This document discusses hyperparameter optimization using approximate gradients. It introduces the problem of optimizing hyperparameters along with model parameters. While model parameters can be estimated from data, hyperparameters require methods like cross-validation. The document proposes using approximate gradients to optimize hyperparameters more efficiently than costly methods like grid search. It derives the gradient of the objective with respect to hyperparameters and presents an algorithm called HOAG that approximates this gradient using inexact solutions. The document analyzes HOAG's convergence and provides experimental results comparing it to other hyperparameter optimization methods.

1. HYPERPARAMETER OPTIMIZATION WITH
APPROXIMATE GRADIENT
Fabian Pedregosa
Chaire Havas-Dauphine
Paris-Dauphine / École Normale
Supérieure

2. HYPERPARAMETERS
Most machine learning models depend on at least one
hyperparameter to control for model complexity. Examples
include:
Amount of regularization.
Kernel parameters.
Architecture of a neural network.
Model parameters
Estimated using some
(regularized) goodness of
t on the data.
Hyperparameters
Cannot be estimated using
the same criteria as model
parameters (over tting).

3. HYPERPARAMETER SELECTION
Criterion to for hyperparameter selection:
Optimize loss on unseen data: cross-validation.
Minimize risk estimator: SURE, AIC/BIC, etc.
Example: least squares with regularization.ℓ2
loss =
Costly evaluation function,
non-convex.
Common methods: grid
search, random search, SMBO.
( − X(λ)∑n
i=1
bi ai )
2

4. GRADIENT-BASED HYPERPARAMETER OPTIMIZATION
Compute gradients with respect to hyperparameters
[Larsen 1996, 1998, Bengio 2000].
Hyperparameter optimization as nested or bi-level
optimization:
arg min
λ∈
s.t. X(λ)
⏟model parameters
f (λ) ≜ g(X(λ), λ)
  loss on test set
∈ arg min
x∈ℝp
h(x, λ)
⏟loss on train set

5. GOAL: COMPUTE ∇f (λ)
By chain rule,
Two main approaches: implicit differentiation and iterative
differentiation [Domke et al. 2012, Macaulin 2015]
Implicit differentiation [Larsen 1996, Bengio 2000]:
formulate inner optimization as implicit equation.
∇f = ⋅+
∂g
∂λ
∂g
∂X
  known
∂X
∂λ
⏟unknown
X(λ) ∈ arg min h(x, λ) ⟺ h(X(λ), λ) = 0∇1
  implicit equation for X

6. GRADIENT-BASED HYPERPARAMETER OPTIMIZATION
∇f = g − g∇2 ( h)∇2
1,2
T
( h)∇2
1
−1
∇1
Possible to compute gradient w.r.t. hyperparameters, given
Solution to the inner optimization
Solution to linear system
X(λ)
g( h)∇2
1
−1
∇1
computationally expensive.⟹

7. HOAG: HYPERPARAMETER OPTIMIZATION WITH APPROXIMATE
GRADIENT
Loose approximation
Cheap iterations, might
diverge.
Precise approximation
Costly iterations,
convergence to stationary
Replace by an approximate solution of the inner
optimization.
Approximately solve linear system.
Update using
Tradeoff
X(λ)
λ ≈ ∇fpk

8. point.
HOAG At iteration perform the following:k = 1, 2, …
i) Solve the inner optimization problem up to tolerance , i.e. nd
such that
ii) Solve the linear system up to tolerance . That is, nd such
that
iii) Compute approximate gradient as
iv) Update hyperparameters:
εk
∈xk ℝp
∥X( ) − ∥ ≤ .λk xk εk
εk qk
∥ h( , ) − g( , )∥ ≤ .∇2
1
xk λk qk ∇1 xk λk εk
pk
= g( , ) − h( , ,pk ∇2 xk λk ∇2
1,2
xk λk )
T
qk
=
(
− )
.λk+1 P λk
1
L
pk

9. ANALYSIS - GLOBAL CONVERGENCE
Assumptions:
(A1). Lipschits and .
(A2). non-singular
(A3). Domain is bounded.
∇g h∇2
h(X(λ), λ)∇2
1

Corollary: If , then converges to a
stationary point :
if is in the interior of then
< ∞∑∞
i=1
εi λk
λ∗
⟨∇f ( ), α − ⟩ ≥ 0 , ∀α ∈ λ∗
λ∗
⟹ λ∗ 
∇f ( ) = 0λ∗

10. EXPERIMENTS
How to choose tolerance ?εk
Different strategies for the tolerance decrease. Quadratic:
, Cubic: , Exponential:= 0.1/εk k
2
0.1/k
3
0.1 × 0.9
k
Approximate-gradient strategies achieve much faster
decrease in early iterations.

11. EXPERIMENTS I
Model: -regularized
logistic regression.
1 Hyperparameter.
Datasets:
20news (18k 130k )
real-sim (73k 20k)
ℓ2
×
×

12. EXPERIMENTS II
Kernel ridge regression.
2 hyperparameters.
Parkinson dataset: 654
17
Multinomial Logistic
regression with one
hyperparameter per feature
[Maclaurin et al. 2015]
784 10
hyperparameters
MNIST dataset: 60k
784
×
×
×

13. CONCLUSION
Hyperparameter optimization with inexact gradient:
can update hyperparameters before model parameters
have fully converged.
independent of inner optimization algorithm.
convergence guarantees under smoothness
assumptions.
Open questions.
Non-smooth inner optimization (e.g. sparse models)?
Stochastic / online approximation?

14. REFERENCES
[Y. Bengio, 2000] Bengio, Yoshua. "Gradient-based optimization of
hyperparameters." Neural computation 12.8 (2000): 1889-1900.
[J. Bergstra, Y. Bengio 2012] Bergstra, James, and Yoshua Bengio. "Random
search for hyper-parameter optimization." The Journal of Machine
Learning Research 13.1 (2012): 281-305.
[J. Snoek et al., 2015] Snoek, J. et al. Scalable Bayesian Optimization Using
Deep Neural Networks. (2015). at
[K. Swersky et al., 2014] Swersky, K., Snoek, J. & Adams, R. Freeze-Thaw
Bayesian Optimization. arXiv Prepr. arXiv1406.3896 1–12 (2014). at
[F. Hutter et al., 2013] Hutter, F., Hoos, H. & Leyton-Brown, K. An
evaluation of sequential model-based optimization for expensive blackbox
functions.
http://arxiv.org/abs/1502.05700a
http://arxiv.org/abs/1406.3896

15. REFERENCES 2
[M. Schmidt et al., 2013] Schmidt, M., Roux, N. & Bach, F. Minimizing nite
sums with the stochastic average gradient. arXiv Prepr. arXiv1309.2388
1–45 (2013). at
[J. Domke et al., 2012] Domke, J. Generic Methods for Optimization-Based
Modeling. Proc. Fifteenth Int. Conf. Artif. Intell. Stat. XX, 318–326 (2012).
[M. P. Friedlander et al., 2012] Friedlander, M. P. & Schmidt, M. Hybrid
Deterministic-Stochastic Methods for Data Fitting. SIAM J. Sci. Comput.
34, A1380–A1405 (2012).
http://arxiv.org/abs/1309.2388

16. EXPERIMENTS - COST FUNCTION

17. EXPERIMENTS
Comparison with other hyperparameter optimization
methods
Random = Random search, SMBO = Sequential Model-Based
Optimization (Gaussian process), Iterdiff = reverse-mode
differentiation .

18. EXPERIMENTS
Comparison in terms of a validation loss.
Random = Random search, SMBO = Sequential Model-Based
Optimization (Gaussian process), Iterdiff = reverse-mode
differentiation .