Random Search is one of the most widely-used method for Hyperparameter Optimization, and is critical to the success of deep learning models. Despite its astonishing performance, little non-heuristic theory has been developed to describe the underlying working mechanism. This paper gives a theoretical accounting of Random Search. We introduce the concept of \emph{scattering dimension} that describes the landscape of the underlying function, and quantifies the performance of random search. We show that, when the environment is noise-free, the output of random search converges to the optimal value in probability at rate $ \widetilde{\mathcal{O}} \left( \left( \frac{1}{T} \right)^{ \frac{1}{d_s} } \right) $, where $ d_s \ge 0 $ is the scattering dimension of the underlying function. When the observed function values are corrupted by bounded $iid$ noise, the output of random search converges to the optimal value in probability at rate $ \widetilde{\mathcal{O}} \left( \left( \frac{1}{T} \right)^{ \frac{1}{d_s + 1} } \right) $. In addition, based on the principles of random search, we introduce an algorithm, called BLiN-MOS, for Lipschitz bandits in doubling metric spaces that are also endowed with a Borel measure, and show that BLiN-MOS achieves a regret rate of order $ \widetilde{\mathcal{O}} \left( T^{ \frac{d_z}{d_z + 1} } \right) $, where $d_z$ is the zooming dimension of the problem instance. Our results show that under certain conditions, the known information-theoretical lower bounds for Lipschitz bandits $\Omega \left( T^{\frac{d_z+1}{d_z+2}} \right)$ can be improved.
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One of the fundamental steps toward understanding a complex system is identifying variation at the scale of the system's components that is most relevant to behavior on a macroscopic scale. Mutual information is a natural means of linking variation across scales of a system due to its independence of the particular functional relationship between variables. However, estimating mutual information given high-dimensional, continuous-valued data is notoriously difficult, and the desideratum -- to reveal important variation in a comprehensible manner -- is only readily achieved through exhaustive search. Here we propose a practical, efficient, and broadly applicable methodology to decompose the information contained in a set of measurements by lossily compressing each measurement with machine learning. Guided by the distributed information bottleneck as a learning objective, the information decomposition sorts variation in the measurements of the system state by relevance to specified macroscale behavior, revealing the most important subsets of measurements for different amounts of predictive information. Additional granularity is achieved by inspection of the learned compression schemes: the variation transmitted during compression is composed of distinctions among measurement values that are most relevant to the macroscale behavior. We focus our analysis on two paradigmatic complex systems: a Boolean circuit and an amorphous material undergoing plastic deformation. In both examples, specific bits of entropy are identified out of the high entropy of the system state as most related to macroscale behavior for insight about the connection between micro- and macro- in the complex system. The identification of meaningful variation in data, with the full generality brought by information theory, is made practical for the study of complex systems.
Model-Based Reinforcement Learning (RL) is widely believed to have the potential to improve sample efficiency by allowing an agent to synthesize large amounts of imagined experience. Experience Replay (ER) can be considered a simple kind of model, which has proved effective at improving the stability and efficiency of deep RL. In principle, a learned parametric model could improve on ER by generalizing from real experience to augment the dataset with additional plausible experience. However, given that learned value functions can also generalize, it is not immediately obvious why model generalization should be better. Here, we provide theoretical and empirical insight into when, and how, we can expect data generated by a learned model to be useful. First, we provide a simple theorem motivating how learning a model as an intermediate step can narrow down the set of possible value functions more than learning a value function directly from data using the Bellman equation. Second, we provide an illustrative example showing empirically how a similar effect occurs in a more concrete setting with neural network function approximation. Finally, we provide extensive experiments showing the benefit of model-based learning for online RL in environments with combinatorial complexity, but factored structure that allows a learned model to generalize. In these experiments, we take care to control for other factors in order to isolate, insofar as possible, the benefit of using experience generated by a learned model relative to ER alone.
The central limit theorem (CLT) is one of the most fundamental results in probability; and establishing its rate of convergence has been a key question since the 1940s. For independent random variables, a series of recent works established optimal error bounds under the Wasserstein-p distance (with p>=1). In this paper, we extend those results to locally dependent random variables, which include m-dependent random fields and U-statistics. Under conditions on the moments and the dependency neighborhoods, we derive optimal rates in the CLT for the Wasserstein-p distance. Our proofs rely on approximating the empirical average of dependent observations by the empirical average of i.i.d. random variables. To do so, we expand the Stein equation to arbitrary orders by adapting the Stein's dependency neighborhood method. Finally we illustrate the applicability of our results by obtaining efficient tail bounds.
In this paper, we propose a general approach for explicit a posteriori error representation for convex minimization problems using basic convex duality relations. Exploiting discrete orthogonality relations in the space of element-wise constant vector fields as well as a discrete integration-by-parts formula between the Crouzeix-Raviart and the Raviart-Thomas element, all convex duality relations are transferred to a discrete level, making the explicit a posteriori error representation -- initially based on continuous arguments only -- practicable from a numerical point of view. In addition, we provide a generalized Marini formula for the primal solution that determines a discrete primal solution in terms of a given discrete dual solution. We benchmark all these concepts via the Rudin-Osher-Fatemi model. This leads to an adaptive algorithm that yields a (quasi-optimal) linear convergence rate.
Consider a diffusion process X=(X_t), with t in [0,1], observed at discrete times and high frequency, solution of a stochastic differential equation whose drift and diffusion coefficients are assumed to be unknown. In this article, we focus on the nonparametric esstimation of the diffusion coefficient. We propose ridge estimators of the square of the diffusion coefficient from discrete observations of X and that are obtained by minimization of the least squares contrast. We prove that the estimators are consistent and derive rates of convergence as the size of the sample paths tends to infinity, and the discretization step of the time interval [0,1] tend to zero. The theoretical results are completed with a numerical study over synthetic data.
A fundamental problem in manifold learning is to approximate a functional relationship in a data chosen randomly from a probability distribution supported on a low dimensional sub-manifold of a high dimensional ambient Euclidean space. The manifold is essentially defined by the data set itself and, typically, designed so that the data is dense on the manifold in some sense. The notion of a data space is an abstraction of a manifold encapsulating the essential properties that allow for function approximation. The problem of transfer learning (meta-learning) is to use the learning of a function on one data set to learn a similar function on a new data set. In terms of function approximation, this means lifting a function on one data space (the base data space) to another (the target data space). This viewpoint enables us to connect some inverse problems in applied mathematics (such as inverse Radon transform) with transfer learning. In this paper we examine the question of such lifting when the data is assumed to be known only on a part of the base data space. We are interested in determining subsets of the target data space on which the lifting can be defined, and how the local smoothness of the function and its lifting are related.
The problem of comparing probability distributions is at the heart of many tasks in statistics and machine learning and the most classical comparison methods assume that the distributions occur in spaces of the same dimension. Recently, a new geometric solution has been proposed to address this problem when the measures live in Euclidean spaces of differing dimensions. Here, we study the same problem of comparing probability distributions of different dimensions in the tropical geometric setting, which is becoming increasingly relevant in computations and applications involving complex, geometric data structures. Specifically, we construct a Wasserstein distance between measures on different tropical projective tori - the focal metric spaces in both theory and applications of tropical geometry - via tropical mappings between probability measures. We prove equivalence of the directionality of the maps, whether starting from the lower dimensional space and mapping to the higher dimensional space or vice versa. As an important practical implication, our work provides a framework for comparing probability distributions on the spaces of phylogenetic trees with different leaf sets.
Due to the diffusion of IoT, modern software systems are often thought to control and coordinate smart devices in order to manage assets and resources, and to guarantee efficient behaviours. For this class of systems, which interact extensively with humans and with their environment, it is thus crucial to guarantee their correct behaviour in order to avoid unexpected and possibly dangerous situations. In this paper we will present a framework that allows us to measure the robustness of systems. This is the ability of a program to tolerate changes in the environmental conditions and preserving the original behaviour. In the proposed framework, the interaction of a program with its environment is represented as a sequence of random variables describing how both evolve in time. For this reason, the considered measures will be defined among probability distributions of observed data. The proposed framework will be then used to define the notions of adaptability and reliability. The former indicates the ability of a program to absorb perturbation on environmental conditions after a given amount of time. The latter expresses the ability of a program to maintain its intended behaviour (up-to some reasonable tolerance) despite the presence of perturbations in the environment. Moreover, an algorithm, based on statistical inference, is proposed to evaluate the proposed metric and the aforementioned properties. We use two case studies to the describe and evaluate the proposed approach.
In this monograph, I introduce the basic concepts of Online Learning through a modern view of Online Convex Optimization. Here, online learning refers to the framework of regret minimization under worst-case assumptions. I present first-order and second-order algorithms for online learning with convex losses, in Euclidean and non-Euclidean settings. All the algorithms are clearly presented as instantiation of Online Mirror Descent or Follow-The-Regularized-Leader and their variants. Particular attention is given to the issue of tuning the parameters of the algorithms and learning in unbounded domains, through adaptive and parameter-free online learning algorithms. Non-convex losses are dealt through convex surrogate losses and through randomization. The bandit setting is also briefly discussed, touching on the problem of adversarial and stochastic multi-armed bandits. These notes do not require prior knowledge of convex analysis and all the required mathematical tools are rigorously explained. Moreover, all the proofs have been carefully chosen to be as simple and as short as possible.
Pre-trained deep neural network language models such as ELMo, GPT, BERT and XLNet have recently achieved state-of-the-art performance on a variety of language understanding tasks. However, their size makes them impractical for a number of scenarios, especially on mobile and edge devices. In particular, the input word embedding matrix accounts for a significant proportion of the model's memory footprint, due to the large input vocabulary and embedding dimensions. Knowledge distillation techniques have had success at compressing large neural network models, but they are ineffective at yielding student models with vocabularies different from the original teacher models. We introduce a novel knowledge distillation technique for training a student model with a significantly smaller vocabulary as well as lower embedding and hidden state dimensions. Specifically, we employ a dual-training mechanism that trains the teacher and student models simultaneously to obtain optimal word embeddings for the student vocabulary. We combine this approach with learning shared projection matrices that transfer layer-wise knowledge from the teacher model to the student model. Our method is able to compress the BERT_BASE model by more than 60x, with only a minor drop in downstream task metrics, resulting in a language model with a footprint of under 7MB. Experimental results also demonstrate higher compression efficiency and accuracy when compared with other state-of-the-art compression techniques.
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