by one, and the predictor is allowed to update his state of mind after each time step. We identify two ways of predicting by MDL for this setup, namely a static} and a dynamic one. (A third variant, hybrid MDL, will turn out inferior.) We will prove that under the only assumption that the data is generated by a distribution contained in the model class, the MDL predictions converge to the true values almost surely. This is accomplished by proving finite bounds on the quadratic, the Hellinger, and the Kullback-Leibler loss of the MDL learner, which are however exponentially worse than for Bayesian prediction. We demonstrate that these bounds are sharp, even for model classes containing only Bernoulli distributions. We show how these bounds imply regret bounds for arbitrary loss functions. Our results apply to a wide range of setups, namely sequence prediction, pattern classification, regression, and universal induction in the sense of Algorithmic Information Theory among others.Comment: 34 page

Minimum Description Length Control

We propose a novel framework for multitask reinforcement learning based on the minimum description length (MDL) principle. In this approach, which we term MDL-control (MDL-C), the agent learns the common structure among the tasks with which it is faced and then distills it into a simpler representation which facilitates faster convergence and generalization to new tasks. In doing so, MDL-C naturally balances adaptation to each task with epistemic uncertainty about the task distribution. We motivate MDL-C via formal connections between the MDL principle and Bayesian inference, derive theoretical performance guarantees, and demonstrate MDL-C's empirical effectiveness on both discrete and high-dimensional continuous control tasks