Source-linked AI summary
Conditional Neural Processes
Marta Garnelo, Dan Rosenbaum, Chris J. Maddison, Tiago Ramalho, David Saxton, Murray Shanahan, Yee Whye Teh, Danilo J. Rezende, S. M. Ali Eslami
TL;DR
The paper addresses the tension between data-hungry task-specific neural networks and computationally expensive Bayesian function models. It proposes Conditional Neural Processes, which learn permutation-invariant conditional distributions over functions using neural networks and gradient-based training. CNPs support regression, classification, and image completion while making predictions from few context points and scaling linearly with observations and targets, though the presented implementations have important limitations.
Problem
Deep networks require large datasets and task-specific training, while Bayesian approaches such as Gaussian Processes become computationally intractable as datasets or dimensionality grow.
Method
CNPs use permutation-invariant neural networks to parameterize conditional distributions over functions given observations, with O(n + m) test-time scaling.
Results
CNPs perform regression, classification, and image completion, with image predictions close to ground truth when conditioned on less than 10% of pixels and stronger performance than GPs and kNNs with small context.
Takeaways & Limitations
A trained CNP extracts high-level statistics from a family of functions that can be reused across multiple tasks.
Takeaways & Limitations
The factored mean-and-variance model cannot produce coherent samples over multiple targets, and the presented implementations are simple proofs-of-concept.
Abstract
from arXiv · showhide
Deep neural networks excel at function approximation, yet they are typically trained from scratch for each new function. On the other hand, Bayesian methods, such as Gaussian Processes (GPs), exploit prior knowledge to quickly infer the shape of a new function at test time. Yet GPs are computationally expensive, and it can be hard to design appropriate priors. In this paper we propose a family of neural models, Conditional Neural Processes (CNPs), that combine the benefits of both. CNPs are inspired by the flexibility of stochastic processes such as GPs, but are structured as neural networks and trained via gradient descent. CNPs make accurate predictions after observing only a handful of training data points, yet scale to complex functions and large datasets. We demonstrate the performance and versatility of the approach on a range of canonical machine learning tasks, including regression, classification and image completion.
1. Introduction
The paper frames supervised learning as function approximation and contrasts task-specific deep networks with Bayesian function models. It then introduces CNPs as neural, end-to-end trained models combining neural networks with Gaussian-process-like features.
- Problem formulation: Supervised learning can be formulated as approximating an unknown function from finite observations, with held-out inputs treated as targets.This framing covers classification, regression, dynamics modeling, and image generation.
- Motivation: Deep neural networks often require large datasets and fail when training data is scarce because learning is not shared across tasks.A two-phase approach can instead learn generic domain statistics before fitting a specific task from few points.
- Bayesian alternative: Gaussian Processes encode prior assumptions through a distribution over functions and perform Bayesian inference, but become computationally intractable as dataset size or dimensionality grows.Kernel parameters can express assumptions such as function smoothness.
- Conditional Neural Processes: CNPs combine neural networks with Gaussian-process-like features and learn conditional distributions over functions given observed data.Their observation dependence is parameterized by a permutation-invariant neural network and trained end-to-end.
- Training and scaling: CNP training samples random datasets, observations, and targets, then maximizes the conditional likelihood across settings with few or many observations relative to targets.The proposed architecture scales as O(n + m) at test time, where n and m are the numbers of observations and targets.
2. Model
CNPs model conditional distributions over functions using permutation-invariant neural representations of observations, trading stochastic-process guarantees for flexibility and O(n + m) prediction scaling. They are trained by predicting observations from subsets of observations, with task-specific output distributions.
- Conditional Neural Processes (CNPs): CNPs directly parametrize conditional distributions over functions without requiring consistency with a prior stochastic process.This trades mathematical guarantees associated with stochastic processes for functional flexibility and scalability.
- Conditional Neural Processes (CNPs): The model factors the conditional distribution over target outputs, while non-factored distributions remain possible and are examined experimentally.For regression, outputs parameterize means and variances; for classification, they parameterize class-probability logits.
- Conditional Neural Processes (CNPs): A CNP represents observations with a fixed-dimensional embedding and is permutation invariant in both observations and targets.The embedding is formed with a commutative aggregation operation over encoded observation pairs.
- Training CNPs: CNPs are trained by predicting all observations from a randomly chosen subset and minimizing negative conditional log probability.The targets include both observed and unobserved values, and gradients are estimated by sampling functions and subset sizes.
- Training CNPs: The training procedure learns empirical conditional distributions from data, but does not guarantee a consistent set of conditionals across observation sets.This shifts prior specification from an analytic prior to empirical data.
- Conditional Neural Processes (CNPs): CNP conditional prediction runs in O(n + m) time for m targets given n observations.The architecture also supports streaming observations with minimal overhead because aggregated representations can be updated incrementally.
- Conditional Neural Processes (CNPs): CNP implementations can adapt the encoder to the data type, using multilayer perceptrons for low-dimensional data and convolutions for higher-dimensional inputs with spatial correlations.The described factored setup cannot produce coherent samples.
3. Related research
Related work combines deep learning, Gaussian processes, and meta-learning to address scalability, expressivity, or few-shot prediction. CNPs are positioned among approaches that learn from data while supporting targeted predictions and, in some cases, linear prediction complexity.
- Gaussian-process approaches: Sparse GPs address Gaussian-process scaling issues, while Deep GPs address limited expressivity from functional restrictions.Deep Kernel Learning combines neural networks with GPs but uses the network to learn expressive kernels within a GP.
- Deep learning and meta-learning: Deep learning learns features and prior knowledge directly from data and scales well, but is generally less flexible to input size and order.It also typically approximates one function rather than distributions over functions.
- Deep learning and meta-learning: Meta-learning shares CNPs’ motivation by addressing prediction across tasks, including few-shot estimation and applications such as reinforcement learning.Generative Query Networks predict new 3D-scene viewpoints from context observations, while neural processes extend few-shot prediction beyond scene understanding.
- Few-shot classification: Matching networks compare target samples with observations in feature space and can require O(n × m) all-to-all comparisons.They can be modified to achieve O(n + m) complexity like CNPs.
- Related generative models: The neural statistician and variational homoencoder estimate densities of observed data but do not support targeted sampling at specified input positions.They instead generate independent and identically distributed samples from the estimated density.
4. Experimental Results
Across regression, image completion, and classification tasks, CNPs make predictions from sparse, flexible observations while estimating uncertainty and incorporating domain-specific structure. Their performance improves with more context, and their scalable architecture supports varied inputs and targets.
- 4.1. Function Regression: CNPs regress fixed- and switching-kernel functions from a few context points, with accuracy improving and estimated uncertainty decreasing as context increases.They achieve similarly good performance on the switching-kernel task, which is not trivial for GPs.
- 4.2.1. MNIST: MNIST predictions approach the underlying digit as context increases, while uncertainty shifts from broadly distributed to digit-specific edges.With one non-informative context point, predictions average over the dataset’s digits; complete observations still do not yield pixel-perfect reconstruction because of a representation bottleneck.
- 4.2.1. MNIST: Selecting the highest-variance MNIST pixel for observation produces better prediction results than selecting conditioning pixels at random.This provides a simple uncertainty-guided active-exploration strategy.
- 4.2.2. CELEBA: On CelebA, predictions conditioned on less than 10% of pixels are close to ground truth and increasingly capture face orientation, expression, and edge-focused uncertainty.With few context points, the model averages over possible faces; additional context reveals image-specific details.
- 4.2.2. CELEBA: CNPs accept arbitrary observation patterns and unseen target coordinates, including subpixel queries and different resolutions.The model can use global spatial properties such as symmetry while querying multiple resolutions with one forward pass.
- 4.2.2. CELEBA: With few context points, CNPs outperform kNNs and GPs on CelebA and remain effective when context points are supplied in order.When most of the image is observed, exact methods such as GPs and kNNs perform better; CNP test-time computation scales as O(n + m), rather than O(nm).
5. Discussion
CNPs combine test-time flexibility with prior knowledge extracted from training data, supporting regression, classification, and image completion. The paper presents them as a step toward reusable high-level abstractions, while noting that the implementations are simple proofs of concept.
- CNPs perform regression, classification, and image completion while combining test-time flexibility with prior knowledge extracted from training data.
- The specific CNP implementations are simple proofs of concept that could be substantially extended with more elaborate architectures.
- CNPs encode high-level statistics of a family of functions rather than remaining tied to one constrained statistical context.
- This reusable representation is presented as a step toward learning high-level abstractions for multiple tasks.