EnsLoss: Stochastic Calibrated Loss Ensembles for Preventing Overfitting in Classification

Ben Dai (CUHK)

Background

The objective of binary classification is to categorize each instance into one of two classes.

• Data: \( \mathbf{X} \in \mathbb{R}^d \to Y \in \{-1,+1\} \)
• Classifier: \( f(\mathbf{X}): \mathbb{R}^d \to \mathbb{R} \)
• Predicted label: \( \widehat{Y} = \text{sgn}(f(\mathbf{X})) \)
• Evaluation via Misclassification error (risk):

where \(\mathbf{1}(\cdot)\) is an indicator function.

Aim. To obtain the Bayes classifier or the best classifier:

$$R(\hat{f}_n) \to R(f^*) \quad f^* := \argmin R(f)$$

$$R(f) = 1 - \text{Acc}(f) = \mathbb{E}( \mathbf{1}(Y f(\mathbf{X}) \leq 0) ),$$

Background

Due to the discontinuity of the indicator function:

$$R(f) = 1 - \text{Acc}(f) = \mathbb{E}( \mathbf{1}(Y f(\mathbf{X}) \leq 0) ),$$

the zero-one loss is usually replaced by a convex and classification-calibrated loss \(\phi\) to facilitate the empirical computation (Lin, 2004; Zhang, 2004; Bartlett et al., 2006):

$$ R_{\phi}(f) = \mathbb{E}( \phi(Y f(\mathbf{X})) ) $$

For example, the hinge loss for SVM, exponential loss for AdaBoost, and logistic loss for logistic regression all follow this framework.

Background

Due to the discontinuity of the indicator function:

$$R(f) = 1 - \text{Acc}(f) = \mathbb{E}( \mathbf{1}(Y f(\mathbf{X}) \leq 0) ),$$

(If we optimize with respect to \(\phi\), will the resulting solution still be the function \(f^*\) that we need?)

That's why we need the loss \(\phi\) to be calibrated?

the zero-one loss is usually replaced by a convex and classification-calibrated loss \(\phi\) to facilitate the empirical computation (Lin, 2004; Zhang, 2004; Bartlett et al., 2006):

$$ R_{\phi}(f) = \mathbb{E}( \phi(Y f(\mathbf{X})) ) $$

For example, the hinge loss for SVM, exponential loss for AdaBoost, and logistic loss for logistic regression all follow this framework.

Background

Definition 1 (Bartlett et al. (2006)). A loss function \(\phi(\cdot)\) is classification-calibrated, if for every sequence of measurable function \(f_n\) and every probability distribution on \( \mathcal{X} \times \{\pm 1\}\),

$$R_{\phi}(f_n) \to \inf_{f} R_{\phi}(f) \ \text{ implies that } \ R(f_n) \to \inf_{f} R(f).$$

A calibrated loss function \(\phi\) guarantees that any sequence \(f_n\) that optimizes \(R_\phi\) will eventually also optimize \(R\), thereby ensuring consistency in maximizing classification accuracy.

Background

Definition 1 (Bartlett et al. (2006)). A loss function \(\phi(\cdot)\) is classification-calibrated, if for every sequence of measurable function \(f_n\) and every probability distribution on \( \mathcal{X} \times \{\pm 1\}\),

$$R_{\phi}(f_n) \to \inf_{f} R_{\phi}(f) \ \text{ implies that } \ R(f_n) \to \inf_{f} R(f).$$

Theorem 1 (Bartlett et al. (2006)) Let \(\phi\) be convex. Then \(\phi\) is classification-calibrated iff it is differentiable at 0 and \(\phi'(0) < 0\).

A series of studies (Lin, 2004; Zhang, 2004; Bartlett et al., 2006) culminates in the following theorem for iff conditions of calibration:

Classification ERM framework

(i) Select a convex and calibrated (CC) loss function \(\phi\)

Classification ERM framework

$$ \widehat{f}_{n} = \argmin_{f \in \mathcal{F}} \ \widehat{R}_{\phi} (f), \quad \widehat{R}_{\phi} (f) := \frac{1}{n} \sum_{i=1}^n \phi \big( y_i f(\mathbf{x}_i) \big).$$

(i) Select a convex and calibrated (CC) loss function \(\phi\)

(ii) Directly minimizes the ERM of \(R_\phi\) to obtain \(f_n\)

(SGD is widely adopted for its scalability and generalization when dealing with large-scale datasets and DL models)

 $$\pmb{\theta}^{(t+1)} = \pmb{\theta}^{(t)} - \gamma \frac{1}{B} \sum_{i \in \mathcal{I}_B} \nabla_{\pmb{\theta}} \phi( y_{i} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}) )$$

$$= \pmb{\theta}^{(t)} - \gamma \frac{1}{B} \sum_{i \in \mathcal{I}_B} y_i \partial \phi( y_{i} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}) ) \nabla_{\pmb{\theta}} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}),$$

The ERM paradigm with calibrated losses, when combined with ML/DL models and optimized using SGD, has achieved tremendous success in numerous real-world applications.

Loss selection in Practice

However, in practical applications, determining which loss function performs better is a very challenging problem, as it is typically unknown and can vary across datasets.

We examine two of the most popular losses: BCE and Hinge loss. We provide experimental results on 45 CIFAR2 datasets, which also confirms that the superiority of the loss function is not consistent across different models / datasets.

Loss selection in Practice

However, in practical applications, determining which loss function performs better is a very challenging problem, as it is typically unknown and can vary across datasets.

Current approaches in theoretical analysis include:

• Convergence rate of \(R_{\phi}(\hat{f}_n) \to R^*_{\phi}\)
• Excess risk bounds (how \(R_{\phi}(\hat{f}_n) \to R^*_{\phi}\) implies \(R(\hat{f}_n) \to R^*\))

Current limitations in theoretical analysis are:

• Rates are difficult to characterize under finite sample situations
• Most existing theories are distribution-free, whereas in practice data is given but its distribution is unknown

Similar issues for model hyper-parameter tuning

Ensemble idea

Mitchell, et al. "Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time." ICML, 2022.

Ensemble idea

Mitchell, et al. "Model soups: averaging weights of multiple fine-tuned models improves accuracy without increasing inference time." ICML, 2022.

Limitation

• Computational Intensity: Bagging requires training multiple base models (often hundreds), making it computationally expensive compared to single models.

• Some methods require additional validation sets.

Let's Dropout!

Nitish, et al. "Dropout: a simple way to prevent neural networks from overfitting." JMLR, 2014

 $$\pmb{\theta}^{(t+1)} = \pmb{\theta}^{(t)} - \gamma \frac{1}{B} \sum_{i \in \mathcal{I}_B} \nabla_{\pmb{\theta}} \phi( y_{i} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}) )$$

EnsLoss: Calibrated Loss Ensembles

Inspired by Dropout

(model ensemble over one training process)

 $$\pmb{\theta}^{(t+1)} = \pmb{\theta}^{(t)} - \gamma \frac{1}{B} \sum_{i \in \mathcal{I}_B} \nabla_{\pmb{\theta}} \phi( y_{i} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}) )$$

EnsLoss: Calibrated Loss Ensembles

Inspired by Dropout

(model ensemble over one training process)

$$= \pmb{\theta}^{(t)} - \gamma \frac{1}{B} \sum_{i \in \mathcal{I}_B} \partial \phi( y_{i} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}) ) \nabla_{\pmb{\theta}} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}),$$

EnsLoss: Calibrated Loss Ensembles

 $$\pmb{\theta}^{(t+1)} = \pmb{\theta}^{(t)} - \gamma \frac{1}{B} \sum_{i \in \mathcal{I}_B} \nabla_{\pmb{\theta}} \phi( y_{i} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}) )$$

$$= \pmb{\theta}^{(t)} - \gamma \frac{1}{B} \sum_{i \in \mathcal{I}_B} \partial \phi( y_{i} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}) ) \nabla_{\pmb{\theta}} f_{\pmb{\theta}^{(t)}}(\mathbf{x}_{i}),$$

EnsLoss

Experiments

CIFAR

We construct binary CIFAR (CIFAR2), by selecting all possible pairs from CIFAR10, resulting:

10 x 9 / 2 = 45

CIFAR2 datasets.

PCam

PCam is a binary image classification dataset comprising 327,680 96x96 images from histopathologic scans of lymph node sections

Experiments

OpenML

We applied a filering:

n >= 1000

d >= 1000

at least one official run

resulting 14 datasets

no dataset cherry pick

CIFAR2

OpenML

PCam

EnsLoss is a more desirable option compared to fixed losses in image data;

and it is a viable option worth considering in tabular data.

Epoch-level performance

Compatibility of prevent-overfitting methods

EnsLoss consistently outperforms the fixed losses across epochs;

and it is compatible with other methods, and their combination yields additional improvement.

Summary

The primary motivation of EnsLoss behind consists of two components: “ensemble” and the “CC” of the loss functions.

This concept can be extensively applied to various ML problems, by identify the specific conditions for loss consistency or calibration.

Thank you!

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