The minimum description length (MDL) principle is extended to supervised learning. The MDL principle is a philosophy that the shortest description of given data leads to the best hypothesis about the data source. One of the key theories for the MDL principle is Barron and Cover's theory (BC theory), which mathematically justifies the MDL principle based on two-stage codes in density estimation (unsupervised learning). Though the codelength of two-stage codes looks similar to the target function of penalized likelihood methods, parameter optimization of penalized likelihood methods is done without quantization of parameter space. Recently, Chatterjee and Barron have provided theoretical tools to extend BC theory to penalized likelihood methods by overcoming this difference. Indeed, applying their tools, they showed that the famous penalized likelihood method `lasso' can be interpreted as an MDL estimator and enjoys performance guarantee by BC theory. An important fact is that their results assume a fixed design setting, which is essentially the same as unsupervised learning. The fixed design is natural if we use lasso for compressed sensing. If we use lasso for supervised learning, however, the fixed design is considerably unsatisfactory. Only random design is acceptable. However, it is inherently difficult to extend BC theory to the random design regardless of whether the parameter space is quantized or not. In this paper, a novel theoretical tool for extending BC theory to supervised learning (the random design setting and no quantization of parameter space) is provided. Applying this tool, when the covariates are subject to a Gaussian distribution, it is proved that lasso in the random design setting can also be interpreted as an MDL estimator, and that lasso enjoys the risk bound of BC theory. The risk/regret bounds obtained have several advantages inherited from BC theory. First, the bounds require remarkably few assumptions. Second, the bounds hold for any finite sample size $n$ and any finite feature number $p$ even if $n\ll p$ . Behavior of the regret bound is investigated by numerical simulations. We believe that this is the first extensions of BC theory to supervised learning (random design).

Cardinality of typical sets and the smallest high probability set for discrete memoryless sources and stationary Markov sources are considered. Usually, its width is fixed in the definition of both sets, but sometimes it is assumed that the width converges to zero as the length of sequence n goes to infinity. In such setting, some condition for the width is necessary to make the cardinality near 2nH, where H denotes the entropy rate of the information source. In this article, we give sufficient conditions for the above propositions. We also show sufficient conditions for that they don’t hold for DMS and Markov sources under a certain restriction.

We study Barron and Cover’s theory (BC theory) in supervised learning. The original BC theory can be applied to supervised learning only approximately and limitedly. Though Barron (2008) and Chatterjee and Barron (2014) succeeded in removing the approximation, their idea cannot be essentially applied to supervised learning in general. By solving this issue, we propose an extension of BC theory to supervised learning. The extended theory has several advantages inherited from the original BC theory. First, it holds for finite sample number n. Second, it requires remarkably few assumptions. Third, it gives a justification of the MDL principle in supervised learning. We also derive new risk and regret bounds of lasso with random design as its application. The derived risk bound hold for any finite n without boundedness of features in contrast to past work. Behavior of the regret bound is investigated by numerical simulations. We believe that this is the first extension of BC theory to general supervised learning without approximation.

For the additive white Gaussian noise channel with average power constraint, sparse superposition codes with least squares decoding are proposed by Barron and Joseph in 2010. The codewords are designed by using a dictionary each entry of which is drawn from a Gaussian distribution. The error probability is shown to be exponentially small for all rates up to the capacity. This paper proves that when each entry of the dictionary is drawn from a Bernoulli distribution, the error probability is also exponentially small for all rates up to the capacity. The proof is via a central limit theorem-type inequality, which we show for this analysis.

For the additive white Gaussian noise channel with average power constraint, sparse superposition codes with least squares decoding were proposed by Barron and Joseph in 2010. The codewords are designed by using a dictionary which is drawn from a Gaussian distribution. The error probability is shown to be exponentially small in code length for all rates up to the capacity. This paper proves that when the dictionary is drawn from a Bernoulli distribution, the error probability is also exponentially small for all rates up to the capacity.

This paper proposes a modification of model selection criterion Direct Eigenvalue Estimator (DEE) for small sample regression proposed by Chapelle et al. (2002). The derivation of DEE requires neither an asymptotic assumption nor an assumption that the variance of noise is known in advance. Chapelle et al. (2002) reported that their model selection procedure performed well even when the data number was small but a little worse than ADJ (Schuurmans, 1997) on experiments. We point out, however, the derivation of DEE includes two mistakes. We further propose a slight modification to correct those mistakes. Our experiments verify that the modified DEE gives more stable model selection than the original DEE.

We propose a local boosting method in classification problems borrowing from an idea of the local likelihood method. Our proposal, local boosting, includes a simple device for localization for computational feasibility. We proved the Bayes risk consistency of the local boosting in the framework of Probably approximately correct learning. Inspection of the proof provides a useful viewpoint for comparing ordinary boosting and local boosting with respect to the estimation error and the approximation error. Both boosting methods have the Bayes risk consistency if their approximation errors decrease to zero. Compared to ordinary boosting, local boosting may perform better by controlling the trade-off between the estimation error and the approximation error. Ordinary boosting with complicated base classifiers or other strong classification methods, including kernel machines, may have classification performance comparable to local boosting with simple base classifiers, for example, decision stumps. Local boosting, however, has an advantage with respect to interpretability. Local boosting with simple base classifiers offers a simple way to specify which features are informative and how their values contribute to a classification rule even though locally. Several numerical studies on real data sets confirm these advantages of local boosting.