Deep End-to-End Survival Analysis with Temporal Consistency

Mariana Vargas Vieyra; Pascal Frossard

cs.LGOct 9, 2024

Deep End-to-End Survival Analysis with Temporal Consistency

Authors: Mariana Vargas Vieyra, Pascal Frossard

Organizations: EPFL

Abstract

In this study, we present a novel Survival Analysis algorithm designed to efficiently handle large-scale longitudinal data. Our approach draws inspiration from Reinforcement Learning principles, particularly the Deep Q-Network paradigm, extending Temporal Learning concepts to Survival Regression. A central argument in our method relies on temporal consistency, a recursive consistency condition between survival predictions made at consecutive states. Our framework introduces a delayed target network that decouples the generation of temporally consistent training targets from model updates, providing a more stable training signal while enabling mini-batch optimization and end-to-end training. Additionally, it supports arbitrarily complex architectures, enabling the modeling of intricate temporal dependencies. Through numerous experiments we provide empirical evidence demonstrating our framework's ability to exploit temporal consistency across datasets of varying sizes. Moreover, our algorithm outperforms benchmarks on datasets with long sequences, demonstrating its ability to capture long-term patterns. Finally, ablation studies show that the degree of target delay has a substantial effect on predictive performance and training variability. Across two distinct survival model families, delayed target updates consistently improve over the same-timescale baseline.

Figures & tables

Figure 1: Performance of event prediction in small datasets. We present results in terms of the CI (higher is better) and IBS (lower is better). The error bars represent the mean and standard deviation over thirty random seeds.

DATASET	SIZE	#STATIC	#TIME-VAR.	$H$
PBC2	312	3	12	16
AIDS	467	4	1	5
SmallRW	–	0	20	11
LargeRW	10k	0	50	100
LastFM	992	0	10	226
NASA	200	0	16	361

Table 1: Statistics of all datasets.

METRIC	DATASET	CELLS	LH	Inc-TCSR	D-TCSR	D $>$ Inc
$C(t)$	NASA	9	0 (0)	0 (0)	9 (5)	9 (9)
	Scania	9	0 (0)	0 (0)	9 (4)	9 (9)
	LastFM	9	0 (0)	0 (0)	9 (7)	9 (7)
	Large-RW	7	0 (0)	0 (0)	7 (7)	7 (7)
	Total	34	0 (0)	0 (0)	34 (23)	34 (32)
Brier	NASA	9	1 (0)	0 (0)	8 (5)	9 (9)

Table 2: Results across landmark–horizon configurations for LH. For the three-way comparison, entries report the number of configurations won by each method, with the number of significant wins in parentheses. A significant win means that the winning method outperforms both other methods in paired

t

-tests over 30 seeds (

p<0.05

). The final column reports the direct comparison between D-TCSR and Inc-TCSR as better (significant) . Higher

C(t)

and lower Brier score are better.

METRIC	DATASET	CELLS	DDH	Inc-TCSR	D-TCSR	D $>$ Inc
$C(t)$	NASA	9	0 (0)	0 (0)	9 (9)	9 (9)
	Scania	9	8 (2)	0 (0)	1 (0)	8 (6)
	LastFM	9	3 (1)	0 (0)	6 (4)	8 (8)
	Large-RW	7	2 (2)	0 (0)	5 (2)	7 (4)
	Total	34	13 (5)	0 (0)	21 (15)	32 (27)
Brier	NASA	9	0 (0)	0 (0)	9 (9)	9 (9)

Table 3: Results across landmark–horizon configurations for DDH. For the three-way comparison, entries report the number of configurations won by each method, with the number of significant wins in parentheses. A significant win means that the winning method outperforms both other methods in paired

t

-tests over 30 seeds (

p<0.05

). The final column reports the direct comparison between D-TCSR and Inc-TCSR as better (significant) . Higher

C(t)

and lower Brier score are better.

$H$	METRIC	$\tau=.01$	$\tau=.05$	$\tau=.1$	$\tau=.25$	$\tau=.5$	$\tau=1.0$
30	CI $\uparrow$	.785 $\pm$ .007	.786 $\pm$ .007	.803 $\pm$ .005	.805 $\pm$ .003	.802 $\pm$ .004	.795 $\pm$ .013
30	IBS $\downarrow$	.216 $\pm$ .009	.152 $\pm$ .003	.140 $\pm$ .003	.139 $\pm$ .002	.142 $\pm$ .003	.149 $\pm$ .008
50	CI $\uparrow$	.760 $\pm$ .012	.771 $\pm$ .008	.788 $\pm$ .004	.789 $\pm$ .003	.782 $\pm$ .005	.766 $\pm$ .011
50	IBS $\downarrow$	.261 $\pm$ .005	.162 $\pm$ .004	.150 $\pm$ .002	.149 $\pm$ .002	.156 $\pm$ .004	.173 $\pm$ .015
100	CI $\uparrow$	.682 $\pm$ .018	.712 $\pm$ .006	.726 $\pm$ .003	.711 $\pm$ .005	.701 $\pm$ .017	.659 $\pm$ .044
100	IBS $\downarrow$	.285 $\pm$ .007	.192 $\pm$ .005	.172 $\pm$ .002	.175 $\pm$ .002	.185 $\pm$ .008	.199 $\pm$ .014

Table 4: Ablation study over the target learning rate

\tau

for random-walk datasets with horizons

H\in\{30,50,100\}

. We report the mean and standard deviation over 30 runs of CI (higher is better) and IBS (lower is better). For each horizon and metric, the best mean, and every value not significantly different from it according to a paired

t

-test with significance level

\alpha=0.05

, is highlighted in bold.

Panel of Figure 2 — Figure 2: Variability of estimates for different values of $\tau$ with $\lambda=0.1$ for datasets with varying horizons: (a) $H=30$ , (b) $H=50$ , and (c) $H=100$ . The red line indicates the mean value.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

ALGORITHM	$\ell$ in MLE	$\lambda$	$\tau$	BATCH?
SA Init St.	$0$	–	–	$\surd$
SA Landm.	$>0$	$1$	–	$\surd$
TCSR (fitted)	$>0$	$\in[0,1]$	–	$\times$
Inc-TCSR	$>0$	$\in[0,1]$	$1$	$\surd$
D-TCSR	$>0$	$\in[0,1]$	$\in(0,1]$	$\surd$

Appendix

Table 5: Comparison of main hyperparameters of each algorithm. Inc-TCSR can be deemed a special case of D-TCSR where

\tau=1

Figure 3: Concordance Index (higher is better) of event prediction in big datasets for the LH model for each landmark and predictive horizon. All the results are based on best validation score and reported over 30 random seeds. Results that are significantly better than the baselines based on a paired

t

-test are highlighted with stars.

Figure 4: Brier score (lower is better) of event prediction in big datasets for the LH model for each landmark and predictive horizon. All the results are based on best validation score and reported over 30 random seeds. Results that are significantly better than the baselines based on a paired

t

-test are highlighted with stars.

Figure 5: Concordance Index (higher is better) of event prediction in big datasets for the DDH model for each landmark and predictive horizon. All the results are based on best validation score and reported over 30 random seeds. Results that are significantly better than the baselines based on a paired

t

-test are highlighted with stars.

Figure 6: Brier score (lower is better) of event prediction in big datasets for the DDH model for each landmark and predictive horizon. All the results are based on best validation score and reported over 30 random seeds. Results that are significantly better than the baselines based on a paired

t

-test are highlighted with stars.

	LH	DDH
Encoder	causal transformer (pre-norm, residual)	GRU + temporal attention
Width	64 (of 64, 128)	128 (of 64, 128)
Depth	2 layers (of 2, 4)	1 GRU layer
Attention	4 heads, key size 16	additive, width 16, over past states
Dropout	0.2	–
Output head	linear $\to$ $H$ hazard logits	linear $\to$ $H$ hazard logits

Appendix

Table 6: Selected architectures. Italics: values searched in the pilot. Both families use AdamW (learning rate

10^{-2}

, weight decay

10^{-4}

) and early stopping on the validation loss.

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