Efﬁcient and Modular Implicit Differentiation
Abstract
Automatic differentiation (autodiff) has revolutionized machine learning. It allows to express complex computations by composing elementary ones in creative ways and removes the burden of computing their derivatives by hand. More recently, differentiation of optimization problem solutions has attracted widespread attention with applications such as optimization layers, and in bi-level problems such as hyper-parameter optimization and meta-learning.
However, so far, implicit differentiation remained difﬁcult to use for practitioners, as it often required case-by-case tedious mathematical derivations and implementations. In this paper, we propose automatic implicit differentiation, an efﬁcient and modular approach for implicit differentiation of optimization problems. In our approach, the user deﬁnes directly in Python a function F capturing the optimality conditions of the problem to be differentiated.
Once this is done, we leverage autodiff of F and the implicit function theorem to automatically differentiate the optimization problem. Our approach thus combines the beneﬁts of implicit differentiation and autodiff. It is efﬁcient as it can be added on top of any state-of-the-art solver and modular as the optimality condition speciﬁcation is decoupled from the implicit differentiation mechanism.
We show that seemingly simple principles allow to recover many existing implicit differentiation methods and create new ones easily.
We demonstrate the ease of formulating and solving bi-level optimization problems using our framework.
We also showcase an application to the sensitivity analysis of molecular dynamics.
Introduction
Automatic differentiation (autodiff) is now an inherent part of machine learning software.
It allows to express complex computations by composing elementary ones in creative ways and removes the tedious burden of computing their derivatives by hand. In parallel, the differentiation of optimization problem solutions has found many applications. A classical example is bi-level optimization, which typically involves computing the derivatives of a nested optimization problem in order to solve an outer one.
Examples of applications in machine learning include hyper-parameter optimization [23, 77, 70, 38, 12, 13], neural networks [59], and meta-learning [39, 72]. Another line of active research involving differentiation of optimization problem solutions are optimization layers [55, 6, 65, 31, 46], which can be used to encourage structured outputs, and implicit deep networks [8, 36, 43, 44, 73], which have a smaller memory footprint than backprop-trained networks.
Since optimization problem solutions typically do not enjoy an explicit formula in terms of their inputs, autodiff cannot be used directly to differentiate these functions.
In recent years, two main approaches have been developed to circumvent this problem. The ﬁrst one consists of unrolling the iterations of an optimization algorithm and using the ﬁnal iteration as a proxy for the optimization problem solution [83, 32, 29, 39, 1].
This allows to explicitly construct a computational graph relating the algorithm output to the inputs, on which autodiff can then be used transparently. However, this requires a reimplementation of the algorithm using the autodiff system, and not all algorithms are necessarily autodiff friendly. Moreover, forward-mode autodiff has time complexity that scales linearly with the number of variables and reverse-mode autodiff has memory complexity that scales linearly with the number of algorithm iterations.
In contrast, a second approach consists in implicitly relating an optimization problem solution to its inputs using optimality conditions.
In a machine learning context, such implicit differentiation has been used for stationarity conditions [11, 59], KKT conditions [23, 45, 6, 67, 66] and the proximal gradient ﬁxed point [65, 12, 13]. An advantage of implicit differentiation is that a solver reimplementation is not needed, allowing to build upon decades of state-of-the-art software. Although implicit differentiation has a long history in numerical analysis [48, 10, 57, 20], so far, it remained difﬁcult to use for practitioners, as it required a case-by-case tedious mathematical derivation and implementation.
CasADi [7] allows to differentiate various optimization and root ﬁnding problem algorithms provided by the library.
However, it does not allow to easily add implicit differentiation on top of existing solvers from optimality conditions expressed by the user, as we do. A recent tutorial explains how to implement implicit differentiation in JAX [34]. However, the tutorial requires the user to take care of low-level technical details and does not cover a large catalog of optimality condition mappings as we do.
Other work [2] attempts to address this issue by adding implicit differentiation on top of cvxpy [30]. This works by reducing all convex optimization problems to a conic program and using conic programming’s optimality conditions to derive an implicit differentiation formula.
While this approach is very generic, solving a convex optimization problem using a conic programming solver—an ADMM-based splitting conic solver [68] in the case of cvxpy—is rarely state-of-the-art for every problem instance.
In this work, we ambition to achieve for optimization problem solutions what autodiff did for computational graphs. We propose automatic implicit differentiation , a simple approach to add implicit differentiation on top of any existing solver.
In this approach, the user deﬁnes directly in
Python a mapping function F capturing the optimality conditions of the problem solved by the algorithm. Once this is done, we leverage autodiff of F combined with the implicit function theorem to automatically differentiate the optimization problem solution.
Our approach is generic , yet it can exploit the efﬁciency of state-of-the-art solvers. It therefore combines the beneﬁts of implicit differentiation and autodiff.
To summarize, we make the following contributions • We describe our framework and its JAX [21, 42] implementation ( https://github.com/ google/jaxopt/ ).
Our framework signiﬁcantly lowers the barrier to use implicit differentiation, thanks to the seamless integration in JAX, with low-level details all abstracted away.
• We instantiate our framework on a large catalog of optimality conditions (Table 1), recovering existing schemes and obtaining new ones, such as the mirror descent ﬁxed point based one.
• On the theoretical side, we provide new bounds on the Jacobian error when the optimization problem is only solved approximately, and empirically validate them. • We implement four illustrative applications , demonstrating our framework’s ease of use. Beyond our software implementation in JAX, we hope this paper provides a self-contained blueprint for creating an efﬁcient and modular implementation of implicit differentiation in other frameworks.
Notation.
We denote the gradient and Hessian of f : R d → R evaluated at x ∈ R d by ∇ f ( x ) ∈ R d and ∇ 2 f ( x ) ∈ R d × d . We denote the Jacobian of F : R d → R p evaluated at x ∈ R d by ∂F ( x ) ∈
R p × d .
When f or F have several arguments, we denote the gradient, Hessian and Jacobian in the i th argument by ∇ i , ∇ 2 i and ∂ i , respectively.
The standard probability simplex is denoted by (cid:52) d :=
{ x ∈ R d : (cid:107) x (cid:107) 1 = 1 , x ≥ 0 } .
For any set C ⊂ R d , we denote the indicator function {+I C : R d → R ∪ { + ∞} where I C ( x ) = 
 if x ∈ C+} , I C ( x ) = + ∞ otherwise. For a vector or matrix A , we note (cid:107)
Automatic implicit differentiation 2.1 General principles
Overview.
Contrary to autodiff through unrolled algorithm iterations, implicit differentiation typically involves a manual, sometimes complicated, mathematical derivation. For instance, numerous works [23, 45, 6, 67, 66] use Karush–Kuhn–Tucker (KKT) conditions in order to relate a constrained optimization problem’s solution to its inputs, and to manually derive a formula for its derivatives.
The derivation and implementation in these works are typically case-by-case.
X_train, y_train = load_data() # Load features and labels def f(x, theta): # Objective function residual = jnp
[Figure] dot(X_train, x)
[Figure]
y_
[Figure]
return (jnp.sum(residual ** 2) + theta * jnp.sum(x ** 2)) / # Since f is differentiable and unconstrained, the optimality # condition F is simply the gradient of f in the 1st argument
F = jax.grad(f, argnums=0)
[Figure]
(F
[Figure]
_solver(init_x, theta):
[Figure]
[Figure]
In this work, we propose a generic way to easily add implicit differentiation on top of existing solvers.
In our approach, the user deﬁnes directly in Python a mapping function F capturing the optimality conditions of the problem solved by the algorithm.
We provide reusable building blocks to easily express such F .
The provided F is then plugged into our Python decorator @custom_root , which we append on top of the solver declaration we wish to differentiate. Under the hood, we combine the implicit function theorem and autodiff of F to automatically differentiate the optimization problem solution.
A simple illustrative example is given in Figure 1.
Differentiating a root. Let F : R d × R n → R d be a user-provided mapping, capturing the optimality conditions of a problem. An optimal solution, denoted x (cid:63) ( θ ) , should be a root of F :
[Equation] We can see x (cid:63) ( θ ) as an implicitly deﬁned function of θ ∈ R n , i.e., x (cid:63) : R n → R d .
More precisely, from the implicit function theorem [48, 57], we know that for ( x 0 , θ 0 ) satisfying F ( x 0 , θ 0 ) =with a continuously differentiable F , if the Jacobian ∂ 1 F evaluated at ( x 0 , θ 0 ) is a square invertible matrix, then there exists a function x (cid:63) ( · ) deﬁned on a neighborhood of θ 0 such that x (cid:63) ( θ 0 ) =
x 0 . Furthermore, for all θ in this neighborhood, we have that F ( x (cid:63) ( θ ) , θ ) = 0 and ∂x (cid:63) ( θ ) exists. Using the chain rule, the Jacobian ∂x (cid:63) ( θ ) satisﬁes
[Equation]
Computing ∂x (cid:63) ( θ ) therefore boils down to the resolution of the linear system of equations [Equation] When (1) is a one-dimensional root ﬁnding problem ( d = 1 ), (2) becomes particularly simple since we then have ∇ x (cid:63) ( θ )
=
B (cid:62) /A , where A is a scalar value. We will show that existing and new implicit differentiation methods all reduce to this simple principle.
We call our approach automatic implicit differentiation as the user can freely express the optimization solution to be differentiated through the optimality conditions F . Our approach is efﬁcient as it can be added on top of any state-of-the-art solver and modular as the optimality condition speciﬁcation is decoupled from the implicit differentiation mechanism.
This contrasts with existing works, where the derivation and implementation are speciﬁc to each optimality condition. Differentiating a ﬁxed point. We will encounter numerous applications where x (cid:63) ( θ ) is instead implicitly deﬁned through a ﬁxed point :
[Equation]
where T : R d × R n → R d . This can be seen as a particular case of (1) by deﬁning the residual
[Equation]
In this case, when T is continuously differentiable, using the chain rule, we have A = −
[Equation]
1 F ( x (cid:63) ( θ ) , θ ) = I − ∂ 1 T ( x (cid:63) ( θ ) , θ ) and B = ∂ 2 F ( x (cid:63) ( θ ) , θ ) = ∂ 2 T ( x (cid:63) ( θ ) , θ ) .
Computing JVPs and VJPs.
In most practical scenarios, it is not necessary to explicitly form the Jacobian matrix, and instead it is sufﬁcient to left-multiply or right-multiply by ∂ 1 F and ∂ 2 F .
These are called vector-Jacobian product (VJP) and Jacobian-vector product (JVP), and are useful for integrating x (cid:63) ( θ ) with reverse-mode and forward-mode autodiff, respectively. Oftentimes, F will be explicitly deﬁned. In this case, computing the VJP or JVP can be done via autodiff.
In some cases, F may itself be implicitly deﬁned, for instance when F involves the solution of a variational problem. In this case, computing the VJP or JVP will itself involve implicit differentiation. The right-multiplication (JVP) between J = ∂x (cid:63) ( θ ) and a vector v , Jv , can be computed efﬁciently by solving A ( Jv ) =
Bv .
The left-multiplication (VJP) of v (cid:62) with J , v (cid:62) J , can be computed by ﬁrst solving A (cid:62) u
= v .
Then, we can obtain v (cid:62) J by v (cid:62) J = u (cid:62) AJ = u (cid:62) B . Note that when B changes but A and v remain the same, we do not need to solve A (cid:62) u
= v once again.
This allows to compute the VJP w.r.t. different variables while solving only one linear system. To solve these linear systems, we can use the conjugate gradient method [51] when A is symmetric positive semi-deﬁnite and GMRES [75] or BiCGSTAB [81] otherwise. These algorithms are all matrix-free: they only require matrix-vector products.
Thus, all we need from F is its JVPs or VJPs. An alternative to GMRES/BiCGSTAB is to solve the normal equation AA (cid:62) u
= Av using conjugate gradient. This can be implemented using JAX’s transpose routine jax.linear_transpose [41].
In case of non-invertibility, a common heuristic is to solve a least squares min J (cid:107) AJ − B (cid:107) 2 instead. Pre-processing and post-processing mappings. Oftentimes, the goal is not to differentiate θ per se, but the parameters of a function producing θ . One example of such pre-processing is to convert the parameters to be differentiated from one form to another canonical form, such as a quadratic program [6] or a conic program [2].
Another example is when x (cid:63) ( θ ) is used as the output of a neural network layer, in which case θ is produced by the previous layer. Likewise, x (cid:63) ( θ ) will often not be the ﬁnal output we want to differentiate. One example of such post-processing is when x (cid:63) ( θ ) is the solution of a dual program and we apply the dual-primal mapping to recover the solution of the primal program. Another example is the application of a loss function, in order to reduce x (cid:63) ( θ ) to a scalar value.
We leave the differentiation of such pre/post-processing mappings to the autodiff system, allowing to compose functions in complex ways.
Implementation details. When a solver function is decorated with @custom_root , we use jax.custom_jvp and jax.custom_vjp to automatically add custom JVP and VJP rules to the function, overriding JAX’s default behavior. As mentioned above, we use linear system solvers based on matrix-vector products and therefore we only need access to F through the JVP or VJP with ∂ 1 F and ∂ 2 F . This is done by using jax.jvp and jax.vjp , respectively.
Note that, as in Figure 1, the deﬁnition of F will often include a gradient mapping ∇ 1 f ( x, θ ) . Thankfully, JAX supports secondorder derivatives transparently.
For convenience, our library also provides a @custom_fixed_point decorator, for adding implicit differentiation on top of a solver, given a ﬁxed point iteration T ; see code examples in Appendix B.
2.2 Examples
We now give various examples of mapping F or ﬁxed point iteration T , recovering existing implicit differentiation methods and creating new ones. Each choice of F or T implies different trade-offs in terms of computational oracles ; see Table 1. Source code examples are given in Appendix B. Stationary point condition.
The simplest example is to differentiate through the implicit function
[Equation]
where f : R d × R n → R is twice differentiable, ∇ 1 f is continuously differentiable, and ∇ 21 f is invertible at ( x (cid:63) ( θ ) , θ ) . In this case, F is simply the gradient mapping [Equation]
[Table]
We then have ∂ 1 F ( x, θ ) = ∇ 21 f ( x, θ ) and ∂ 2 F ( x, θ ) = ∂ 2 ∇ 1 f ( x, θ ) , the Hessian of f in its ﬁrst argument and the Jacobian in the second argument of ∇ 1 f ( x, θ ) . In practice, we use autodiff to compute Jacobian products automatically.
Equivalently, we can use the gradient descent ﬁxed point
[Equation]
[Equation]
[Equation]
(5) for all η > 0 . Using (3), it is easy to check that we obtain the same linear system since η cancels out.
KKT conditions. As a more advanced example, we now show that the KKT conditions, manually differentiated in several works [23, 45, 6, 67, 66], ﬁt our framework. As we will see, the key will be to group the optimal primal and dual variables as our x (cid:63) ( θ ) .
Let us consider the general problem argmin z ∈ R p f ( z, θ ) subject to G ( z, θ ) ≤ 0 , H ( z, θ ) = 0 , where z ∈ R p is the primal variable, f : R p × R n → R , G : R p × R n → R r and H : R p × R n → R q are twice differentiable convex functions, and ∇ 1 f , ∂ 1 G and ∂ 1 H are continuously differentiable.
The stationarity, primal feasibility and complementary slackness conditions give
[Equation]
where ν ∈ R q and λ ∈ R r + are the dual variables, also known as KKT multipliers.
The primal and dual feasibility conditions can be ignored almost everywhere [34]. The system of (potentially nonlinear) equations (6) ﬁts our framework, as we can group the primal and dual solutions as x (cid:63) ( θ ) =
( z (cid:63) ( θ ) , ν (cid:63) ( θ ) , λ (cid:63) ( θ )) to form the root of a function F ( x (cid:63) ( θ ) , θ ) , where F : R d × R n → R d and d = p + q + r .
The primal and dual solutions can be obtained from a generic solver, such as an interior point method.
In practice, the above mapping F will be deﬁned directly in Python (see Figure 7 in Appendix B) and F will be differentiated automatically via autodiff. Proximal gradient ﬁxed point.
Unfortunately, not all algorithms return both primal and dual solutions. Moreover, if the objective contains non-smooth terms, proximal gradient descent may be more efﬁcient. We now discuss its ﬁxed point [65, 12, 13].
Let x (cid:63) ( θ ) be implicitly deﬁned as
[Equation] where f : R d × R n → R is twice-differentiable convex and g : R d × R n → R is convex but possibly non-smooth.
Let us deﬁne the proximity operator associated with g by
[Equation]
To implicitly differentiate x (cid:63) ( θ ) , we use the ﬁxed point mapping [69, p.150]
[Equation]
for any step size η > 0 . The proximity operator is 1 -Lipschitz continuous [64]. By Rademacher’s theorem, it is differentiable almost everywhere.
If, in addition, it is continuously differentiable in a neighborhood of ( x (cid:63) ( θ ) , θ ) and if I − ∂ 1 T ( x (cid:63) ( θ ) , θ ) is invertible, then our framework to differentiate x (cid:63) ( θ ) applies.
Similar assumptions are made in [13].
Many proximity operators enjoy a closed form and can easily be differentiated, as discussed in Appendix C. An implementation is given in Figure 2.
[Figure]
Projected gradient ﬁxed point.
As a special case, when g ( x, θ ) is the indicator function I C ( θ ) ( x ) , where C ( θ ) is a convex set depending on θ , we obtain
[Equation]
The proximity operator prox g becomes the Euclidean projection onto C ( θ )
[Equation]
and (7) becomes the projected gradient ﬁxed point [Equation]
Compared to the KKT conditions, this ﬁxed point is particularly suitable when the projection enjoys a closed form. We discuss how to compute the JVP / VJP for a wealth of convex sets in Appendix C.
Current limitations.
While we have not observed issues in practice, we note that the approach developed in this section theoretically only applies to settings where the implicit function theorem is valid, namely, where optimality conditions satisfy the differentiability and invertibility conditions stated in §2.1.
While this covers a wide range of situations even for non-smooth optimization problems (e.g., under mild assumptions the solution of a Lasso regression can be differentiated a.e. with respect to the regularization parameter, see Appendix E), an interesting direction for future work is to extend the framework to handle cases where the differentiability and invertibility conditions are not satisﬁed, using, e.g., the theory of nonsmooth implicit function theorems [25, 19].
In practice, either by the limitations of ﬁnite precision arithmetic or because we perform a ﬁnite number of iterations, we rarely reach the exact solution x (cid:63) ( θ ) . Instead, we reach an approximate solution ˆ x and apply the implicit differentiation equation (2) at this approximate solution.
This motivates the need for precision guarantees of this approach. We introduce the following formalism.
Deﬁnition 1.
Let F : R d × R n → R d be a continously differentiable optimality criterion mapping.
Let
A := − ∂ 1 F and B := ∂ 2 F .
We deﬁne the Jacobian estimate at ( x, θ ) , when A ( x, θ ) is invertible, as the solution to the linear equation
A ( x, θ ) J ( x, θ ) =
B ( x, θ ) .
It is a function J : R d × R n → R d × n . It holds by construction that J ( x (cid:63) ( θ ) , θ ) =
∂x (cid:63) ( θ ) .
Computing J (ˆ x, θ ) for an approximate solution ˆ x of x (cid:63) ( θ ) therefore allows to approximate the true Jacobian ∂x (cid:63) ( θ ) . In practice, an algorithm used to solve (1) depends on θ .
Note however that, what we compute is not the Jacobian of ˆ x ( θ ) , unlike works differentiating through unrolled algorithm iterations, but an estimate of ∂x (cid:63) ( θ ) .
We therefore use the notation ˆ x , leaving the dependence on θ implicit.
We develop bounds of the form (cid:107) J (ˆ x, θ ) − ∂x (cid:63) ( θ ) (cid:107) <
C (cid:107)
ˆ x − x (cid:63) ( θ ) (cid:107) , hence showing that the error on the estimated Jacobian is at most of the same order as that of ˆ x as an approximation of x (cid:63) ( θ ) .
These bounds are based on the following main theorem, whose proof is included in Appendix D.
Theorem 1.
Let F : R d × R n → R d be continuously differentiable. If there are α, β, γ, ε, R >
[Table]
Under these conditions, when (cid:107) ˆ x − x (cid:63) ( θ ) (cid:107) ≤ ε , we have
[Equation]
This result is inspired by [52, Theorem 7.2], that is concerned with the stability of solutions to inverse problems. As a difference, we consider that A ( · , θ ) is uniformly well-conditioned, rather than only at x (cid:63) ( θ ) . This does not affect the ﬁrst order in ε of this bound, and makes it valid for all ˆ x .
Our goal with Theorem 1 is to provide a result that works for general F but can be tailored to speciﬁc cases. In particular, for the gradient descent ﬁxed point (5), this yields
[Equation]
By specializing Theorem 1 for this ﬁxed point, we obtain Jacobian precision guarantees with conditions directly on f rather than F ; see Corollary 1 in Appendix D. These guarantees hold for instance for the dataset distillation experiment in Section 4. Our analysis reveals in particular that Jacobian estimation by implicit differentiation gains a factor of t compared to automatic differentiation , after t iterations of gradient descent in the strongly-convex setting [1, Proposition 3.2]. While our guarantees concern the Jacobian of x (cid:63) ( θ ) , we note that other studies [47, 54, 13] give guarantees on hypergradients (i.e., the gradient of an outer objective).
[Equation]
[Figure]
Unrolling achieves a much worse error for
{+comparable iterate error We illustrate these results on ridge regression, where x (cid:63) ( θ ) = argmin x (cid:107) Φ x − y (cid:107) 
 ++}
(cid:80)
i θ i x 2 i . This problem has the merit that the solution x (cid:63) ( θ ) and its Jacobian ∂x (cid:63) ( θ ) are available in closed form. By running gradient descent for t iterations, we obtain an estimate ˆ x of x (cid:63) ( θ ) and an estimate J (ˆ x, θ ) of ∂x (cid:63) ( θ ) ; cf. Deﬁnition 1.
By doing so for different numbers of iterations t , we can graph the relation between the error (cid:107) x
(cid:63) ( θ ) − ˆ x (cid:107) 2 and the error (cid:107) ∂x (cid:63) ( θ ) − J (ˆ x, θ ) (cid:107) 2 , as shown in Figure 3, empirically validating Theorem 1.
The results in Figurewere obtained using the diabetes dataset from [35], with other datasets yielding a qualitatively similar behavior.
We derive similar guarantees in Corollary 2 in Appendix
D for proximal gradient descent.
Experiments
In this section, we demonstrate the ease of solving bi-level optimization problems with our framework.
We also present an application to the sensitivity analysis of molecular dynamics.
4.1 Hyperparameter optimization of multiclass SVMs
In this example, we consider the hyperparameter optimization of multiclass SVMs [27] trained in the dual.
Here, x (cid:63) ( θ ) is the optimal dual solution, a matrix of shape m × k , where m is the number of training examples and k is the number of classes, and θ ∈ R
+ is the regularization parameter.
The challenge in differentiating x (cid:63) ( θ ) is that each row of x (cid:63) ( θ ) is constrained to belong to the probability simplex (cid:52) k . More formally, let X tr ∈ R m × p be the training feature matrix and Y tr ∈ { 0 , 1 }
m × k be the training labels (in row-wise one-hot encoding).
Let W ( x, θ ) :
= X (cid:62) tr ( Y tr − x ) /θ ∈
R p × k be the dual-primal mapping. Then, we consider the following bi-level optimization problem min θ = [Equation] [Equation]
subject to
[Equation]
where C = (cid:52) k × · · · × (cid:52) k is the Cartesian product of m probability simplices. We apply the change of variable θ = exp( λ ) in order to guarantee that the hyper-parameter θ is positive. The matrix W ( x (cid:63) ( θ ) , θ ) ∈
R p × k contains the optimal primal solution, the feature weights for each class.
The outer loss is computed against validation data X val and Y val .
While KKT conditions can be used to differentiate x (cid:63) ( θ ) , a more direct way is to use the projected gradient ﬁxed point (9).
The projection onto C can be easily computed by row-wise projections on the
[Figure]
simplex. The projection’s Jacobian enjoys a closed form (Appendix C). Another way to differentiate x (cid:63) ( θ ) is using the mirror descent ﬁxed point (13).
Under the KL geometry, projections correspond to a row-wise softmax. They are therefore easy to compute and differentiate. Figure 4 compares the runtime performance of implicit differentiation vs. unrolling for the latter two ﬁxed points.
Dataset distillation [82, 59] aims to learn a small synthetic training dataset such that a model trained on this learned data set achieves a small loss on the original training set.
Formally, let X tr ∈ R m × p and y tr ∈ [ k ] m denote the original training set.
The distilled dataset will contain one prototype example for each class and therefore θ ∈ R k × p .
The dataset distillation problem can then naturally be cast as a bi-level problem, where in the inner problem we estimate a logistic regression model x (cid:63) ( θ ) ∈ R p × k trained on the distilled images θ ∈ R k × p , while in the outer problem we want to minimize the loss achieved by x (cid:63) ( θ ) over the training set:
[Equation]
subject to x (cid:63)
[Equation]
argmin
[Equation]
where f ( W, X ; y ) : = (cid:96) ( y, XW ) , (cid:96) denotes the multiclass logistic regression loss, and ε = 10 − 3 is a regularization parameter that we found had a very positive effect on convergence.
In this problem, and unlike in the general hyperparameter optimization setup, both the inner and outer problems are highdimensional, making it an ideal test-bed for gradient-based bi-level optimization methods. For this experiment, we use the MNIST dataset.
The number of parameters in the inner problem is p = 28 2 = 784 . while the number of parameters of the outer loss is k × p = 7840 .
We solve this problem using gradient descent on both the inner and outer problem, with the gradient of the outer loss computed using implicit differentiation, as described in §2.
This is fundamentally different from the approach used in the original paper, where they used differentiation of the unrolled iterates instead. For the same solver, we found that the implicit differentiation approach was 4 times faster than the original one.
The obtained distilled images θ are visualized in Figure 5.
4.3 Task-driven dictionary learning
Task-driven dictionary learning was proposed to learn sparse codes for input data in such a way that the codes solve an outer learning problem [60, 78, 85].
Formally, given a data matrix X tr ∈ R m × p
[Table]
and a dictionary of k atoms θ ∈ R k × p , a sparse code is deﬁned as a matrix x (cid:63) ( θ ) ∈ R m × k that minimizes in x a reconstruction loss f ( x, θ ) : = (cid:96) ( X tr , xθ ) regularized by a sparsity-inducing penalty g ( x ) .
Instead of optimizing the dictionary θ to minimize the reconstruction loss, [60] proposed to optimize an outer problem that depends on the code. Given a set of labels Y tr ∈ { 0 , 1 } m , we consider a logistic regression problem which results in the bilevel optimization problem:
[Equation]
subject to x
[Equation]
argmin
[-B REAST C ANCER  7.2 × 
 BODYFAT  0.021 S YNTHETIC  0. Train-test split. For every dataset, we only use the train set, where the split is given by the libsvmtools 4 project.-][Equation]
[Equation]
When (cid:96) is the squared Frobenius distance between matrices, and g the elastic net penalty, [60, Eq.
21] derive manually, using optimality conditions (notably the support of the codes selected at the optimum), an explicit re-parameterization of x (cid:63) ( θ ) as a linear system involving θ .
This closedform allows for a direct computation of the Jacobian of x (cid:63) w.r.t.
θ .
Similarly, [78] derive ﬁrst order conditions in the case where (cid:96) is a β -divergence, while [85] propose to use unrolling of ISTA iterations.
Our approach bypasses all of these manual derivations, giving the user more leisure to focus directly on modeling (loss, regularizer) aspects. We illustrate this on breast cancer survival prediction from gene expression data. We frame it as a binary classiﬁcation problem to discriminate patients who survive longer than 5 years ( m 1 = 200 ) vs patients who die within 5 years of diagnosis ( m 0 = 99 ), from p = 1 , 000 gene expression values.
As shown in Table 2, solving (11) (Task-driven DictL) reaches a classiﬁcation performance competitive with state-of-the-art L 1 or L 2 regularized logistic regression with 100 times fewer variables.
with respect to increasing the diameter of the blue particles Many physical simulations require solving optimization problems, such as energy minimization in molecular [76] and continuum [9] mechanics, structural optimization [53] and data assimilation [40]. In this experiment, we revisit an example from JAX-MD [76], the problem of ﬁnding energy minimizing conﬁgurations to a system of k packed particles in a 2 -dimensional box of size (cid:96)
[Equation]
where x (cid:63)
( θ ) ∈
R k × 2 are the optimal coordinates of the k particles,
U ( x i,j , θ ) is the pairwise potential energy function, with half the particles at diameter 1 and half at diameter θ = 0 .
6 , which we optimize with a domain-speciﬁc optimizer [15].
Here we consider sensitivity of particle position with respect to diameter ∂x (cid:63) ( θ ) , rather than sensitivity of the total energy from the original experiment.
Figure 6 shows results calculated via forward-mode implicit differentiation (JVP).
Whereas differentiating the unrolled optimizer happens to work for total energy, here it typically does not even converge, due to the discontinuous optimization method.
Conclusion
We proposed in this paper an approach for automatic implicit differentiation, allowing the user to freely express the optimality conditions of the optimization problem whose solutions are to be differentiated, directly in Python. The applicability of our approach to a large catalog of optimality conditions is shown in the non-exhaustive list of Table 1, and illustrated by the ease with which we can solve bi-level and sensitivity analysis problems.
Appendix A More examples of optimality criteria and ﬁxed points
To demonstrate the generality of our approach, we describe in this section more optimality mapping F or ﬁxed point iteration T .
Mirror descent ﬁxed point.
We again consider the case when x (cid:63) ( θ ) is implicitly deﬁned as the solution of (8).
We now generalize the projected gradient ﬁxed point beyond Euclidean geometry.
Let the Bregman divergence D ϕ : dom( ϕ ) × relint(dom( ϕ )) → R + generated by ϕ be deﬁned by
[Equation]
We deﬁne the Bregman projection of y onto C ( θ ) ⊆ dom( ϕ ) by
[Equation]
Deﬁnition (12) includes the mirror map ∇ ϕ ∗ ( y ) for convenience. It can be seen as a mapping from R d to dom( ϕ ) , ensuring that (12) is well-deﬁned.
The mirror descent ﬁxed point mapping is then
[Equation]
Because T involves the composition of several functions, manually deriving its JVP/VJP is error prone.
This shows that our approach leveraging autodiff allows to handle more advanced ﬁxed point mappings.
A common example of ϕ is ϕ
( x ) = (cid:104) x, log x − 1 (cid:105) , where dom( ϕ ) = R d + . In this case, D ϕ is the Kullback-Leibler divergence.
An advantage of the Kullback-Leibler projection is that it sometimes easier to compute than the Euclidean projection, as we detail in Appendix C.
Newton ﬁxed point.
Let x be a root of G ( · , θ ) , i.e., G ( x, θ ) = 0 . The ﬁxed point iteration of Newton’s method for root-ﬁnding is
[Equation]
By the chain and product rules, we have
[Equation]
Using (3), we get A = − ∂ 1 F ( x, θ ) = ηI . Similarly,
[Equation]
Newton’s method for optimization is obtained by choosing G ( x, θ
[Equation]
= ∇ 1 f ( x, θ ) , which gives
[Equation]
It is easy to check that we recover the same linear system as for the gradient descent ﬁxed point (5).
A practical implementation can pre-compute an LU decomposition of ∂ 1 G ( x, θ ) , or a Cholesky decomposition if ∂ 1 G ( x, θ ) is positive semi-deﬁnite.
Proximal block coordinate descent ﬁxed point.
We now consider the case when x (cid:63) ( θ ) is implicitly deﬁned as the solution
[Equation]
where g 1 , . .
. , g m are possibly non-smooth functions operating on subvectors (blocks) x 1 , . . . , x m of x . In this case, we can use for i ∈ [ m ] the ﬁxed point
[Equation]
where η 1 , . . . , η m are block-wise step sizes. Clearly, when the step sizes are shared, i.e., η 1 = · · · = η m = η , this ﬁxed point is equivalent to the proximal gradient ﬁxed point (7) with g ( x, θ )
= (cid:80) ni
=1 g i ( x i , θ ) .
Quadratic programming. We now show how to use the KKT conditions discussed in §2.2 to differentiate quadratic programs, recovering Optnet [6] as a special case. To give some intuition, let us start with a simple equality-constrained quadratic program (QP) [Equation]
subject to H ( z, θ ) =
Ez − d = 0 , where Q ∈ R p × p , E ∈ R q × p , d ∈ R q . We gather the differentiable parameters as θ = ( Q, E, c, d ) . The stationarity and primal feasibility conditions give
[Equation] In matrix notation, this can be rewritten as
[Equation]
We can write the solution of the linear system (16) as the root x = ( z, ν ) of a function F ( x, θ ) . More generally, the QP can also include inequality constraints
[Equation]
subject to H ( z, θ ) = Ez − d = 0 , G ( z, θ ) = Mz − h ≤ 0 . where M ∈ R r × p and h ∈ R r .
We gather the differentiable parameters as θ = ( Q, E, M, c, d, h ) . The stationarity, primal feasibility and complementary slackness conditions give [Equation]
In matrix notation, this can be written as [Equation] While x = ( z, ν, λ ) is no longer the solution of a linear system, it is the root of a function F ( x, θ ) and therefore ﬁts our framework.
With our framework, no derivation is needed. We simply deﬁne f , H and G directly in Python.
Conic programming.
We now show that the differentiation of conic linear programs [3, 5], at the heart of differentiating through cvxpy layers [2], easily ﬁts our framework.
Consider the problem
[Equation]
subject to Ez + s =
d, s ∈ K , (17) where λ
=
( c, E, d ) , E ∈ R m × p , d ∈ R m , c ∈ R p and K ⊆
R m is a cone; z and s are the primal and slack variables, respectively.
Every convex optimization problem can be reduced to the form (17). Let us form the skew-symmetric matrix
[Equation]
where N = p + m
+ 1 .
Following [3, 2, 5], we can use the homogeneous self-dual embedding to reduce the process of solving (17) to ﬁnding a root of the residual map
[Equation]
where Π = proj R p ×K ∗ × R + and K ∗ ⊆
R m is the dual cone. The splitting conic solver [68], which is based on ADMM, outputs a solution F ( x (cid:63) ( θ ) , θ ) = 0 which is decomposed as x (cid:63) ( θ ) =
( u (cid:63) ( θ ) , v (cid:63) ( θ ) , w (cid:63) ( θ )) .
We can then recover the optimal solution of (17) using z (cid:63) ( λ ) =
u (cid:63) ( θ ( λ )) and s (cid:63) ( λ ) = proj K ∗ ( v (cid:63) ( θ ( λ ))) − v (cid:63) ( θ ( λ )) . The key oracle whose JVP/VJP we need is therefore Π , which is studied in [4].
The projection onto a few cones is available in our library and can be used to express F .
Frank-Wolfe.
We now consider
[Equation] [Equation]
where C ( θ ) is a convex polytope, i.e., it is the convex hull of vertices v 1 ( θ ) , . .
. , v m ( θ ) .
The FrankWolfe algorithm requires a linear minimization oracle (LMO)
[Equation]
and is a popular algorithm when this LMO is easier to compute than the projection onto C ( θ ) . However, since this LMO is piecewise constant, its Jacobian is null almost everywhere.
Inspired by
SparseMAP [67], which corresponds to the case when f is a quadratic, we rewrite (19) as
[Equation]
where V ( θ ) is a d × m matrix gathering the vertices v 1 ( θ ) , . . . , v m ( θ ) . We then have x (cid:63) ( θ ) = V ( θ ) p (cid:63) ( θ ) .
Since we have reduced (19) to minimization over the simplex, we can use the projected gradient ﬁxed point to obtain [Equation] We can therefore compute the derivatives of p (cid:63) ( θ ) by implicit differentiation and the derivatives of x (cid:63) ( θ ) by product rule. Frank-Wolfe implementations typically maintain the convex weights of the vertices, which we use to get an approximation of p (cid:63) ( θ ) .
Moreover, it is well-known that after t iterations, at most t vertices are visited.
We can leverage this sparsity to solve a smaller linear system.
Moreover, in practice, we only need to compute VJPs of x (cid:63) ( θ ) .
B Code examples B.1 Code examples for optimality conditions
Our library provides several reusable optimality condition mappings F or ﬁxed points T . We nevertheless demonstrate the ease of writing some of them from scratch. KKT conditions. As a more advanced example, we now describe how to implement the KKT conditions (6).
The stationarity, primal feasibility and complementary slackness conditions read [Equation]
Using jax.vjp to compute vector-Jacobian products, this can be implemented as grad = jax.grad(f) def F(x, theta): z, nu, lambd = x theta_f, theta_H, theta_G = theta _, H_vjp = jax.vjp(H, z, theta_H) stationarity =
(grad(z, theta_f) + H_vjp(nu)[0]) primal_feasability = H(z, theta_H) _, G_vjp = jax.vjp(G, z, theta_G) stationarity += G_vjp(lambd)[0] comp_slackness = G(z, theta_G)
* lambd return stationarity, primal_feasability, comp_slackness Similar mappings F can be written if the optimization problem contains only equality constraints or only inequality constraints. Mirror descent ﬁxed point.
Letting η = 1 and denoting θ = ( θ f , θ proj ) , the ﬁxed point (13) is [Equation] We can then implement it as follows.
[Equation]
return bregman_projection(y, theta_proj)
Although not considered in this example, the mapping ∇ ϕ could also depend on θ if necessary. B.2 Code examples for experiments We now sketch how to implement our experiments using our framework. In the following, jnp is short for jax.numpy .
In all experiments, we only show how to compute gradients with the outer objective. We can then use these gradients with gradient-based solvers to solve the outer objective. Multiclass SVM experiment. X_tr, Y_tr, X_val, Y_val = load_data() def W(x, theta): # dual-primal map return jnp.dot(X_tr.
T, Y_tr - x) / theta def f(x, theta): # inner objective return (0.5 * theta * jnp.sum(W(x, theta) ** 2) + jnp.vdot(x, Y_tr)) grad = jax.grad(f)
proj = jax.vmap(projection_simplex) # row-wise projections def T(x, theta): return proj(x - grad(x, theta) ) @custom_fixed_point(T) defmsvm_dual_solver(init_x, theta):
# [...] return x_star # solution of the dual objective def outer loss(lambd): theta = jnp.exp(lambd) x_star = msvm_dual_solver(init_x, theta) # inner solution Y_pred = jnp.dot(W x_star, theta), X_val) return 0.5 * jnp.sum((Y pred Y_val)
** 2) print(jax.grad(outer_loss)(lambd)) Task- X_tr, y_tr = load_data() def f(x, theta): # dictionary loss residual =
X_tr - jnp.dot(x, theta) return huber_loss(residual) grad = jax.grad(f) def T(x, theta): # proximal gradient fixed point return prox_lasso(x - grad(x, theta))
@custom_fixed_point(T)
defsparse_coding(init_x, theta): # inner objective # [...] return x_star # lasso solution def outer_loss(theta, w): # task-driven loss x_star = sparse_coding(init_x, theta)
# sparse codes y_pred = jnp.dot(x_star, w) return logloss(y_tr, y_pred) print(jax.grad(outer_loss, argnums=(0,1)))
Dataset distillation experiment.
X_tr, y_tr =
load_data() logloss = jax.vmap(loss.multiclass_logistic_loss) def f(x, theta, l2reg=1e-3): # inner objective scores
=
jnp.dot(theta, x) distilled_labels = jnp.arange(10) penalty =
l2reg * jnp.sum(x * x) return jnp.mean(logloss(distilled_labels, scores))
+ penalty
F = jax.grad(f) @custom_root(F) deflogreg_solver(init_x, theta):
# [...] return x_star def outer_loss(theta): x_star =
logreg_solver(init_x, theta) # inner solution scores = jnp.dot(X_tr, x_star) return jnp.mean(logloss(y_tr, scores)) print(jax.grad(
[Figure]
Molecular dynamics experiment.
energy_fn = soft_sphere_energy_fn(diameter) init_fn, apply_fn = jax_md.minimize.fire_descent( energy_fn, shift_fn) x0 = random.uniform(key, (N, 2))
R0 =
L * x # transform to physical coordinates
R = lax.fori_loop( 0,num_optimization_steps, body_fun=lambdat,state:apply_fn(state, t=t), init_val=
init_fn(R0)).position x_star = R / L def F(x, diameter): # normalized forces energy_fn = soft_sphere_energy_fn(diameter) normalized_energy_fn = lambda x: energy_fn(L * x) return -jax.grad(normalized_energy_fn)(x) dx = root_jvp(F, x_star, diameter, 1.0, solve=linear_solve.solve_bicgstab) print(dx) C Jacobian products
Our library provides numerous reusable building blocks. We describe in this section how to compute their Jacobian products. As a general guideline, whenever a projection enjoys a closed form, we leave the Jacobian product to the autodiff system. C.1 Jacobian products of projections
We describe in this section how to compute the Jacobian products of the projections (in the Euclidean and KL senses) onto various convex sets. When the convex set does not depend on any variable, we simply denote it C instead of C ( θ ) . Non-negative orthant. When C is the non-negative orthant, C = R d + , we obtain proj C
( y ) = max( y, 0) , where the maximum is evaluated element-wise. This is also known as the ReLu function.
The projection in the KL sense reduces to the exponential function, proj ϕ C ( y ) = exp( y ) . Box constraints.
When C ( θ ) is the box constraints C ( θ ) = [ θ 1 , θ 2 ] d with θ ∈ R 2 , we obtain proj C
( y, θ ) =
clip ( y, θ 1 , θ 2 ) : = max(min ( y, θ 2 ) , θ 1 ) . This is trivially extended to support different boxes for each coordinate, in which case θ ∈ R d × 2 .
Probability simplex. When C is the standard probability simplex, C = (cid:52) d , there is no analytical solution for proj C ( y ) . Nevertheless, the projection can be computed exactly in O ( d ) expected time or O ( d log d ) worst-case time [22, 63, 33, 26].
The Jacobian is given by diag ( s ) − ss (cid:62) / (cid:107) s (cid:107) 1 , where s ∈ { 0 , 1 } d is a vector indicating the support of proj C ( y ) [62].
The projection in the KL sense, on the other hand, enjoys a closed form: it reduces to the usual softmax proj ϕ C ( y ) =
exp( y ) / (cid:80) dj
=1 exp( y j ) .
Box sections.
Consider now the Euclidean projection z (cid:63) ( θ ) =
proj C
( y, θ ) onto the set C ( θ )
=
{ z ∈ R d : α i ≤ z i ≤ β i , i ∈ [ d ]; w (cid:62)
z
= c } , where θ =
( α, β, w, c ) .
This projection is a singlyconstrained bounded quadratic program.
It is easy to check (see, e.g., [66]) that an optimal solution satisﬁes for all i ∈
[ d ]
[Equation]
where L : R × R n → R d is the dual-primal mapping and x (cid:63) ( θ ) ∈ R is the optimal dual variable of the linear constraint, which should be the root of
[Equation]
The root can be found, e.g., by bisection.
The gradient ∇ x (cid:63) ( θ ) is given by ∇ x (cid:63) ( θ ) =
B (cid:62) /A and the
Jacobian ∂z (cid:63) ( θ ) is obtained by application of the chain rule on L .
Norm balls.
When C ( θ ) = { x ∈ R d : (cid:107) x (cid:107) ≤ θ } , where (cid:107) · (cid:107) is a norm and θ ∈ R + ,
proj C
( y, θ ) becomes the projection onto a norm ball.
The projection onto the (cid:96) 1 -ball reduces to a projection onto the simplex, see, e.g., [33].
The projections onto the (cid:96) 2 and (cid:96) ∞ balls enjoy a closed-form, see, e.g., [69, §6.5].
Since they rely on simple composition of functions, all three projections can therefore be automatically differentiated.
Afﬁne sets.
When C ( θ ) = { x ∈ R d : Ax = b } , where A ∈ R p × d , b ∈ R p and θ = ( A, b ) , we get proj C ( y, θ ) = y − A
[Equation]
Ay
[Equation] b ) where A † is the Moore-Penrose pseudoinverse of A . The second equality holds if p < d and A is full rank. A practical implementation can pre-compute a factorization of the Gram matrix AA (cid:62) .
Alternatively, we can also use the KKT conditions. Hyperplanes and half spaces. When C ( θ ) = { x ∈ R d : a (cid:62) x = b } , where a ∈ R d and b ∈ R and θ = ( a, b ) , we get
[Equation]
When C ( θ ) = { x ∈ R d : a (cid:62) x ≤ b } , we simply replace a (cid:62) y − b in the numerator by max( a (cid:62) y − b, 0) .
Transportation and Birkhoff polytopes.
When C ( θ ) = { X ∈ R p × d : X 1 d = θ 1 , X (cid:62) 1 p
= θ 2 , X ≥ 0 } , the so-called transportation polytope, where θ 1 ∈ (cid:52) p and θ 2 ∈ (cid:52) d are marginals, we can compute approximately the projections, both in the Euclidean and KL senses, by switching to the dual or semi-dual [17].
Since both are unconstrained optimization problems, we can compute their
Jacobian product by implicit differentiation using the gradient descent ﬁxed point.
An advantage of the KL geometry here is that we can use Sinkhorn [28], which is a GPU-friendly algorithm.
The
Birkhoff polytope, the set of doubly stochastic matrices, is obtained by ﬁxing θ 1 = θ 2 = 1 d /d .
Order simplex.
When C ( θ ) = { x ∈ R d : θ 1 ≥ x 1 ≥ x 2 ≥ · · · ≥ x d ≥ θ 2 } , a so-called order simplex [49, 16], the projection operations, both in the Euclidean and KL sense, reduce to isotonic optimization [58] and can be solved exactly in O
( d log d ) time using the Pool Adjacent Violators algorithm [14].
The Jacobian of the projections and efﬁcient product with it are derived in [31, 18].
Polyhedra.
More generally, we can consider polyhedra, i.e., sets of the form C ( θ ) = { x ∈ R d : Ax = b, Cx ≤ d } , where A ∈ R p × d , b ∈ R p , C ∈ R m × d , and d ∈ R m .
There are several ways to differentiate this projection.
The ﬁrst is to use the KKT conditions as detailed in §2.2.
A second way is consider the dual of the projection instead, which is the maximization of a quadratic function subject to non-negative constraints [69, §6.2].
That is, we can reduce the projection on a polyhedron to a problem of the form (8) with non-negative constraints, which we can in turn implicitly differentiate easily using the projected gradient ﬁxed point, combined with the projection on the non-negative orthant.
Finally, we apply the dual-primal mapping , which enjoys a closed form and is therefore amenable to autodiff, to obtain the primal projection.
C.2 Jacobian products of proximity operators
We provide several proximity operators, including for the lasso (soft thresholding), elastic net and group lasso (block soft thresholding).
All satisfy closed form expressions and can be differentiated automatically via autodiff.
For more advanced proximity operators, which do not enjoy a closed form, recent works have derived their Jacobians.
The Jacobians of fused lasso and OSCAR were derived in [65].
For general total variation, the Jacobians were derived in [80, 24]. D Jacobian precision proofs Proof of Theorem 1.
To simplify notations, we note A (cid:63) :
= A ( x (cid:63) , θ ) and ˆ A :=
A (ˆ x, θ ) , and similarly for B and J . We have by deﬁnition of the Jacobian estimate function A (cid:63) J (cid:63) =
B
[Equation]
and ˆ A
ˆ J = ˆ B .
Therefore we have
[Equation]
For any invertible matrices M
[Equation]
M
[Equation]
{+it holds that M − 
+}
[Equation]
Therefore,
[Equation]
As a consequence, the second term in J (ˆ x, θ ) − ∂x (cid:63) ( θ ) can be upper bounded and we obtain
[Equation]
which yields the desired result. Corollary 1 (Jacobian precision for gradient descent ﬁxed point) . Let f be such that f ( · , θ ) is twice differentiable and α -strongly convex and ∇ 21 f ( · , θ ) is γ -Lipschitz (in the operator norm) and ∂ 2 ∇ 1 f ( x, θ ) is β -Lipschitz and bounded in norm by R . The estimated Jacobian evaluated at ˆ x is then given by
[Equation]
For all θ ∈ R n , and any ˆ x estimating x (cid:63) ( θ ) , we have the following bound for the approximation error of the estimated Jacobian
[Equation]
Proof of Corollary 1.
This follows from Theorem 1, applied to this speciﬁc A ( x, θ ) and B ( x, θ ) . For proximal gradient descent, where T ( x, θ ) = prox ηg ( x − η ∇ 1 f ( x, θ ) , θ ) , this yields
[Equation]
We now focus in the case of proximal gradient descent on an objective f ( x, θ ) + g ( x ) , where g is smooth and does not depend on θ .
This is the case in our experiments in §4.3.
Recent work also exploits local smoothness of solutions to derive similar bounds [13, Theorem 13]
Corollary 2 (Jacobian precision for proximal gradient descent ﬁxed point) .
Let f be such that f ( · , θ ) is twice differentiable and α -strongly convex and ∇ 21 f ( · , θ ) is γ -Lipschitz (in the operator norm) and ∂ 2 ∇ 1 f ( x, θ ) is β -Lipschitz and bounded in norm by R .
Let g : R d → R be a twicedifferentiable µ -strongly convex (with special case µ = 0 being only convex), for which the function
Γ η ( x, θ ) = ∇ 2 g ( prox ηg ( x − η ∇ 1 f ( x, θ )) is κ η -Lipschitz in it ﬁrst argument. The estimated Jacobian evaluated at ˆ x is then given by
[Equation]
For all θ ∈ R n , and any ˆ x estimating x (cid:63) ( θ ) , we have the following bound for the approximation error of the estimated Jacobian [Equation] Proof of Corollary 2. First, let us note that prox ηg
( y, θ ) does not depend on θ , since g itself does not depend on θ , and is therefore equal to classical proximity operator of ηg which, with a slight overload of notations, we denote as prox
ηg ( y ) (with a single argument). In other words, [Equation] Regarding the ﬁrst claim (expression of the estimated Jacobian evaluated at ˆ x ), we ﬁrst have that prox ηg
( y ) is the solution to ( x (cid:48) − y ) + η ∇ g ( x (cid:48) ) {+= 
 in x (cid:48) - by ﬁrst-order condition for a smooth convex function.+}
We therefore have that
[Equation]
the ﬁrst I and inverse being functional identity and inverse, and the second I d and inverse being in the matrix sense, by inverse rule for Jacobians ∂h ( z ) = [ ∂h − 1 ( h ( z ))] − 1 (applied to the prox). As a consequence, we have, for Γ η ( x, θ ) = ∇ 2 g
( prox ηg ( x − η ∇ 1 f ( x, θ ))
that [Equation]
As a consequence, for all x ∈ R d , we have that
[Equation]
In the following, we modify slightly the notation of both A and B , writing [Equation]
With the current hypotheses, following along the proof of Theorem 1, we have that ˜
A is ( α + µ ) well-conditioned, and ( γ + κ η ) -Lipschitz in its ﬁrst argument, and ˜ B is β -Lipschitz in its ﬁrst argument and bounded in norm by R . The same reasoning yields [Equation] − x (cid:63) ( θ ) (cid:107) .
Our approach to differentiate the solution of a root equation F ( x, θ ) = 0 is valid as long as the smooth implicit function theorem holds, namely, as long as F is continuously differentiable near a solution ( x 0 , θ 0 ) and ∇ 1 F ( x 0 , θ 0 ) is invertible. While the ﬁrst assumption is easy to check when F is continuously differentiable everywhere , it does not always hold when this is not the case, e.g., when F involves the proximity operator of a non-smooth function.
In such cases, one may therefore have to study theoretically the properties of the function F near the solutions ( x ( θ ) , θ ) to justify differentiation using the smooth implicit function theorem. Here, we develop such an analysis to justify the use of our approach to differentiate the solution of a Lasso regression problem with respect to the regularization parameter. We note that the smooth implicit function theorem has already been used for this problem [12, 13]; here we justify why it is a valid approach, even though F itself is not continuously differentiable everywhere. More precisely, we consider the Lasso problem:
[Equation]
where Φ ∈ R m × d and b ∈ R m Note that we parameterize the regularization parameter as e θ to ensure that it remains strictly positive for θ ∈ R , but the analysis below does not depend on this particular choice. This is a typical non-smooth optimization problem where we may want to use a proximal gradient ﬁxed point equation (9) to differentiate the solution. In this case we have f ( x, θ ) =
(1 / 2) (cid:107) Φ x − b (cid:107) 22 , hence ∇ 1 f ( x, θ ) =
Φ (cid:62) (Φ x − b ) , and g ( x, θ ) = e θ (cid:107) x (cid:107) 1 , hence prox ηg ( y, θ ) =
ST ( y, ηe θ ) where ST : R d × R → R d is the soft-thresholding operator: ST ( a, b ) i
= sign ( a i ) × max( | a i | − b, 0) . The root equation that characterizes the solution is therefore, for any η > 0 : [Equation] It is well-known that, under mild assumptions, (20) has a unique solution for any θ ∈ R , and that the solution path { x (cid:63) ( θ ) : θ ∈ R } is continuous and piecewise linear, with a ﬁnite number of nondifferentiable points (often called “kinks”)
Since the function θ (cid:55)→ x (cid:63) ( θ ) is not differentiable at the kinks, the smooth implicit function theorem using (21) is not applicable at those points (as any other methods to compute the Jacobian of x (cid:63) ( θ ) , such as unrolling).
Interestingly, though, the following result shows that, under mild assumptions, the smooth implicit function theorem using (21) is valid on all other points of the solution path, thus justifying the use of our approach to compute the
Jacobian of x (cid:63) ( θ ) whenever it exists.
Note that we state this theorem under some assumption on the matrix Φ that is sufﬁcient to ensure that the solution to the lasso is unique [79, Lemma 4], but that weaker assumptions could be possible (see proof).
Theorem 2.
If the entries of Φ are drawn from a continuous probability distribution on R m × d , then the smooth implicit function theorem using the root equation (21) holds at any point ( x (cid:63) ( θ ) , θ ) that is not a kink of the solution path, with probability one.
Proof.
We ﬁrst show that F η is continuously differentiable in a neighborhood of ( x (cid:63) ( θ ) , θ ) , for any θ that is not a kink.
Since the ST operator is continuously differentiable everywhere except on the closed set:
[Equation]
it sufﬁces to show from (21) that ( x (cid:63) ( θ ) − η Φ (cid:62) (Φ x (cid:63) ( θ ) − b ) , ηe θ ) / ∈ S .
For that purpose, we characterize x (cid:63) ( θ ) using the subgradient of (20) as follows: there exists γ ∈ R d such that
[Equation]
where
[Equation]
if
[Equation]
γ
[Equation]
otherwise.
We thus need to show that ( x (cid:63) ( θ ) + ηe θ γ, ηe θ ) / ∈ S .
We ﬁrst see that, for any i ∈ [1 , d ] such that x (cid:63) (
[Equation]
i
[Equation]
[Equation]
The case x (cid:63) ( θ ) i (cid:54) = 0 requires more care, since the property γ i ∈ [ − 1 , 1] is not sufﬁcient to show that | x (cid:63) ( θ ) i + ηe θ γ i | = ηe θ | γ
i | is not equal to ηe θ : we need to show that, in fact, | γ i | < 1 .
For that purpose, let E ( θ ) =
{ i ∈ [1 , d ] : | γ i | = 1 } .
Denoting Φ E ( θ ) the matrix made of the columns of Φ in
E ( θ ) , we know that, under the assumptions of Theorem 2, with probability one the matrix Φ (cid:62)E ( θ ) Φ E ( θ ) is invertible and the lasso problem has a unique solution given by x (cid:63) ( θ ) E ( θ ) C
= 0 and
[Equation]
where s ( θ ) = sign (Φ (cid:62) ( b − Φ x (cid:63) ( θ ))))
∈ {− 1 , 0 , 1 } d [79].
Furthermore, we know that E ( θ ) is constant between two successive kinks [61], so if x (cid:63) ( θ ) is not a kink then there is a neighborhood
[ θ 1 , θ 2 ] of θ such as E ( θ (cid:48) )
=
E ( θ ) and s ( θ (cid:48) )
E ( θ (cid:48) )
= s ( θ ) E ( θ ) ,
for any θ (cid:48) ∈ [ θ 1 , θ 2 ] .
Let us now assume that Φ is such that for any E ⊂ [1 , d ] and s ∈ {− 1 , 1 } |E| , Φ (cid:62)E Φ E is invertible and (Φ (cid:62)E Φ E ) − 1 s has no coordinate equal to zero. This happens with probability one under the assumptions of Theoremsince the set of singular matrices is measure zero.
Then we see from (22) that, for θ (cid:48) ∈
[ θ 1 , θ 2 ] and i ∈
E ( θ ) , x (cid:63) ( θ (cid:48) ) i is an afﬁne and non-constant function of e θ (cid:48) . Since in addition x (cid:63) ( θ 1 ) i and x (cid:63) ( θ 2 ) i are either both nonnegative or nonpositive, then necessarily x (cid:63) ( θ ) i is positive or negative, respectively.
In other words, we have shown that | γ i | = 1 = ⇒ x (cid:63) ( θ ) i (cid:54) = 0 , or equivalently that x (cid:63) ( θ ) {+i = 
+}
= ⇒ | γ i | < 1 .
From this we deduce that for any i ∈ [1 , d ] such that x (cid:63) ( θ ) i = 0 ,
[Equation]
This concludes the proof that ( x (cid:63) ( θ ) + ηe θ γ, ηe θ ) / ∈ S , and therefore that F η is continuously differentiable in a neighborhood of ( x (cid:63) ( θ ) , θ ) .
The second condition for the smooth implicit theorem to hold, namely, the invertibility of ∇ 1 F η ( x (cid:63) ( θ ) , θ ) , is easily obtained by explicit computation [12, 13, Proposition 1] F Experimental setup and additional results
Our experiments use JAX [21], which is Apache2-licensed and scikit-learn [71], which is BSDlicensed. F.1 Hyperparameter optimization of multiclass SVMs Experimental setup. Synthetic datasets were generated using scikit-learn ’s sklearn.datasets.make_classification
[71], following a model adapted from [50].
All datasets consist of m = 700 training samples belonging to k = 5 distinct classes.
To simulate problems of different sizes, the number of features is varied as p ∈ { 100 , 250 , 500 , 750 , 1000 , 2000 , 3000 , 4000 , 5000 , 7500 , 10000 } , with 10% of features being informative and the rest random noise. In all cases, an additional m val = 200 validation samples were generated from the same model to deﬁne the outer problem. For the inner problem, we employed three different solvers: (i) mirror descent, (ii) (accelerated) proximal gradient descent and (iii) block coordinate descent. Hyperparameters for all solvers were individually tuned manually to ensure convergence across the range of problem sizes.
For mirror descent, a stepsize of 1 . 0 was used for the ﬁrst 100 steps, following a inverse square root decay afterwards up to a total of 2500 steps. For proximal gradient descent, a stepsize of 5 · 10 − 4 was used for 2500 steps. The block coordinate descent solver was run for 500 iterations. All solvers used the same initialization, namely, x init =
1 k 1 m × k , which satisﬁes the dual constraints. For the outer problem, gradient descent was used with a stepsize of 5 · 10 − 3 for the ﬁrst 100 steps, following a inverse square root decay afterwards up to a total of 150 steps. Conjugate gradient was used to solve the linear systems in implicit differentiation for at mostiterations. All results reported pertaining CPU runtimes were obtained using an internal compute cluster.
GPU results were obtained using a single NVIDIA P100 GPU with 16GB of memory per dataset. For each dataset size, we report the average runtime of an individual iteration in the outer problem, alongside a 90% conﬁdence interval estimated from the corresponding 150 runtime values. Additional results Figure 13 compares the runtime of implicit differentiation and unrolling on GPU.
These results highlight a fundamental limitation of the unrolling approach in memory-limited systems such as accelerators, as the inner solver suffered from out-of-memory errors for most problem sizes ( p ≥ 2000 for mirror descent, p ≥ 750 for proximal gradient and block coordinate descent).
While it might be possible to ameliorate this limitation by reducing the maximum number of iterations in the inner solver, doing so might lead to additional challenges [84] and require careful tuning.
Figure 14 depicts the validation loss (value of the outer problem objective function) at convergence.
It shows that all approaches were able to solve the outer problem, with solutions produced by different
[Figure] [Figure]
d a t i o
[Figure] multiple problem sizes setting as Figure 4) As be seen, all methods performed similarly in terms of validation loss. This conﬁrms that the faster runtimes for implicit differentiation compared to unrolling shown in Figure 4 (CPU) and Figure(GPU) are not at the cost of worse validation loss.
[Figure]
The ground-truth solution x (cid:63) ( θ ) is computed using the liblinear solver [37] available in scikit-learn [71] with a very low tolerance of 10 − 9 . Unlike in Figure 3, which was on ridge regression, the ground-truth Jacobian ∂x (cid:63) ( θ ) cannot be computed in closed form, in the more difﬁcult setting of multiclass SVMs. We therefore use a ﬁnite difference to approximately compute ∂x (cid:63) ( θ ) . Our results nevertheless conﬁrm similar trends as in Figure 3.
approaches being qualitatively indistinguishable from each other across the range of problem sizes considered. Figure 15 shows the Jacobian error achieved as a function of the solution error, when varying the number of features. F.2 Task-driven dictionary learning We downloaded from http://acgt.cs.tau.ac.il/multi_omic_benchmark/download.html a set of breast cancer gene expression data together with survival information generated by the TCGA
Research Network ( https://www.cancer.gov/tcga ) and processed as explained by [74].
The gene expression matrix contains the expression value for p=20,531 genes in m=1,212 samples, from which we keep only the primary tumors (m=1,093). From the survival information, we select the patients who survived at least ﬁve years after diagnosis ( m 1 = 200 ), and the patients who died before ﬁve years ( m 0 = 99 ), resulting in a cohort of m = 299 patients with gene expression and binary label. Note that non-selected patients are those who are marked as alive but were not followed foryears. To evaluate different binary classiﬁcation methods on this cohort, we repeated 10 times a random split of the full cohort into a training (60%), validation (20%) and test (20%) sets.
For each split and each method, 1) the method is trained with different parameters on the training set, 2) the parameter that maximizes the classiﬁcation AUC on the validation set is selected, 3) the method is then re-trained on the union of the training and validation sets with the selected parameter, and 4) we measure the AUC of that model on the test set. We then report, for each method, the mean test AUC over the 10 repeats, together with a 95% conﬁdence interval deﬁned a mean ± 1.96 × standard error of the mean. We used Scikit Learn’s implementation of logistic regression regularized by (cid:96) 1 (lasso) and (cid:96) 2 (ridge) penalty from sklearn.linear_model. LogisticRegression , and varied the C regularization parameter over a grid of 10 values: { 10 − 5 , 10 − 3 , . .
For the unsupervised dictionary learning experiment method, we estimated a dictionary from the gene expression data in the training and validation sets, using sklearn.decomposition.
DictionaryLearning(n_components=10, alpha=2.0) , which produces sparse codes in k = 10 dimensions with roughly 50% nonzero coefﬁcients by minimizing the squared Frobenius reconstruction distance with lasso regularization on the code.
We then use sklearn.linear_model.
LogisticRegression to train a logistic regression on the codes, varying the ridge regularization parameter C over a grid of 10 values { 10 − 1 , 10 0 , . .
Finally, we implemented the task-driven dictionary learning model (11) with our toolbox, following the pseudo-code in Figure 10.
Like for the unsupervised dictionary learning experiment, we set the dimension of the codes to k = 10 , and a ﬁxed elastic net regularization on the inner optimization problem to ensure that the codes have roughly 50% sparsity.
For the outer optimization problem, we solve an (cid:96) 2 regularized ridge regression problem, varying again the ridge regularization parameter
C over a grid of 10 values { 10 − 1 , 10 0 , . . . Because the outer problem is non-convex, we minimize it using the Adam optimizer [56] with default parameters. F.3 Dataset Distillation Experimental setup.
For the inner problem, we used gradient descent with backtracking linesearch, while for the outer problem we used gradient descent with momentum and a ﬁxed step-size. The momentum parameter was set to 0 . 9 while the step-size was set to 1 . Figure 5 was produced after 4000 iterations of the outer loop on CPU (Intel(R) Xeon(R)
Platinum
P-8136 CPU @ 2.00GHz), which took 1h55. Unrolled differentiation took instead 8h:05 ( 4 times more) to run the same number of iterations. As can be seen in Figure 16, the output is the same in both approaches. F.4 Molecular dynamics
Our experimental setup is adapted from the JAX-MD example notebook available at https:// github.com/google/jax-md/blob/master/notebooks/meta_optimization.ipynb .
We emphasize that calculating the gradient of the total energy objective, f ( x, θ ) = (cid:80) ij U ( x i,j , θ ) , with respect to the diameter θ of the {+smaller particles, ∇ 1 f ( x, θ ) , does not require implicit differentiation or unrolling. This is because ∇ 1 f ( x, θ ) = 
+}
at x = x
[Equation]
( θ ) :
[Equation]
This is known as Danskin’s theorem or envelope theorem.
Thus instead, we consider sensitivities of position ∂x (cid:63) ( θ ) directly, which does require implicit differentiation or unrolling. Our results comparing implicit and unrolled differentiation for calculating the sensitivity of position are shown in Figure 17. We use BiCGSTAB [81] to perform the tangent linear solve. Like in the original JAX-MD experiment, we use k = 128 particles in m = 2 dimensions.
[Figure]
for 40 different random initial conditions (different colored lines).
Gradients through the unrolled FIRE optimizer [15] for many initial conditions do not converge, in contrast to implicit differentiation.