[Paper Review] Deep Gamblers: Learning to Abstain with Portfolio Theory

Liu Ziyin, Zhikang Wang+4 more|arXiv (Cornell University)|Jun 29, 2019

Financial Markets and Investment Strategies61 references33 citations

TL;DR

The paper reframes selective classification as a gambling problem, using portfolio theory to learn abstention end-to-end, achieving competitive results on SVHN, CIFAR-10, and Cat-vs-Dog with a rejection mechanism.

ABSTRACT

We deal with the extit{selective classification} problem (supervised-learning problem with a rejection option), where we want to achieve the best performance at a certain level of coverage of the data. We transform the original $m$-class classification problem to $(m+1)$-class where the $(m+1)$-th class represents the model abstaining from making a prediction due to disconfidence. Inspired by portfolio theory, we propose a loss function for the selective classification problem based on the doubling rate of gambling. Minimizing this loss function corresponds naturally to maximizing the return of a extit{horse race}, where a player aims to balance between betting on an outcome (making a prediction) when confident and reserving one's winnings (abstaining) when not confident. This loss function allows us to train neural networks and characterize the disconfidence of prediction in an end-to-end fashion. In comparison with previous methods, our method requires almost no modification to the model inference algorithm or model architecture. Experiments show that our method can identify uncertainty in data points, and achieves strong results on SVHN and CIFAR10 at various coverages of the data.

Motivation & Objective

• Address selective classification with a rejection option to balance accuracy and data coverage.
• Transform m-class classification into m+1 classes by adding an abstain option.
• Introduce a loss based on the doubling rate from portfolio theory that encourages abstention when uncertain.
• Enable end-to-end training with minimal modifications to standard architectures and inference.
• Provide theoretical insights linking classification to gambling and quantify conditions for abstention strategies.

Proposed method

• Recast classification as a horse race with an abstention (m+1) category.
• Define a loss based on the doubling rate W(b,p) = sum_i p_i log2(b_i o_i + b_{m+1}) for mini-batches, with o as a payoff parameter.
• Interpret the (m+1)-th class as the abstention decision, enabling a tunable coverage via threshold h on g(x) = f_w(x)_{m+1}.
• Link portfolio theory to learning by showing that maximizing doubling rate corresponds to maximizing prediction likelihood under abstention constraints.
• Provide theoretical results (Theorems 1-3) to justify the gambling formulation and the effect of the payoff parameter o on optimal strategy.
• Demonstrate the approach with synthetic and real-data experiments, comparing against entropy, Bayes dropout, and SelectiveNet.

Experimental results

Research questions

• RQ1 Can abstention be learned through a gambling-based loss that connects to portfolio theory?
• RQ2 How should the abstention payoff parameter o affect the optimal betting (prediction) strategy and absten­tion behavior?
• RQ3 Is end-to-end training feasible for selective classification without sampling or architectural changes?
• RQ4 How does the proposed method perform relative to state-of-the-art selective classification methods across datasets and coverage levels?

Key findings

• The method achieves competitive, often state-of-the-art performance on SVHN across coverages, including best per-coverage results.
• Strong results on CIFAR-10 and Cat-vs-Dog datasets, especially in the 0.8–1.0 coverage range.
• The approach identifies uncertainty effectively, outperforming entropy-based baselines in identifying hard or out-of-distribution examples.
• The loss is end-to-end trainable with no sampling or retraining for different coverage levels, and requires no architectural changes.
• t-SNE analyses suggest the learned representation has better class separation under the deep gambler loss, indicating a qualitatively different learned feature space.