Efficient Learning of Survival Analysis Models Under Outcome-Dependent Truncation and Censoring

arXiv:2610.12538 · cs.LG · Submitted 2026-10-08 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "Efficient Learning of Survival Analysis Models Under Outcome-Dependent Truncation and Censoring".

Jane: The gist Modern machine learning models are typically trained on data assumed to be representative of the predictive tasks the model will face in the future,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we're looking at this paper called "Efficient Learning of Survival Analysis Models Under Outcome-Dependent Truncation and Censoring". It’s written by Anagnostides, Zilberstein, Kilic, and Sandholm. Jane, what is the basic problem they are setting out to solve here?

Jane: Well, the authors start by pointing out that modern machine learning models are usually trained on data that they assume is representative of what's coming in the future. But in real life, especially with practical datasets, this assumption often breaks down because observations get picked non-randomly. This paper focuses on outcome-dependent truncation where an observation only appears in the training data based on its label or value.

Lu: Essentially, they are connecting two separate areas of research: efficient learning when you have truncation and survival analysis. Existing methods for truncated survival analysis usually just rely on iterative procedures like expectation maximization, but this paper is trying to find a way to characterize the optimization landscape instead.

Meng: I’m wondering how much effort is needed here because these selection biases can really mess with high-stakes predictions, like treatment decisions or organ allocation. If you ignore the truncation mechanism during training, you can get serious bias propagated into your model’s results.

Lalam: This paper addresses that by proving that Cox regression can actually be learned efficiently even when the data is truncated and right-censored, which is a pretty big deal for real-world application.

The paper's summary: Tom: So, what’s the main finding they present in this paper about how to learn Cox regression under these specific conditions? They’re suggesting that first-order optimization can actually be used efficiently when the baseline hazard is known, even though there are some tricky non-convexities involved.

Jane: That sounds like a technical piece, but what does it mean for us as people who aren't deep in survival analysis? It’s about moving away from those slow, iterative methods like expectation maximization and finding a more direct way to get to the right parameters faster.

Lu: The core idea is that they show that even though the setup is non-convex, there’s a unique stationary point for the population negative log-likelihood at the true parameter value. And they prove something important about how much you move away from it.

Meng: What are we talking about when we talk about "population negative log-likelihood" and "stationary point"? Is this just some fancy way of saying the model is converging to the right answer? We need to know if this translates into actual predictive accuracy improvements for us.

Lalam: It does, because they show that when you use projected stochastic gradient descent, it converges directly to the true optimum. That’s a strong guarantee that we can trust this learning process more than before.

The paper's improvements: Tom: The paper highlights a few key things they've shown as improvements over previous approaches. They show that even if you correctly specify the model and there’s no censoring, jointly estimating the regression coefficient and the baseline hazard can create a suboptimal local minimum.

Jane: Okay, so they are showing that just having two parameters—the regression coefficient and the baseline hazard—can actually lead you away from the best possible result if you don't handle it right during joint estimation. That’s a warning for anyone building survival models.

Lu: They also show that when you look at a specific covariate vector, say x, the one-dimensional population negative log-likelihood function has a unique global minimizer and unique stationary point at the true parameter value. This means there’s no ambiguity about where the best solution is for that specific setup.

Meng: That uniqueness is good because it helps us avoid getting stuck in bad local minima when we're training models on large datasets, which is where I usually run into trouble with standard optimization routines.

Lalam: And they also show how their theoretical guarantees actually translate into real practical gains in experiments. They tested this on four clinical datasets, and the truncated neural network performed the best among those tested.

Conclusion: Tom: So, to wrap up, the main point of this paper is that efficient learning of a survival model under outcome-dependent truncation is actually possible when you know the baseline hazard. And they showed that first-order methods like projected SGD can converge efficiently to the true parameter values in this scenario.

Jane: What does this mean for someone just listening, someone who isn't working on these models every day? It means we have a better tool now to handle selection bias in survival analysis without getting bogged down in complicated iterative procedures that take forever.

Lu: They’ve provided a characterization of when Cox regression is efficient under truncation, which helps us know exactly when we can use these techniques versus when we should stick with other methods.

Meng: From an engineering standpoint, the speedup they show is pretty significant. They found that their first-order methods are substantially faster than standard expectation maximization, showing speedups of around one hundred times on datasets that are already quite large.

Lalam: And they also showed this correction works even for neural networks empirically, which is a big step because it means we can apply these insights to those deep learning models too.

Itai Zilberstein, Ioannis Anagnostides, Arman Kilic, Tuomas Sandholm

Department of Computer Science, Carnegie Mellon University · Department of Surgery, Division of Cardiothoracic Surgery, Medical University of South Carolina

cs.LG

Submitted: 2026-10-08

Updated: 2026-10-13

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 91/100

The gist: The gist Modern machine learning models are typically trained on data assumed to be representative of the predictive tasks the model will face in the future, but practical datasets violate this

Key concepts

Outcome-Dependent Truncation: This occurs when an observation's presence in the training data depends on its outcome label. For example, only patients who experience a specific event are included in the sample. This violates standard assumptions about how training data is collected for machine learning models.
Survival Analysis: This field uses statistical methods to analyze time-to-event data, such as how long it takes for an event (like death) to occur. The paper connects this analysis with machine learning optimization techniques to find the best predictive model parameters.
Baseline Hazard: The baseline hazard represents the underlying risk of an event occurring at any given time, independent of any covariates. In this study, knowing this function allows for more efficient learning of the survival model parameters.
First-Order Optimization: This is a method used to find the best set of parameters in a complex model by moving in the direction of steepest ascent or descent (the gradient). The paper shows that for known baseline hazards, this simple method can efficiently find the true optimal parameter values even when the problem is non-convex.

Summary

The gist Modern machine learning models are typically trained on data assumed to be representative of the predictive tasks the model will face in the future, but practical datasets violate this assumption Our work connects two largely distinct literatures: efficient learning under truncation and survival analysis We address this gap and prove that Cox regression can be efficiently learned under truncated and right-censored data using first-order optimization when the baseline hazard is known (despite nonconvexity).

Key Contributions

The paper studies whether Cox regression can be learned efficiently from outcome-dependent truncated and right-censored data. The main positive result shows that, despite nonconvexity, the population negative log-likelihood has a unique stationary point at the true parameter and its gradient always has a component pointing away from it when the baseline hazard is known (Theorem 7). We then show that even when the model is correctly specified, identifiable, and there is no censoring, jointly estimating the regression coefficient and baseline can create a suboptimal local minimum (Theorem 9). Finally, we illustrate that our theoretical guarantees translate into practical gains on semi-synthetic experiments with four clinical datasets.

Theoretical Framework and Results

The analysis connects Cox regression to weighted truncated exponential distributions through the transformation of event times by the cumulative baseline hazard H0(t). Lemma 4 establishes that conditional on sample inclusion, H0(T∗i) is distributed as TruncExp e x⊤iθ, with parameters derived from the intervals and retention probabilities. The single-observation negative log-likelihood gradient is expressed as ∇θfi(θ) = e x⊤iθ [δiH0(Ti) + (1 − δi)µi(Ti θ) − µi(θ)] xi.

Optimization Guarantees

The paper establishes that for a fixed covariate vector x, the one-dimensional population negative log-likelihood Fx(z) has z∗ as its unique global minimizer and unique stationary point. Theorem 7 further proves that the population negative log-likelihood F(θ) satisfies ⟨∇θF(θ), θ − θ∗⟩ > 0 for every θ ≠ θ∗, implying that θ∗ is the unique stationary point of F. Corollary 8 uses this property to prove that projected SGD converges to the true optimum.

Impossibility Results for Unknown Baselines

When the baseline hazard is unknown, non-identifiability can occur (Example 3). Even when identifiable, joint estimation of the regression coefficient and baseline can create a suboptimal local minimum (Theorem 9). Furthermore, after optimizing over a class of baseline hazards for each regression coefficient, the resulting profiled population loss can be flat and suboptimal over an entire interval of incorrect parameters (Theorem 10).

Practical Validation

Experiments with four survival datasets demonstrate that accounting for truncation reduces the prediction bias induced by naive Cox regression and extends to neural networks. The first-order methods are substantially faster compared to standard expectation maximization (EM), showing orders of magnitude speedups on enlarged datasets. Specifically, Adam and projected SGD achieve comparable MSE to EM but are around 100× faster on HEARTS-Large. The truncated neural network performs best in the experiments.

Conclusion

Efficient learning of a survival model under outcome-dependent truncation is possible when the baseline hazard is known, allowing projected SGD to converge efficiently to the true parameter values. The work provides a characterization of when and when not Cox regression is efficiently learnable under truncation. The results show that first-order methods can effectively optimize truncated Cox likelihoods, reducing prediction error relative to models that ignore selection. The same correction also improved neural networks empirically. Efficiently learning a survival model under truncation is an important capability to scale models to higher dimensions.

Acknowledgments

Tuomas Sandholm and his PhD students Ioannis Anagnostides and Itai Zilberstein are supported by NIH award A240108S001, the Vannevar Bush Faculty Fellowship ONR N00014-231-2876, and National Science Foundation grant RI-2312342. Itai Zilberstein is also supported by the NSF Graduate Research Fellowship Program under grants DGE2140739 & DGE2631988. Arman Kilic is supported by NIH RO1 grant 5R01HL162882-03 which contributed to the funding for completion of this project. All additional authors have no financial relationships to disclose.

References

A. M. Alaa and M. Van Der Schaar. Bayesian inference of individualized treatment effects using multi-task gaussian processes. I. Anagnostides, I. Zilberstein, Z. W. Sollie, A. Kilic, and T. Sandholm Position: Machine learning for heart transplant allocation policy optimization should account for incentives. J. Berrevoets, J. Jordon, I. Bica, and M. van der Schaar OrganITE: Optimal transplant donor organ offering using an individual treatment effect. R. Breen Regression models: Censored, sample selected, or truncated data Number 111 Sage 1996. S. H. Chiou, M. D. Austin, J. Qian, and R. A Betensky Transformation model estimation of survival under dependent truncation and independent censoring Statistical Methods in Medical Research 28(12):3785–3798 2019. A. F. Connors, N. V. Dawson, N. A. Desbiens, W. J Fulkerson, L Goldman, W A Knaus, J Lynn R K Oye M Bergner A Damiano et al A controlled trial to improve care for seriously ill hospitalized patients The study to understand prognoses and preferences for outcomes and risks of treatments Journal of the American Medical Association JAMA 274(20):1591–1598 1995. D. R. Cox Regression models and life-tables Journal of the Royal Statistical Society Series B (Methodological) 34(2):187–202 1972. C Curtis, S P Shah, S F Chin, G Turashvili O M Rueda M J Dunning D Speed A G Lynch S Samarajiwa Y Yuan et al The genomic and transcriptomic architecture of 2000 breast tumours reveals novel subgroups Nature 486(7403):346–352 2012. C Daskalakis T Gouleakis C Tzamos and M Zampetakis Computationally and statistically efficient truncated regression Conference on Learning Theory COLT 2019. J P Dickerson and T Sandholm FutureMatch: Combining human value judgments and machine learning to match in dynamic environments Conference on Artificial Intelligence AAAI 2015. A Dispenzieri J A Katzmann R A Kyle D R Larson T M Therneau C L Colby R J Clark G P Mead S Kumar L J Melton III et al Use of nonclonal serum immunoglobulin free light chains to predict overall survival in the general population Mayo Clinic Proceedings volume 87 pages 517–523 2012.

Improvements for AI systems

Bold optimization of Cox regression under truncation: The system can efficiently learn Cox regression using first-order optimization when baseline hazard is known, as Cox regression can be efficiently learned under truncated and right-censored data using first-order optimization when the baseline hazard is known (despite nonconvexity). This provides a computationally faster alternative to standard expectation maximization methods.
Neural network risk score correction: The system can effectively optimize neural network survival models by incorporating the truncation mechanism into the loss function, as The truncated neural network instead conditions this likelihood on inclusion, which empirically shows that the truncated neural network performs best. This improves prediction accuracy compared to standard models that ignore truncation.
Scalable first-order optimization: The system can achieve significant speedups in training time by replacing standard expectation maximization with first-order methods, as our first-order methods are substantially faster, reducing training time by roughly 100× on enlarged datasets compared to EM.
Guaranteed convergence for known baselines: When the baseline hazard is known, the system's projected stochastic gradient descent (SGD) is guaranteed to converge to the true parameter because the population negative log-likelihood has a unique stationary point at the true parameter and its gradient always has a component pointing away from it when the baseline hazard is known (Theorem 7).
Identification of suboptimal local minima: The system can be designed to detect and avoid suboptimal solutions in complex models by leveraging theoretical results showing that when the baseline hazard is learned jointly, we construct identifiable and correctly specified models whose negative log-likelihood contains strict suboptimal local minima.

Abstract

Modern machine learning models are typically trained on data assumed to be representative of the predictive tasks the model will face in the future, but practical datasets violate this assumption. We focus on outcome-dependent truncation in which an observation appears in the training data depending on its label. Our work connects two largely distinct literatures: efficient learning under truncation and survival analysis. Existing methods for truncated survival analysis do not characterize the optimization landscape and instead rely on iterative procedures such as expectation maximization. We address this gap and prove that Cox regression can be efficiently learned under truncated and right-censored data using first-order optimization when the baseline hazard is known (despite nonconvexity). In contrast, when the baseline hazard is learned jointly, we construct identifiable and correctly specified models whose negative log-likelihood contains strict suboptimal local minima. On experiments with four survival datasets, we show that accounting for truncation reduces the prediction bias induced by naive Cox regression and extends to neural networks. We also show orders of magnitude speedups compared to standard expectation maximization.