Conformalized Online Learning: Online Calibration Without a Holdout Set

We develop a framework for constructing uncertainty sets with a valid coverage guarantee in an online setting, in which the underlying data distribution can drastically -- and even adversarially -- shift over time. The technique we propose is highly flexible as it can be integrated with any online learning algorithm, requiring minimal implementation effort and computational cost. A key advantage of our method over existing alternatives -- which also build on conformal inference -- is that we do not need to split the data into training and holdout calibration sets. This allows us to fit the predictive model in a fully online manner, utilizing the most recent observation for constructing calibrated uncertainty sets. Consequently, and in contrast with existing techniques, (i) the sets we build can quickly adapt to new changes in the distribution; and (ii) our procedure does not require refitting the model at each time step. Using synthetic and real-world benchmark data sets, we demonstrate the validity of our theory and the improved performance of our proposal over existing techniques. To demonstrate the greater flexibility of the proposed method, we show how to construct valid intervals for a multiple-output regression problem that previous sequential calibration methods cannot handle due to impractical computational and memory requirements.

Weisfeiler and Leman Go Infinite: Spectral and Combinatorial Pre-Colorings

Graph isomorphism testing is usually approached via the comparison of graph invariants. Two popular alternatives that offer a good trade-off between expressive power and computational efficiency are combinatorial (i.e., obtained via the Weisfeiler-Leman (WL) test) and spectral invariants. While the exact power of the latter is still an open question, the former is regularly criticized for its limited power, when a standard configuration of uniform pre-coloring is used. This drawback hinders the applicability of Message Passing Graph Neural Networks (MPGNNs), whose expressive power is upper bounded by the WL test. Relaxing the assumption of uniform pre-coloring, we show that one can increase the expressive power of the WL test ad infinitum. Following that, we propose an efficient pre-coloring based on spectral features that provably increase the expressive power of the vanilla WL test. The above claims are accompanied by extensive synthetic and real data experiments. The code to reproduce our experiments is available at https://github.com/TPFI22/Spectral-and-Combinatorial

Calibrated Multiple-Output Quantile Regression with Representation Learning

We develop a method to generate predictive regions that cover a multivariate response variable with a user-specified probability. Our work is composed of two components. First, we use a deep generative model to learn a representation of the response that has a unimodal distribution. Existing multiple-output quantile regression approaches are effective in such cases, so we apply them on the learned representation, and then transform the solution to the original space of the response. This process results in a flexible and informative region that can have an arbitrary shape, a property that existing methods lack. Second, we propose an extension of conformal prediction to the multivariate response setting that modifies any method to return sets with a pre-specified coverage level. The desired coverage is theoretically guaranteed in the finite-sample case for any distribution. Experiments conducted on both real and synthetic data show that our method constructs regions that are significantly smaller (sometimes by a factor of 100) compared to existing techniques.

Improving Conditional Coverage via Orthogonal Quantile Regression

We develop a method to generate prediction intervals that have a user-specified coverage level across all regions of feature-space, a property called conditional coverage. A typical approach to this task is to estimate the conditional quantiles with quantile regression -- it is well-known that this leads to correct coverage in the large-sample limit, although it may not be accurate in finite samples. We find in experiments that traditional quantile regression can have poor conditional coverage. To remedy this, we modify the loss function to promote independence between the size of the intervals and the indicator of a miscoverage event. For the true conditional quantiles, these two quantities are independent (orthogonal), so the modified loss function continues to be valid. Moreover, we empirically show that the modified loss function leads to improved conditional coverage, as evaluated by several metrics. We also introduce two new metrics that check conditional coverage by looking at the strength of the dependence between the interval size and the indicator of miscoverage.