Motivated from the theory of non-parametric regression, we introduce a \emph{new variational constraint} that enforces the comparator sequence to belong to a discrete $k^{th}$ order Total Variation ball of radius $C_n$. This variational constraint models comparators that have piece-wise polynomial structure which has many relevant practical applications [Tibshirani2015]. By establishing connections to the theory of wavelet based non-parametric regression, we design a \emph{polynomial time} algorithm that achieves the nearly \emph{optimal dynamic regret} of $\tilde{O}(n^{\frac{1}{2k+3}}C_n^{\frac{2}{2k+3}})$. The proposed policy is \emph{adaptive to the unknown radius} $C_n$. Further, we show that the same policy is minimax optimal for several other non-parametric families of interest.

Active learning is the task of using labelled data to select additional points to label, with the goal of fitting the most accurate model with a fixed budget of labelled points. In binary classification active learning is known to produce faster rates than passive learning for a broad range of settings. However in regression restrictive structure and tailored methods were previously needed to obtain theoretically superior performance. In this paper we propose an intuitive tree based active learning algorithm for non-parametric regression with provable improvement over random sampling. When implemented with Mondrian Trees our algorithm is tuning parameter free, consistent and minimax optimal for Lipschitz functions.

We consider the framework of non-stationary stochastic optimization [Besbes et al., 2015] with squared error losses and noisy gradient feedback where the dynamic regret of an online learner against a time varying comparator sequence

Motivated from the theory of non-parametric regression, we introduce a \emph{new variational constraint} that enforces the comparator sequence to belong to a discrete k {th} order Total Variation ball of radius C_n . This variational constraint models comparators that have piece-wise polynomial structure which has many relevant practical applications [Tibshirani2015]. The proposed policy is \emph{adaptive to the unknown radius} C_n . Further, we show that the same policy is minimax optimal for several other non-parametric families of interest.

We address the regression problem for a general function $f:[-1,1]^d\to \mathbb R$ when the learner selects the training points $\{x_i\}_{i=1}^n$ to achieve a uniform error bound across the entire domain. In this setting, known historically as nonparametric regression, we aim to establish a sample complexity bound that depends solely on the function's degree of smoothness. Assuming periodicity at the domain boundaries, we introduce PADUA, an algorithm that, with high probability, provides performance guarantees optimal up to constant or logarithmic factors across all problem parameters. Notably, PADUA is the first parametric algorithm with optimal sample complexity for this setting. Due to this feature, we prove that, differently from the non-parametric state of the art, PADUA enjoys optimal space complexity in the prediction phase. To validate these results, we perform numerical experiments over functions coming from real audio data, where PADUA shows comparable performance to state-of-the-art methods, while requiring only a fraction of the computational time.

Yet, I still think that their results (including those appearing in their response) are not showing significant gains as seen in other Active Learning settings. The paper addresses the problem of active learning for regression with random trees. The authors introduce a simple'oracle' algorithm and a risk variance is provided for this criterion. Authors provide a risk minimizing query criteria and claim it is better than the random selection criterion An estimation algorithm is presented that provides the necessary statistics at its first stage to allow an approximation to the'oracle' algorithm. Numerical results are presented in which improved performance over random and uncertainty sample is provided for simulated data, and no improvement for real UCI data.

This learning problem arises frequently in many application areas ranging from signal processing, computer vision, over robotics to computer graphics. We present a new algorithmic scheme for the solution of this general learning problem based on regularized empirical risk minimization. The regularization functional takes into account the geometry of input and output manifold, and we show that it implements a prior which is particularly natural. Moreover, we demonstrate that our algorithm performs well in a difficult surface registration problem.

This learning problem arises frequently in many application areas ranging from signal processing, computer vision, over robotics to computer graphics. We present a new algorithmic scheme for the solution of this general learning problem based on regularized empirical risk minimization. The regularization functional takes into account the geometry of input and output manifold, and we show that it implements a prior which is particularly natural. Moreover, we demonstrate that our algorithm performs well in a difficult surface registration problem. Papers published at the Neural Information Processing Systems Conference.

Active learning is the task of using labelled data to select additional points to label, with the goal of fitting the most accurate model with a fixed budget of labelled points. In binary classification active learning is known to produce faster rates than passive learning for a broad range of settings. However in regression restrictive structure and tailored methods were previously needed to obtain theoretically superior performance. In this paper we propose an intuitive tree based active learning algorithm for non-parametric regression with provable improvement over random sampling. When implemented with Mondrian Trees our algorithm is tuning parameter free, consistent and minimax optimal for Lipschitz functions.

Having a regression model, we are interested in finding two-sided intervals that are guaranteed to contain at least a desired proportion of the conditional distribution of the response variable given a specific combination of predictors. We name such intervals predictive intervals. This work presents a new method to find two-sided predictive intervals for non-parametric least squares regression without the homoscedasticity assumption. Our predictive intervals are built by using tolerance intervals on prediction errors in the query point's neighborhood. We proposed a predictive interval model test and we also used it as a constraint in our hyper-parameter tuning algorithm. This gives an algorithm that finds the smallest reliable predictive intervals for a given dataset. We also introduce a measure for comparing different interval prediction methods yielding intervals having different size and coverage. These experiments show that our methods are more reliable, effective and precise than other interval prediction methods.