Title: Generalized Fast Exact Conformalization
Venue: NeurIPS 2024
Accepted: yes
Keywords: Conformal Prediction

Generalized Statistical Estimation

Numerical Optimization
Abstract: Conformal prediction converts nearly any point estimator into a prediction interval under standard assumptions while ensuring valid coverage. However, the extensive computational demands of full conformal prediction are daunting in practice, as it necessitates a comprehensive number of trainings across the entire latent label space. Unfortunately, existing efforts to expedite conformalization often carry strong assumptions and are developed specifically for certain models, or they only offer approximate solution sets. To address this gap, we develop a method for fast exact conformalization of generalized statistical estimation. Our analysis reveals that the structure of the solution path is inherently piecewise smooth, and indicates that utilizing second-order information of difference equations suffices to approximate the entire solution spectrum arbitrarily. We provide a unified view that not only encompasses existing work but also attempts to offer geometric insights. Practically, our framework integrates seamlessly with well-studied numerical solvers. The significant speedups of our algorithm as compared to the existing standard methods are demonstrated across numerous benchmarks.

Generalized Fast Exact Conformalization

Abstract

Conformal prediction converts nearly any point estimator into a prediction interval under standard assumptions while ensuring valid coverage. However, the extensive computational demands of full conformal prediction are daunting in practice, as it necessitates a comprehensive number of trainings across the entire latent label space. Unfortunately, existing efforts to expedite conformalization often carry strong assumptions and are developed specifically for certain models, or they only offer approximate solution sets. To address this gap, we develop a method for fast exact conformalization of generalized statistical estimation. Our analysis reveals that the structure of the solution path is inherently piecewise smooth, and indicates that utilizing second-order information of difference equations suffices to approximate the entire solution spectrum arbitrarily. We provide a unified view that not only encompasses existing work but also attempts to offer geometric insights. Practically, our framework integrates seamlessly with well-studied numerical solvers. The significant speedups of our algorithm as compared to the existing standard methods are demonstrated across numerous benchmarks.

1 Introduction

In modern algorithmic practice, quantifying uncertainty is crucial for accurate and reliable model predictions. Conformal prediction [1] serves as a powerful statistical tool that leverages the observed data to construct prediction intervals containing the outcome with a predefined probability level. It enjoys model-free coverage guarantee regardless of the underlying distribution of the data. In recent years, conformal prediction has gained increasing attention from the community of machine learning [2-5], data mining [6-8] and computer vision [9, 10]. This growing interest is attributed to the attractive properties that it operates under the assumption of exchangeability, which is a weaker condition than independence and identical distribution, allowing for a wider range of applications in real-world scenarios where data may not meet strict statistical assumptions. Meanwhile, conformal prediction can be combined with almost any existing point estimators, even when the model is potentially misspecified [5].

While exhibits appealing properties, the application of conformal prediction often comes at a high computational cost [11, 12]. Kindly note that in this paper we refer to the full conformal prediction that does not discard training points as opposed to the split conformal prediction, as the latter involves only one single fitting. From a numerical perspective, when constructing a conformal prediction set, one needs to exhaustively search all points (potential candidates) in the label space, where for each point the learning model needs to be refitted and the conformity score needs to be re-calculated. In many scenarios like regression, the number of possible candidates is infinite as the latent label can take an uncountable number of possible values. Conventional conformal prediction works by a grid-search type method to loop over the label space [13, 14], which discretize the interval of interest and subsequently solve a sequence of individual optimization subproblems. To improve such brute-force approach, there have been many efforts in community that devoted to develop better algorithms for computing the prediction set. A natural idea is to generate the set of all solutions

indexed by the latent label candidate using numerical continuation ( a.k.a. homotopy) method, and we name this set of optimal solutions as (exact) solution path , as illustrated in Figure 1.

Despite extensive theoretical and empirical efforts, the understanding of conformalization path remains rather deficient. For some simple cases, closed-form characterizations of conformal prediction sets are available, such as k -nearest neighbors, least squares regression [15], and ridge regression [16] with quadratic loss. The study by [12] presents an exact solution path for conformalized Lasso and elastic net using their statistical property and glyph[lscript] 1 -sparsity analysis. Investigations into more general objectives as discussed in [17, 18] incorporate linear interpolation to approximate the intrinsic piecewise smooth structure, yielding prediction sets lacking of finite-sample calibration. In other terms, [17, 18] offer only an upper bound for the approximation error and fail to control the degree of approximation relative to the optimal

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solutions. We summarize these relevant prior studies in Table 1. Given that existing exact algorithms are either tailored to a select few models, or turn out to be grid-search type approaches that take a very black-box approach, it prompts the following question:

Can we 'open the black-box' by developing a methodology that better exploits the structure of the path?

We answer this question positively by introducing a differential equation perspective to analyze the ground truth solution path, which enables us to better reveal and exploit the fundamental path structure. This more profound understanding enables us to build more generalized conformalization algorithm and present improved computational guarantees.

1.1 Our contributions

The main contributions brought by this paper are summarized as follows.

Generalizable framework This study aims to extend the application of fast exact conformalization into generalized statistical estimation. We relax the assumptions by considering a cost function that is no longer globally differentiable, but rather piecewise differentiable , while introducing constraints on the weight vector to fit more statistical models. In our analysis, the Clarke subdifferential of the objective is derived, and the set of nonsmooth points of path can essentially be described as a level set of certain smooth functions. By adopting a reparameterization regime, our framework effectively broadens the scope of several existing algorithms in Table 1 and provides a unified view of them.

Theoretical insights We analyze the underlying structure of path through the first-order optimality conditions of the regularized problem, identifying sufficient conditions for a local path to be smooth around a given point and reveal that the structure of the path is inherently piecewise smooth. More precisely, if conditions are met, then the path is locally the projection of a higher-dimensional smooth manifold onto the optimization space, thus offering preliminary geometric intuitions. Our

investigation further suggests that leveraging the second-order information of difference equations can approximate the solution spectrum arbitrarily.

Practical efficiency Practically, our framework is not only straightforward to implement but also computationally efficient. With theoretical analysis, we present explicit expressions for the gradient flows of objective, which is homogenized and well aligned with standard forms used by mainstream numerical libraries for ordinary differential equations (ODE), thereby easing the programming efforts. When crossing potential kinks (or nonsmooth points), the computations are facilitated by boundary conditions pre-set in the numerical solver. Notably, our algorithm eliminates the need for extensive iterations for computing the entire solution spectrum, contrasting sharply with conventional baselines. In experiments, we demonstrate the significant computational speed advantages of our algorithm over existing baselines, without compromising on accuracy.

2 Background

Notation For a set J ⊆ R p , we denote by cl( J ) the closure and by int( J ) the interior of J w.r.t. the natural topology on R p . Define J ( i ) the i -th element of J and conv( J ) be the convex hull of J , or conv( J ) = { v : v = ∑ k i =1 ˆ θ i u i , u i ∈ J , ˆ θ i ∈ R > 0 , ∑ k i =1 ˆ θ i = 1 } . The w ( i ) is the i -th element of vector w and I ( · ) is an indicator function. Let ˙ z ( t ) be the derivative dz ( t ) dt of function z ( t ) . Let O be the zero matrix and P {·} be the event probability. The sgn( · ) is the sign function sgn( x ) = x | x | ( x = 0) or 0 ( x = 0) that applied entrywise.

glyph[negationslash]

2.1 Model and assumptions

Given the dataset { x i , y i } n i =1 , where x i ∈ R p is the sample (covariate) and y i is the i -th label (response) live in label space Y ⊆ R 1 . We consider the generalized parametric estimation problem

where η w is the model prediction function, L i is the loss and Ω j is the regularizer. The g i , h j are constraints on w , and parameter λ j ∈ R > 0 controls the degree of regularity. In the following, we will define the piecewise differentiability and state main assumptions that used throughout this work.

Definition 1 ( PC r Function) . Let f : U → R be continuous on the open set U ⊆ R p and f i : U → R , i ∈ { 1 , . . . , k } be a set of r -times continuously differentiable (or C r ) functions for r ∈ N ∪ {∞} . If f ( x ) ∈ { f i ( x ) } i ∈{ 1 ,...,k } holds for all x ∈ U , then f is an r -times piecewise continuously differentiable (or PC r ) function. The { f 1 , . . . , f k } is a set of selection functions of f .

When working with PC r -functions in a local sense, it is useful to only consider the selection functions that have an impact on the local behavior around a given point.

Definition 2 (Essentially Active Set) . Let f : U → R be a PC r function on the open set U ⊆ R p with a set of selection functions { f 1 , . . . , f k } . Denote I K = { 1 , . . . , k } . Then given a x 1 ∈ U , I a f ( x 1 ) glyph[defines] { i ∈ I K : f ( x 1 ) = f i ( x 1 ) } is called the active set at x 1 , and the I e f ( x 1 ) glyph[defines] { i ∈ I K : x 1 ∈ cl ( int ( { x 2 ∈ U : f ( x 2 ) = f i ( x 2 ) } )) } is the essentially active set at x 1 .

Assumption 2. We assume that Ω j and L i are non-differentiable at w with multiple active selection functions, where j ∈ { 1 , . . . , m } , i ∈ { 1 , . . . , n + 1 } . We further assume that I a Ω j ( w ) ≡ I e Ω j ( w ) , I a L i ( w ) ≡ I e L i ( w ) holds for all w considered and all Ω j , L i in the following.

Assumption 1. We assume that L i and Ω j in (1) are PC r functions each with a set of selection functions ⋃ k ∈ I e L i { D k L i } and ⋃ k ∈ I e Ω j { D k Ω j } , respectively.

Assumption 2 ensures that all selection functions we consider are actually relevant for the representation of Ω j , L i . i.e. , it does not matter if we consider the active or the essentially active set in the underlying optimizing space, which allows for an easier representation of D k Ω j and D k L i .

2.2 Conformal prediction

Definition 3 (Symmetrical Algorithm) . A deterministic algorithm A : ( x 1 , . . . , x n ) → A glyph[star] is symmetric if for any permutation τ of { 1 , . . . , n } , A ( x 1 , . . . , x n ) a.s. = A ( x τ (1) , . . . , x τ ( n ) ) .

Definition 4 (Conformity Score) . The conformity score function A , symmetric in its first n inputs, is defined as A ( glyph[vector] x 1 , . . . , glyph[vector] x n ; glyph[vector] x n +1 ): R ( p +1) × ( n +1) → R , where glyph[vector] x i glyph[defines] ( x i , y i ) .

Conformal prediction starts from a conventional model fitting stage, followed by the evaluation of conformity score and the construction of prediction set. The score function A serves as a measure of deviation or conformity, assessing the extent to which the new input x n +1 aligns with the previously fitted model. A higher conformity score indicates a better match between x n +1 and the model [1, 14]. For a new instance x n +1 where the prediction region is desired, the conformalization method operates by assigning a p -value to each latent y n +1 ∈ Y , formalized as

where A i glyph[defines] A ( glyph[vector] x 1 , .., glyph[vector] x i -1 , glyph[vector] x i +1 , .., glyph[vector] x n +1 ; glyph[vector] x i ) . Specifically, in density estimation, A is defined as η w glyph[star] ( x n +1 ) , where η w glyph[star] is the density function estimated from the augmented dataset. For regression tasks, A might be set as -| y n +1 -η w glyph[star] ( x n +1 ) | , with η w glyph[star] being the regression function trained by the dataset including n + 1 samples. Under the assumption of exchangeability among the pairs { x i , y i } n +1 i =1 , the ˆ p y n +1 returned by (2) has been demonstrated to be statistically valid [13, 14]. To generate the prediction set Γ( · ) , one thresholds these p -values at a prescribed error level α ∈ (0 , 1) , resulting in

Theorem 1. [19] Suppose that { x i , y i } n +1 i =1 are exchangeable and the fitting algorithm A is symmetric. Conformal prediction applied on { x i , y i } n i =1 ∪ { x n +1 } outputs a set Γ( · ) such that

where y glyph[star] n +1 is the ground truth ( n +1) -th label.

Theorem 1 ( a.k.a. coverage guarantee) requires only exchangeability of input data and symmetry of the conformity score function, which are met by nearly all prevalent model fitting algorithms. In alignment with the analysis taken in prior research, we treat y n +1 := y n +1 ( z ) as a function of scalar variable z . We utilize it to facilitate the traversal of y n +1 across the entire label space Y , and compute the homotopy solution path { w glyph[star] ( z ) : z min ≤ z ≤ z max } (also shown in Figure 1). We assume that y n +1 ( · ): [ z min , z max ] →Y is continuously differentiable in terms of z for simplicity.

3 Main results

We present our main results for fast exact conformalization. The discussions here focus on positive z , 1 but the derivation extends easily to include negative values as well, which will be discussed later.

3.1 Surrogate function

Lemma 1. Let f : U → R be a PC r -function on the open set U ⊆ R p and let C r -functions { f 1 , . . . , f k } be a set of selection functions of f . Then for any x ∈ U , there exists an open neighborhood U ′ ⊆ U of x on which f is also a continuous selection of { f i : i ∈ I e f ( x ) } .

Drawing insights from Lemma 1, we minimize the surrogate function of (1) as

where ρ ∈ R > 0 and y n +1 = y n +1 ( z ) . This definition of E z ( w ) is meaningful regardless of whether the contributing functions are convex. Denote model estimation terms ∑ i L i + ∑ j λ j Ω j as L M ( w | z ) , and the minimizer of (5) as w glyph[star] ( z ) . It is interesting to compare E z ( w ) to the Lagrangian function

which captures the behavior of L M ( · ) near the optimum. At a constrained minimum w glyph[star] , the Lagrangian satisfies the stationarity condition ∇L ( w glyph[star] ) = 0 ; its inequality multipliers ˜ µ j are nonnegative and satisfy the complementary slackness ˜ µ j h j ( w glyph[star] ) = 0 . In the penalized (5), one usually takes

Theorem 2. Our surrogate function E z ( w ) is increasing in ρ . Furthermore, E z ( w ) is strictly convex for one ρ > 0 if and only if it is strictly convex for all ρ > 0 . Likewise, it is coercive for one ρ > 0 if and only if it is coercive for all ρ > 0 . Finally, if L M ( · ) is strictly convex (or coercive), then E z ( w ) for all ρ are strictly convex (or coercive).

which creates the favorable circumstances: ( i ) L z ( w ) ≤ E z ( w ) for all w , ( ii ) L z ( w ) ≤ L M ( w | z ) = E z ( w ) for all feasible w , ( iii ) L z ( w glyph[star] ) = L M ( w glyph[star] | z ) = E z ( w glyph[star] ) with profound consequences.

Given Theorem 2, several classical results [20] state that minimizing E z ( w ) is effective in minimizing L M ( · ) subject to the constraints if we choose ρ by (7).

3.2 Conformal path characterization

Definition 5 (Clarke Subdifferential) . For continuous function f : U → R on the open set U ⊆ R p , let Θ ⊆ U be the set of points in which f is not differentiable. The Clarke subdifferential of f at x ∈ U is defined as

Specifically, if f is continuously differentiable in x , ∂f ( x ) = {∇ f ( x ) } . While PC r -functions are generally non-smooth, we can use the Clarke subdifferential to obtain first-order optimality of (5),

where ˆ θ k Ω j , ˆ θ k L i is the k -th auxiliary parameter for convex hull of each Ω j , L i . The (8) is accompanied by the active sets conditions and the subdifferentials conditions, rewritten in detail as

where r j , l i is randomly selected from I a Ω j , I a L i and being fixed, with coefficients satisfying ˆ θ g i ∈

constraint functions g i (1 ≤ i ≤ r ) and h j (1 ≤ j ≤ s ) are affine , i.e. , the gradients ∇ g i ( w ) , ∇ h j ( w ) are constant. 2 We define g i and h j as constraint residuals g i ( w ) := v glyph[latticetop] i w -d i , h j ( w ) := ω glyph[latticetop] j w -e j .

We keep track of the following index sets determined by signs of constraint residuals:

To characterize the path, we further introduce a reparameterization in terms of an auxiliary variable t ≥ 0 (thought of as time ), whereby for a given T > 0 we introduce functions z ( · ) : [0 , T ] → [ z min , z max ] and ξ ( · ) : [ z min , z max ] → R such that: ξ ( · ) is Lipschitz, z ( · ) is differentiable on (0 , T ) , and we have ˙ z = ξ ( z ( t )) for all t ∈ (0 , T ) . In a slight abuse of notation, we define the path w.r.t. t as { w glyph[star] ( t ) glyph[defines] w glyph[star] ( z ( t )) : t ∈ [0 , T ] } . To enhance structural clarity, we denote n Z = |Z E ∪Z I | and represent the certain matrix inversion as

where the rows of matrix U Z are the constant differentials, v glyph[latticetop] i for i ∈ Z E and ω glyph[latticetop] j for j ∈ Z I , and

Now we are fully equipped to unveil the structure of homotopy path, as shown in next subsection.

3.3 Piecewise smooth structure & Unified view

To describe the structure, we define the kink (non-smooth point) as the point that ˆ θ k Ω j , ˆ θ k L i hit the restriction bound in (9), or set in (10) is violated so the entire structure changes.

Theorem 3. Given an optimum ( w glyph[star] 0 , z 0 ) at t 0 , and assume that t 0 is not a kink, then there exists an open neighborhood of t 0 such that w glyph[star] ( t ) is a C 1 function of t and satisfies the following autonomous system

where P ( w glyph[star] | z ) is obtained from the definition in (11), and the ˆ θ L i , ˆ θ Ω j can be solved from (8).

The (13) offers an explicit gradient flow of the solution path in underlying optimization space. Solving this ODE numerically would recover all the optimal parametric solutions, which provide the conformity scores and one can obtain the required p -values (2) that used for exact conformal prediction via (3).

Theorem 4. The optimality solution ( w glyph[star] ( t ) , z ( t )) has a unique trajectory for t ∈ (0 , T ) . The w glyph[star] ( t ) is continuous if selections D k L i ( · ) , D k Ω j ( · ) are convex. Furthermore, if gradients of constraints {∇ g i ( w ) : ∇ g i ( w ) = 0 }∪{∇ h j ( w ) : ∇ h j ( w ) = 0 } are affinely independent at the solution w glyph[star] ( z ) over an open neighborhood of z , then the coefficient paths ˆ θ g i , ˆ θ h j are unique and continuous at z .

Theorem 5. On a optimality path with set configuration (10), the coefficients for constraints satisfies

where Q ( w glyph[star] ) is from (11), and

Although Theorem 3, 4, and 5 are highly technical and may difficult to grasp on first glance, they lay the groundwork for practical application, as illustrated later in Section 4. Specifically, Theorem 4 ensures continuity, which is fundamental for the numerical ODE solvers. Theorem 5, on the other hand, provides a rule for handling constraints.

Theorem 6. Suppose that D k L i ( · ) is µ -strongly convex for µ ≥ 0 , D k Ω j ( · ) is σ -strongly convex for σ > 0 , and P ( w glyph[star] | z ) , ∂ z D k L n +1 ( · ) , ∂ z ∇ D k L n +1 ( · ) are all locally glyph[lscript] -Lipschitz continuous. Suppose further that ξ ( · ) is Lipschitz continuous on [ z min , z max ] and satisfies | ξ ( z ) | ≤ ¯ C for all z ∈ [ z min , z max ] . Then, it holds that Υ( · , · ) defined in (13) is uniformly glyph[lscript] Υ -Lipschitz continuous with glyph[lscript] Υ = ¯ Cglyph[lscript] 2 + 2 ¯ Cglyph[lscript] ( n +1) µ + ∑ j m =1 λ j σ for any z ∈ [ z min , z max ] when the active selections I L , I Ω are fixed.

The essence of Theorem 6 lies in its suggestion that the dynamics of w glyph[star] ( t ) is piecewise continuous, i.e. , w glyph[star] ( t ) maintains smoothness between two adjacent kinks. By considering specific choices of ξ ( · ) and y n +1 ( z ) , our system (13) generalizes some previously studied methodologies in fast conformalization. First, consider the scenario with an equally spaced discretization of the interval [0 , T ] , namely t k = k · h ′ for some fixed step-size h ′ > 0 . Thus, the sequence z k := z ( t k ) is approximately given by z k +1 ≈ z k + h ′ · ξ ( z k ) . Intuitively, the choice of ξ ( · ) controls the dynamic of z ( · ) and generalizes some previously considered sequences { z k } for the problem (5). For example, letting ξ ( z ) := 1 we recover the arithmetic sequence in [12] and letting ξ ( z ) := -z we recover the geometric sequence in [17].

4 Fast conformalization algorithm

The complete algorithm on the fast exact conformalization is outlined in the plate referred as Algorithm 1.

Dataset

Model (Parameter)

Grid1

/

Grid2

SCP

Exact

Coverage

Length

Coverage

Length

Coverage

Length

fried

NLS

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7

2097

4194

6291

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10485

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Size of dataset (delta)

0

5

10

15

20

25

30

Running Time of NLS (s)

Grid1

Grid2

Exact

900

1800

2700

3600

4500

5400

6300

7200

Size of dataset (fried)

0

5

10

15

20

25

30

Running Time of GFM (s)

Grid1(

1 =0.01)

Grid1(

1 =0.03)

1 =0.01)

1 =0.03)

Exact(

1 =0.01)

Exact(

1 =0.03)

4500

9000

13500

18000

22500

27000

31500

36000

Size of dataset (fried)

0

5

10

15

20

25

Running Time of IGR (s)

Grid1(

1, 2 =0.1, 0.02)

Grid1(

1, 2 =0.2, 0.03)

1, 2 =0.1, 0.02)

1, 2 =0.2, 0.03)

Exact(

1, 2 =0.1, 0.02)

Exact(

1, 2 =0.2, 0.03)

4.1 On solving (11)

To avoid the direct computation of matrix inversion (11), our practical algorithm involves sweep operator [22, 23]. Suppose A is an p × p symmetric matrix, sweeping on the k -th diagonal entry A kk = 0 of A results in a matrix A with entries glyph[negationslash]

glyph[negationslash]

glyph[negationslash]

Since the sweeping (15) preserves symmetry, all operations can be performed solely on either the lower or upper-triangular part of A to ease the computational burden [24]. To begin with, we initiate with a sweeping tableau as [ ˜ H ( w glyph[star] | z ) * U Z O ] , and further sweeping of diagonal entries of block ˜ H yields M glyph[defines] [ M 11 * M 21 M 22 ] . Then we reinitialize our new tableau in the form of [ -M 22 M 21 * -M 11 ] , and further sweeping of diagonal entries of block M 22 makes [ R Q glyph[latticetop] ( w glyph[star] | z ) * P ( w glyph[star] | z ) ] . Compared to direct ( p 2 + n 2 )

inversion, it also decreases a O Z storage space.

4.2 Efficiency

Algorithm 1 shows a very favourable behavior empirically, and converges remarkably faster than the standard grid-search type algorithm. We argue that this observation is actually quite natural. Indeed, our algorithm can follow the ground truth solution path, and the numerical integration process is fully deterministic, which avoids large fluctuations between the iteration steps like stochastic gradient descent. Therefore, the solving process of Algorithm 1 exhibits a more stable character being completely deterministic and has no extensive loops, which explains the much faster convergences observed in practice. In contrast, the standard grid-search type method would cost N -times the original batch iterative algorithms, where N is the number of grid points.

5 Numerical experiments

We provide experimental results on real-world benchmarks to validate our derived algorithm. All experiments presented in this study were conducted on a workstation running the Ubuntu 18 . 04 operating system, equipped with Intel Xeon Gold 5218R CPU × 64 and 60 . 9 GB of RAM. We integrate a system of ordinary differential equations using lsoda from the FORTRAN library, where an glyph[negationslash]

interface for SciPy is available using the odepack . The concrete parameter settings of ODE solver are shown in the Table 3, wherein the numerical solver exploit the Runge-Kutta method of order 4 or 5 . The parameterizers are set to y n +1 ( z ) = 4 z , ξ ( z ) = 1 , respectively.

Model For evaluation, here we employ 3 specific forms of (1), i.e. , Nonnegative Least Squares ( NLS ) [25], Graph-guided Fused Model ( GFM ) [26], and Inverse Gaussian Regression ( IGR ) [27].

Conformalization A conformal prediction set with target coverage level 0 . 8 ( α = 0 . 2 ) is calculated for each sample in testing set using each of the 4 methods, i.e. , the standard grid-point evaluation method ( Grid1 ) [1], grid-point method with warm-restart strategy ( Grid2 ) [28], the split conformal prediction method ( SCP ) [4], and our exact conformalization method in Algorithm 1 ( Exact ). We use the conformity score function A i = -| y i -η w glyph[star] ( x i ) | . Conventionally, the interval [ y min n +1 , y max n +1 ] (part of the input) can be chosen simply as [ y [1] , y [ n ] ] , where y [1] ≤ y [2] ≤ · · · ≤ y [ n ] are the order statistics of the response variable. In experiments we set the search range even more conservatively, enlarging the sample range by 50% of length [ y min n +1 , y max n +1 ] := [ y [1] -0 . 25( y [ n ] -y [1] ) , y [ n ] +0 . 25( y [ n ] -y [1] ) ] .

Dataset Our experiments were conducted using real-world datasets. We employ real-world datasets from OpenML [29] and UCI repository [30] in simulations. We randomly partition the dataset into training set, testing set, and calibration set (used in SCP ) with 70%, 10%, and 20% of the total samples. To facilitate optimization, we have standardized the entire original dataset by removing the mean and scaling to unit variance for the features, and adjusting the mean of all labels to 0 .

Setup Our central claim is twofold, encompassing both accuracy and efficiency.

42

84

126

168

210

252

294

336

Number of grids (fried dataset)

0

5

10

15

20

25

30

Running Time of GFM (s)

Grid1(

1 =0.03)

Grid1(

1 =0.05)

1 =0.03)

1 =0.05)

Exact(

1 =0.03)

Exact(

1 =0.05)

0

1

2

3

0.0

0.2

0.4

0.6

0.8

1.0

Probability

0.971

0.027

0.002

0.0

7

6

5

4

Running time (

log

(

s

))

=1e-2

100

200

300

400

500

k

7

6

5

4

=2e-2

λ

1

λ

1

We first report the average empirical coverage, average length of the prediction set in Table 2. Regarding running efficiency, we present average training time per dataset in Figure 2 while varying the scale of training set. In Figure 3, we compare the training times when different grid numbers are used in Grid1 and Grid2 . We further plot the histogram of kink numbers, and the running time against various y n +1 ( · ) in Figure 4.

Results & Analysis From Table 2 we observe that all these methods provide valid and nearly perfect coverage. The grid and exact method give similar lengths, where the slight difference is due

to the rounding between neighboring grid points. The SCP produces wider intervals due to a less efficient use of data. Given Figure 2, our exact method is much faster than the baselines, with same solid performance. We can also learn from the Figure 3 that the time of Algorithm 1 still compares favorably against the grid-search type method, even when the grid is sparse. Figure 4 found that there exists tiny number of kinks in majority runnings, which offers hope for the future expansion of our algorithm, and indicates that the choice of y n +1 can make a difference in efficiency, as it will determine the solving interval and the total query times of gradient.

6 Conclusion

In this work, we present a unified framework and an elaborate algorithm with statistical analysis for fast exact conformalization regarding generalized parametric estimation. We illustrate the strong and competitive performance of proposed methods in a series of benchmarks.

In future work, a potential direction is to consider scenarios where labels are multidimensional, such as multi-task learning, in which the label space Y would be indexed by multiple independent parameters ( z 1 , . . . , z K ) . Under such conditions, the homotopy solution path would extend to a solution surface, and our ODE system could be reformulated as a corresponding system of partial differential equations. Additionally, compared to some previous work, Algorithm 1 offers increased speed but at the cost of higher memory requirements. It necessitates storing all training samples throughout the process for gradient queries. We believe it would be interesting to use recent advancements like the Kroneckerfactored approximate method to potentially enhance the memory scalability.

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Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors study the *solution path* of constrained optimization problems when one training label varies in ${\\mathbb R}$. In the Conformal Prediction framework, this is equivalent to studying the output of a Full-CP algorithm. The paper extends the approach to optimization problems with general constraints and a piece-wise differentiable objective.\n\nSTRENGTHS:\n- Characterizing the solution path of a series of smoothly varying optimization problems is relevant, even beyond the CP framework.\n- The Lagrangian reparametrization approach is interesting.\n- The proposed method seems to be more efficient than Split CP.\n\nWEAKNESSES:\n- It would be helpful to clarify from the start what \"opening the black box\" means, e.g. by listing the assumptions needed to obtain the solution path efficiently. \n- The authors should justify better why they need \ni) general constraints and \nii) piece-wise differentiable objectives. \nA warm-up section describing the unconstrained case would also help. Has the unconstrained and globally differentiable case been considered elsewhere?\n- The method seems to apply in a convex neighbor of the optimum. Assuming the algorithm is initialized there, the optimization problem is quadratic. The authors should comment on the difference between their method and the standard full-CP approach in the presence of several local optima.\n\nQUESTIONS:\n- In the abstract, you mention the computational burden of full-CP algorithms. What about Split-CP? Does \"strong assumptions\" mean data inefficiency?\n- Is the solution path unique? What happens for overparametrized models?\n- In what sense the prediction sets of [18] and [19] do not have statistical guarantees? Can't one combine the upper bound on the estimation error and standard CP validity?\n- Is the formalism developed in Section 3 only needed because of the general constraints and piece-wise assumption in  Equation 1? \n- Intuitively, what does produce a kink in $w_*(t)$? \n- Why do you fix $\\alpha=0.8$?\n- In Algorithm 1, does $z$ need to be discretized?\n- Why Split CP does not appear in Figures 3 and 4?\n- Why is the standard approach, i.e. ``Grid1/Grid2`` in Table 2, outperformed by Split CP? Normally full-CP is expected to be more efficient.\n\nLIMITATIONS:\nThe authors do not discuss the limitations of their work. They may have added a few lines about the applicability of their assumption and the limited choice of predictors in the experiments.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-04"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a new method to accelerate the computation of full conformal prediction sets, a task that traditionally requires a computationally intensive grid-search approach. The proposed method aims to streamline this process, potentially reducing the need to refit the predictive model for each test point and each possible outcome value. However, the paper suffers from poor writing, heavy mathematical notation, and an overly dense presentation, while also employing tricks like small font sizes in figures and text wrapping to extend beyond the standard page limits. Due to these significant issues, I believe that a thorough review of this paper is not feasible at this time. I would consider this a 'desk rejection' from my perspective. I hope the authors will not be offended or discouraged, as this is not a judgment on the quality of their research, which I was unable to carefully assess, but rather an invitation to improve the presentation to facilitate the review process.\n\nSTRENGTHS:\n- This paper studies an important problem in conformal inference, namely to speed up computations for full-conformal prediction.\n- The proposed method could potentially reduce the computational burden associated with conventional grid-search approaches.\n\nWEAKNESSES:\n- The paper is not very well written, it's very dense, and difficult to understand, even for those familiar with the broader field.\n- The paper is too long and only fits within the page limits due to some \"tricks\" such as using very small font sizes in figures and wrapping text around figures.\n\nQUESTIONS:\n- Could you try to enhance the writing quality and clarity of the paper to make it more accessible to a broader audience?\n- Would you consider shortening the paper, or alternatively, submitting it to a venue that accommodates longer articles? This paper might turn out to be much easier to understand in an extended version. My philosophy is that if the content cannot be effectively conveyed within 9 pages, it is better suited for a different format. Forcing it to fit within this limit does a disservice to the readers by compromising clarity and comprehensiveness.\n\nLIMITATIONS:\nThe primary limitation of this paper is its poor writing quality and extremely dense presentation, making it difficult to follow and understand. In its current form, the potential audience of this paper might at best be limited to a very narrow group of readers, which unfortunately does not include this reviewer.", "TITLE": "Official Review", "RECOMMENDATION": 3, "REVIEWER_CONFIDENCE": 1, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-08"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper introduces a method to compute exact conformal prediction intervals that have statistical guarantees. The method improves from previous work in three ways:\n* The conformal interval has exact guarantees instead of approximative\n* The loss function family covered is larger than convex, same for the regularizer family\n* It works for linearly constrained problem \n\nThe core idea is to build a path from  value $t\\in [0,1]$ to targets $y$ and find how the parameters of the function $\\mathbf{w}$ evolve as we vary $t$ (and therefore $y$). The weights $\\mathbf{w}(\\cdot)$ as a function of $t$ satisfy an ODE that can be solve with standard solvers when the functions in the loss function and constrains are piecewise continuously differentiable.\n\nThe authors then show the improvement in terms of coverage, dataset size and more importantly speed on several datasets.\n\nSTRENGTHS:\n* The examples are compelling are show very well the improvement\n* The contribution is pretty significant as it widens the possible families of loss functions\n\nWEAKNESSES:\n* My main issue is that the paper is very hard to read. It is too notation heavy and dense. For example, why use $\\hat \\theta$ instead of $\\theta$? The $\\hat {}$ make them look like estimators when it is not the case here\n* Classification is a big application of conformal prediction, it is unfortunate that we do not have an example .\n* Figure 1 might be a bit misleading. It seems to imply that the method works for higher dimensions of $\\mathcal{Y}$ but it does not seem the case\n\nQUESTIONS:\n* Does the method require more memory than the baseline methods?\n* I am not convinced by the explanation given for classification(1067-1074). This might be valid for some ranking tasks but not when there is no order in the classes, in my opinion. Could you illustrate it, even with a toy example?\n\nLIMITATIONS:\n* You mention not implementing [13] because it uses underlying structure and therefore might be faster. It would still be interesting to see if the gain of speed from using [13] instead of your method is significant or not, in cases where it applies. \n* (see comment on classification)", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 3, "SOUNDNESS_CORRECTNESS": 2, "DATE": "2024-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nIn the manuscript \"Towards fast exact Confomralization of Generalised Parametric Estimation\" the authors provide a very interesting generalisation of an approach, fundamental even if a bit disregarded in the mainstream literature on Conformal Prediction, that aims at computing the whole solution path of a regression algorithm in order to render more efficient the estimation of (full) conformal prediction sets.\nAfter having described the very general framework in which they operate, the authors proceed to obtain theoretical insights about their methodology. More specifically, they characterise the underlying structure of the path to be an inherently piecewise smooth function.\nAfter doing so they propose a practical methodology to compute such path, essentially by solving an ODE.\nThe method is then evaluated with an extensive empirical study, showing its remarkable performance\n\nSTRENGTHS:\n- The work is extremely thorough and deep, and the discussion about the different peculiarities of their approach\n- The referencing work is quite remarkable, as it depicts very clearly the literature landscape where the authors are moving\n- While building on previous works, the authors provide a very novel take on the subject, proposing a completely new methodology, of great practical usability\n\nWEAKNESSES:\n- The authors are not tackling an easy task in terms of communication, having to tackle, and in a fairly deep way, many different topics. For this reason I find the presentation a bit \"nebulous\" and vague at times\n- No code is provided, which has prevented me from verifying some of the claims.\n\nQUESTIONS:\n- Some claims are a bit vague. I understand why the authors focus on Full conformal, but I doubt a reader not fully aware of Conformal Prediction will be able to do it (Paragraph 2 of the introduction). I suggest to the authors to be more explicit about the reasons\n- Footnote of page 1 - the authors need to clarify that symmetricity in this case is intended in the specific sense used in the CP literature, which in fact is defined later on... or omit the consideration which seems rather tangential\n- Rather then \"inexact\" I would rather talk about lack of (finite sample) calibration.\n- I don't understand why the already present approaches are deemed to be \"black boxes\". moreover I don't think that \"opening the black box\" is the right narrative to use in this case. Maybe something along the line of \"let's generalise...\"\n- The label space is a generic $\\mathcal{Y}$. I am wondering if the authors can be more specific about, for instance, possible extensions of their approach to the relatively rich field of Conformal Prediction for multivariate and complex data.\n\nLIMITATIONS:\nThe authors state relevant open problems on the subject.\nSince the very clear applicative interest this paper may have, I believe that adding something on the methodological side (such as the need for extensions to the multivariate case, or some ideas about possible extensions to a dependent data setting) could be very interesting.", "TITLE": "Official Review", "RECOMMENDATION": 8, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-10"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe authors introduce an algorithm to compute prediction intervals for conformal prediction in the context of empirical risk minimization  (ERM) with constraints on the parameters. To do so, the authors derive a differential equation solved by the ERM estimator as a function of the last label to compute a solution path and generate the conformity scores for each possible label.\nIn their numerical experiments, the method is compared to the baseline (which simply iterates over a grid of labels) and split conformal prediction (SCP). The authors observe that their method provide the correct coverage and roughly the same interval length as the baseline for a fraction of the numerical cost.\n\nSTRENGTHS:\n- The method introduced by the authors works for general scenarios, extending beyond the cases of Ridge / Lasso.\n- On the numerical side, their methods appear to provide a significant improvement (in terms of running time) over the baseline while having the same interval size. As a by product, the authors show that their method can be used to update the estimator when a single label is changed in the training data.\n\nWEAKNESSES:\n- Clarity : I found the mathematical details of section 3 to be difficult to follow and check (for the parts in the Appendix). I feel like some parts, while they are important for the rigor of the paper, could be explained after the introduction of Algorithm 1 or put in the Appendix. For instance Theorem 4 and Theorem 6.\n\nQUESTIONS:\n- It would be nice to have a comparison with the running time of split conformal prediction (while the computations are not exactly the same) as in Figure 3 to get a sense of the order of magnitude of the computation time.\n- For simple settings such as Lasso, it would have been nice to compare the computation time with methods from the literature e.g. [Lei, 2017].\n- How does the method scale (in terms of running time) with the dimensionality ? \n\nMisc. questions : \n- In equation (12), is there a difference between Q(w*) and Q(w* | z) from (9) or is the conditioning on z implicit ?\n\nLIMITATIONS:\nSome limitations are discussed in Appendix F. As the paper is quite theoretical, the question of societal impact is not applicable here.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 2, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-12"}