Conditional mean embeddings as regressors

By Steffen Grünewälder, Guy Lever, Arthur Gretton, Luca Baldassarre, Sam Patterson, Massimiliano Pontil. 2012

165 citations recorded by the source corpus.

We demonstrate an equivalence between reproducing kernel Hilbert space (RKHS) embeddings of conditional distributions and vector-valued regressors. This connection introduces a natural regularized loss function which the RKHS embeddings minimise, providing an intuitive understanding of the embeddings and a justification for their use. Furthermore, the equivalence allows the application of vector-valued regression methods and results to the problem of learning conditional distributions. Using this link we derive a sparse version of the embedding by considering alternative formulations. Further, by applying convergence results for vector-valued regression to the embedding problem we derive minimax convergence rates which are O(\log(n)/n) -- compared to current state of the art rates of O(n^{-1/4}) -- and are valid under milder and more intuitive assumptions. These minimax upper rates coincide with lower rates up to a logarithmic factor, showing that the embedding method achieves nearly optimal rates. We study our sparse embedding algorithm in a reinforcement learning task where the algorithm shows significant improvement in sparsity over an incomplete Cholesky decomposition.

Subfields: Statistics and Probability, Representation & Self-Supervised Learning

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Statistics and Probability

In Representation & Self-Supervised Learning

Citations in the corpus

Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, methodology): “Nevertheless, Grünewälder et al. (2012) gives a more rigorous account of this perspective.”
K2-ABC: Approximate Bayesian Computation with Infinite Dimensional Summary Statistics via Kernel Embeddings (2015, methodology): “The problem is equivalent to learning a regression function in the RKHSs of s(y) and θ induced by their respective kernels (Grünewälder et al., 2012).”
Hilbert Space Embeddings of Predictive State Representations (2013, methodology): “tional embedding operator is W YjX def= C YXC 1 XX (9) and for all g2Gwe have E[g(Y) jx] = hg;W YjX˚X(x)i G Given Ti.i.d. pairs (x t;y t) from P[X;Y], we can estimate W YjX by kernel ridge regression [18, 19]: Wc YjX= (1=T) Y (1=T) X y where the regularized pseudoinverse y is given by 1y = ( +I) ”
Kernel Mean Estimation and Stein's Effect (2013, positive): “Furthermore, the F-KMSE is of a similar form as the conditional mean embedding used in [23], which can be viewed more generally as a regression problem in RKHS with smooth operators [24].”
Learning from Conditional Distributions via Dual Embeddings (2016, methodology): “as (Silver et al., 2014). See appendix F for detailed settings. Results are averaged over 10 independent trials. Navigation. The navigation in an unbounded room task extending the discretized MDP in (Grunewalder et al., 2012b) to continuous-state continuous-action MDP. Specically, the reward is R(s)”
Non-parametric Policy Search with Limited Information Loss (2017, methodology): “On the one hand, we consider the non-parametric value-function based methods introduced by Grünewälder et al. (2012b) and Pazis and Parr (2011).”
Kernel Bayes' rule: Bayesian inference with positive definite kernels (2013, methodology): “An alternative way of obtaining the regularized conditional mean embedding is as the solution to a vector-valued ridge regression problem, as proposed by Grünewälder et al. (2012) and Grünewälder et al. (2013).”
Kernel Embeddings of Conditional Distributions: A Unified Kernel Framework for Nonparametric Inference in Graphical Models (2013, background): “The conditional mean embedding operator can alternatively be found as the solution to a function-valued least squares regression problem [24]: ĈY |X = arg min C:F7→F ∑m i=1 ‖φ(yi)− Cφ(xi)‖ 2 F + λ ‖C‖2HS , where ‖C‖HS denotes the Hilbert-Schmidt norm (generalized Frobenius norm) of operator C (See F”
A Generalized Kernel Approach to Structured Output Learning (2012, background): “Also, it has been shown recently that these spaces are appropriate for learning conditional mean embeddings (Grunewalder et al., 2012).”
The Nonparametric Kernel Bayes Smoother (2016, methodology): “The nKSR algorithm is obtained by a function-valued kernel ridge regression [13].”
Learning of Non-Parametric Control Policies with High-Dimensional State Features (2015, methodology): “In this approach by Grünewälder et al. [2012b], the value function is assumed to be an element of the chosen RKHS.”
Smooth Operators (2013, methodology): “, 2010; Grünewälder et al., 2012b; Nishiyama et al., 2012). We like to think that the main text of this paper is readable with a basic knowledge of functional analysis and scalar valued RKHS theory. Obviously, we also use techniques from the vector-valued RKHS literature, however this is kept to a m”
Monte Carlo Filtering Using Kernel Embedding of Distributions (2014, methodology): “In such cases, algorithms such as the conditional kernel embedding (Song et al. 2009; Grünewälder et al. 2012a) and Kernel Bayes’ rule (Fukumizu, Song, and Gretton 2011; 2014), which we will use in the proposed method, can be used for computing the weights wi in (2).”
Learning from Conditional Distributions via Dual Kernel Embeddings (2016, methodology): “as (Silver et al., 2014). See appendix F for detailed settings. Results are averaged over 10 independent trials. Navigation. The navigation in an unbounded room task extending the discretized MDP in (Grunewalder et al., 2012b) to continuous-state continuous-action MDP. Specically, the reward is R(s)”
Causal Inference via Kernel Deviance Measures (2018, methodology): “In particular, due to the equivalence of conditional mean embeddings and vector-valued regressors [12], this is analogous to the requirement for constraining the assumed functional class in structural equation models in order to ensure identifiability [25].”