This proposal aims to allow us to load balance properly between Guard,
Middle, and Exit nodes with the addition of padding traffic to the
network.
Canonical proposal current lives in my load_balancing-squashed branch:
https://gitweb.torproject.org/user/mikeperry/torspec.git/tree/proposals/xxx-load-balancing-with-overhead.txt?h=load_balancing-squashed
Here is the text in-line, for ease of commenting on-list:
==============================================================
Filename: xxx-load-balancing-with-overhead.txt
Title: Load Balancing with Overhead Parameters
Authors: Mike Perry
Created: 01 January 2016
Status: Draft
0. Motivation
In order to properly load balance in the presence of padding and
non-negligible amounts of directory and hidden service traffic, the load
balancing equations in Section 3.8.3 of dir-spec.txt are in need of some
modifications.
In addition to supporting the idea of overhead, the load balancing
equations can also be simplified by treating Guard+Exit nodes as Exit
nodes in all cases. This causes the 9 sub-cases of the current load
balancing equations to consolidate into a single solution, which also
will greatly simplify the consensus process, and eliminate edge cases
such as #16255[1].
1. Overview
For padding overhead due to Proposals 251 and 254, and changes to hidden
service path selection in Proposal 247, it will be useful to be able to
specify a pair of parameters that represents the additional traffic
present on Guard and Middle nodes due to these changes.
The current load balancing equations unfortunately make this excessively
complicated. With overhead factors included, each of the 9 subcases goes
from being a short solution to over a page of calculations for each
subcase.
Moreover, out of 8751 hourly consensus documents produced in 2015[2],
only 78 of them had a non-zero weight for using Guard+Exit nodes in the
Guard position (weight Wgd), and most of those were well under 1%. The
highest weight for using Guard+Exits in the Guard position recorded in
2015 was 2.62% (on December 10th, 2015). This means clients that chose a
Guard node during that particular hour used only 2.62% of Guard+Exit
flagged nodes' bandwidth when performing a bandwidth-weighted Guard
selection. All clients that chose a Guard node during any other hour did
not consider Guard+Exit nodes at all as potential candidates for their
Guards.
This indicates that we can greatly simplify these load balancing
equations with little to no change in diversity to the network.
2. Simplified Load Balancing Equations
Recall that the point of the load balancing equations in section 3.8.3
of dir-spec.txt is to ensure that an equal amount of client traffic is
distributed between Guards, Middles, Exits, and Guard+Exits, where each
flag type can occupy one or more positions in a path. This allocation is
accomplished by solving a system of equations for weights for flag
position selection to ensure equal allocation of client traffic for each
position in a circuit.
If we ignore overhead for the moment and treat Guard+Exit nodes as Exit
nodes, then this allows the simplified system of equations to become:
Wgg*G == M + Wme*E + Wmg*G # Guard position == middle position
Wgg*G == Wee*E # Guard position == equals exit position
Wmg*G + Wgg*G == G # Guard allocation weights sum to 1
Wme*E + Wee*E == E # Exit allocation weights sum to 1
This system has four equations and four unknowns, and by transitivity we
ensure that allocated capacity for guard, middle, and exit positions are
all equal. Unlike the equations in 3.8.3 of dir-spec.txt, there are no
special cases to the solutions of these equations because there is no
shortage of constraints and no decision points for allocation based on
scarcity. Thus, there is only one solution. Using SymPy's symbolic
equation solver (see attached script) we obtain:
E + G + M E + G + M 2*E - G - M 2*G - E - M
Wee: ---------, Wgg: ---------, Wme: -----------, Wmg: ------------
3*E 3*G 3*E 3*G
For the rest of the flags weights, we will do the following:
Dual-flagged (Guard+Exit) nodes should be treated as Exits:
Wgd = 0, Wmd = Wme, Wed = Wee
Directory requests use middle weights:
Wbd=Wmd, Wbg=Wmg, Wbe=Wme, Wbm=Wmm
Handle bridges and strange exit policies:
Wgm=Wgg, Wem=Wee, Weg=Wed
2.1. Checking for underflow and overflow
In the old load balancing equations, we required a case-by-case proof to
guard against overflow and underflow, and to decide what to do in the
event of various overflow and underflow conditions[3]. Even still, the
system proved fragile to changes, such as the implementation of Guard
uptime fractions[1].
Here, with the simplified equations, we can plainly see that the only
time that a negative weight can arise is in Wme and Wmg, when 2*E < G+M
or when 2*G < E+M. In other words, only when Exits or Guards are scarce.
Similarly, the only time that a weight exceeding 1.0 can arise is in Wee
and Wgg, which also happens when 2*E < G+M or 2*G < E+M. This means that
parameters will always overflow in pairs (Wee and Wme, and/or Wgg and
Wmg).
In both these cases, simply clipping the parameters at 1 and 0 provides
as close of a balancing condition as is possible, given the scarcity.
3. Load balancing with Overhead Parameters
Intuitively, overhead due to padding and path selection changes can be
represented as missing capacity in the relevant position. This means
that in the presence of a Guard overhead fraction of G_o and a Middle
overhead fraction of M_o, the total fraction of actual client traffic
carried in those positions is (1-G_o) and (1-M_o), respectively.
Then, to achieve a balanced allocation of traffic, we consider only the
actual client capacity carried in each position:
# Guard position minus overhead matches middle position minus overhead:
(1-G_o)*(Wgg*G) == (1-M_o)*(M + Wme*E + Wmg*G)
# Guard position minus overhead matches exit position:
(1-G_o)*(Wgg*G) == 1*(Wee*E)
# Guard weights still sum to 1:
Wmg*G + Wgg*G == G
# Exit weights still sum to 1:
Wme*E + Wee*E == E
Solving this system with SymPy unfortunately yields some unintuitively
simplified results. For each weight, we first show the SymPy solution,
and then factor that solution into a form analogous to Section 2:
-(G_o - 1)*(M_o - 1)*(E + G + M)
Wee: ---------------------------------------
E*(G_o + M_o - (G_o - 1)*(M_o - 1) - 2)
(1 - G_o)*(1 - M_o)*(E + G + M)
Wee: ---------------------------------------
E*(2 - G_o - M_o + (1 - G_o)*(1 - M_o))
(M_o - 1)*(E + G + M)
Wgg: ---------------------------------------
G*(G_o + M_o - (G_o - 1)*(M_o - 1) - 2)
(1 - M_o)*(E + G + M)
Wgg: ---------------------------------------
G*(2 - G_o - M_o + (1 - G_o)*(1- M_o))
-E*(M_o - 1) + G*(G_o - 1)*(-M_o + 2) - M*(M_o - 1)
Wmg: ---------------------------------------------------
G*(G_o + M_o - (G_o - 1)*(M_o - 1) - 2)
(2 - M_o)*G*(1 - G_o) - M*(1 - M_o) - E*(1 - M_o)
Wmg: ---------------------------------------------------
G*(2 - G_o - M_o + (1 - G_o )*(1 - M_o))
E*(G_o + M_o - 2) + G*(G_o - 1)*(M_o - 1) + M*(G_o - 1)*(M_o - 1)
Wme: -----------------------------------------------------------------
E*(G_o + M_o - (G_o - 1)*(M_o - 1) - 2)
(2 - G_o - M_o)*E - G*(1 - G_o)*(1 - M_o) - M*(1 - G_o)*(1 - M_o)
Wme: -----------------------------------------------------------------
E*(2 - G_o - M_o + (1 - G_o)*(1 - M_o))
A simple spot check with G_o = M_o = 0 shows us that with zero overhead,
these solutions become identical to the solutions in Section 2 of this
proposal.
The final condition that we need to ensure is that these weight values
never become negative or greater than 1.0[3].
3.1. Ensuring against underflow and overflow
Note that if M_o = G_o = 0, then the solutions and the overflow
conditions are the same as in Section 2.
Unfortunately, SimPy is unable to solve multivariate inequalities, which
prevents us from directly deriving overflow conditions for each variable
independently (at least easily and without mistakes). Wolfram Alpha is
able to derive closed form solutions to some degree for this, but they
are more complicated than checking the weights for underflow and
overflow directly.
However, for all overflow and underflow cases, simply warning in the
event of overflow or underflow in the weight variable solutions above is
equivalent anyway. Optimal load balancing given this scarcity should
still result if we clip the resulting solutions to [0, 1.0].
It will be wise in the implementation to test the overflow conditions
with M_o = G_o = 0, and with their actual values. This will allow us to
know if the overflow is a result of inherent unbalancing, or due to
input overhead values that are too large (and need to be reduced by, for
example, reducing padding).
4. Consensus integration
4.1. Making use of the Overhead Factors
In order to keep the consensus process simple on the Directory
Authorities, the overhead parameters represent the combined overhead
from many factors.
The G_o variable is meant to account for sum of directory overhead,
netflow padding overhead, future two-hop padding overhead, and future
hidden service overhead (for cases where Guard->Middle->Exit circuits
are not used).
The M_o variable is meant to account for multi-hop padding overhead,
hidden service overhead, as well as an overhead for any future two-hop
directory connections (so that we can consolidate Guards and Directory
guard functionality into a single Guard node).
There is no need for an E_o variable, because even if there were
Exit-specific overhead, it could be represented by an equivalent
reductions in both G_o and M_o instead.
Since all relevant padding and directory overhead information is
included in the extra-info documents for each relay, the M_o and G_o
variables could be computed automatically from these extra-info
documents during the consensus process. However, it is probably wiser to
keep humans in the loop and set them manually as consensus parameters
instead, especially since we have not previously had to deal with
serious adversarial consequences from malicious extra-info reporting.
For clarity, though, it may be a good idea to separate all of the
components of M_o and G_o into separate consensus parameters, and
combine them (via addition) in the final equations. That way it will be
easier to pinpoint the source of any potential overflow issues. This
separation will also enable us to potentially govern padding's
contribution to the overhead via a single tunable value.
4.2 Integration with Guardfraction
The GuardFraction changes in Proposal 236 and #16255 should continue to
work with these new equations, so long as the total T, G, and M values
are counted after the GuardFraction multiplier has been applied.
4.3. Guard flag assignment
Ideally, the Guard flag assignment process would also not count
Exit-flagged nodes when determining the Guard flag uptime and bandwidth
cutoffs, since we will not be using Guard+Exit flagged nodes as Guard
nodes at all when this change is applied. This will result in more
accurate thresholds for Guard node status, as well as better control
over the true total amount of Guard bandwidth in the consensus.
1. https://trac.torproject.org/projects/tor/ticket/16255
2. https://collector.torproject.org/archive/relay-descriptors/consensuses/
3. http://tor-dev.torproject.narkive.com/17H9FewJ/correctness-proof-for-new-bandwidth-weights-bug-1952
Appendix A: SymPy Script for Balancing Equation Solutions
#!/usr/bin/python
from sympy.solvers import solve
from sympy import simplify, Symbol, init_printing, pprint
# Sympy variable declarations
(G,M,E,D) = (Symbol('G'),Symbol('M'),Symbol('E'),Symbol('D'))
(Wgd,Wmd,Wed,Wme,Wmg,Wgg,Wee) = (Symbol('Wgd'),Symbol('Wmd'),Symbol('Wed'),
Symbol('Wme'),Symbol('Wmg'),Symbol('Wgg'),
Symbol('Wee'))
(G_o, M_o) = (Symbol('G_o'),Symbol('M_o'))
print "Current Load Balancing Equation Solutions, Case 1:"
pprint(solve(
[Wgg*G + Wgd*D - (M + Wmd*D + Wme*E + Wmg*G),
Wgg*G + Wgd*D - (Wee*E + Wed*D),
Wed*D + Wmd*D + Wgd*D - D,
Wmg*G + Wgg*G - G,
Wme*E + Wee*E - E,
Wmg - Wmd,
3*Wed - 1],
Wgd, Wmd, Wed, Wme, Wmg, Wgg, Wee))
print
print "Case 1 with guard and middle overhead: "
pprint(solve(
[(1-G_o)*(Wgg*G + Wgd*D) - (1-M_o)*(M + Wmd*D + Wme*E + Wmg*G),
(1-G_o)*(Wgg*G + Wgd*D) - (Wee*E + Wed*D),
Wed*D + Wmd*D + Wgd*D - D,
Wmg*G + Wgg*G - G,
Wme*E + Wee*E - E,
Wmg - Wmd,
3*Wed - 1],
Wgd, Wmd, Wed, Wme, Wmg, Wgg, Wee))
print "\n\n"
print "Elimination of combined Guard+Exit flags (no overhead): "
pprint(solve(
[(Wgg*G) - (M + Wme*E + Wmg*G),
(Wgg*G) - 1*(Wee*E),
Wmg*G + Wgg*G - G,
Wme*E + Wee*E - E],
Wme, Wmg, Wgg, Wee))
print
print "Elimination of combined Guard+Exit flags (Guard+middle overhead): "
combined = solve(
[(1-G_o)*(Wgg*G) - (1-M_o)*(M + Wme*E + Wmg*G),
(1-G_o)*(Wgg*G) - 1*(Wee*E),
Wmg*G + Wgg*G - G,
Wme*E + Wee*E - E],
Wme, Wmg, Wgg, Wee)
pprint(combined)
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