AAAI Conferences

Conditional Value at Risk (CVaR) is a prominent risk measure that is being used extensively in various domains. We develop a new formula for the gradient of the CVaR in the form of a conditional expectation. Based on this formula, we propose a novel sampling-based estimator for the gradient of the CVaR, in the spirit of the likelihood-ratio method. We analyze the bias of the estimator, and prove the convergence of a corresponding stochastic gradient descent algorithm to a local CVaR optimum. Our method allows to consider CVaR optimization in new domains. As an example, we consider a reinforcement learning application, and learn a risk-sensitive controller for the game of Tetris.

Reverse iterative volume sampling for linear regression

arXiv.org Machine Learning

We study the following basic machine learning task: Given a fixed set of $d$-dimensional input points for a linear regression problem, we wish to predict a hidden response value for each of the points. We can only afford to attain the responses for a small subset of the points that are then used to construct linear predictions for all points in the dataset. The performance of the predictions is evaluated by the total square loss on all responses (the attained as well as the hidden ones). We show that a good approximate solution to this least squares problem can be obtained from just dimension $d$ many responses by using a joint sampling technique called volume sampling. Moreover, the least squares solution obtained for the volume sampled subproblem is an unbiased estimator of optimal solution based on all n responses. This unbiasedness is a desirable property that is not shared by other common subset selection techniques. Motivated by these basic properties, we develop a theoretical framework for studying volume sampling, resulting in a number of new matrix expectation equalities and statistical guarantees which are of importance not only to least squares regression but also to numerical linear algebra in general. Our methods also lead to a regularized variant of volume sampling, and we propose the first efficient algorithms for volume sampling which make this technique a practical tool in the machine learning toolbox. Finally, we provide experimental evidence which confirms our theoretical findings.

Unbiased estimates for linear regression via volume sampling

Neural Information Processing Systems

Given a full rank matrix X with more columns than rows consider the task of estimating the pseudo inverse $X^+$ based on the pseudo inverse of a sampled subset of columns (of size at least the number of rows). We show that this is possible if the subset of columns is chosen proportional to the squared volume spanned by the rows of the chosen submatrix (ie, volume sampling). The resulting estimator is unbiased and surprisingly the covariance of the estimator also has a closed form: It equals a specific factor times $X^+X^{+\top}$. Pseudo inverse plays an important part in solving the linear least squares problem, where we try to predict a label for each column of $X$. We assume labels are expensive and we are only given the labels for the small subset of columns we sample from $X$. Using our methods we show that the weight vector of the solution for the sub problem is an unbiased estimator of the optimal solution for the whole problem based on all column labels. We believe that these new formulas establish a fundamental connection between linear least squares and volume sampling. We use our methods to obtain an algorithm for volume sampling that is faster than state-of-the-art and for obtaining bounds for the total loss of the estimated least-squares solution on all labeled columns.

Exact expressions for double descent and implicit regularization via surrogate random design

arXiv.org Machine Learning

Double descent refers to the phase transition that is exhibited by the generalization error of unregularized learning models when varying the ratio between the number of parameters and the number of training samples. The recent success of highly over-parameterized machine learning models such as deep neural networks has motivated a theoretical analysis of the double descent phenomenon in classical models such as linear regression which can also generalize well in the over-parameterized regime. We build on recent advances in Randomized Numerical Linear Algebra (RandNLA) to provide the first exact non-asymptotic expressions for double descent of the minimum norm linear estimator. Our approach involves constructing what we call a surrogate random design to replace the standard i.i.d. design of the training sample. This surrogate design admits exact expressions for the mean squared error of the estimator while preserving the key properties of the standard design. We also establish an exact implicit regularization result for over-parameterized training samples. In particular, we show that, for the surrogate design, the implicit bias of the unregularized minimum norm estimator precisely corresponds to solving a ridge-regularized least squares problem on the population distribution.

Linear Algebra Formulas for Econometrics


Econometrics is fundamental to many of the problems that data scientists care about, and it requires many skills. There's philosophical skill, for thinking about whether fixed effects or random effects models are more appropriate, for example, or what the direction of causality in a particular problem is. There's some coding, including knowing the right commands to interact with statistical programs like Stata or R, and how to interpret their output. There's the intuition to know which policy issues are worth researching, the political skill to obtain data or grant money, even the writing skill to communicate ideas. And "beneath" it all there is linear algebra: matrix formulas for the estimators that are reported, interpreted, and acted on.