Representation Learning: A Review and New Perspectives <s> Learning Approximate Inference <s> Prior work has shown that features which appear to be biologically plausible as well as empirically useful can be found by sparse coding with a prior such as a laplacian (L1) that promotes sparsity. We show how smoother priors can preserve the benefits of these sparse priors while adding stability to the Maximum A-Posteriori (MAP) estimate that makes it more useful for prediction problems. Additionally, we show how to calculate the derivative of the MAP estimate efficiently with implicit differentiation. One prior that can be differentiated this way is KL-regularization. We demonstrate its effectiveness on a wide variety of applications, and find that online optimization of the parameters of the KL-regularized model can significantly improve prediction performance. <s> BIB001 </s> Representation Learning: A Review and New Perspectives <s> Learning Approximate Inference <s> In Sparse Coding (SC), input vectors are reconstructed using a sparse linear combination of basis vectors. SC has become a popular method for extracting features from data. For a given input, SC minimizes a quadratic reconstruction error with an L1 penalty term on the code. The process is often too slow for applications such as real-time pattern recognition. We proposed two versions of a very fast algorithm that produces approximate estimates of the sparse code that can be used to compute good visual features, or to initialize exact iterative algorithms. The main idea is to train a non-linear, feed-forward predictor with a specific architecture and a fixed depth to produce the best possible approximation of the sparse code. A version of the method, which can be seen as a trainable version of Li and Osher's coordinate descent method, is shown to produce approximate solutions with 10 times less computation than Li and Os-her's for the same approximation error. Unlike previous proposals for sparse code predictors, the system allows a kind of approximate "explaining away" to take place during inference. The resulting predictor is differentiable and can be included into globally-trained recognition systems. <s> BIB002 </s> Representation Learning: A Review and New Perspectives <s> Learning Approximate Inference <s> Divide-and-conquer is key to building sophisticated learning machines: hard problems are solved by composing a network of modules that solve simpler problems (LeCun et al., 1998; Rohde, 2002; Bradley, 2009). Many such existing systems rely on learning algorithms which are based on simple parametric gradient descent where the parametrization must be predetermined, or more specialized per-application algorithms which are usually ad-hoc and complicated. We present a novel approach for training generic modular networks that uses two existing techniques: the error propagation strategy of backpropagation and more recent research on descent in spaces of functions (Mason et al., 1999; Scholkopf & Smola, 2001). Combining these two methods of optimization gives a simple algorithm for training heterogeneous networks of functional modules using simple gradient propagation mechanics and established learning algorithms. The resulting separation of concerns between learning individual modules and error propagation mechanics eases implementation, enables a larger class of modular learning strategies, and allows per-module control of complexity/regularization. We derive and demonstrate this functional backpropagation and contrast it with traditional gradient descent in parameter space, observing that in our example domain the method is significantly more robust to local optima. <s> BIB003 </s> Representation Learning: A Review and New Perspectives <s> Learning Approximate Inference <s> We present a new approximate inference algorithm for Deep Boltzmann Machines (DBM’s), a generative model with many layers of hidden variables. The algorithm learns a separate “recognition” model that is used to quickly initialize, in a single bottom-up pass, the values of the latent variables in all hidden layers. We show that using such a recognition model, followed by a combined top-down and bottom-up pass, it is possible to efficiently learn a good generative model of high-dimensional highly-structured sensory input. We show that the additional computations required by incorporating a top-down feedback plays a critical role in the performance of a DBM, both as a generative and discriminative model. Moreover, inference is only at most three times slower compared to the approximate inference in a Deep Belief Network (DBN), making large-scale learning of DBM’s practical. Finally, we demonstrate that the DBM’s trained using the proposed approximate inference algorithm perform well compared to DBN’s and SVM’s on the MNIST handwritten digit, OCR English letters, and NORB visual object recognition tasks. <s> BIB004 </s> Representation Learning: A Review and New Perspectives <s> Learning Approximate Inference <s> Graphical models are often used \inappropriately," with approximations in the topology, inference, and prediction. Yet it is still common to train their parameters to approximately maximize training likelihood. We argue that instead, one should seek the parameters that minimize the empirical risk of the entire imperfect system. We show how to locally optimize this risk using back-propagation and stochastic metadescent. Over a range of synthetic-data problems, compared to the usual practice of choosing approximate MAP parameters, our approach signicantly reduces loss on test data, sometimes by an order of magnitude. <s> BIB005
Let us now consider from closer how a representation is computed in probabilistic models with latent variables, when iterative inference is required. There is a computation graph (possibly with random number generation in some of the nodes, in the case of MCMC) that maps inputs to representation and, in the case of deterministic inference (e.g., MAP inference or variational inference), that function could be optimized directly. This is a way to generalize PSD that has been explored in recent work on probabilistic models at the intersection of inference and learning BIB001 , BIB002 , BIB003 , BIB004 , BIB005 ,  , where a central idea is that instead of using a generic inference mechanism, one can use one that is learned and is more efficient, taking advantage of the specifics of the type of data on which it is applied.