SlowGuess/NeurIPS_2023_OCR huggingface dataset:
where  $(i_{*},j_{*})$  denotes the test point. Effectively,  $\Delta$  is quantifying the difference between the "oracle" weights,  $w_{ij}^{*}$ , defined in (5), and the estimated weights,  $\widehat{w}_{ij}$ , defined in (4), except that  $\Delta$  is defined relative to a specific test point, while the weights  $\widehat{\pmb{w}}$  (and  $\pmb{w}^*$ , for the oracle version) provide a "one-shot" procedure that is universal across all possible test points, and is thus slightly more conservative.
Theorem 3.2. Let  $\widehat{\mathbf{M}}$ ,  $\widehat{\mathbf{s}}$ , and  $\widehat{\mathbf{P}}$  be estimates constructed using any algorithms, which depend on the data only through the training samples  $\mathbf{M}_{S_{\mathrm{tr}}}$  at locations  $S_{\mathrm{tr}}$ . Then, under the notation and definitions above, conformalized matrix completion (cmc) satisfies
$$
\mathbb {E} \left[ \operatorname {A v g C o v} (\widehat {C}; \mathbf {M}, \mathcal {S}) \right] \geq 1 - \alpha - \mathbb {E} [ \Delta ].
$$
In the homogeneous case, where we are aware that sampling is uniform (i.e.,  $p_{ij} \equiv p$  for some potentially unknown  $p \in (0,1)$ ), our error in estimating the weights is given by  $\Delta \equiv 0$  (since  $h_{ij}$  is constant across all  $(i,j)$ , and same for  $\widehat{h}_{ij}$  —and therefore, the true and estimated weights are all equal to  $\frac{1}{n_{\mathrm{cal}} + 1}$ ). In this setting, therefore, we would achieve the exact coverage goal (2), with  $\mathbb{E}\big[\mathrm{AvgCov}(\widehat{C};\mathbf{M},\mathcal{S})\big]$  guaranteed to be at least  $1 - \alpha$ .
3.3 Examples for missingness
To characterize the coverage gap explicitly, we present the following concrete examples for  $\mathbf{P}$  and show how the coverage gap  $\Delta$  can be controlled. Technical details are shown in the Supplementary Material.
3.3.1 Logistic missingness
Suppose that the missingness follows a logistic model, with  $\log (p_{ij} / (1 - p_{ij})) = -\log (h_{ij}) = u_i + v_j$  for some  $\mathbf{u}\in \mathbb{R}^{d_1}$  and  $\mathbf{v}\in \mathbb{R}^{d_2}$ , where we assume  $\mathbf{u}^{\top}\mathbf{1} = 0$  for identifiability. This model is closely related to logistic regression with a diverging number of covariates (Portnoy, 1988; He and Shao, 2000; Wang, 2011). Following Chen et al. (2023), we estimate  $\mathbf{u},\mathbf{v}$  via maximum likelihood,
$$
\widehat {\boldsymbol {u}}, \widehat {\boldsymbol {v}} = \arg \max  _ {\boldsymbol {u}, \boldsymbol {v}} \mathcal {L} (\boldsymbol {u}, \boldsymbol {v}) \qquad \text {s u b j e c t t o} \quad \boldsymbol {1} ^ {\top} \boldsymbol {u} = 0.
$$
Here the log-likelihood is defined by
$$
\mathcal {L} (\boldsymbol {u}, \boldsymbol {v}) = \sum_ {(i, j) \in [ d _ {1} ] \times [ d _ {2} ]} \left\{- \log \left(1 + \exp (- u _ {i} - v _ {j})\right) + \mathbf {1} _ {(i, j) \in S _ {\mathrm {t r}} ^ {c}} \log \left(1 - q + \exp (- u _ {i} - v _ {j})\right) \right\}.
$$
Consequently, we define the estimate of  $p_{ij}$  as  $\widehat{p}_{ij} = (1 + \exp (-\widehat{u}_i - \widehat{v}_j))^{-1}$ , for which one can show that  $\mathbb{E}[\Delta] \lesssim \sqrt{\frac{\log(\max\{d_1,d_2\})}{\min\{d_1,d_2\}}}$ . The proof of this upper bound is shown in Section A.3.1.
3.3.2 Missingness with a general link function
More broadly, we can consider a general link function, where we assume  $\log (p_{ij} / (1 - p_{ij})) = -\log (h_{ij}) = \phi (A_{ij})$ , where  $\phi$  is a link function and  $\mathbf{A}\in \mathbb{R}^{d_1\times d_2}$  is the model parameter. The logistic model we introduced above corresponds to the case when  $\phi$  is the identity function and  $\mathbf{A} = \mathbf{u}\mathbf{1}^{\top} + \mathbf{1}\mathbf{v}^{\top}$  is a rank-2 matrix. In this general setup, if  $\mathrm{rank}(\mathbf{A}) = k^{*}$  and  $\| \mathbf{A}\|_{\infty}\leq \tau$ , we can apply one-bit matrix completion (Davenport et al., 2014) to estimate  $\{p_{ij}\}$ . More precisely, we define the log-likelihood
$$
\mathcal {L} _ {\mathcal {S} _ {\mathrm {t r}}} (\mathbf {B}) = \sum_ {(i, j) \in \mathcal {S} _ {\mathrm {t r}}} \log (\psi (B _ {i j})) + \sum_ {(i, j) \in \mathcal {S} _ {\mathrm {t r}} ^ {c}} \log (1 - \psi (B _ {i j})),
$$
where  $\psi (t) = q(1 + e^{-\phi (t)})^{-1}$  (here rescaling by the constant  $q$  is necessary to account for subsampling the training set  $S_{\mathrm{tr}}$  from the observed data indices  $\mathcal{S}$ ), and solve the following optimization problem
$$
\widehat {\mathbf {A}} = \arg \max  _ {\mathbf {B}} \mathcal {L} _ {\mathcal {S} _ {\mathrm {t r}}} (\mathbf {B}) \quad \text {s u b j e c t t o} \quad \| \mathbf {B} \| _ {*} \leq \tau \sqrt {k ^ {*} d _ {1} d _ {2}}, \| \mathbf {B} \| _ {\infty} \leq \tau , \tag {7}
$$
where  $\| \mathbf{B}\|_{*}$  is the nuclear norm of  $\mathbf{B}$ . Consequently, we let  $\widehat{h}_{ij} = \exp (-\phi (\widehat{A}_{ij}))$ , which leads to  $\mathbb{E}[\Delta ]\lesssim \min \{d_1,d_2\}^{-1 / 4}$ . The proof of this upper bound is shown in Section A.3.2.
4 Simulation studies
In this section, we conduct numerical experiments to verify the coverage guarantee of conformalized matrix completion (cmc) using both synthetic and real datasets. As the validity of cmc is independent of the choice of base algorithm, we choose alternating least squares (als) (Jain et al., 2013) as the base algorithm (results using a convex relaxation (Fazel et al., 2004; Candes and Recht, 2012; Chen et al., 2020) are shown in the Supplementary Material). We use cmc-als to refer to this combination. We use  $\mathrm{AvgCov}(\widehat{C})$  (the average coverage, defined in (3)), and average confidence interval length,  $\mathrm{AvgLength}(\widehat{C}) = \frac{1}{|\mathcal{S}^c|}\sum_{(i,j)\in \mathcal{S}^c}\mathrm{length}\big(\widehat{C} (i,j)\big)$ , to evaluate the performance of different methods. All the results in this section can be replicated with the code available at https://github.com/yugjerry/conf-mc.
4.1 Synthetic dataset
Throughout the experiments, we set the true rank  $r^* = 8$ , the desired coverage rate  $1 - \alpha = 0.90$  and report the average results over 100 random trials.
Data generation. In the synthetic setting, we generate the data matrix by  $\mathbf{M} = \mathbf{M}^{*} + \mathbf{E}$ , where  $\mathbf{M}^{*}$  is a rank  $r^*$  matrix, and  $\mathbf{E}$  is a noise matrix (with distribution specified below). The low-rank component  $\mathbf{M}^{*}$  is generated by  $\mathbf{M}^{*} = \kappa \mathbf{U}^{*}\mathbf{V}^{*^{\top}}$ , where  $\mathbf{U}^{*} \in \mathbb{R}^{d_{1} \times r^{*}}$  is the orthonormal basis of a random  $d_{1} \times r^{*}$  matrix consisting of i.i.d. entries from a certain distribution  $\mathcal{P}_{u,v}$  (specified below) and  $\mathbf{V}^{*} \in \mathbb{R}^{d_{2} \times r^{*}}$  is independently generated in the same manner. In addition, the constant  $\kappa$  is chosen such that the average magnitude of the entries  $|M_{ij}^{*}|$  is 2. Following the observation model (1), we only observe the entries in the random set  $\mathcal{S}$ .
Implementation. First, when implementing a1s, we adopt  $r$  as the hypothesized rank of the underlying matrix. For model-based methods (Chen et al., 2019), by the asymptotic distributional characterization  $\widehat{M}_{ij} - M_{ij}^{*} \xrightarrow{d} \mathcal{N}(0, \theta_{ij}^{2})$  and  $E_{ij} \perp (\widehat{M}_{ij} - M_{ij}^{*})$ , one has  $\widehat{M}_{ij} - M_{ij} \xrightarrow{d} \mathcal{N}(0, \theta_{ij}^{2} + \sigma^{2})$ . Consequently, the confidence interval for the model-based methods can be constructed as
$$
\widehat {M} _ {i j} \pm q _ {1 - \alpha / 2} \sqrt {\widehat {\theta} _ {i j} ^ {2} + \widehat {\sigma} ^ {2}},
$$
where  $q_{\beta}$  is the  $\beta$ th quantile of  $\mathcal{N}(0,1)$ . The estimates are obtained via  $\widehat{\sigma}^2 = \|\mathbf{M}_{S_{\mathrm{tr}}} - \widehat{\mathbf{M}}_{S_{\mathrm{tr}}}\|_{\mathrm{F}}^2 / |S_{\mathrm{tr}}|$  and  $\widehat{\theta}_{ij}^2 = (\widehat{\sigma}^2 / \widehat{p})(\|\widehat{\mathbf{U}}_{i,\cdot}\|^2 + \|\widehat{\mathbf{V}}_{j,\cdot}\|^2)$  with the SVD  $\widehat{\mathbf{M}} = \widehat{\mathbf{U}}\widehat{\boldsymbol{\Sigma}}\widehat{\mathbf{V}}^\top$ . For cmc-als, we obtain  $\widehat{\mathbf{M}}, \widehat{\mathbf{s}}$  from  $\mathbf{M}_{S_{\mathrm{tr}}}$  by running a1s on this training set (and taking  $\widehat{s}_{ij} = (\widehat{\theta}_{ij}^2 + \widehat{\sigma}^2)^{1/2}$ ). The probabilities  $\widehat{p}_{ij}$ 's are estimated via  $\widehat{p}_{ij} \equiv (d_1 d_2 q)^{-1} |S_{\mathrm{tr}}|$  for this setting where the  $p_{ij}$ 's are homogeneous.
4.1.1 Homogeneous missingness
In the homogeneous case, we consider the following four settings with  $d_{1} = d_{2} = 500$ :
- Setting 1: large sample size + Gaussian noise.  $p = 0.8$ ,  $\mathcal{P}_{u,v} = \mathcal{N}(0,1)$ , and  $E_{ij} \sim \mathcal{N}(0,1)$ .
- Setting 2: small sample size + Gaussian noise.  $p = 0.2$ ,  $\mathcal{P}_{u,v} = \mathcal{N}(0,1)$ , and  $E_{ij} \sim \mathcal{N}(0,1)$ .
- Setting 3: large sample size + heavy-tailed noise.  $p = 0.8$ ,  $\mathcal{P}_{u,v} = \mathcal{N}(0,1)$ , and  $E_{ij} \sim 0.2 \, \mathrm{t}_{1,2}$ , where  $t_\nu$  denotes the  $t$  distribution with  $\nu$  degrees of freedom.
- Setting 4: violation of incoherence.  $p = 0.8$ ,  $\mathcal{P}_{u,v} = t_{1,2}$ , and  $E_{ij} \sim \mathcal{N}(0,1)$ .
Figure 1 displays the results (i.e., the coverage rate and the interval length) for both cmc-als and als when we vary the hypothesized rank  $r$  from 2 to 40. The oracle length is the difference between the  $(1 - \alpha / 2)$ -th and  $(\alpha / 2)$ -th quantiles of the underlying distribution for  $E_{ij}$ . As can be seen, cmc-als achieves nearly exact coverage regardless of the choice of  $r$  across all four settings. When the hypothesized rank  $r$  is chosen well, cmc-als is often able to achieve an average length similar to that of the oracle.
For the model-based method a1s, when we underestimate the true rank  $(r < r^{*})$ , we may still see adequate coverage since  $\widehat{\sigma}$  tends to over-estimate  $\sigma$  by including the remaining  $r^{*} - r$  factors as noise. However, if we overestimate the rank  $(r > r^{*})$ , then a1s tends to overfit the noise with additional factors and under-estimate  $\sigma$  regardless of the choice of  $r$ , leading to significant undercoverage
(a) Setting 1: large sample size + Gaussian noise
(b) Setting 2: small sample size + Gaussian noise
(c) Setting 3: large sample size + heavy-tailed noise
(d) Setting 4: violation of incoherence
(a) Large sample size  $^+$  Gaussian noise:  $r = r^{*} = 8$
Figure 1: Comparison between cmc-als and als.
(b) Large sample size  $^+$  Gaussian noise:  $r^{*} = 8 <   50 = r$
(c) Violation of incoherence:  $r = 6 < 8 = r^{*}$
Figure 2: Histogram of standardized scores for als and prediction lower and upper bounds for 50 distinct unobserved entries.
in Settings 1, 2 and 4. In Setting 3 with heavy-tailed noise, a1s tends to be conservative due to overestimating  $\sigma$ . Overall, Figure 1 confirms our expectation that the validity of the model-based estimates relies crucially on a well specified model, and it fails to hold when sample size is small (cf. Figure 1b), when noise is heavy tailed (cf. Figure 1c), and when the underlying matrix is not incoherent (cf. Figure 1d).
In Figure 2, we present the histogram of standardized scores  $(\widehat{M_{ij}} - M_{ij}) / \sqrt{\widehat{\theta}_{ij}^2 + \widehat{\sigma}^2}$  and the plot of the upper and lower bounds for three settings. In Figure 2a, when the model assumptions are met and  $r = r^*$ , the scores match well with the standard Gaussian and the prediction bounds produced by a1s and cmc-als are similar. With the same data generating process, when the rank is overparametrized, the distribution of scores cannot be captured by the standard Gaussian, thus the quantiles are misspecified. As we can see from the confidence intervals, a1s tends to have smaller intervals which lead to the undercoverage. In the last setting, the underlying matrix is no longer incoherent. When the rank is underestimated, the  $r^* - r$  factors will be captured by the noise term and the high heterogeneity in the entries will further lead to overestimated noise level. As a result, the intervals by a1s are much larger while the conformalized intervals are more adaptive to the magnitude of entries.
4.1.2 Heterogeneous missingness
Now we move on to the case with heterogeneous missingness, where the observed entries are no longer sampled uniformly. To simulate this setting, we draw  $\{a_{il}:i\leq d_1,l\leq k^*\}$  i.i.d. from Unif(0,1) and  $\{b_{lj}:l\leq k^{*},j\leq d_{2}\}$  i.i.d. from Unif(-0.5,0.5), and define sampling probabilities by
$$
\log \left(p _ {i j} / \left(1 - p _ {i j}\right)\right) = \sum_ {l = 1} ^ {k ^ {*}} a _ {i l} b _ {l j}.
$$
(a) Gaussian noise:  $k^{*} = 1$ .
(c) Adversarial heterogeneous noise:  $k^{*} = 1$ .
(e) Random heterogeneous noise:  $k^{*} = 1$ .
Figure 3: Comparison between cmc with true and estimated weights under heterogeneous missingness.
(b) Gaussian noise:  $k^{*} = 5$ .
(d) Adversarial heterogeneous noise:  $k^{*} = 5$ .
(f) Random heterogeneous noise:  $k^{*} = 5$ .
We consider two settings with  $k^{*} = 1$  and  $k^{*} = 5$ , respectively. In both settings,  $\widehat{p}_{ij}$  is constructed by estimating  $\widehat{a}_{il}$  and  $\widehat{b}_{lj}$  via constrained maximum likelihood estimation as is shown in Example 3.3.2.
To analyze the effect of estimation error predicted by Theorem 3.2, we consider three matrix generation processes for each choice of  $k^*$ , where  $\mathbf{M}^*$  is generated in the same way as in Section 4.1.1 with  $d_1 = d_2 = d = 500$ ,  $\mathcal{P}_{u,v} = \mathcal{N}(0,1)$ :
- Gaussian homogeneous noise:  $E_{ij} \sim \mathcal{N}(0,1)$  independently.
- Adversarial heterogeneous noise:  $E_{ij} \sim \mathcal{N}(0, \sigma_{ij}^2)$  independently, where we take  $\sigma_{ij} = 1 / 2p_{ij}$ . For this setting, note that the high-noise entries (i.e., those entries that are hardest to predict) occur in locations  $(i,j)$  that are least likely to be observed during training.
- Random heterogeneous noise:  $E_{ij} \sim \mathcal{N}(0, \sigma_{ij}^2)$  independently, where  $(\sigma_{ij})_{(i,j) \in [d_1] \times [d_2]}$  is drawn from the same distribution as  $(1/2p_{ij})_{(i,j) \in [d_1] \times [d_2]}$ .
We write  $\mathrm{cmc}^*$ -als to denote the conformalized method (with als as the base algorithm) where we use oracle knowledge of the true  $p_{ij}$ 's for weighting, while  $\mathrm{cmc}$ -als uses estimates  $\widehat{p}_{ij}$ . The results are shown in Figures 3, for the settings  $k^* = 1$  and  $k^* = 5$  (i.e., the rank of  $\mathbf{P}$ ). Under both homogeneous noise and random heterogeneous noise, both  $\mathrm{cmc}^*$ -als and  $\mathrm{cmc}$ -als achieve the correct coverage level, with nearly identical interval length. For adversarial heterogeneous noise, on the other hand,  $\mathrm{cmc}^*$ -als achieves the correct coverage level as guaranteed by the theory, but  $\mathrm{cmc}$ -als shows some undercoverage due to the errors in the estimates  $\widehat{p}_{ij}$  (since now the variance of the noise is adversarily aligned with these errors); nonetheless, the undercoverage is mild. In Section B.2 in the appendix, the local coverage of  $\mathrm{cmc}$ -als conditioning  $p_{ij} = p_0$  for varying  $p_0$  is presented.
4.2 Real data application
We also compare conformalized approaches with model-based approaches using the Rossmann sales dataset $^{4}$  (Farias et al., 2022). This real data provides the underlying fixed matrix  $\mathbf{M}$ ; the missingness pattern is generated artificially, as detailed below. We focus on daily sales (the unit is 1K dollar) of 1115 drug stores on workdays from Jan 1, 2013 to July 31, 2015 and hence the underlying matrix has dimension  $1115 \times 780$ . The hypothesized rank  $r$  is varied from 5 to 60 with step size 5 and we set
<sup>4</sup>https://www.kaggle.com/c/rossmann-store-sales
(a)als vs cmc-als: homogeneous missingness.
Figure 4: Comparison between conformalized and model-based matrix completion with sales dataset.
(b)als vs cmc-als: logistic missingness.
the target level to be  $1 - \alpha = 0.9$ . We apply random masking for 100 independent trials and report the average for AvgCov and AvgLength in Figure 4.
- Homogeneous  $p_{ij}$ . For each entry, we first apply random masking  $1 - Z_{ij} \sim \mathrm{Bern}(0.2)$  independently. In Figure 4a, conformalized approach has exact coverage at  $1 - \alpha$  but a1s tends to be conservative when  $r$  is small and loses coverage when  $r$  increases to 50.
- Heterogeneous  $p_{ij}$ . After drawing  $p_{ij}$ 's from  $\log(p_{ij} / (1 - p_{ij})) = -\log(h_{ij}) = \phi(A_{ij})$  in Section 4.1.2 with  $k^* = 1$ ,  $\widehat{\mathbf{P}}$  is obtained via the constrained maximum likelihood estimator in Example 3.3.2. In Figure 4b, cmc-als has coverage around  $1 - \alpha$  and in comparison, when  $r$  is greater than 40, als fails to guarantee the coverage and AvgCov decreases to 0.75.
In this paper, we introduce the conformalized matrix completion method cmc, which offers a finite-sample coverage guarantee for all missing entries on average and requires no assumptions on the underlying matrix or any specification of the algorithm adopted, relying only on an estimate of the entrywise sampling probabilities. Given an estimate of the entrywise sampling probability, we provide an upper bound for the coverage gap and give the explicit form with examples of the logistic as well as the general one-bit low-rank missingness. In the implementation, an efficient one-shot weighted conformal approach is proposed with the provable guarantee and achieves nearly exact coverage.
We can compare our findings to a few related results in the literature. The work of Chernozhukov et al. (2021) applies conformal prediction to counterfactual and synthetic control methods, where they include matrix completion as an example with a regression-type formulation. However, their approach relies on a test statistic that is a function of estimated residuals. Consequently, their method requires the assumption of stationarity and weak dependence of errors. Furthermore, the validity of their approach is contingent upon the estimator being either consistent or stable. Additionally, Wieczorek (2023) studies conformal prediction with samples from a deterministic and finite population, but the validity under the sampling without replacement remains an open question in their work.
Our results suggest a number of questions for further inquiry. In the setting of matrix completion, while the results are assumption-free with regard to the matrix  $\mathbf{M}$  itself, estimating the sampling probabilities  $\mathbf{P}$  that determine which entries are observed remains a key step in the procedure; while our empirical results suggest the method is robust to estimation error in this step, studying the robustness properties more formally is an important open question. More broadly, our algorithm provides an example of how the conformal prediction framework can be applied in a setting that is very different from the usual i.i.d. sampling regime, and may lend insights for how to develop conformal methodologies in other applications with non-i.i.d. sampling structure.
R.F.B. was partially supported by the Office of Naval Research via grant N00014-20-1-2337, and by the National Science Foundation via grant DMS-2023109. C.M. was partially supported by the National Science Foundation via grant DMS-2311127.
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A Proofs 14
A.1 Proof of Lemma 3.1 14
A.2 Proof of Theorem 3.2 16
A.3 Proofs for examples in Section 3.3 17