CW-ERM Improving Autonomous Driving Planning with Closed-loop Weighted Empirical Risk Minimization Eesha Kumar1 Yiming Zhang2 Stefano Pini1 Simon Stent1

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CW-ERM: Improving Autonomous Driving Planning with

Closed-loop Weighted Empirical Risk Minimization

Eesha Kumar1,∗, Yiming Zhang2, Stefano Pini1, Simon Stent1,

Ana Ferreira2, Sergey Zagoruyko1, Christian S. Perone1,∗

Abstract— The imitation learning of self-driving vehicle poli-

cies through behavioral cloning is often carried out in an open-

loop fashion, ignoring the effect of actions to future states.

Training such policies purely with Empirical Risk Minimization

(ERM) can be detrimental to real-world performance, as it

biases policy networks towards matching only open-loop behav-

ior, showing poor results when evaluated in closed-loop. In this

work, we develop an efﬁcient and simple-to-implement principle

called Closed-loop Weighted Empirical Risk Minimization (CW-

ERM), in which a closed-loop evaluation procedure is ﬁrst

used to identify training data samples that are important for

practical driving performance and then we these samples to

help debias the policy network. We evaluate CW-ERM in a

challenging urban driving dataset and show that this procedure

yields a signiﬁcant reduction in collisions as well as other non-

differentiable closed-loop metrics.

I. INTRODUCTION

Learning effective planning policies for self-driving vehi-

cles (SDVs) from data such as human demonstrations remains

one of the major challenges in robotics and machine learning.

Since early works such as ALVINN [1], Imitation Learning

has seen major recent developments using modern Deep

Neural Networks (DNNs) [2]–[7]. Imitation Learning (IL),

and especially Behavioral Cloning (BC), however, still face

fundamental challenges [8], including causal confusion [9]

(later identiﬁed as a feedback-driven covariate shift [10]) and

dataset biases [8], to name a few.

There is one particular limitation of IL policies trained

with BC that is, however, often overlooked: the mismatch

between training and inference-time execution of the policy

actions. Most of the time, BC policies are trained in an open-

loop fashion, predicting the next action given the immediate

previous action and optionally conditioned on recent past

actions [2]–[5], [7]. These policies, however, when executed

in real-world, impact the future states. Small prediction errors

can drive covariate shift and make the network predict in an

out-of-distribution regime.

In this work, we address the mismatch between training

and inference through the development of a simple training

principle. Using a closed-loop simulator, we ﬁrst identify and

then reweight samples that are important for the closed-loop

performance of the planner. We call this approach

CW-ERM

Author is with Woven Planet United Kingdom Limited,

114-116 Curtain Road, London, United Kingdom, EC2A 3AH.

firstname.lastname@woven-planet.global

Author is with Woven Planet North America, Inc.,

900 Arastradero Rd, Palo Alto, CA, USA 94304.

firstname.lastname@woven-planet.global

∗Equal contribution.

(Closed-loop Weighted Empirical Risk Minimization), since

we use Weighted ERM [11] to correct the training distribution

in favour of closed-loop performance. We extensively evaluate

this principle on real-world urban driving data and show that

it can achieve signiﬁcant improvements on planner metrics

that matter for real-world performance (e.g. collisions).

Our contributions are therefore the following:

•

We motivate and propose Closed-loop Weighted Em-

pirical Risk Minimization (CW-ERM), a technique that

leverages closed-loop evaluation metrics acquired from

policy rollouts in a simulator to debias the policy network

and reduce the distributional differences between training

(open-loop) and inference time (closed-loop);

•

we evaluate CW-ERM experimentally on a challenging

urban driving dataset in a closed-loop fashion to show

that our method, although simple to implement, yields

signiﬁcant improvements in closed-loop performance

without requiring complex and computationally expen-

sive closed-loop training methods;

•

we also show an important connection of our method

to a family of methods that addresses covariate shift

through density ratio estimation.

In Section II, we detail the proposed CW-ERM and in

Section IV we show the CW-ERM experiments and compare

them against ERM.

II. METHODOLOGY

A. Problem Setup

The traditional formulation of supervised learning for

imitation learning, also called behavioral cloning (BC), can

be formulated as ﬁnding the policy ˆπBC :

ˆπBC = argmin

π∈Π

Es∼dπ∗,a∼π∗(s)[`(s, a, π)] (1)

where the state

is sampled from the expert state dis-

tribution

dπ∗

induced when following the expert policy

π∗

Actions

are sampled from the expert policy

π∗(s)

. The loss

is also known as the surrogate loss that will ﬁnd the policy

ˆπBC

that best mimics the unknown expert policy

π∗(s)

. In

practice, we only observe a ﬁnite set of state-action pairs

(si, a∗

i)m

i=1

, so the optimization is only approximate and we

then follow the Empirical Risk Minimization (ERM) principle

to ﬁnd the policy πfrom the policy class Π.

If we let

Es∼dπ∗,a∼π∗(s)[`(s, a, π)] = 

, then it follows

that

J(π)≤J(π∗) + T2

as shown by the proof in [13],

where

is the total cost and

is the task horizon. As we

can see, the total cost can grow quadratically in T.

arXiv:2210.02174v2 [cs.LG] 11 Oct 2022

Training

Scenes

Upsampled

Training Scenes

Identiﬁcation

Policy

Final

Policy

Traditional open-loop ERM training

Closed-loop Weighted Empirical Risk Minimization (CW-ERM)

Closed-loop

Simulation

Error set

construction

Fig. 1: High-level overview of our proposed Closed-loop Weighted Empirical Risk Minimization (CW-ERM) method. In

steps

(1-2)

we train an identiﬁcation policy

ˆπERM

using traditional ERM [12] on a set of training data samples or driving

"scenes". In step

(3)

, we perform closed-loop simulation of the policy

ˆπERM

and collect metrics to construct the error set in

step

(4)

. With the error set in hand, we upsample scenes in the training set as shown in step

(5)

. We train the ﬁnal policy

ˆπCW-ERM using CW-ERM as shown in step (6) with the upsampled Dup set.

When the policy

ˆπBC

is deployed in the real-world,

it will eventually make mistakes and then induce a state

distribution

dˆπBC

different than the one it was trained on

(

dπ∗

). During closed-loop evaluation of driving policies,

non-imitative metrics such as collisions and comfort are also

evaluated. However, they are often ignored in the surrogate

loss or only implicitly learned by imitating the expert due

to the difﬁculty of overcoming differentiability requirements,

as smooth approximations of these metrics are still different

than the non-differentiable counterparts often used. These

policies can often show good results in open-loop training, but

perform poorly in closed-loop evaluation or when deployed

in a real SDV due to the differences between

dˆπBC

and

dπ∗

where the estimator is no longer consistent.

B. Closed-loop Weighted Empirical Risk Minimization

In our method, called “Closed-loop Weighted Empirical

Risk Minimization” (CW-ERM), we seek to debias a policy

network from the open-loop performance towards closed-

loop performance, making the model rely on features that

are robust to closed-loop evaluation. Our method consists of

three stages: the training of an identiﬁcation policy, the use

of that policy in closed-loop simulation to identify samples,

and the training of a ﬁnal policy network on a reweighted

data distribution. More explicitly:

Stage 1 (identiﬁcation policy)

: train a traditional BC policy

network in open-loop using ERM, to yield ˆπERM.

Stage 2 (closed-loop simulation)

: perform rollouts of the

ˆπERM

policy in a closed-loop simulator, collect closed-loop

metrics and then identify the error set below:

EˆπERM ={(si, ai)s.t. C(si, ai)>0},(2)

where

is a training data sample, or “scene” with a ﬁxed

number of timesteps from the training set,

is the action

performed during the roll-out and

C(·)

is a cost such as the

number of collisions found during closed-loop rollouts.

Stage 3 (ﬁnal policy)

: train a new policy using weighted

ERM where the scenes belonging to the error set

EˆπERM

are

upweighted by a factor w(·), yielding the policy ˆπCW-ERM:

argmin

π∈Π

Es∼dπ∗,a∼π∗(s)[w(EˆπERM , s)`(s, a, π)] (3)

As we can see, the CW-ERM policy in Equation 3 is

very similar to the original BC policy trained with ERM in

Equation 1, with the key difference of a weighting term based

on the error set from closed-loop simulation in Stage 2. In

practice, although statistically equivalent, we upsample scenes

by a ﬁxed factor rather than reweighting, as it is known to

be more stable and robust [14].

By training a policy using CW-ERM, we expect it to up-

sample scenes that perform poorly in closed-loop evaluation,

making the policy network robust to the covariate shift seen

during inference time while unrolling the policy.

We describe the complete CW-ERM training procedure in

Algorithm 1 and in Figure 1 we show a high-level overview

of our method.

C. Relationship to covariate shift adaptation with density

ratio estimation

One important connection of our method is with covariate

shift correction using density ratio estimation [11]. To correct

for the covariate shift, the negative log-likelihood is often

weighted by the density ratio r(s):

argmin

π∈Π

Es∼dπ∗,a∼π∗(s)[r(s)`(s, a, π)] (4)

where

r(s)

is deﬁned as the density ratio between test and

training distributions:

r(s) = ptest(s)

ptrain(s)(5)

In practice,

r(s)

is difﬁcult to compute and is thus

estimated. The density ratio will be higher when the sample is

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CW-ERM:ImprovingAutonomousDrivingPlanningwithClosed-loopWeightedEmpiricalRiskMinimizationEeshaKumar1;,YimingZhang2,StefanoPini1,SimonStent1,AnaFerreira2,SergeyZagoruyko1,ChristianS.Perone1;AbstractTheimitationlearningofself-drivingvehiclepoli-ciesthroughbehavioralcloningisoftencarriedoutinanopen-...

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CW-ERM Improving Autonomous Driving Planning with Closed-loop Weighted Empirical Risk Minimization Eesha Kumar1 Yiming Zhang2 Stefano Pini1 Simon Stent1

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