Step sizes in neural network training are largely determined using predetermined rules such as fixed learning rates and learning rate schedules, which require user input to determine their functional form and associated hyperparameters. Global optimization strategies to resolve these hyperparameters are computationally expensive. Line searches are capable of adaptively resolving learning rate schedules. However, due to discontinuities induced by mini-batch sampling, they have largely fallen out of favor. Notwithstanding, probabilistic line searches have recently demonstrated viability in resolving learning rates for stochastic loss functions. This method creates surrogates with confidence intervals, where restrictions are placed on the rate at which the search domain can grow along a search direction. This paper introduces an alternative paradigm, Gradient-Only Line Searches that are inexact (GOLS-I), as an alternative strategy to automatically resolve learning rates in stochastic cost functions over a range of 15 orders of magnitude without the use of surrogates. We show that GOLS-I is a competitive strategy to reliably resolve step sizes, adding high value in terms of performance, while being easy to implement. Considering mini-batch sampling, we open the discussion on how to split the effort to resolve quality search directions from quality step size estimates along a search direction.

We consider Hadamard product parametrization as a change-of-variable (over-parametrization) technique for solving least square problems in the context of linear regression. Despite the non-convexity and exponentially many saddle points induced by the change-of-variable, we show that under certain conditions, this over-parametrization leads to implicit regularization: if we directly apply gradient descent to the residual sum of squares with sufficiently small initial values, then under proper early stopping rule, the iterates converge to a nearly sparse rate-optimal solution with relatively better accuracy than explicit regularized approaches. In particular, the resulting estimator does not suffer from extra bias due to explicit penalties, and can achieve the parametric root-$n$ rate (independent of the dimension) under proper conditions on the signal-to-noise ratio. We perform simulations to compare our methods with high dimensional linear regression with explicit regularizations. Our results illustrate advantages of using implicit regularization via gradient descent after over-parametrization in sparse vector estimation.

In classification problems (i.e. when the labels Y are discrete), the scaling of the test error with respect to n is determined by characterizations of the VC-dimension [2]/Rademacher complexity [3] of the function class, which in the worst case increases with its number of parameters. In regression (i.e. when the labels Y are continuous), the mean-squared error of the ordinary least-squares estimate is characterized by the condition number of the regression matrix, which is reasonable for appropriate ratios of d/n but tends to increase astronomically as d approaches n. The qualitative fear is the same: if the function class is too complex, it starts to overfit noise and can generalize poorly to unseen test data. But there is a gap between "can" and "will" -- and indeed this conventional wisdom has been challenged by the recent advent of deeper and deeper neural networks. In particular, a thought-provoking paper [4] noted that several deep neural networks generalize well despite achieving zero or close to zero training error, and being so expressive that they even have the ability to fit pure noise. As they put it, "understanding deep learning requires rethinking generalization". How can we reconcile the fact that good interpolative solutions exist with the classical bias-variance tradeoff? These phenomena are being actively investigated in a statistical sense [5,6] and a computational sense [7-9] in classification problems and/or noiseless models.

This paper proposes using a linear function approximator, rather than a deep neural network (DNN), to bias a Monte Carlo tree search (MCTS) player for general games. This is unlikely to match the potential raw playing strength of DNNs, but has advantages in terms of generality, interpretability and resources (time and hardware) required for training. Features describing local patterns are used as inputs. The features are formulated in such a way that they are easily interpretable and applicable to a wide range of general games, and might encode simple local strategies. We gradually create new features during the same self-play training process used to learn feature weights. We evaluate the playing strength of an MCTS player biased by learnt features against a standard upper confidence bounds for trees (UCT) player in multiple different board games, and demonstrate significantly improved playing strength in the majority of them after a small number of self-play training games.

We analyze SVAG, a variance reduced stochastic gradient method with SAG and SAGA as special cases. Our convergence result for SVAG is the first to simultaneously capture both the biased low-variance method SAG and the unbiased high-variance method SAGA. In the case of SAGA, it matches previous upper bounds on the allowed step-size. The SVAG algorithm has a parameter that decides the bias-variance trade-off in the stochastic gradient estimate. We provide numerical examples demonstrating the intuition behind this bias-variance trade-off.

This work is motivated by multimodality breast cancer imaging data, which is quite challenging in that the signals of discrete tumor-associated microvesicles (TMVs) are randomly distributed with heterogeneous patterns. This imposes a significant challenge for conventional imaging regression and dimension reduction models assuming a homogeneous feature structure. We develop an innovative multilayer tensor learning method to incorporate heterogeneity to a higher-order tensor decomposition and predict disease status effectively through utilizing subject-wise imaging features and multimodality information. Specifically, we construct a multilayer decomposition which leverages an individualized imaging layer in addition to a modality-specific tensor structure. One major advantage of our approach is that we are able to efficiently capture the heterogeneous spatial features of signals that are not characterized by a population structure as well as integrating multimodality information simultaneously. To achieve scalable computing, we develop a new bi-level block improvement algorithm. In theory, we investigate both the algorithm convergence property, tensor signal recovery error bound and asymptotic consistency for prediction model estimation. We also apply the proposed method for simulated and human breast cancer imaging data. Numerical results demonstrate that the proposed method outperforms other existing competing methods.

Time series models such as dynamical systems are frequently fitted to a cohort of data, ignoring variation between individual entities such as patients. In this paper we show how these models can be personalised to an individual level while retaining statistical power, via use of multi-task learning (MTL). To our knowledge this is a novel development of MTL which applies to time series both with and without control inputs. The modelling framework is demonstrated on a physiological drug response problem which results in improved predictive accuracy and uncertainty estimation over existing state-of-the-art models.

This work addresses the problem of learning sparse representations of tensor data using structured dictionary learning. It proposes learning a mixture of separable dictionaries to better capture the structure of tensor data by generalizing the separable dictionary learning model. Two different approaches for learning mixture of separable dictionaries are explored and sufficient conditions for local identifiability of the underlying dictionary are derived in each case. Moreover, computational algorithms are developed to solve the problem of learning mixture of separable dictionaries in both batch and online settings. Numerical experiments are used to show the usefulness of the proposed model and the efficacy of the developed algorithms.

Determining the optimal initial dose for warfarin is a critically important task. Several factors have an impact on the therapeutic dose for individual patients, such as patients' physical attributes (Age, Height, etc.), medication profile, co-morbidities, and metabolic genotypes (CYP2C9 and VKORC1). These wide range factors influencing therapeutic dose, create a complex environment for clinicians to determine the optimal initial dose. Using a sample of 4,237 patients, we have proposed a companion classification model to one of the most popular dosing algorithms (International Warfarin Pharmacogenetics Consortium (IWPC) clinical model), which identifies the appropriate cohort of patients for applying this model. The proposed model functions as a clinical decision support system which assists clinicians in dosing. We have developed a classification model using Support Vector Machines, with a polynomial kernel function to determine if applying the dose prediction model is appropriate for a given patient. The IWPC clinical model will only be used if the patient is classified as "Safe for model". By using the proposed methodology, the dosing mode's prediction accuracy increases by 15 percent in terms of Root Mean Squared Error and 17 percent in terms of Mean Absolute Error in dose estimates of patients classified as "Safe for model".

Decision trees are ubiquitous in machine learning for their ease of use and interpretability; however, they are not typically implemented in reinforcement learning because they cannot be updated via stochastic gradient descent. Traditional applications of decision trees for reinforcement learning have focused instead on making commitments to decision boundaries as the tree is grown one layer at a time. We overcome this critical limitation by allowing for a gradient update over the entire tree structure that improves sample complexity when a tree is fuzzy and interpretability when sharp. We offer three key contributions towards this goal. First, we motivate the need for policy gradient-based learning by examining the theoretical properties of gradient descent over differentiable decision trees. Second, we introduce a regularization framework that yields interpretability via sparsity in the tree structure. Third, we demonstrate the ability to construct a decision tree via policy gradient in canonical reinforcement learning domains and supervised learning benchmarks.