Lipschitzness, which we make precise in Section 2 and in Assumption 3.1.

As the set of solutions to Eq. (3.4) is a line parallel to the subspace A.2 Proof of Lemma 2 For every θ E, we have Φ θ null= e . The auxiliary algorithm (A.1) can be rewritten in the following vector form Θ Bellman operator H is indifferent, i.e., H ( Q + x) H (Q) E, x E So it is impossible to apply the finite time analysis in the literature to establish the convergence of the iterates to some fix point. Then the following properties hold. Lemma 4.a) implies that (c So the Lemma 4.b) implies c Proposition 2. If M is L-smooth with respect to null null Now let's analyze the iterates generated by the following stochastic approximation scheme for solving We make the following assumptions regarding the function H and its stochastic sample ˆ H . Assumption 4. 1. H A and B . 3. There exist a fixed equivalent class, i.e., x Now we study the last term. Now let's focus on the last term in Notice that the monotonicity of infimal convolution (Lemma 4.a) and Lemma 4.b)) implies By update rule (B.5), we have E[ null null x Let's consider the decreasing stepsize first.

Bregman divergence properties that have been pervasive in machine learning.

Linear regression used to be a mainstay of statistics, and remains one of our most important tools for data analysis [Hastie et al., 2009].

Feature hashing, also known as the hashing trick, introduced by Weinberger et al. (2009), is one of the key techniques used in scaling-up machine learning algorithms.

Feature hashing, also known as the hashing trick, introduced by Weinberger et al. (2009), is one of the key techniques used in scaling-up machine learning algorithms.

We prove that the improvement in the relative error of the solution can be tightly characterized in terms of the spectrum of the data matrix, and provide probabilistic upper-bounds. We then illustrate the consequences of our theory with data matrices of different spectral decay.

Feature hashing and other random projection schemes are commonly used to reduce the dimensionality of feature vectors.