Local Minimax Learning of Approximately Polynomial Functions

Suppose we have a number of noisy measurements of an unknown real-valued function $f$ near point of interest $mathbf{x}_0 in mathbb{R}^d$. Suppose also that nothing can be assumed about the noise distribution, except for zero mean and bounded covariance matrix. We want to estimate $f$ at $mathbf{x=x}_0$ using a general linear parametric family $f(mathbf{x};mathbf{a}) = a_0 h_0 (mathbf{x}) ++ a_q h_q (mathbf{x})$, where $mathbf{a} in mathbb{R}^q$ and $h_i$'s are bounded functions on a neighborhood $B$ of $mathbf{x}_0$ which contains all points of measurement. Typically, $B$ is a Euclidean ball or cube in $mathbb{R}^d$ (more generally, a ball in an $l_p$-norm). In the case when the $h_i$'s are polynomial functions in $x_1,ldots,x_d$ the model is called locally-polynomial. In particular, if the $h_i$'s form a basis of the linear space of polynomials of degree at most two, the model is called locally-quadratic (if the degree is at most three, the model is locally-cubic, etc.). Often, there is information, which is called context, about the function $f$ (restricted to $B$ ) available, such as that it takes values in a known interval, or that it satisfies a Lipschitz condition. The theory of local minimax estimation with context for locally-polynomial models and approximately locally polynomial models has been recently initiated by Jones. In the case of local linearity and a bound on the change of $f$ on $B$, where $B$ is a ball, the solution for squared error loss is in the form of ridge regression, where the ridge parameter is identified; hence, minimax justification for ridge regression is given together with explicit best error bounds. The analysis of polynomial models of degree above 1 leads to interesting and difficult questions in real algebraic geometry and non-linear optimization. We show that in the case when $f$ is a probability function, the optimal (in the minimax sense) estimator is effectively computable (with any given precision), thanks to Tarski's elimination principle.