Empirically, each successive iteration produces diminishing additional improvement. c)In the normalized space of 𝑦, the corresponding densities 𝑝(𝑦2|𝑦1) are identical. d) The marginal𝑝(𝑦2) is identical to the conditional, soI (𝑦1;𝑦2) =0. So in this case a conditional affine transformation removed the dependencies. a) An autoregressive flow preprocesses a data set x1:𝑇 to produce a new set y1:𝑇 with reduced temporal dependencies. Train and test negative log-likelihood histograms for (a)SLVM and (b)SLVM + 1-AF on KTH actions. c) The generalization gap for SLVM + 1-AF remains small for varying amounts of KTH training data, while it becomes worse in the low data regime for SLVM.
Cybernetics
At the lowest level, this thesis presents two core techniques for improving probabilistic modeling, applied to both state estimation, or perception, and action selection, or control. At a higher level, this thesis strengthens a bridge between neuroscience and machine learning through the theory of predictive coding.
Neuroscience and Machine Learning, Convergence and Divergence . 3
Beheer
The Future of Feedback & Feedforward
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Background
Introduction
Probabilistic Models
From this perspective, hierarchical latent variable models are an iterative application of the latent variable technique to generate increasingly complex empirical antecedents (Efron and Morris, 1973). We can also consider the inclusion of latent variables in sequential (autoregressive) probability models, which results in the creation of sequential models of latent variables.
Variational Inference
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Discussion
Frembringer wake-sleep-algoritmen gode tæthedsestimatorer?" I:Advances in neural information processing systems, s. Estimation of non-normalized statistic models by score matching. I: Journal of Machine Learning Research6.4.
Introduction
Although amortization has dramatically improved the efficiency of variational inference, the direct inference models (or “encoders”) typically used raise their own issues (see Section 3.2). In this way, inference again becomes an iterative optimization algorithm, using negative feedback to minimize errors or gradient sizes, but now with the efficiency of amortization.
Issues with Direct Inference Models
Iterative Amortized Inference
Feedback
Iterative Inference in Latent Gaussian Models
Iterative inference models based on approximate posterior gradients naturally account for both bottom-up and top-down influences (Figure 3.4). Since iterative inference models already learn to perform approximate posterior updates, it is natural to ask whether errors provide a sufficient signal to optimize inference more quickly.
Experiments
ELBO for direct and iterative inference models on binarized MNIST for (a) additional inference iterations during training and (b) additional samples. To verify this, we train direct and iterative inference models on binarized MNIST using 1,5,10, and 20 estimated posterior samples.
Discussion
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Amortized Variational Filtering
Introduction
In this chapter, we extend damped iterative inference to perform filtering on deep sequential latent variable models, providing a general-purpose algorithm for performing approximate Bayesian filtering. We then describe an example of this algorithm, implementing inference optimization at each step with iterative damped inference.
Background
To scale this algorithm, stochastic gradients can be used for inference (Ranganath, Gerrish, & Blei, 2014) and learning (Hoffman et al., 2013). Finally, several recent works have used particle filtering in conjunction with damped inference to provide a tighter log-likelihood bound for sequential models (Maddison et al., 2017; Naesseth et al., 2018; Le et al., 2018).
Variational Filtering
Sampling from the approximate posterior (blue) produces a conditional probability (green), 𝑝𝜃(x𝑡|x<𝑡,z≤𝑡), which is evaluated at the observation (grey),x𝑡, to calculate the prediction error. However, the filtering setting dictates that the optimization of the approximate posterior can be conditioned only on the past and present variables at each step, i.e. steps from 1 to 𝑡.
Experiments
4.11, where the estimated posterior parameters and their gradients are encoded at each inference iteration at each step. This aspect differs from many baseline filtering methods, which directly produce the estimated posterior result at each step.
Discussion
Appendix: Filtering ELBO Derivation
Model
Improving Sequential Latent Variable Models with Autoregressive
Introduction
This chapter explores an additional technique that integrates predictions directly at the data level, via conditional probability, thus bypassing inference entirely. Feedforward processing can be thought of as learning a frame of reference to help model the data by preprocessing the input at each time step.
Method
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Experiments
Visualization In Figure 5.3, we visualize the preprocessing procedure for SLVM + 1-AF in BAIR Robot Pushing. In Figure 5.4 and the Appendix, we present visualizations on complete sequences, where we see that different models remove different scales of temporal structure.
Discussion
Appendix: ELBO Derivation
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- Introduction
- Background
- Iterative Amortized Policy Optimization
- Experiments
- Discussion
Sequential Autoregressive Flow-Based Policies
Introduction
In this chapter, we complete the table outlined in Chapter 1 by applying learned feedback processing to control. In formulating learned supply control policies, we again consider sequential autoregressive flows, as introduced in Chapter 5.
Autoregressive Flow-Based Policies
The affine transformation (above) acts as a feedback policy, purely conditioned on previous actions, while the basis distribution (below) acts as a feedback policy. Conversely, given a sequence,a1:𝑇, the affine transformation provides a mechanism for removing temporal dependencies, i.e. I (u1:𝑇) ≤ I (a1:𝑇), as seen in Chapter 5, thereby simplifying estimation in the space of u1. :𝑇.
Experiments
In Figure 7.3, we present the performance results in these latter environments with different autoregressive window sizes. On the left side of Figure 7.5, we plot the base distribution,𝜋(u|s), as well as the affine autoregressive transform,δ𝜃±β𝜃, and the operations (before applying the tanh transform).
Discussion
Comparing the left and right columns, we see that the feedforward policies are generally more consistent, which is due to the limited temporal window of the affine flow. The autoregressive nature of the policy thus provides an inductive bias, resulting in policies with a more regular rhythmic structure.
Introduction
Predictive Coding
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Connections
As we have seen, a simplified version of sequential autoregressive flows, which uses the prior variable as an affine shift, extracts an approximation of the time derivatives (Section 5.2). Thus, given a sufficiently short time step, Δ𝑡 =𝑡1−𝑡0, sequential autoregressive flows provide a technique for automatically learning the generalized coordinate approximation.
Correspondences
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Discussion
What does the free energy principle tell us about the brain?" In:arXiv preprint arXiv:1901.07945. Can the "whitening" theory explain the central surround properties of retinal ganglion cell receptive fields? In: Vision Research 46.18, p.
Conclusion
Iterative Estimation
Combining Generative Perception & Control
Controlling Perception
Input
Bridging Neuroscience and Machine Learning
This shows that a limited set of computational operations, suitably combined, is able to adapt to the specific tasks of perception and control of the environment. Similarly, default neural circuit architectures, particularly in the cortex, are capable of adapting to the specific perceptual and control tasks an organism encounters during its lifetime.
