When \(p\) is comparable to or larger than the number of observed events, the partial likelihood is no longer stabilized by classical information inversion. Sparsity and restricted risk-set geometry replace ordinary fixed-dimensional identifiability.
Penalized partial likelihood
Definition. Let \[ \mathcal L_n(\beta)=-\frac1n\ell_p(\beta). \] The Cox lasso is \[ \widehat\beta \in\arg\min_{\beta\in\mathbb R^p} \left\{\mathcal L_n(\beta)+\lambda\|\beta\|_1\right\}. \] An elastic-net version adds \(\gamma\|\beta\|_2^2/2\); group penalties replace \(\|\beta\|_1\) when variables enter in structured blocks.
Proposition (KKT conditions). A vector \(\widehat\beta\) solves the Cox lasso if and only if \[ \nabla\mathcal L_n(\widehat\beta)+\lambda\widehat z=0, \] where \[ \widehat z_j= \begin{cases} \operatorname{sign}(\widehat\beta_j),&\widehat\beta_j\ne0,\\ \text{an element of }[-1,1],&\widehat\beta_j=0. \end{cases} \] Hence every excluded coordinate satisfies \(|\nabla_j\mathcal L_n(\widehat\beta)|\le\lambda\).
Proof. The objective is convex. Fermat's rule gives \(0\in\nabla\mathcal L_n(\widehat\beta)+\lambda\partial\|\widehat\beta\|_1\). The coordinatewise subdifferential of the absolute value gives the displayed conditions.
Score concentration
Theorem. Suppose covariates are uniformly bounded by \(K\), the true Cox model holds, and risk-set denominators stay uniformly positive. Then for constants \(c_1,c_2>0\), \[ P\left( \|\nabla\mathcal L_n(\beta_0)\|_\infty >c_1K\sqrt{\frac{\log(p/\delta)}{n}} \right)\le\delta \] whenever \(\log(p/\delta)\le c_2n\). Thus \(\lambda\asymp\sqrt{\log p/n}\) dominates the stochastic score.
Proof outline. Each score coordinate is a normalized martingale integral \[ -\nabla_j\mathcal L_n(\beta_0) =\frac1n\sum_i\int \{Z_{ij}(t)-\bar Z_j(\beta_0,t)\}\,\mathrm{d}M_i(t). \] Its jumps are bounded by \(2K/n\), and its predictable variation is \(O(K^2/n)\). Freedman's martingale inequality gives an exponential tail for one coordinate. A union bound over \(p\) coordinates yields the result. Unbounded covariates require sub-Gaussian or truncation arguments.
Restricted curvature
Definition. Let \(S=\operatorname{supp}(\beta_0)\), \(|S|=s\), and \[ \mathcal C(S,3) =\{\delta:\|\delta_{S^c}\|_1\le3\|\delta_S\|_1\}. \] A compatibility constant \(\kappa>0\) satisfies \[ \delta^\top\nabla^2\mathcal L_n(\widetilde\beta)\delta \ge\frac{\kappa^2}{s}\|\delta_S\|_1^2 \] for \(\delta\in\mathcal C(S,3)\) and \(\widetilde\beta\) on a neighborhood of \(\beta_0\).
The Hessian is an event-weighted risk-set covariance. Compatibility therefore means that sparse linear combinations of covariates retain variation among subjects who are compared at event times. Ordinary covariance of baseline covariates is not the relevant geometry.
Oracle inequalities
Theorem. On the event \[ \|\nabla\mathcal L_n(\beta_0)\|_\infty\le\lambda/2, \] and under the compatibility condition, \[ \|\widehat\beta-\beta_0\|_1 \le C\frac{s\lambda}{\kappa^2}, \qquad (\widehat\beta-\beta_0)^\top \nabla^2\mathcal L_n(\widetilde\beta) (\widehat\beta-\beta_0) \le C\frac{s\lambda^2}{\kappa^2} \] for a universal constant \(C\). Under a restricted eigenvalue condition, \[ \|\widehat\beta-\beta_0\|_2 \lesssim\frac{\sqrt s\,\lambda}{\kappa}. \]
Proof. Put \(\delta=\widehat\beta-\beta_0\). Optimality gives the basic inequality \[ \mathcal L_n(\beta_0+\delta)-\mathcal L_n(\beta_0) +\lambda\|\beta_0+\delta\|_1 \le\lambda\|\beta_0\|_1. \] Convexity and the score event imply \[ \text{curvature} +\frac{\lambda}{2}\|\delta_{S^c}\|_1 \le\frac{3\lambda}{2}\|\delta_S\|_1, \] so \(\delta\in\mathcal C(S,3)\). Apply compatibility and \(\|\delta_S\|_1\le\sqrt s\|\delta_S\|_2\), then solve the resulting quadratic inequality. The restricted eigenvalue version directly controls \(\|\delta\|_2\).
Corollary. If \(\lambda\asymp\sqrt{\log p/n}\) and \(s\log p/n\to0\), then the prediction curvature tends to zero and \[ \|\widehat\beta-\beta_0\|_1 =O_P\left(s\sqrt{\frac{\log p}{n}}\right). \]
Support recovery
Estimation consistency does not imply exact variable selection. Sign recovery additionally requires:
- a beta-min condition \[ \min_{j\in S}|\beta_{0j}|\gg\lambda; \]
- an irrepresentability condition controlling how inactive covariates are represented by active ones in the risk-set information matrix;
- accurate model specification and stable censoring support.
These conditions can fail in highly correlated scientific designs even when prediction remains accurate. Selected-variable \(p\)-values computed as though the selected model were fixed are then invalid.
Debiased Cox inference
Definition. Let \[ \widehat\Sigma=\nabla^2\mathcal L_n(\widehat\beta) \] and let \(\widehat\Theta_j\) estimate the \(j\)-th row of \(\Sigma^{-1}\), for example by nodewise lasso or the Dantzig constraint \[ \|\widehat\Sigma\theta-e_j\|_\infty\le\mu. \] The debiased coordinate estimator is \[ \widetilde\beta_j =\widehat\beta_j -\widehat\Theta_j^\top \nabla\mathcal L_n(\widehat\beta). \]
Theorem. Suppose the target row of the inverse information is sparse, lasso and inverse-information estimation errors satisfy the required product-rate conditions, and typically \[ s\frac{\log p}{\sqrt n}\to0. \] Then \[ \sqrt n(\widetilde\beta_j-\beta_{0j}) =-\widehat\Theta_j^\top \sqrt n\,\nabla\mathcal L_n(\beta_0)+o_P(1) \Rightarrow N(0,\sigma_j^2). \] Under correct specification, \(\sigma_j^2=\Theta_j^\top\Sigma\Theta_j\).
Proof. Taylor expansion gives \[ \nabla\mathcal L_n(\widehat\beta) =\nabla\mathcal L_n(\beta_0) +\widehat\Sigma^\star(\widehat\beta-\beta_0). \] Substitute this into \(\widetilde\beta_j\). The leading estimation bias is \[ \{e_j^\top-\widehat\Theta_j^\top\widehat\Sigma^\star\} (\widehat\beta-\beta_0), \] bounded by an inverse-information approximation error times the lasso \(\ell_1\)-error; the product-rate conditions make it \(o_P(n^{-1/2})\). The remaining score term is a martingale integral, so its standardized version is asymptotically Gaussian.
High-dimensional baseline prediction
For \[ \widehat\Lambda_0(t) =\int_0^t \frac{\mathrm{d}N(u)} {\sum_iY_i(u)e^{\widehat\beta^\top Z_i(u)}}, \] the plug-in error contains \[ B(t)^\top(\widehat\beta-\beta_0). \] An \(\ell_1\)-consistent lasso may be adequate for risk prediction but not for root-\(n\) inference on absolute survival. Orthogonalized or debiased functionals are needed, often with sample splitting when flexible nuisance learners are used. The key requirement is that first-order sensitivity to nuisance-estimation error be removed before applying a central limit theorem.
Tuning and validation under censoring
Cross-validation loss must respect risk sets; splitting rows from one subject across folds leaks information. Possible criteria include held-out partial deviance, inverse-censoring-weighted Brier score, and time-dependent log score. The concordance index measures ranking and cannot by itself tune calibrated absolute-risk predictions. Clustered or recurrent-event data should be split by subject or cluster.
The underlying oracle-inequality, compatibility, and debiasing tools are developed more generally in statistical learning.