Selective Prediction via Training Dynamics
Authors: Stephan Rabanser
Anvith Thudi
Kimia Hamidieh
Adam Dziedzic
Israfil Bahceci
Akram Bin Sediq
HAMZA SOKUN
Nicolas Papernot
Venue: Accepted by TMLR ()

Abstract: Selective Prediction is the task of rejecting inputs a model would predict incorrectly on. This involves a trade-off between input space coverage (how many data points are accepted) and model utility (how good is the performance on accepted data points). Current methods for selective prediction typically impose constraints on either the model architecture or the optimization objective; this inhibits their usage in practice and introduces unknown interactions with pre-existing loss functions. In contrast to prior work, we show that state-of-the-art se- lective prediction performance can be attained solely from studying the (discretized) training dynamics of a model. We propose a general framework that, given a test input, monitors metrics capturing the instability of predictions from intermediate models (i.e., checkpoints) obtained during training w.r.t. the final model’s prediction. In particular, we reject data points exhibiting too much disagreement with the final prediction at late stages in training. The proposed rejection mechanism is domain-agnostic (i.e., it works for both discrete and real-valued prediction) and can be flexibly combined with existing selective prediction approaches as it does not require any train-time modifications. Our experimental evaluation on image classification, regression, and time series problems shows that our method beats past state-of-the-art accuracy/utility trade-offs on typical selective prediction benchmarks.

Published in Transactions on Machine Learning Research (02/2025)
Selective Prediction via Training Dynamics
Stephan Rabanser
stephan@cs.toronie edu
University of Toronto & Vector Institute
Anvith Thudi University of Toronto Vector Institute
anwith. cosith.duudd@mmil..uooooo utoronto..c
Kimia Hamidich
hamidich edu
Massachusetts Institute of Technology
Adam Dziedzic
dolemidiiedziieeiii Scispa.
CISPA Helmholtz Center for Information Security
Israfil Bahceci
Bricsson
Akram Bin Sediq
dbram.bin..eiiggeericsoo.cm
Bricsson
Hamza Sokun
hamnzz..sokknn@errresso..ccm
Ericsson
Nicolas Papernot
nicolas. ponolassppperroo@@uuuoootocccc
University of Toronto & Vector Institute
Reviewed on OpenReview: https: // openreview. net/ net/forum?
forum? id= 2ugnep@jyF
Abstract
Selective Prodiction is the task of rejecting inputs a model would predict incorrectly on. This involves a trade-off between input space coverage (how many data points are accepted) and model utility (how good the performance on accepted data points). Current methods for selective prediction typically impose constraints on either the model architecture or the opti- mization objective; this inhibits their usage in practice and introduces unknown interactions with pre-existing loss functions. In contrast to prior work, we show that state-of-the-arl se- lective prediction performance can be attained solely from studying the (discretized) training dynamics of a model. We propose a general framework that, given a test input, monitors metrics capturing the instability of predictions from intermediate models (i.e., checkpoints) obtained during training w.r.t. the final model's prediction. In particular, we reject data. points exhibiting too much disagreement with the final prediction at late stages in training. The proposed rejection mechanism is domain-agnostin (i.e., it works for both discrete and real-valued prediction) and can be flexibly combined with existing selective prediction ap- proaches as it docs not require any train-time modifications. Our experimental evaluation on image classification, regression, and time series problems shows that our hod beats past state-of- otate--f-the-art the-art accuracy cccurcy/utilliyy /utility trade-offs on typical selective prediction benchmarks.
1 Introduction
Machine learning (ML) increasingly deployed in high-stakes docision-making environments with strong reliability and safety requirements. One of these requirements is the detection of inputs for which the ML model produces erroneous prediction. This is particularly important when deploying deep neural networks

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Training stage
Testing stage
Truining set
Model training
Intermediate models
Test points
Intermediate models
Large with Final cliss
S
agreement with final class -
2
Epochs
Chazkgunn
Figure 1: Our proposed SPTD method for a classification example. We store checkpoints of inter- mediate models during model training. At inference time, given test input, compute various metrics capturing the stability of intermediate predictions with respect to the final model prediction. Data points with high stability are accepted, data points with low stability are rejected.
(DNNs) for applications with low tolerances for falsc-positives (i.e., classifying with a wrong label), such as healthcare (Challen et al., 2019 Mozannar & Sontag 2020), self-driving (Ghodsi et al. 202I), and law (Vieira et al. 2021 This problem setup is captured by the Selective Prediction (SP) framework, which introduces accept/reject function (a so-called gating mechonism) to abstain from predicting on individual test points in the presence of high prediction uncertainty (Geifman & El-Yaniv, 2017). Specifically, SP aims to (i) only accept inpuls on which the ML model would achieve high utility, while (ii) maintaining high coverage (i.e., correctly accepting as many imputs as possible).
Current selective prediction techniques take one of two directions: (i) augmentation of the architccture of the underlying ML model (Geilman & El-Yaniv 2019); or (ii) training the model using purposefully adapted loss function (Liu et al. 2019 Huang et al. 2020 Gangrade et al. 2021p. The unifying principle behind these methods is to modify the training stage in order to accommodate selective prediction. While many ad-hoc experimentation setups are amenable to these changes, productionalized environments often impose data pipeline constraints which limit the applicability of existing methods. Such constraints include, but are not limited to, data access revocation, high (re)-training costs, or pre-existing architecture/ modifications whose interplay with selective prediction adaptations are unexplored. As a result of theses limitations, selective prediction approaches are hard to deploy in production environments.
We instead show that these modifications are unnecessary. That is, method, which establishes new SOTA results for selective prediction across a variely of datasets, not only oul perforrmmm existing work but our method can be easily applied on top of all existing models, unlike past methods. Moreover, our mothod is not restricted to classification problems but can be applicd for real-valued prediction problems, too, like regression and time series prediction tasks. This is an important contribution as recent SP approaches have solely focused on improving selective classification.
Our approach builds the following observation: typical DNNs are trained using an iterative optimization procedure, e.g., using Stochastic Gradient Descent (SGD). Due to the sequential nature of this optimization process, as training goes on, the optimization process yields sequence of intermediate models. Current selective prediction methods rely only on the final model, ignoring valuable statistics available from the model's training sequence. In this work, however, we propose to take advantage of the information contained in these optimization trajectories for the purpose of selective prediction. By studying the usefulness of these trajectorics, we observe that instability in SGD convergence is often indicative of high alcatoric uncertainty (i.e., irreducible data noise such as overlap between distinct data classes). Furthermore, past work on example difficulty (Jiang et al. 2020; Toneva et al.) 2018 Hooker et al. 2019 Agarwal et al. 2020) has highlighted faster convergence as indicative of easy-to-learn training examples (and conversely slow convergence of hard- to-learn training examples). We hypothesize that such training time correlations with uncertainty also hold for test points and studying how test time predictions evolve over the intermediate checkpoints is useful for reliable uncertainty quantification.
With this hypothesis, we derive the first framework for Selective Prediction based on neural network Training Dynamics (SPTD, Figure for an example using classification setup). Through formalization
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of this particular neural network training dynamics problem, first note that useful property of the intermediate models' predictions for a test point is whether they converge ahead of the final prediction. This convergence can be measured by deriving a prediction instability score measuring how strongly predictions of intermodiate models agree with the final model. While the exact specifics of how we mcasure instability differs between domains (classification vs regression), our resulting score generalizes across application domains and measures weighted prediction instability. This weighting allows us emphasize instability late training which we deem indicative of points that should be rejected. Note that this approach is transparent w.r.t. the training stage: method only requires that intermediate checkpoints were recorded when model was trained, which is an established practice (especially when operating in shared computing environments such as GPU clusters). Moreover, when compared to competing ensembling- baste methods, such as Deep Ensembles (Lakshminarayanan et al. 2017), our approach can match the same inference-lime cost while being significantly cheaper to train.
To summarize, our main contributions are as follows:
1. We present a motivating synthetic example using a linear model, showcasing the effectiveness of training dynamics information in the presence of a challenging classification task (Section 3.1
2. We propose novel method for selective prediction based on training dynamics (SPTD, Section 3.2 To that end, we devise effective scoring mechanism capturing weighted prediction instability of intermediate models with the final prediction for individual test points. Our methods allow for selective classification, selective regression, and selective time series prediction. Moreover, SPTD can be applied to all existing models whose checkpoints were recorded during training.
3. We highlight an in-depth connection between our SPTD approach and forging (Thudi al. (2022) Section (33), which has shown that optimizing a model distinct datasets lead to the same sequence of checkpoints. This connection demonstrates that our metrics can be motivated from a variely of different perspectives.
We perform a comprehensive set of empirical experiments on established selective prediction bench- marks spanning over classification, regression, and time series prediction problems (Section 4 Our results obtained from all instances of SPTD demonstrate highly favorable utility/coverage trade- offs, establishing new state-of-the-art results in the field at a fraction of the training time cost of competitive prior approaches.
2 Background on Selective Prediction
Supervised Learning Sctup. Our work considers the standard supervised learning setup. We assume access to a dataset D = [ ) consisting of M data points (x, v) with æ E X' and E y. We refer to := Rdd as the covariate space (or input//ata space) of dimensionality d. For classification problems, we define y = [C] = 21. C} as the label space consisting C classes. For regression and time series problems (such as demand forecasting) we instead define y == R and ) := R respectively (with R being the prediction horizon). All data points (ar, v) are sampled independently from the underlying distribution p defined over the joint covariate and label spaces X' x ). Our goal is to learn a prediction function f x which minimizes the risk R(fo) with respect to the underlying data distribution and an appropriately chosen loss function é: 2: x IR. We can derive the optimal parameters i via empirical risk minimization which approxinates the true risk R(fo) through sampling, thereby ensuring that 0 s Ô for sufficiently large amount of samples:
0' arg min R(So) = arg min E Epta. ((Fo(()))))))))))
(1)
0
θ arg min R(So) = arg min
1 M
Σ ((fo(()))))))
(2)
In the following, drop the explicit dependence of I on O and simply denote the predictive function by
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Selective Prediction Setup. Selective prediction alters the standard supervised learning setup by intro- ducing a rejection state 1. through galing mechanism (ElYaniis & Wiener 2010). Tn particular, such a mechanism introduces a selection function g X' + R which determines if a model should predict on a data point æ. Given an acceptance threshold T, the resulting predictive model can summarized as:
{1 f(x) 1
g(x) T
(f.g((z)
otherwise.
Selective Prediction Evaluation Metrics. Prior work evaluates the performance of a selective predictor (f.g) based on two metrics: the coverage of (f.9) (i.e., what fraction of points we predict on) and the selective utility gf.g) on the accepted points. Note that the exact utility metric depends on the type of the underlying selective prediction task (e.g. accuracy for classification, R2 for regression, and a quantile- basod loss for time scries forecasting). Successful SP methods aim obtain both strong selective utility and high coverage. Note that these Iwo metries to lower coverage and vere-versa. The complete performance profile of model can be specified using the risk-coverage curve, which defines the risk as a function of coverage El-Y Y-aiviv & Wiener 2010). These metrics can be formally defined as follows:
at odds with each other: naïvely improving utility leads
coverage(S.g) Mr
(4)
M
utility(f.g)
Σ u(f(x),y)
(5)
((.))):(()
Here, -) corresponds to the specifically used utility function, == [xx : g(x,) S r] corresponds
the number of accepted data points threshold т, and 1[:] corresponds to the indicator function. We define the following utility functions to be used based on the problem domain:
1. Classification: We use accuracy on accepted points as our utility function for classification:
M.
Accuracy = M
1
Σ 1k f(x;) = ys]
(6)
2. Regression: We the cocfficient of determination (R² score, which is scaled version of the mean squared error) on accepted points our utility function for regression:
Mr E (f(x,) y2)
R
ΣE
M. (14
Σラ 9,)
M,
3. Time Series Forecasting: We use the Mean Scaled Interval Score (MSIS) Gneiting & Raftery (2007) on accepted series as our utility function for time series forecasting
a (Me,r 4.
2 (h., vi.r) 1/8i, 4.rr]
M.
+ (Ji.r us.r) 1 1yyer > werr
MSIS
M,
where refers to specific predictive quantile, n to the conditioning length of the time series, to the length of the seasonal period, and uyr and 4r to the upper and lower bounds the prediction range, respectively.
2.1 Past & Related Work
Softmax Response Bascline (classification).
The first work on selective classification was the softmax
response (SR) mechanism (Hendrycks & Gimpel 2016 Geifman & El-Yaniv 2017 A classification model
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typically has softmax output layer which takes in unnormalized activations in 2, E RC (referred to logits) from a linear model or deep neural net. These activations are mapped through the softmax function which normalizes all entries
ea:
Oz2)
(9)
Σ e
to the interval [0 and further ensures that Σi= 0z); 1. As a result, the softmax output can be inter- preted as a conditional probability distribution which we denote flylar). The softmax response mechanism applies a threshold T to the maximum response of the softmax layer: maxyey f(y(z). Given a confidence parameter 8 and desired risk Rf), SR constructs (f.g) with test error no larger than R(f)) with probability 21 1 6. 8. While this approach is simple to implement, it has been shown to produce over-confident. results due to poor calibration of deep neural networks (Guo et al. 2017
Loss Modifications (mostly classification). The first work to deliberately address selective classifica- tion via architecture modification was SelectiveNet (Geifman & El- Yaniv 2019), which trains a model to jointly optimize for classification and rejection. A loss penalty is added to enforce particular coverage constraint using a variant of the interior point method Potra & Wright (2000) which is often used for solving linear and non-linear convex optimization problems. To optimize selective accuracy over the full coverage spectrum in single training run, Deep Gamblers (Liu et al. 2019) transform the original C-class problem into a (C + cllclass problem where the additional class represents model abstention. A similar approach is given by Self-Adaplive Training (SAT) (Huang el al.) 2020) which also uses (C 1))class setup but instead incorporates an exponential average of intermediate predictions into the loss function. Other similar ap- proaches include: performing statistical inference for the marginal prediction-set coverage rates using model ensembles (Feng et al., 2021; p confidence prediction using earlier snapshot of the model (Geifman et al. 2018), estimating the gap between classification regions corresponding to each class (Gangrade et al. 2021] and complete precision by classifying only when models consistent with the training data predict the same output Khani et al. 2016).
Uncertainty Quantification (classification + regression). It was further shown by Lakkshmi- marayanan et al., 2017; Zaoui et al. 2020) that deep model ensembles (i.c., a collection of multiple models trained with different hyper-parameters until convergence) can provide state-of-the-art uncertainty quantifi- cation, task closely related to selective prediction. This however raises the need to train multiple models from scratch. To reduce the cost of training multiple models, (Gal & Ghahramani 2016 proposed absten- tion based on the variance statistics from several dropout Srivastava et al. (2014) enabled forward passes at test time. Another popular technique for uncertainly quantification, especially for regression and time series forecasting, is given by directly modeling the output distribution (Alexandrov et al. 2019 in a parametric fashion. Training with a parametric output distribution however can lead to additional training instability, often requiring extensive prprr-paraeeter tuning and distributional assumptions. On the other hand, our approach does not require any architecture or other training-time modifications. Finally, we note that selec- tive prediction and uncertainty are also strongly related to the field of automated model evaluation which relies on the construction a proximal prediction pipeline of the testing performance without the presence of ground-truth labels (Peng et. al. 2023 2024
Training Dynamics Approaches (classification. Checkpoint and snapshot ensembles Huang et al. 2017: Chen et al.] 2017) constitute the first usage of training dynamics to boost model utility. Our work is closest in spirit to recent work on dataset cartography (Swayamdipta et al.) (020) which relies on using training dynamics from an example difficulty viewpoint by considering the variance of logits. However, their approach does not consider selective prediction and further requires access to true label information (which is not available in the selective prediction setting). Recent work on out-of-distribution delection (Adila & Kang 2022), closely related yet distinct application scenario from selective prediction, harness similar training dynamics based signals.
Under miscalibration, model's prediction frequency events does not match the true observed frequency of events.
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Example Difficulty. A related line of work to selective prediction is identifying difficult examples, or how well a model can generalize to given unseen example. Recent work Jiang et al. (2020) has demonstrated that the probability of predicting the ground truth label with models trained on data sub-samples of different sizes can be estimated via a per-instance empirical consistency score. Unlike our approach, however, this requires training a large number of models. Example difficulty can also be quantified through the lens of a forgetting event Toneva et al. (2018) which the example is misclassified after being correctly classified. Instead, the metrics that we introduce in Section 3 are based on the disagreement of the label at each checkpoint with the final predicted label. Other approaches estimate the example difficulty by: prediction depth of the first layer at which a k-NN classifier correctly classifies example (Baldock et al. 2021] the impact of quantization and compression on model predictions of a given sample (Hooker et al. 2019 and est estimating the leave-one-out. influence of each (raining example on the accuracy of an algorithm by using inftuence functions (Feldman & Zhang, 2020). Closest to our method, the work of Agarwal et a.. (2020) utilizes gradients of intermediate models during training to rank examples by difficulty. In particular, they average pixel-wise variance of gradients for each given input image. Notably, this approach is more costly and less practical than our approach and also does not study the utility /everraee trade-off which is of paramount importance to selective prediction.
Disagreement Our SPTD method heavily relies on the presence of disagreements between intermediate models. Past work on (dis- Jaireeetge has studied the connection between generalization and disagreement of full SGD runs ІJiang et al.) 2021) as well as correlations between in-distribution and out-of-distribution agreement across models (Back et al. 2022
3 Selective Prediction via Neural Network Training Dynamics
We now introduce our selective prediction algorithms based on neural network training dynamics. We start by presenting a motivating example showcasing the effectiveness analyzing training trajectorics for a linear classification problem. Following this, we formalize our selective prediction scoring rule based on training-time prediction disagreements. We refer to the class of methods we propose as SPTD.
3.1 Method Intuition: Prediction Disagreements Generalize Softmax Response
Stochastic iterative optimization procedures, such as Stochastic Gradient Descent (SGD), yield a sequence of models that iteratively derived by minimizing a loss function E(;,-) on randomly selected mini-batch (X,, y:) from the training set. The iterative update rule can be expressed as follows
0.41 0, aE(f(X,),
(10)
00,
where the learning rate controls the speed of optimization and t E {l,.... T) represents a particular time-step during the optimization process.
Current methods for selective prediction disregard the properties of this itcrative process and only rely on the final set of parameters 0r. However, the optimization trajectories contain information that we can use to determine prediction reliability. In particular, on hard optimization tasks, the presence of stochasticity from SGD and the potential ambiguity of the data often leads noisy optimization behavior. As result, intermediate predictions produced over the course of training might widely disagree in what the right pre- diction would be for a given data point. Our class of selective prediction approaches explicitly make use of these training dynamics by formalizing rejection scores based on the observed frequency of prediction disagreements with the final model throughout training.
To illustrate and reinforce this intuition that training dynamics contain meaningfully more useful information for selective prediction than the final model, we present a synthetic logistic regression example. First, we generate a mixture of two Gaussians each consisting of 1000 samples: D ((x,, 0)) {(0,, 1))
where D; ~ N((a 0] T_ I) and xy ~ N([[a 0] I). Note that a controls the distance between the two 2-dimensional Gaussian clusters, allowing us to specify the difficulty of the learning task. Then, train
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Figure Synthetic example of anomaly scoring based on SR vs SPTD. The first row shows a test set from the generative Gaussian model as well as the learned decision boundary separating the two Gaussians. For small a, the decision boundary is overconfident. The second row shows the same data set but instead depicts the scores yielded by applying SPTD to the full domain. SPTD highlights rightful rejection regions more clearly than the overconfident SR score: larger regions are flagged as exhibiting noisy training dynamics (with stronger shades of green indicating stronger disagreement) as + 0. The bottom row shows the distribution of the SR and SPTD scores, clearly showing that SPTD leads to improved uncertainty under stronger label ambiguity.
1,5). Finally, we compute both linear classification model using SGD for 1000 epochs for each E (0, 1, the softmax response score (SR) score, the typical baseline for selective classification, as well our SPTD score (details in Section 3.2
We showcase the results from this experiment in Figure 2 We see that if the data is linearly separable (a = 5) the learned decision boundary is optimal and the classifier's built-in confidence score SR reflects well-calibratod uncertainty. Morcover, the optimization process stable as SPTD yields low scores over the full domain. However, as we move the two Gaussians closer together (i.e., by reducing a) we see that the SR bascline increasingly suffers from overconfidence: large parts of the domain are very confidently classified as either (red) or (blue) with only a small ambiguous region (white). However, the optimization trajectory is highly unstable with the decision boundary changing abruptly between successive optimization steps. SPTD identifies the region of datapoints exhibiting large prediction disagreement due to this training instability and correctly rejects them (as those are regions also subject to label ambiguity in this casc). In summary, we observe that. SPTD provides improved uncertainly quantification in ambiguous classification regions (which induce training instability) and reduces to the SR solution as the classification task becomes easier. Hence, we expect SPTD to generalize SR performance, which supported by this logistic regression experiment.
5)
3.2 Method Overview: Measuring Prediction Instability During Training
We procoed to describe the statistics we collect from intermediate checkpoints which we later devise our scores for deciding which inputs to reject on. The objective of these statistics is to capture how unstable the prediction for a datapoint was over the training checkpoints. Let [f1, f2...., fr.....f) fr] be a sequence of intermediate chockpoints, and Durain Dest be the set of all data points. We define prediction disagroement score at time & (1...., T} as some function a, : R with aı(z)= = if f((x) fr()). Note that the exact ae(-) we depends the problem domain (classification vs regression) and we define our choices below.
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Algorithm 1: SPTD for classification
A Algorithm 2: SPTD for regression
Require: Intermediate models [fi...., frl, query
Require: Intermediate models ff.....ffi query
point x, weighting parameter [0, oc).
point æ, weighting parameter is E [0, oo).
for / [T] do
1: for i [T] do
2:
if ft(x)= fT) then ar + 0 else or t
2:
+ [1f((x) - fr('))l
3: Vt= + -())
3: 0 + (#)*
4: end for
4: end for
9 - E. vrag
5: g - Σ 12at
if g S then ((x) fr())) else f(x))
6: if 0 then f(x) (x) else f(x)) 1
In the following, when conditioning æ is understood from context, we drop the explicit dependence on æ and write aç.
For a fixed data point æ; our approach takes a given sequence of prediction disagreements [a],..., @r] and associates a weight v, to each disagreement or to capture how severe a disagreement at step is. To derive this weighting we ask: How indicative of æ being incorrectly classified is a disagroement step t? Relatod work in the example difficulty literature (see Section 2.1 for details) found that easy-to-optimize samples are learned early in training and converge faster. While prior work specifically derived these convergence insights for training points only, the novelty of our method is to show such conclusions for training points also generalize to test points. Hence, we propose to use the weighting 2 = (1)² f for k E [0, oo) to penalize late prediction disagreements as more indicative of a test point we will not predict correctly on. With this weighting, our methods compute a weighted sum of the prediction disagroements, which effectively forms our selection function g(-):
g(x)) Σ ((0((z)
(11)
Instability for Classification. For discrete prediction problems (i.e., classification) we define the label disagreement score as a; (16),(())fr((e) where 8 is the Dirac-delta function: a, is hence 1 if the intermediate prediction fi at checkpoint t disagrees with the final prediction fr for æ, else 0. The resulting algorithm using this definition of a, for classification is given in Algorithm [ We remark that continuous metrics such as the maximum softmax score, the predictive entropy (i.e., the entropy of the predictive distribution f(y/=)), or the gap between the two most confiden classes could be usod as alternate measures for monitoring stability (see Appendix A for discussion). However, these measures only provide noisy proxy and observing a discrete deviation in the predicted class provides the most direct signal for potential mis-classification.
Instability for Regression. One key advantage of our method over many previous ones is that it is applicable to any predictive model, including regression. Here, we propose the following prediction disagreement score mcasuring the distance of intermediate predictions to the final model's prediction: ar = [I]((x)) (( [ The resulting algorithm using this definition of o, for regression is given in Al- gorithm 2 We again highlight the fact that Algorithm 2 only differs from Algorithm I in the computation of the prodiction disagrcement a, (linc highlighted in both algorithms).
Instability for Time Series Prediction. We can further generalize the instability sequence used for regression time series prediction problems by computing the regression score for all time points on the prediction horizon. In particular, we compute 0r,r [=])())), fr (x))|| for all E {1...., R). Recall that for time series problems fe(z) returns vector of predictions y E R² and we use the subscript on f((x), to denote the vector indexing operation. Our selection function is then given by computing Equation II for cach and summing up the instabilities over the prediction horizon: g(x) E, (x). The full a algorithm therefore shares many conceptual similarities with Algorithm 2 and we provide the detailed algorithm as part of Algorithm 3 Note that the presented generalization for time series applicable to any setting in
We explored a more robust normalization by averaging predictions computed over the last checkpoints: a = [If((x) 1 Eeef (T- -141........ fe(x)]]- Across many 1, we found the obtained results to be statistically indistinguishable from the n results obtained by normalizing w.r.t. the last. checkpoint fr.
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Algorithm 3: SPTD for time series forecasting
Require: Intermediate models .f.....:fe frl, query point æ, weighting E [0, oo), prediction horizon R. for E [T] do
for r [R] do
at, + (af (z)x1)
end for
5: 1t (+)*
6: end for
7: g - Σ, E, Uraer
g S then f(x) = L clsc f(x) = 1
which the variability of predictions can be computed. As such, this formalism can extend to application scenarios beyond time series prediction such as object detection or segmentation.
3.3 Selective Prediction and Forging
While our SPTD method primarily motivated from the example difficulty view point, we remark that the scores SPTD computes to docide which points reject can be derived from multiple different perspectives. To showcase this, we provide formal treatment on the connection between selective classification and forging (Thudi et al. 2022). which ultimately leads to the same selection function g(-) above.
Previous work has shown that running SGD on different datasets could lead to the same final model (Hardt et al., 2016 Bassily et al. 2020; Thudi et al.) 2022). For example, this is intuitive when two datasets were sampled from the same distribution. We would then expect that training on either dataset should not significantly affect the model returned by SGD, For our selective prediction problem, this suggests an approach to decide which points the model is likely to predict correctly on: identify the datasets that could have been trained (in lieu of the training set it
actually trained on). Any point from the datasets the model could have trained would then be likely to be predicted on correctly by the model. Recent work forging Thudi et al. (2022) solves this problem of identifying datasets the model could have trained on by brute-force searching through different manihbacches to determine if a mini batih in the alternative dataset can be used to reproduce one of the original training steps. Even then, this is only a sufficient condition to show a datapoint could have plausibly been used to train: if the brute-force fails, does not mean the datapoint could not have been used obtain the final model. As an alternative, we propose to instead characterize the optimization behaviour of training on dataset as a probabilistic necessary condition, i.e, a condition most datapoints that were (plausibly) trained would satisfy based training dynamics. Our modified hypothesis is then that the set of datapoints we optimized for (which contains the forgeable points) coincides significantly with the set of points the model predicts correctly on.
3.3.1 A Framework for Being Optimized
In this section we derive an upper-bound on the probability that a datapoint could have been used to obtain the model's checkpointing sequence. This yields probabilistically necessary (though not sufficient) characterization of the points we explicitly optimized for. This bound, and the variables depends on, informs what we characterize as 'optimizing' for a datapoint, and, hence, our selective classification methods.
Let us denote the set of all datapoints as D, and let D D be the training set. We are interested in the setting where model f is plausibly sequentially trained on D (e.g., with stochastic gradient descent). We thus also have access to sequence of T intermediate states for f. which we denote [fi...., In this sequence, note that fr is exactly the final model f.
Now, let Pt represent the random variable for outputs on D given by an intermediate model fi where the outputs have been binarized: we have 0 if the output agrees with the final prediction and if not. In other words, Pt the distribution of labels given by first drawing æ D and then outputting 1 ff where 5 denotes the Dirac delta function. Note that we always have both well-defined mean and variance
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for Pt as it is bounded. Furthermore, we always have the variances and expectations of {pt} converge to 0 with increasing L: as Pr = O always and the sequence is finite convergence trivially occurs. To state this formally, let V/- = Væ>D[Pi] [px] and let ét = Bæ [Pt] denote the variances and expectations over points in D. In particular, we remark that er = 0, 2r = 0, so both e, and i converge. More formally, for all e > 0 there exists an N E fl,...,77 7 such that v, < e for all > N. Similarly, for all > 0 there exists a (possibly
different) N E (1, (I....,, T such that e << c for all t> N.
However, the core problem is that we do not know how this convergence in the variance and expectation occurs. More specifically, if we knew the exact values of e, and vt, we could use the following bound on the fraction of training data points producing given [a1.".', -al as a reject option for points that are not optimized for. We consequently introduce the notation [ai,..., ar) where oe = 1 - fr())))r()))) which we call the 'label disagreement (at t)', Note that the az are defined with respect to a given input, while pt represent the distribution of ar over all inputs in D.
Lemma 1. Given a datapoint a, let {ai,..., oT where at (0)())))r)())) Assuming not all a = et then the probability x E D is S mine, a. agtes Tas-
Proof. By Chebyshev's inequality we have the probability of a particular sequence {ai,..., aT} occurring for a training point is S Te.- for every (a bound on any of the individual o occurring as that event is in the event [pe ex| |at- ex| occurs). By taking the minimum over all these upper-bounds we obtain our upper-bound.
We do not guarantee Lemma] 1 is tight. Though we do take a minimum to make it tighter, this is a minimum over inequalities all derived from Chebyshev's 's inequality Despite this potential looseness, using the bound from Lemma T we can design a naïve selective classification protocol based on the 'optimized = correct (often)' hypothesis and use the above bound on being a plausible training datapoint as our characterization of optimization; for a test input æ, if the pouer-bound on the probability of being datapoint in D is lower than some threshold I reject, else accept. However, the following question prevents us from readily using this method: How do E[pe] and Wipi] evolve during training?
To answer this question, we propose to examine how the predictions on plausible training points evolve during training. Informally, the evolution of E(p,] represents knowing how often we predict the final label at step t, while the evolution of Vipdl represents knowing how we become more consistent as we continue training. Do note that the performance of this optinization-baseed approach to selective classification will depend on how unoptimized incorrect test points are. In particular, our hypothesis is that incorrect points often appear sufficiently un-optimized, yielding distinguishable patterns for E(ps] and V/p.] when compared to optimized points. We verily this behavior in Section 4 where we discuss the distinctive label evolution patterns of explicitly optimized, correct, and incorrect datapoints.
3.3.2 Last Disagreement Model Score For Discrete Prediction (SMAx)
Here, we propose selective classification approach based on characterizing optimizing for a datapoint based off of Lemma [ Recall the bound given in Lemma D depends on expected values and variances for the Ps (denoted e and o respectively). In Section 4 we observe that e quickly converge to O, and so by assuming & = 0 alway:] the frequentist bound on how likely a datapoint is a training point becomes Fi min; s.1 a,=1 jer- = min; a.=1 Uy. Using this result for selective classification, we would impose accep-
tance if min; s.t a,=1 2 r. Moreover, in Section g we further observe that i2 monotonically decreases in a convex manner (after an initial burn-in phase). Hence, imposing min, s.t a.m1 Vt 2 T simply imposes a last checkpoint that can have a disagreement with the final prediction.
Based on these insights, we propose the following selective classification score: Smax Note that this score directly follows from the previous discussion but flips the thresholding direction from min s.t a,=1 8/ to max, s. a,=1 1 < t for consistency with the anomaly scoring literature [Ruff et al. i,
max, s. a,=1 m
2018). Finally, we choose to approximate the empirical trend of i2 as observed in Section with Uy = 1
9nne could potentially use information about the distribution of points not in D refine this bound.
We tried removing this assumption and observed similar performance.
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for E (o). Based on the choice of k, this approximation allows to (i) avoid explicit estimation of from validation data; and (ii) enables us to flexibly specify how strongly we penalize model disagreements late in training.
Hence, our first algorithm for selective classification is:
Denote = fo (x)), i.. the label our final model predicts.
1 2. If 36 s.t = 1 then compute smax max; sat a==1 et as per the notation in Section 3.3.1 (i.c = iff fe(r) * L), else accept a with prediction L.
3. If Smax accept with prediction clse reject (1).
Note again, all our candidate increase, the algorithm imposes last intermediate model which output prediction that disagrees with the final prediction: hereafter, the algorithm must output models that consistently agree with the final prediction.
3.3.3 Overall Disagreement Model Score (sSuM)
Note that the previous characterization of optimization, defined by the score SMAX, could be sensitive to stochasticity in training and hence perform sub-optimally. That is, the exact time of the last disagreement, which SMAX relies on, is subject to high noise across randomized training runs. In light of this potential lim- itation we propose the following "summation" algorithm which computes a weighted sum over training-timg disggreements to get a more consistent statistic. Do note that typically to get a vorrinvere statistic one would take an average, but multiplying by scalars can be replaced by correspondingly scaling the threshold we use. Hence, our proposed algorithm
1. Denote L = fr()), i.e. the label our final model predicts.
31 0r compute Saum - 0, else accept æ with prediction
3. 8eum I f accept x with prediction L, else reject (1).
Recalling our previous candidates for vt, have the ssUM places higher weight on late disagreements. This gives us biased average of the disagreements which intuitively approximates the expected last disagreement but now is less susceptible to noise. More generally, this statistic allows us to perform selective classification by utilizing information from all the disagreements during training. In Appendix [3.2.7 we experimentally show that SSUM leads to more robust selective classification results compared to SMAX. We remark that
the sum score SSUM corresponds exactly our score g(-) proposed as part of SPTD (recall Equation 11 from Section 3.2). showcasing the strong connection of our method to forging.
4 Empirical Evaluation
We present comprehensive empirical study demonstrating the effectiveness of SPTD across domains. Our rosults show that computing and thresholding the proposed weighted instability score from SPTD provides strong score for selective classification, regression, and time series prediction.
4.1 Classification
Key Research Goals. As part of our experiments we:
Study the accuracy/coverage trade-off with comparison past work, showing that SPTD outper- forms existing work.
Present exemplary datasets.
label evolution curves for individual examples from all
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Examine our method's sensitivity to the checkpoint selection strategy and the weighting parameter k.
Evaluate the detection performance of out-of-distribution and adversarial examples, showing that SPTD can be applicd beyond the i..d. assumption of scloctive prodiction.
Provide a detailed cost vs performance tradeoff of SPTD and competing selective prediction meth- ods.
Analyze distributional training dynamics patterns of both correct and incorrect data points, the separation of which enables performative selective classification.
Datasets & Training. We evaluate SPTD on image dataset benchmarks that are common in the selective classification literature: 1EFAR-10/IFAAR-.
Krizhevsky et al., 2009), StanfordCars (Krausc et al. 2013). and Food101 (Bossard et al. 2014 For each dataset, we train a deep neural network following the ResNet-1 18 architecture (He et al. 2016) and checkpoint each model after processing 50 mini-batches of size 128. All models are trained over 200 epochs (400 epochs for StanfordCars) using the SGD optimizer with an initial learning rate of 10- momentum 0.9, and weight decay 10 10-4, 4, Across all datasets, we decay the learning rate by factor of 0.5 in 25-epoch intervals.
Baselines. We compare our method (SPTD) to common SC techniques previously introduced in Section 2 Softmax Response (SR) and Self-Adaptive Training (SAT). Based on recent insights from Feng al. (2023) we (i) train SAT with additional entropy regularization and (ii) derive SAT's score by applying Softmax Response (SR) to the underlying classifier (instead of thresholding the abstention class). We refer to this method as SAT+ER+SR. We do not include results for SclectiveNet, Deep Gamblers, or Monto-Carlo Dropout as previous works (Huang el al. 2020; Feng et al. (023) have shown that SAT+ER+SR strictly dominates these methods. In contrast to recent SC works, we do however include results with Deep Ensembles (DE) [Takshminarayanan et al. 2017), ; a relevant bascline from the uncertainty quantification literature. Our hyper-parameter tuning procedure documented in Appendix B.1
Accuracy/ / /Coverage Trade-off. Consistent with standard evaluation schemes for selective classification, our main experimental results examine the accuracy caaracyycoverrag /coverage trade-off of SPTD. We present our perfor- mance results with comparison to past work in Table [ where we demonstrate SPTD's effectiveness on CIFAR-10, CIFAR-100, StanfordCars, and Food101. We document the results obtained by SPTD, SAT, SR, and DE across the full coverage spectrum. We see that SPTD outperforms both SAT and SR and performs similarly as DE. To further boost performance across the accuracy / /coverage spectrum, we bine SPTD and DE by applying SPTD on each ensemble member from DE and then average their scores. More concretely, we estimate DE+SPTD = 1 Σ SPTD,, where SPTD,, computes g on each ensemble member m [M]. This combination leads to new state-of-the-art selective classification performance and showcases that SPTD can be flexibly applied on top of established training pipelines. Further evidence towards this flexibility is provided in Appendix B.2.3 where we show that applying SPTD on top of SAT also improves selective prediction performance.
Individual Evolution Plots. To analyze the effectiveness of our disagreement metric proposed in Sec- tion 3 we examine the evolution curves of our indicator variable a, for individual datapoints in Figure 3 In particular, for each dataset, we present the most stable and the most unstable data points from the test sets and plot the associated label disagreement. metric a; over all checkpoints. We observe that easy-to-classify examples only show small degree of oscillation while harder examples show a higher frequency of oscilla- tions, especially towards the end of training. This result matches our intuition: our model should produce correct decisions on data points whose prediction is mostly constant throughout training and should reject data points for which intermediate models predict inconsistently. Moreover, as depicted in Figure 4 we also show that our score g(-) yields distinct distributional patterns for both correctly and incorrectly classified points. This separation enables strong coverage/ eveerageaccuurayy accuracy trade-offs via our thresholding procedure.
Thiis entropy regularization step is designed encourage the model be more confident in its predictions.
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Coverage
SAT+ER+SR
DE
SPTD
DE+SPTD
100
92.9 (=0000
92.9 (=0000
92.9 (=00.0)
92.9 (=0.0)
92.9 (=0.1)
90
96.4 (=0.1)
96.3 (=0.1)
96.8 (=011)
96.5 (=000)
96.7 (=.11)
98.1 (=0.1)
98.1 (#0.1)
98.7 (=0.00
98.4 (10.1)
98.8 (=0.1)
CIPAR-I1
70
98.6 (#0.2)
99.0 (=0.1)
99.4 (#:0.1
99.2 (1000)
99.5 (=000)
60
98.7 (=0.1)
99.4 (+0.0)
99.6 (#0.1)
99.6 (=0.2)
99.8 (=0.0)
50
98.6 (#0.2)
99.7 (#0.1)
99.7 (#0.1)
99.8 (+0.0)
99.9 (=0.0)
98.7 (=0.0)
99.7 (40.0)
99.8 (40.0)
99.8 (10.1)
100.0 (=0.0)
30
98.5 (10.0)
99.8 (40.0)
99.8 (40.0)
99.8 (#0.1)
100.0 (+000)
98.5 (=0.1)
99.8 (=0.1)
99.8 (+0.0)
100.0 (=0.00
100.0 (=0.0)
10
98.7 (=00.0
99.8 (=0.1)
99.8 (+0.1)
100.0 (=0.0)
100.0 (=000)
100
75.1 (=:0.0)
75.1 (=:0.0)
75.1 (+:0.0)
75.1 (=000)
75.1 (=000)
78.2 (+: 0.1)
78.9 (4: 0.1)
80.2 (4: 0.0)
80.4 (d: 0.1)
81.1 (+ 0.1)
80
82.1 (+: 0.0)
82.9 (+ 0.0)
84.7 (+ 0.1)
84.6 (+: 0.1)
85.0 (# 0.2)
CIFAR-100 R-00
86.4 0.1)
87.2 (+ 0.1)
88.6 (+ 0.1)
(#= 0.0)
88.8 (# 0.1)
60
90.0 (+ 0.0)
90.3 (+ 0.2)
90.2 (+ 0.2)
90.1 (+ 0.0)
90.4 (+ 0.1)
92.9 (11
93.3 (+ 0.0)
94.8 (+ 0.0)
94.6 (+ 0.0)
94.9 (+ 0.0)
40
95.1 (+000)
95.2 (+ 0.1)
96.8 (+ 0.1)
96.9 (+ 0.1)
96.9 (# 0.0)
97.2 (+))
97.5 (+ 0.0)
98.4 (= 0.1)
(#= 0.1)
98.5 (# 0.0)
20
97.8 0.1)
98.3 (+ 0.1)
99.0 (= 0.0)
98.8 (+ 0.2)
99.2 0.1)
98.1 (# 0.0)
(+ 0.1)
(+ 0.1)
99.4 (#= 0.1)
99.6 (# 0.1)
100
81.1 (=0000
81.1 (=000)
81.1 (#:00)
81.1 (=000)
81.1 (=0.0)
90
85.3 (=0.1)
85.5 (#0.2)
86.2 (#0.1)
85.7 (#0.0)
86.7 (=0.0)
87.1 (=000)
89.5 (+0.0)
90.3 (#0.0)
89.9 (#0.0)
91.3 (=0.1)
Food101
70
92.1 (#0.1)
92.8 (#0.1)
94.5 (#0.1)
93.7 (#0.0)
94.6 (=0.0)
95.2 (#0.1)
95.5 (#0.1)
97.0 (#0.0)
97.0 (=0.0)
97.0 (=0.0)
50
97.3 (#0.1)
97.5 (40.0)
98.2 (#0.0)
98.3 (=0.2)
98.5 (=0.0)
98.7 (=0.0)
98.7 (#0.2)
99.1 (#0.0)
99.1 (10.1)
99.2 (=0.1)
30
99.5 (=0.0)
99.7 (#0.2)
99.2 (10.0)
99.6 (10.0)
99.7 (=0.0)
20
99.7 (#0.1)
99.7 (40.2)
99.9 (#0.1)
99.8 (#0.0)
99.9 (=0.1)
10
99.8 (1:0.0
99.8 (1001)
99.9 (#001)
99.9 (=0.1)
99.9 (=0.1)
100
77.6 (=000)
77.6 (=00.0)
77.6 (=000)
77.6 (=0.0)
77.6 (=0.0)
90
83.0 (=0.1)
83.0 (+0.2)
83.7 (=0.1)
83.3 (#0.1)
83.7 (=0.2)
87.6 (:10.0)
88.0 (40.1)
88.7 (10.1)
89.3 (+00.)
89.7 (+0.0)
StonfordCors
70
90.8 (10.0)
92.2 (4001)
92.4 (40.1)
93.6 (+000)
93.4 (10.1)
60
93.5 (+00.1)
95.2 (4001)
95.3 (40.0)
96.2 (+0.0)
96.3 (=0.0)
50
95.3 (+0.0)
96.9 (+0.2)
96.4 (+0.1)
97.0 (#0.1)
97.1 (#0.3)
41)
96.8 (+0.0)
97.8 (+0.0)
97.8 (#:0.2)
97.8 (+0.1)
97.8 (=0.0)
30
97.5 (+0.1)
98.2 (+0.2)
98.6 (+:0.0)
98.2 (+0.2)
98.9 (=0.0)
20
98.1 (+0.0)
98.4 (+0.1)
98.9 (#:0.2)
98.6 (+0.0)
99.0 (=0.0)
10
98.2 (+00.)
98.7 (+0.1)
99.5 (=:0.)
98.5 (+0.1)
99.5 (=000)
Table 1: Selective accuracy achieved across coverage levels. We find that SPTD-based methods out- performs current SOTA error rates across multiple datasets with full-coverage accuracy alignment. Numbers are reported with mean values and standard deviation computed over 5 random runs. Bold numbers are best results at a given coverage level across all methods and underlined numbers are best results for hods relying on a single training run only. Datascts are consistent with Feng et al. 2003
Checkpoint Weighting Sensitivity. One important hyper-parameter of our method is the weighting of intermediate predictions. Recall from Scction 3 that SPTD approximates the expected stability for correctly classified points via a weighting function u, = (+)² In Figure 5 in the Appendix, we observe that SPTD is robust to the choice of and that & E [1, 3] performs best. At the same time, we find that increasing k too much leads to a decrease accuracy at medium coverage levels. This result emphasizes that (i) large parts of the training process contain valuable signals for selective classification; and that (ii) early label disagreements arising at the start of optinization should be de-emphasized by our method.
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CIFAR-10
CIFAR- -100
Food101
StanfordCars
Figure 3: Most characteristic examples across datasets. For each dataset, we show the samples with the most stable and most unstable (dis-) agreement with the final label along with their corresponding a indicator function. Correct points are predominantly characterized by disagreements early in training while incorrect points change their class label throughout (but importantly close to the end of) training. We provide additional examples from all datasets in Figure 20 in the Appendix.
CIFAR-10
CIFAR-100
StanfordCars
Food101
10
correct
6
25
incomeet
10
0
0
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5
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0.000
0.00
0.000
0.00
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400
0
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400
0
100
200
0
200
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600
Figure 4: Distribution of g for different datasets and selective classification methods. Since all methods are designed to address the selective prediction problem, they all manage to separate correct from incorrect points (albeit at varying success rates). We see that SPTD spreads the scores for incorrect points over wide range with little overlap. We observe that for SR, incorrect and correct points both have their mode at approximately the same location which hinders performative selective classification. Although SAT and DE show larger bumps at larger score ranges, the separation with correct points is weaker as correct points also result in higher scores more often than for SPTD.
Checkpoint Selection Strategy. The second important ppper-parameter of our method is the checkpoint selection strategy. In particular, to reduce computational cost, we study the sensitivity of SPTD with respect to the checkpointing resolution in Figure 6 Our experiments demonstrate favorable coverage/erro trade-offs between 25 and 50 checkpoints when considering the full coverage spectrum. However, when considering the high coverage regime in particular (which is what most selective prediction works focus on), even sub- sampling 10 intermediate models is sufficient for SOTA selective classification. Hence, with only observing overhead reduces to only 10 forward passes at test time when the goal is to reject at most 30% - 50% of incoming data points. In contrast, DE requires to first train E models (with E = 10 being a typical and also our particular choice for DE) and perform inference on these E models at test time. Further increasing the checkpointing resolution does offer increasingly diminishing returns but also leads to improved accuracy-coverage trade-offs, especially at low coverage.
the training stage, our hod's computationns
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CIFAR-10 -0
CIFAR- 100
Food101
StanfordCars
1.00
1.0
1.0
1.0
Accuracy
5
0.95
10
0.8
0.8
0.8
0.0
1.0
0.5
1.0
0.0
0.5 Coverage
1.0
0.0
0.5 Coverage
Coverage
Coverage
Figure 5: Coverage/erroi trade-off of SPTD for varying checkpoint weighting k as used in ut. We observe strong performance for E [1,3] across datasets.
CIFAR-10
CIFAR- 100
Food101
StanfordCars
1.00
1.0
1.0
Accuracy
0.95
full
200
50
25
0.8
100
10
0.8
0.0
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0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
Coverage
Coverage
Coverage
Coverage
Figure 6: Coverage/error trade-off of SPTD for varying checkpoint counts. SPTD delivers consistent performance independent of the checkpointing resolution at high coverage. At low coverage, more detailed characlerization of training dynamics helps.
Examining the Convergence Behavior of Training and Test Points. The effectiveness of SPTD relies on our hypothesis that correctly classified points and incorrectly classified points exhibit distinct training dynamics. We verify this hypothesis in Figure 7 where we examine the convergence behavior of the disogreement, distributions of correct. (ct²) incorrect (itt) training and correct (ct*) incorrect (1te test points. We observe that the expected disagreement for both correctly classified training of and test points c points converge to over the course of training. The speed of convergence is subject to the difficulty of the optimization problem with more challenging datasets exhibiting slower convergence in predicted label disagreement. We also see that the variancos follow analogous decreasing trend. This indicates that correctly classified points converge to the final label quickly and fast convergence is strongly indicative of correctness. Furthermore, the overlap suggests that correct test points are more likely to be forgeable as their dynamics look indistinguishable to correct training points (recall Section 3.3 on the connection between our method and forging). In contrast, incorrectly classified points it" and it" show significantly larger mean and variance levels. This clear separation in distributional evolution patterns across correct and incorrect points leads to strong selective prediction performance in our SPTD framework.
Detection of Puttof--Distribtion and Adversarial Examples. Out-of-distribution (OOD) and ad- versarial example detection are important disciplines trustworthy ML related to selective prediction. We therefore provide preliminary evidence in Figure [ that our method can be used for detecting OOD and adversarial examples. While these results are encouraging, we remark that adversarial and OOD samples are less well defined as incorrect data points and can come in a variety of different flavors (i.e., various kinds of attacks or various degrees of OOD-ness). As such, we strongly believe that future work is needed to determine whether a Beaining-dynamics-bbe based approach to selective prediction can be reliably used for OOD and adversarial sample identification.
Cost vs Performance Tradcoff. In Table 2 we report both the time and space complexities for all SC methods at training and test time along with their selective classification performance as per our results in Table [ and Figure D We denote with E the number of DE models and with T the number of SPTD chockpoints. Although SR and SAT are the cheapest methods to run, they also perform the poorest SC. SPTD is significantly cheaper train than DE and achieves competitive performance at E. Although DE+SPTD the most expensive model, it also provides the strongest performance.
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d correct training points
f incorrect training points
4 correct test points
f
incorrect test points
CIFAR-I0
CIFAR-100
Food101
StanfordCars
1.0
1.0
0.5
0.5
0.0
0.2
0.2
0.2
0.1
0.0
0.0
0
1000
1000
1000
2000
500
Chockpoints
Checkpoints
Checkpoints
Checkpoints
Figure 7: Monitoring expectations E|| and variances V[:] for correct correct/incorret /incorrect training and test points. We observe that correctly classified points (cold colors) have both their expectations and variances quickly decreasing to 0 training progresses. Incorrectly classified points (warm colors) both exhibit large expectations and variances and stay elevated over large periods.
CIFAR1O OOD
CIFARIO ADV
CIFARIOD OOD
CIFARIO0
ADV
Denett
cod
Density
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adv
Donsity
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0.0
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Filln Preitive Rate
Falen Positive Rate
Palse Positive Rate
Falso Preitive Rote
Figure 8: Performance of SPTD on out-of-distribution (OOD) and adversarial sample detection. The first row shows the score distribution of the in-distribution CIFAR-10/100 test set vs the SVHN OOD test set or set consisting of adversarial samples generated via a PGD atlack in the final model. The second row shows the effectiveness of thresholding mechanism by computing the area under the ROC curve. Our score enables separation of anomalous data points from in-distribution test points.
4.2 Regression Experiments
Datasets. Our experimental suite for regression considers the following datasets: California housing
dataset (Pace & Barry 1997) (N = 20640, D = 8), the concrete strength dataset (Ych 2007 (N = =1030,
D = 9), and the fish toxicity dataset Ballabio et al. 2019 (N 546, D ===99 9).
Model Setup & Baselines. We split all datasets into 80% training and 20% test sets after random shuffle. Then, we train fully connected neural network with layer dimensionalities D + 10 + 4 + 1. Optimization is performed using full-batch gradient descent using the Adam optimizer with Icarning rate 10-2 over 200 epochs and weight decay 10-2 We consider the following baseline methods for rejecting input samples: (i) approximating the predictive variance using deep ensembles (DE) (Lakshminarayanan et al.] 2017; Zaoui et al.] (020)); (ii) SelectiveNet (SN) which explicitly optimizes utility given a desired coverage constraint; and (ii) training the model with Gaussian parametric output distribution (ODIST)
via maximum likelihood maximization (Alexandrov al. 2019).
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M4 Hourly
M4 Daily
M4 Weekly
Hospital
MSIS
50
30
20
5.0
SPTD
25
20
10
2.5
DE
ODIST
0.5 Coverage
1.0
0.5
1.0
0.5
1.0
0.5
1.0
Coverage
Coverage
Coverage
Figure 9: MSIS/ trade-off across various time series prediction datasets, SPTD offers comparable performance to DE but provides improved results at low coverage.
Method
Train Time
Train Space
Inf Time
Inf Space Rank
SR
((1)
0(1)
O(1)
O(1)
5
SAT
0(1)
0(1)
0(1)
0(1)
DE
0(E)
O(E)
O(E)
O(E)
=2
SPTD
((1)
O(T) O(T) O(T)
=2
DE+SPTD
O(b) O O(ET)
O
1
Table 2: Cost vs performance tradeoff in terms of training time/space, inference time/space and the performance rank. SPTD is comparable in performance (at T F) and cheaper to train than DE. DE+SPTD is the most expensive model, but delivers the best performance across datasets.
Main results. We document results in Figure 10 We that the ODIST only delivers subpar results (likely due mis-calibration) and does not provide meaningful signal for selective prediction. On the other hand, DE and SPTD perform comparably with SPTD outperforming DE at low coverage. We stress again that SPTD's training cost is significantly cheaper than DE's while matching the inference-time cost when sub-sampling a reduced set of checkpoints.
4.3 Time Series Experiments
Datasets. As part of our time series experiments, mainly consider the M4 forecasting competition dataset (Makridakis et al. 2020) which contains time series aggregated at various time intervals (e.g., hourly, daily). In addition, we also provide experimentation on the Hospital dataset (Hyndman) 2015
Models & Setup. Our experimentation is carried out using the GluonTS time series framework A Aexan- drov et al. 2019). We carry out our experimentation using the DeepAR. model (Salinas et al. 2020), a recurrent neural network designed for time series forecasting. We train all models over 200 epochs and eval- uate performance using the mean scaled interval score (MSIS) performance metric (Makridakis et al. 2020 Our baselines correspond to the output parameterization using Student-t distribution (ODIST).
presented for regression in Section 4.2 deep ensembles (DE). and
Main results. Our time series results are shown in Figure and are consistent with results for regression: ODIST does not provide mcaningful signal for selective prediction while SPTD and DE perform similarly well. SPTD further improves results over DE at low converge.
5
Conclusion
In this work we have proposed SPTD, selective prediction technique that relies measuring prediction instability of test points over intermediate model states obtained during training. Our method offers several advantages over previous works. In particular (i) it can be applied to all existing models whose checkpoint were recorded (hence the potential for immediate impact); (ii) it is composable with existing selective predic- tion techniques; (iii) can be readily applied to both discrete and real-valued prediction problems; and (iv) it is more computationally efficient than competing ensembling based approaches. We verified the performance of SPTD using extensive empirical evaluation, leading new state-of-the-art performance in the field.
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California Housing
Concrete Strength
Fish Toxicity
0.9
0.95
0.8
SPTD
0.90
DE
0.7
0.8
0.85
ODIST
0.5
1.0
0.5
1.0
0.5 Coverage
1.0
Coverage
Coverage
Figure 10: R²/ /coverage trade-off across various regression datascts. SPTD offers comparable per- formance to DE but provides improved results at low coverage.
Beyond our work, expect training dynamics information to be useful for identifying and mitigating other open problems in trustworthy machine learning such as (un)fairness, privacy, and model interpretability.
Acknowledgements
This work was supported by CIFAR (through Canada CIFAR AI Chair), by NSERC (under the Discovery Program), and by gift from Ericsson. Anvith Thudi is supported by a Vanier Fellowship. We are also grateful to the Vector Institute's sponsors for their financial support. In particular, we thank Roger Grosse, Chris Maddison, Franziska Boenisch, Natalie Dullerud, Mohammad Yaghini, Sierra Wyllie, Jonas Guan, Michael Zhang, and Tom Ginsberg for fruitful discussions. We would also like thank the Vector ops and services teams for making the office such a wonderful place to work, and for the (limitless) free hot chocolate.
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A Alternate Metric Choices
We briefly discuss additional potential metric choices that we investigated but which lead to selective clas- sification performance worse than main method.
A.1
Jump Score Sjmp
We also consider a score which captures the level of disagreement between the prodicted label of two successive intermediate models (i.e., how much jumping occurred over the course of training). For je = 0 iff ft( =
x and 1: = otherwise we can compute the jump score as 8jmp = 1 - Σ Urje and threshold it as in
3.3.2
and $ 3.3.3
A.2 Variance Score Svar for Continuous Metrics
We consider monitoring the evolution of continuous metrics that have been shown to be correlated with example difficulty. These metrics include (but are not limited to):
Confidence (conf): max.ey ft(x)
Confidence gap bet lop 2 most confident classes (gap): maxcey [((x) maxe/g f((x))
f((x), Eutropy (ent)): f log (f(()))c
Jiang et al. 22020) show that example difficulty is correlated with confidence and entropy. Moreover, they find that difficult examples are learned later in the training process. This observation motivates designing score based on these contimous metrics that penalises changes later in the training process more heavily. We consider the maximun softmax class probability known as confidence, the negative entropy and the gap between the most confident classes for each example instead of the model predictions. Assume that any of these metrics given by a sequence 2 f21...., 2T) obtained from T intermediate models. Then we can capture the uniformity of z via a (weighted) variance score Svar E, we( - 11²² for mean μ == 7 Ee ze and [W1,....,yy wT].
increasing weighting sequence
In order to show the effectiveness of the variance score Svar for continuous metrics, provide simple bound on the variance of confidence maxvey fx( in the final checkpoints of the training. Assuming that the model has converged to a local minima with a low learning rate, we can assume that the distribution of model weights can be approximated by a Gaussian distribution.
We consider linear regression problem where the inputs are linearly separable.
Lemma 2. Assume that. we have some Gaussian prior on the model parometers in the logistic regres- sion setting across m final checkpoints. More specifically, given total checkpoints of model parameters {Wi, w WT} we have p(W w.) = NWw μ,, s, checkpoints of the model are sampled from this distribution. We show that the variance of model confidence max ply æz; w:) for a datapoint (æi, classifying this example by the optimal weights.
for E (T m m++1 l.......777 and we assume that final
be upper bounded by a factor of probability of correctly
Proof. We first compute the variance of model predictions p(y, H; W) for a given datapoint (x), Following previous work (Schein & Ungar 2007 Chang et al. 2017). the variance of predictions over these checkpoints can be estimated follows:
Taking two terms in Taylor expansion for model predictions we have p(y, æ; W) p(u: æ, w) +
(w)) (W- - w) where W and w are current and the expected estimate of the parameters and (: = p(u, aa w)(( p(y æ,, w))x; e is the gradient vector. Now we can write the variance with respect to the model prior as:
V (p(s: a, W)) V (g:(w)) (W -w)) (ur) F-1
(=-1g(((()) (u()
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where F is the variance of posterior distribution p(W X,Y) ~ N(W F. This suggests that the
variance of probability of correctly classifying æ, is proportional to p(y: T; w²) p(u: 4; w)) Now
we can bound the variance of maximum class probability or confidence as below:
max ply ai, W.) < V (p(yi æ,, W)) + V (x(- y/ vE(-1.1)
0i, W))
2p(w 2 z., 2l p 3i w)) F x.
We showed that if the probability of correctly classifying an example given the final estimate of model parameters is close to one, the variance of model predictions following a Gaussian prior gets close to zero, we expect similar behaviour for the variance of confidence under samples of this distribution.
B Extension of Empirical Evaluation
B.1 Full
We document full hyper-parameter settings for our method (SPTD) as well as all baseline approaches Table
Table 3: Hyper-parameters used for all algorithms for classification.
Dataset
SC Algorithm
Softmax Response (SR)
N/A P = 100
Self-Adaptive Training (SAT)
CIFAR-10
Deep Ensemblee (DE)
E = 10
Selective Prediction Training Dynamics (SPTD)
T 1600,
Softmax Response (SR)
N/A
Self-Adaptive Training (SAT)
P P= 100
CIFAR-100
Deep Ensembles (DE)
E = 10
Selective Prediction Training Dynamics (SPTD)
T = 1600,
Softmax Response (SR)
N/A
Self-Adaptive Training (SAT)
P P= 100
Food101
Deep Ensembles (DE)
E = 10
Selective Prediction Training Dynamics (SPTD)
T = 2200, =
Softmax Response (SR)
N/A
Self-Adaptive Training (SAT)
P = 100
StanfordCars
Deep Ensembles (DE)
E = 10
Selective Prediction Training Dynamics (SPTD)
T 800, h: =
B.2
Additional Selective Prediction Results
B.2.1
Extended Synthetic Experiments
We extend the experiment from Figure 2 to all tested SC methods Figure result using Bayesian Linear Regression in Figure 12
11
We also provide an extended
B.2.2
CIFAR-100 Results With ResNet-50
We further provide full set of results using the larger ResNet-50 architecture CIFAR-100 in Figure
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Table Selective accuracy achieved across coverage levels for CIFAR-100 with ResNet-50.
Cov.
SR
S
DE
SPTD
DE+SPTD
100
77.0 (=0.0)
77.0 (=0.0)
77.0 (#0.0)
77.0 (=0.0)
77.0 (#0.0)
90
79.2 (# 0.1)
79.9 (# 0.1)
(+
81.4 0.1)
82.1 (# 0.1)
80
83.1 (+ 0.0)
83.9 (# 0.0)
85.7
85.6 85.6 0.1)
86.0 (# 0.2)
70
87.4 (# 0.1)
88.2 (# 0.1)
89.6 0.1)
89.7 (# 0.0)
89.8 (# 0.1)
90.5 (# 0.0)
90.8 0.2)
90.7 0.2)
90.6 (# 0.0)
90.9 (# 0.1)
50
93.4 0.1)
93.8 0.0)
(=
95.1 ( 0.0)
95.4 (# 0.0)
40
95.4 0.0)
95.5 0.1)
97.1 (# 0.1)
97.2 (# 0.1)
97.2 (# 0.0)
30
97.4 (+ 0.2)
97.7 0.0)
98.6 (+ 0.1)
98.6 (# 0.1)
98.7 (# 0.0)
20
97.9 (+ 0.1)
98.4 (# 0.1)
99.0 (+ 0.0)
99.2 (# 0.1)
99.2 (# 0.1)
10
98.1 (+ 0.0)
98.8 0.1)
99.2 (+ 0.1)
99.4 (# 0.1)
99.6 (# 0.1)
B.2.3
Applying SPTD on Top of SAT
Our main set of results suggest that applying SPTD on top of DE further improves performance. The same effect holds when applying SPTD on top of non-ensemble-baset methods such as SAT. We document this result in Figure
1
B.2.4
Ablation on
We provide a comprehensive ablation on the weighting parameter Figures and 14
B.2.5
Comparison With Logit-Variance Approaches
We showease the effectiveness of SPTD against LOGITVAR (Swayamdipta et al., 2020), an approach that also computes predictions of intermediate models but instead computes the variance of the correct prediction. We adapt this method to our selective prediction approach (for which true labels are not available) by computing the variance over the maximum predicted logit instead of the truc logit. In Figure 15 we see that the weighting of intermediate checkpoints introduced by SPTD leads to stronger performance over the LOGITVAR baseline approach.
B.2.6 Estimating - on Validation VS Test Data
Consistent with prior works [Geifman El-Yaniy 2017 Liu et al. 2019 Huang el al. 2020 Feng el al. 2023), we estimate directly on the test set. However, a realistically deployable approach has compute thresholds based validation set for which labels are available. In the case of selective classification, the training, validation, and test sets follow the iid.. assumption, which means that approach that sets the threshold based on validation set should work performantly on test set, too. Under consistent distributional assumptions, estimating thresholds on a validation set functions as an unbiased estimator of accuracy/cowee /coverage tradeoffs on the test set. By the same virtue, setting thresholds directly on the test set and observing the SC performance on that test set should be indicative for additional test samples beyond the actual provided test set. It important to remark that the validation should only be used for setting the thresholds and not for model selection early stopping which would indeed cause a potential divergence between SC performance the validation and test sets. Further note that violations of the i.i.d assumption can lead to degraded performance due to mismatches in attainable coverage explored in Bar-Shalom et al. 2023
To confirm this intuition, we present an experiment in Figure 16 and Figure 17 where we select 50% of the samples from the test set as our validation (and maintain the other 50% samples as our new test set). We first generate 5 distinct such validation-test splits, set the thresholds for 7 based on the validation set, and then evaluate selective classification performance the test set by using the thresholds derived from
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the validation set. We compare these results with our main approach which sets the thresholds based on the test set directly (ignoring the validation set). We provide an additional experiment where we partition the validation set from the training set in Figure 18 We see that the results are statistically indistinguishable from each other, confirming that this evaluation practice valid for the selective classification setup we consider.
B.2.7 Comparing MMAX and 85UM
As per our theoretical framework and intuition provided in Section 3 the sum score 8SUM should offer the most competitive selective classification performance. We confirm this finding in Figure 19 where we plot the accuracy / coverage curves across all datasets for both SMAX and $SUM- Overall, we find that the sum scorc SSUM consistently outperforms the more noisy maximum score SMAX.
B.3 The Importance of Accuracy Alignment
Our results in Table rely on accuracy alignment: We explicitly make sure to compare all methods on an equal footing by disentangling selective prediction performance from gains in overall utility. This is done by early stopping model training when the accuracy of the worst performing model is reached.
We expand on the important point that many previous approaches conflate both (i) generalization perfor-
mance and (ii) selective prediction performance into single score: the area under the eecuracy/coverze curve. This metric can be maximized either by improving generalization performance (choosing different architectures or model classes) by actually improving the ranking of points for selective prediction (accept- ing correct points first and incorrect ones last). As raised by a variety of recent works Geifman et al. (2018) Rabanser et al. (2023): Cattelan & Silva (022)), it is impossible and problematic to truly assess whether a method performs better at selective prediction (i.c., determining the correct acceptance ordering) without normalizing for this inherent difference yielded as a side effect by various SC methods. In other words, an SC method with lower base accuracy (smaller correct set) can still outperform another SC method with higher accuracy (larger correct set) in terms of the selective acceptance ordering (an example of which is given in Table 3 of Liu et al. (2119)) Accuracy normalization allows us to eliminate these confounding effects between full-coverage utility and selective prediction performance by identifying which models are better at ranking correct points first and incorrect ones last. This is of particular importance when comparing selective prediction methods which change the training pipeline in different ways, as is done in the methods presented Table [
However, when just comparing SPTD to one other method, we do not need to worry about accuracy normalization. Showcasing this, we run SPTD top of unnormalized SAT+ER+SR run and provide these experiments in Figure 13 We see that the application of SPTD top of SAT+ER+SR allows to further boost performance (similar to the results where we apply SPTD on top of DE in Table [ So to conclude, experimentally, when using the best model, we see that SPTD still performs better at selective prediction than the relevant baseline for that training pipeline. We wish to reiterate that this issue of accuracy normalization highlights another merit of SPTD, which that can easily be applied on top of any training pipeline (including those that lead to the best model) and allows easy comparison to the selective classification method that training pipeline was intended to be deployed with.
B.4 Evaluation using other performance metrics
We further provide results of summary performance metrics across datasets in Table
The area. under the accuracy-coverage curve (AUACC) as discussed in Geifman al. 20118)
The area under the receiver operating characteristic (AURDC) as suggested by Galil et al. 2023
The accuracy normalized selective classification score (ANSC) from Geifman et al. 2011 and Ra- banser et al. (2023).
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Table 5: Evaluation of SC approaches using various evaluation metrics.
Dataset
Method
1 AUACC
ANSC
AUROC
SR
0.053 + 0.002
0.007 + 0.000
0.918 + 0.002
SPTD
0.048 = 0.001
0.004 + 0.000
0.938 + 0.002
CIFARIO
DE
0.046 = 0.002
0.004 + 0.000
0.939 + 0.003
SAT
0.054 = 0.002
0.006 + 0.000
0.924 + 0.005
DG
0.054 + 0.001
0.006 + 0.000
0.922 + 0.005
SR
0.181 + 0.001
0.041 + 0.001
0.865 + 0.003
SPTD
0.174 = 0.002
0.037 + 0.000
0.872 + 0.002
CIFAR100
DE
0.159 = 0.001
0.030 + 0.001
0.880 + 0.003
SAT
0,180 = 0.001
0.041 + 0.001
0.866 + 0.003
DG
0.182 = 0.001
0.041 + 0.001
0.867 0.002
SR
0.020 = 0.002
0.001 + 0.000
0.986 = 0.003
SPTD
0.019 + 0.002
0.001 + 0.000
0.986 + 0.005
GTSRB
DE
0.015 = 0.001
0.001 + 0.000
0.986 = 0.002
SAT
0.027 = 0.001
0.001 + 0.000
0.984 + 0.002
DG
0.019 = 0.003
0.001 + 0.000
0.986 0.002
SR
0.027 = 0.000
0.006 + 0.001
0.895 + 0.004
SPTD
0.025 = 0.003
0.003 + 0.001
0.932 + 0.005
SVHN
DE
0.021 + 0.001
0.005 + 0.000
0.912 + 0.003
SAT
0.028 + 0.001
0.006 + 0.000
0.895 0.002
DG
0.026 + 0.001
0.007 + 0.000
0.896 0.006
B.5
Evaluation of more competing approaches
We further compare method with two contemporary selective prediction approaches:
AUCOC (Sangalli et al. 2024): : This work uses a custom cost function for multi-class classification that accounts for the trade-off between neural network's accuracy and the amount of data that requires manual inspection from domain expert.
CCL-SC eWu al. 2024): This work proposes optimizing feature layers to reduce intra-class variance contrastive learning.
Across both methods, we find that they do not outperform ensemble-based methods like DE and hence also not outperform SPTD. See Table for detailed results.
26

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Published in Transactions on Machine Learning Research (02/2025)
Table 6: Selective accuracy achieved across coverage levels for AUCOC and CCL-SC. Similar as Table I Neither AUCOC nor CCL-SC is able to outperform DE or SPTD. Bold mumbers are best results at given coverage level across all methods and underlined numbers are best results for methods relying on single training run only.
Coverage
AUCOC
CCL-SC
SPTD
DE
100
92.9 (1:0.)
92.9 (=0.0)
92.9 (+0.0)
92.9 (+0.0)
90
96.0 (+0.1)
95.9 (+0.2)
96.5 (+0.0)
96.8 (+0.1)
80
98.1 (#0.2)
98.0 (#0.3)
98.4 (#0.1)
98.7 (#0.0)
70
99.0 (#0.3)
98.5 (#0.2)
99.2 (+0.1)
99.4 (#0.1)
CIFAR-10
60
99.3 (+0.1)
99.1 (+0.2)
99.6 (+0.2)
99.6 (+0.1)
50
99.4 (#0.2)
99.0 (#0.3)
99.8 (=0.0)
99.7 (#0.0)
40
99.5 (#0.1)
99.4 (#0.2)
99.8 (+0.1)
99.8 (40.0)
30
99.5 (#0.2)
99.2 (#0.3
99.8 (#01)
99.8 (#0.0)
20
99.6 (+0.1)
99.4 (+0.2)
100.0
(+0.0)
99.8 (+0.0)
10
99.7 (#0.0)
99.4 (+0.1)
100.0
(+0.0)
99.8 (+0.0)
100
75.1 (4:0.0
75.1 (+.0.)
75.1
(+000)
75.1 (4.0.0
90
78.7 (+0.2)
76.5 (+0.3)
80.4 (+0.0)
80.2 (+0.0)
80
83.2 (#0.1)
82.2 (#0.2)
84.6 (=0.1)
84.7 (#0.1)
CIFAR-100 100
70
87.4 (#0.1)
86.1 (+0.2)
88.7 (=00))
88.6 (#0.1)
60
89.8 (+0.2)
88.6 (+0.3)
90.1 (+-0.))
90.2 (+0.2)
50
93.3 (#0.1)
92.1 (#0.2)
94.6 (+0.0)
94.8 (#0.0)
40
95.9 (+002)
95.2 (+0.3)
96.9 (+0.1)
96.8 (+0.1)
30
98.2 (#0.1)
96.6 (#0.2)
98.4 (#0.1)
98.4 (#0.1)
20
98.6 (+0.2)
98.4 (+0.3)
98.8 (+0.2)
99.0 (+0.0)
10
98.8 (+0.1)
98.7 (+022)
99.4 (+0.1)
99.2 (+0.1)
27

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Published in Transactions on Machine Learning Research (02/2025)
a
0.5
a
1.00
2.5
2.5
2.3
2.5
0.75
0.0
0.0
0.0
0.0
0.50
0.25
-2.
-25
-2.5
-2.5
0.00
0
5
0
2.5 0.0 2.5
2.50.0 2.5
1.00
2.5
2.5
2.5
2.5
0.75
0.0
0.0
0.0
0.0
0.50
0.25
2.5
2.5
2.5
2.5
0.00
5
5
5
-2. 0.0
2.5
-2.50.0 2.5
2.5
2.5
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2.5
0.75
0.0
0.0
0.0
0.0
0.50
0.25
-25
-25
-2,
2.5
0.00
5
0
5
5
0
2.5 0.0 2.5
2.5000 2.5
1.00
2.5
2.5
2.5
2.5
0.75
0.0
0.0
0.0
0.0
0.50
0.25
2.5
2.5
2.5
2.5
0.00
5
5
5
0
-2.5 0.0
2.5
-2.50.0 2.5
15000
SR
6
4
10000
SPTD
SAT
5000
DE
2
2
2-
0
0
Figure 11: Extended Gaussian experiment. The first row corresponds the anomaly scoring result of SR, the second the result of SAT, the third the result of DE, and the fourth the result of SPTD. The bottom row shows the score distribution for each method over the data points. We see that all methods reliably improve over the SR baseline. At the same time, we notice that SAT and DE still assign higher confidence away from the data due to limited use of decision boundary oscillations. SPTD addresses this limitation and assigns more uniform uncertainty over the full data space.
28

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Published in Transactions on Machine Learning Research (02/2025)
1.00
2.5
2.5
2.5
2.5
0.75
0.0
0.0
0.0
0.0
0.50
-2.5
2.5
-2.
-2.
0.25
0.00
0
2.5 0.0
2.5
2.5000 2.5
Figure 12: Bayesian linear regression experiment on Gaussian data. Results comparable to DE
CIFAR-I0
CIFAR-100
StunfordCars
1.0
Accurccy
SAT
0.8
SPT+
0.0
0.5
1.0
0.0
0.0
0.5
1.0
CIFAR-10
CIFAR-100
StanforCCar
Coverage
Coverage
Covenge
Figure 13: Applying SPTD on top of SAT. Similar as with DE, we observe that the application of SPTD improves performance.
1.0
1.00
Weighting
Acsunacy
unif
AUC
0.986
0.5
0.1
0.2
0.5
0.984
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
UnIF
0.1
02
0.5
Fraction of checkpoints
Coverage
Figure 14: Extended ablation results on i on CIFAR-10. We now also consider E (0, 1] as well as a uniform weighting assigning the same weight to all checkpoints. We confirm that a convex weighting yiclds best performance.
CIFAR-10
CIFAR-100
StanfordCars
1.0
Accurac
0.9
1.DCTIVAR
0.8
0.9
SPID
0.8
0.0
0.5
1.0
0.0
0.5
1.0
0.0
0.5
1.0
CIFAR-10
CIPAR-100
StanfoniCurs
Coverage
Coverage
Covenge
Figure 15: Comparison of LOGITVAR vs SPTD. We observe that SPTD, which incorporates weighting of intermediate checkpoints using V,, outperforms LOGITVAR.
29

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Published in Transactions on Machine Learning Research (02/2025)
CIFAR- 10
CIFAR- --0
StanfordCars
1.0
Accuracy
0.95
val sel
AUC
test sel
0,9
0.8
0.0
0.5
0.5
1.0
0.0
0.5
1.0
CIFAR-U
CIFAR- 100
StanfordCars
Coverage
Coverage
Coverage
Devaity
0.02
0:01
0.01
0.00
0.00
0.00
200
400
0
2001
400
600
100)
200
300
CIFAR 10
CIFAR 100
StanfarCCars
Figure 16: SPTD /coverage trade-offs and score distributions on test data obtained by computing t on a validation set or directly on the test set. The first row shows the obtained accu- racy/coverage trade-offs with the respective AUC scores. In the second row, we show the score distribution for both the picked validation and test sets, along with p-values from a KS-test to determine the statistical closeness of the distributions. Overall, we observe that both methods are statistically indistinguishable from each other.
CIFAR-10
CIFAR-100
StanfordCars
1.00
1.0
1.0
Accuracy
val sel
0.8
ALC
0.9
0.0
0.5
1.0
0.5
1.0
0.0
0.5
1.0
CIFAR-10
CIFAR-1100
StanfordCar
Coverage
Coverage
Coverage
Density
val
0.01
0
25
50
50
100
200
300
CIPAR-10
CIFAR-190
StanfordCars
8
Figure 17: SAT accuracy coverage trade-offs and score distributions on test data obtained by computing validation set directly the test set. Same Figure 16 but with SAT.
CIFAR-10
CIFAR-100
StanfordCars
1.0
1.0
1.0
1.0
Accuracy
val sel (from train)
AUC
test sel
0.9
0.0
0.5
1.0
0.0
0.5 Coverage
1.0
0.0
0.5
1.0
CIFAR-10
CIFAR-
StanfordCars
Coverage
Coverage
Densiy
0.02
0.01
0.00
0.00
0.00
200
400
0
200
400
600
0
100
200
300
CIFAR-10 10
SarnfondCars
CIFAR-
8
Figure 18: SPTD accuracy /coverae trade-offs and score distributions on test data obtained by computing on validation set or directly on the test set.. Same as Figure 16 but with the validation set is taken from the original training set.
30

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Published in Transactions on Machine Learning Research (02/2025)
CIFAR-I0 10
CIFAR- 100
Food101
StanfordCars
1.00
1.0
1.0
1.0
Accuracy
0.95
Sum max
0.8
0.8
0.8
0.0
0.5
1.0
0.5
1.0
0.0
0.5 Coverage
1.0
Coverage
Coverage
Coverage
Figure 19: Comparing MAX and SSUM performance. It is clear that SSUM effectively denoises SMAX
31

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Published in Transactions on Machine Learning Research (02/2025)
CIFAR-10
(b) CIFAR-100
(c) Food101
(d) StanfordCars
Figure 20: Additional individual examples across datasets.
32


Reviews:
[{"review_id": "koKRJdCpmE", "reviewer": "Reviewer_BCan", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths**\n\n1. High Compatibility and Flexibility: Compatible with existing model architectures and training objectives, selective prediction methods without any additional training modifications. In addition, it is competent in SP tasks other than just classification: regression and time series prediction tasks. (I enjoy it! 😁)\n\n2. Rich experimental verification and reasonable conclusions\n\n**Weaknesses**\n\n1. Checkpoint Selection: The fixed schedule for choosing checkpoints may lead to suboptimal model performance.\n\n2. Hyperparameter Complexity: The method involves several hyperparameters, such as the number of checkpoints and weights for calculating the selection function, making it unclear which intermediate checkpoints are optimal for inference.\n\n3. Resource Overhead: The method incurs additional storage and inference costs due to the need for multiple intermediate models\n\n4. Lack of citations from papers in similar fields:\n\n   [1] Assessing Generalization of SGD via Disagreement, ICLR 2022.\n\n   [2] Agreement-on-the-Line Predicting the Performance of Neural Networks under Distribution Shift, Neurips2022.\n\n   [3] Energy-based Automated Model Evaluation, ICLR 2024.\n\n   [4] CAME: Contrastive Automated Model Evaluation, ICCV 2023.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This article addresses the problem of selective prediction by monitoring prediction instability during training. Selective prediction aims to reject inputs that the model may misclassify, balancing input coverage with model utility. The proposed method, SPTD, leverages the dynamics of neural network training. Unlike existing methods that rely solely on the final model, SPTD utilizes intermediate models generated during iterative optimization processes like SGD. A prediction instability score is calculated to measure the reliability of predictions, with higher scores indicating greater uncertainty. This score informs a selection function that decides whether to predict a test point. SPTD is applicable to classification, regression, and time series forecasting problems. Experimental results demonstrate that SPTD achieves new state-of-the-art accuracy/coverage trade-offs across multiple datasets, significantly improving prediction accuracy while maintaining high coverage compared to existing methods.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "Please Check Weakness, by the way, this paper has some confused descriptions, such as \"and disregard the rich set of statistics that are contained (or derivable) from this model sequence.\", please revise the writing of the paper, Make it easier for readers to understand.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "LPZiLcaaoe", "reviewer": "Reviewer_Qa8X", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n- The method extends the application of selective predication to regression and time-series prediction.\n- The ablations are well-justified. The paper is well-written.\n\n\n**Weaknesses:**\n- This paper does not justify the selection of checkpoints as a reasonable approach for measuring uncertainty in the ensemble method of sub-models.\n- In SPTD, saving checkpoints exceeding 1k for uncertainty measurement is not very feasible in practice, especially when the model has a large number of parameters.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a novel method for selective prediction, termed SPTD, which is based on an ensemble approach that incorporates a gating function defined as $g(x) = \\sum_{1 \\leq t \\leq T} v_t a_t(x)$. SPTD demonstrates strong performance across various tasks, including regression, time series analysis, and classification.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- A discussion on the technical differences between SPTD and Deep Ensemble methods.\n- A discussion on the possible feasibility of scaling the performance of SPTD to models such as ViT, which possess significantly more parameters.", "broader_impact_concerns": "None", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "8hf8mPIL1j", "reviewer": "Reviewer_5C8G", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths:\n- The idea of using training dynamics for selective classification is interesting. \n- The experiments are extensive and demonstrate the advantage of the proposed method.\n- Sensitivity of the proposed method to it's hyperparameters is evaluated in ablation studies.\n\nWeakness:\n- Although the paper presents extensive experiments, it misses comparison with some recent works such as [1, 2].\n\n[1] Sangalli et al. Expert load matters: operating networks at high accuracy and low manual effort, Neurips 2023.\n[2] Wu et al. Confidence-aware Contrastive Learning for Selective Classification, ICML 2024\n\n- The paper mentions Agarwal et al. (2020) as work that is closest to the proposed method; however, the proposed method is not compared to it.\n\n- There are many \"loss medications\" based methods in the literature as also mentioned in the paper. And the proposed method can be complementary to those and one can get checkpoints during training with different losses. I think it would be an interesting experiment to show the performance of the method when training with various losses.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper proposes a selective prediction method based on training dynamics. The method saves multiple checkpoints during training. Each test sample is passed through. Then, a score for selective classification is computed based on the consistency between the predictions at each checkpoint and the final prediction. The predictions for the samples that have high uncertainty are rejected. \n\nThe paper proposes different score functions for classification, regression, and time series prediction problems and performs extensive experiments to compare the proposed method with the existing methods on multiple benchmark datasets. The results demonstrate that the method improves the performance of the existing methods. Additionally, ablation studies are performed to show the effect of crucial hyperparameters of the method on the performance.", "claims_and_evidence_yn": "Yes", "claims_explanation": "", "audience_yn": "Yes", "audience_explanation": "", "requested_changes": "- Perform comparisons with more recent works such as Sangalli et al. and Wu et al.\n- Perform comparisons with the works that are close to the proposed method, e.g. Agarwal et al.\n- Demonstrate the performance of the method when the network is trained with different losses tailored for selective classification to demonstrate that the method can be complementary to the existing methods.", "broader_impact_concerns": "I don't have any broader impact concern.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: All reviewers commended the novelty of the proposed method, The paper is also well-written.  Two reviewers officially endorsed the acceptance. While the third reviewer (Reviewer BCan) did not provide the official recommendation, they commented in response to the authors' response that their concerns and questions were well-addressed and recommended its acceptance. 

Based on the reviewers' recommendations and my reading of the paper, I recommend its acceptance.
Decision: Accept (poster)