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Implicit Maximum Likelihood Estimation

Ke Li, Jitendra Malik

arXiv:1809.09087v2cs.LGcs.NEstat.ML

TL;DR

Implicit models can make likelihood intractable to evaluate. The paper proposes a likelihood-free estimator that minimizes distances from data examples to model samples and is equivalent to maximum likelihood under some conditions. It reports encouraging results on MNIST, TFD, and CIFAR-10.

  • Problem

    For implicit models, likelihood terms require an intractable integral, making direct maximum-likelihood estimation difficult.

    Source: §1 · p. 2, §1.1 · p. 2

  • Method

    The method draws model samples and adjusts parameters to bring a sample near each data example, yielding a likelihood-free estimator equivalent to maximum likelihood under some conditions.

    Source: §1.2 · p. 3, §2.1 · p. 4

  • Results

    The method captures full data diversity and achieves encouraging results on MNIST, TFD, and CIFAR-10 when combined with vanilla architectures.

    Source: §7 · p. 12

  • Takeaways & Limitations

    Disallowing mode dropping can make sample-quality improvements indicate an upward shift in the precision-recall curve and density-estimation performance.

    Source: §3 · p. 6

    With finite data, any method using an infinite-capacity model will memorize the examples and overfit.

    Source: §6.2 · p. 11

Abstract

from arXiv · show

Implicit probabilistic models are models defined naturally in terms of a sampling procedure and often induces a likelihood function that cannot be expressed explicitly. We develop a simple method for estimating parameters in implicit models that does not require knowledge of the form of the likelihood function or any derived quantities, but can be shown to be equivalent to maximizing likelihood under some conditions. Our result holds in the non-asymptotic parametric setting, where both the capacity of the model and the number of data examples are finite. We also demonstrate encouraging experimental results.

1 Introduction

Implicit models are naturally specified by sampling, but their likelihoods can be intractable to evaluate. The paper proposes a likelihood-free estimator that is equivalent to maximum likelihood under some conditions, including finite model capacity and finite data.

1 Introduction

  • Implicit models are specified through sampling, while evaluating their likelihood can require an intractable integral and be numerically challenging.Prescribed models instead specify density explicitly, though their partition functions can also be intractable.Source: §1 · pp. 1–2

1.1 Challenges in Parameter Estimation

  • Direct likelihood estimation is computationally difficult, and likelihood-free alternatives such as GANs can face mode dropping, vanishing gradients, and training instability.The paper relates these challenges to finite capacity, finite data, and optimization assumptions.Source: §1.1 · pp. 2–3

1.2 Our Contribution

  • The proposed estimator is likelihood-free and equivalent to maximum likelihood under some conditions, including finite model capacity and a finite number of data examples.It encourages each data example to lie close to a model sample, though some data examples may remain unchosen.Source: §1.2 · p. 3
  • The method could sidestep mode collapse, vanishing gradients, and training instability, and recent nearest-neighbor algorithms are described as enabling scale to large, high-dimensional datasets.The paper’s remaining sections define the estimator, analyze its likelihood equivalence, report experiments, and discuss concerns.Source: §1.2 · p. 3, §1.3 · pp. 3–4

2 Implicit Maximum Likelihood Estimator

IMLE turns likelihood estimation into minimizing each data example’s distance to its nearest model sample. It draws fresh model samples, assigns nearby samples to data examples, and optimizes the resulting sample-based objective.

2.1 Intuition

  • A model assigning high density to each data example should produce samples near those examples, motivating minimization of each example’s nearest-sample distance.The distance is framed as the radius of a neighborhood around a data example that contains a sample.Source: §2.1 · p. 4

2.2 Definition

  • IMLE defines a random distance from each data example to its nearest among m i.i.d. model samples, with m ≥ n, and minimizes these distances.The samples are drawn from the parameterized model distribution, and the estimator is defined from the resulting sample-based objective.Source: §2.2 · p. 4

2.3 Algorithm

  • Each outer iteration draws model samples, selects a data batch, assigns each example its nearest sample, then uses an iterative optimizer to minimize the sample-based objective.The method focuses on Euclidean distance in the natural data representation, while noting extensions to suitable embedded or other metrics.Source: §2.3 · pp. 4–5

3 Why Maximum Likelihood

The paper argues that maximum likelihood’s support behavior matters when learning from finite empirical data. Preserving all data modes connects sample quality to density-estimation performance, whereas allowing mode dropping can make that connection unreliable.

3 Why Maximum Likelihood

  • For finite empirical data, maximum likelihood requires model support to contain every data example, while reverse KL requires model support to stay within the observed examples.The paper says reverse KL therefore forces zero density on unseen examples and effectively prohibits generalization.Source: §3 · p. 5
  • Maximum likelihood preserves all modes, allowing high-capacity models to use all training data; objectives permitting mode dropping leave the choice of omitted examples uncontrolled.The precision-recall framing distinguishes plausible samples from coverage of all data modes.Source: §3 · pp. 5–6
  • When mode dropping is disallowed, improved sample quality implies an upward shift in the precision-recall curve; when it is allowed, sample quality may mislead about density-estimation performance.With mode dropping, higher precision may come at the expense of lower recall.Source: §3 · p. 6

4 Analysis

The analysis connects maximizing likelihood to minimizing expected distance between observations and generated samples, and formalizes when this preserves the maximum-likelihood optimizer. A practical objective drops unknown coefficients and upper-bounds the ideal objective.

4 Analysis

  • For one observation, maximizing likelihood corresponds to maximizing the CDF of sample distance’s area, or equivalently minimizing expected Euclidean distance.The CDF tracks the probability mass inside a ball centered on the observation; its complementary area is the expected distance.Source: §4 · p. 6
  • Under some conditions, minimizing expected distance between each observation and a generated sample preserves the maximum-likelihood optimizer.The theorem considers a family with a unique maximum-likelihood solution and m ≥ 1 i.i.d. samples from the model.Source: §4 · p. 6
  • The stated conditions include smoothness and density-shape requirements; examples suggest they hold for many analytic distributions, location or scale families, and richly parameterized models.The text notes that the final condition is usually satisfied when the parameter dimension d is large.Source: §4 · pp. 6–7
  • Because the ideal objective’s coefficients are typically unknown, the proposed objective omits them and minimizes an upper bound whose tightness depends on likelihood variation at the optimum.The bound is usually reasonably tight when a model with reasonable capacity does not assign very different likelihoods to individual observations at the maximum-likelihood estimate.Source: §4 · p. 7

5 Experiments

Experiments across MNIST, TFD, and CIFAR-10 report reasonably good samples, stable improvement during training, and smooth latent interpolations. Nearest-neighbor comparisons suggest the models did not overfit, while the authors qualify sample-quality comparisons by the full-recall setting.

Experimental setup

  • Experiments used Gaussian-noise feedforward models on MNIST, TFD, and CIFAR-10, including a 20-layer convolutional architecture for CIFAR-10.MNIST and TFD used fully connected networks with two hidden layers; both used 100-dimensional noise, while CIFAR-10 used 1000-dimensional noise.Source: §5 · pp. 7–8

Results

  • The method produced reasonably good samples across MNIST, TFD, and CIFAR-10; high estimated log-likelihood on MNIST and TFD suggests no significant mode dropping.The table reports test-data log-likelihood under Gaussian Parzen-window densities estimated from generated samples; no cell values are supplied here.Source: Table 1 · p. 9, §5 · p. 9

Evaluation

  • The evaluation used randomly chosen samples and estimated log-likelihood; neither metric alone establishes density quality, though reasonable performance on both offers some comfort about precision and recall.The authors note that cherry-picked samples can show maximum rather than mean sample quality.Source: §5 · pp. 8–9

Results

  • Latent-space interpolations appeared plausible and smooth, including on CIFAR-10, suggesting the model support did not collapse to isolated points.The authors say this indicates that the method learns data-manifold geometry without explicitly learning a distance metric.Source: §5 · p. 9, Figure 3 · p. 10
  • Sample quality consistently improved as training progressed, with samples becoming sharper over time.The authors interpret this pattern as evidence of training stability.Source: §5 · p. 9

Caveats and future work

  • The sample-quality results may not be state-of-the-art, but they were obtained under full recall, so they do not establish worse modeling accuracy than methods that may drop modes.The authors leave improving the precision-recall curve through model refinement for future work.Source: §5 · pp. 9–10

Results

  • Generated samples differed from their nearest training-set neighbors, suggesting little overfitting, though the models may be somewhat underfitted.The authors suggest that increasing model capacity may improve sample quality.Source: §5 · p. 9, Figure 4 · p. 10

Caveats and future work

  • The authors present this initial study as a foundation for future work on refining architectures with task-specific insights.They report more plausible CIFAR-10 samples than initial GAN and PixelRNN versions despite their method's minimal sophistication.Source: §5 · p. 10

6 Discussion

The discussion examines whether likelihood training harms sample quality, whether nearest-sample objectives overfit, and how disjoint supports affect maximum-likelihood gradients. It argues that finite capacity and a different loss can qualify these concerns.

6.1 Does Maximizing Likelihood Necessarily Lead to Poor Sample Quality?

  • The proposed sample-quality trade-off assumes fixed capacity; increasing capacity alongside maximum-likelihood training may achieve precision and recall together.The discussion says avoiding mode dropping becomes more important as capacity increases because more parameters must be fit with the available training data.Source: §6.1 · p. 11

6.2 Would Minimizing Distance to the Nearest Samples Cause Overfitting?

  • Infinite-capacity models trained on finitely many examples memorize them, whereas finite-capacity objectives need not have zero minima or necessarily overfit.The passage applies the memorization argument to nearest-sample distance, likelihood, and divergence minimization.Source: §6.2 · p. 11

6.3 Does Disjoint Support Break Maximum Likelihood?

  • With disjoint data and model supports, the proposed loss has distance-scaled gradients, unlike log-likelihood, and may work better when likelihood gradients are tractable.The loss and log-likelihood have the same optima under some conditions.Source: §6.3 · pp. 11–12

7 Conclusion

The conclusion presents nearest-sample parameter updates as a likelihood-free method equivalent to maximum likelihood under some conditions. It reports encouraging results on MNIST, TFD, and CIFAR-10.

7 Conclusion

  • Nearest-sample updates provide likelihood-free parameter estimation equivalent to maximum likelihood under some conditions, with encouraging results on MNIST, TFD, and CIFAR-10.The procedure draws model samples, finds the nearest sample to each data example, and adjusts model parameters to bring samples closer.Source: §7 · p. 12
  • The method is presented as capturing data diversity while avoiding mode collapse, vanishing gradients, and training instability.Source: §7 · p. 12

8 Appendix

The appendix develops lemmas about nearest-sample distances and transformed objectives, then applies them to a theorem concerning a unique maximum-likelihood solution under specified conditions.

Appendix

  • Under the lemma's conditions, minimizing the weighted sum of transformed per-example objectives has a unique minimizer.Source: §8 · p. 15
  • For a density continuous at x0, the right derivative at zero of the scaled nearest-distance CDF equals the density at x0.Source: §8 · p. 16
  • The appendix establishes that Ψ(z) decreases as z increases.Source: §8 · p. 19
  • Theorem 1 considers observations and a model family with a unique maximum-likelihood solution, under stated conditions on the family and nearest-sample objective.The proof applies a lemma using a strictly increasing transform of per-example negative log likelihood.Source: §8 · pp. 19, 21

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